wpmath0000012_10

Pfaffian_constraint.html

  1. A ( q ) q ˙ = 0 A(q)\dot{q}=0

Pfeffer_integral.html

  1. A C G * ACG^{*}
  2. A C G * ACG_{*}
  3. A C G * ACG^{*}

Phase_field_models.html

  1. ϕ \phi
  2. d o d_{o}
  3. F [ e , ϕ ] = d 𝐫 [ K | ϕ | 2 + h 0 f ( ϕ ) + e 0 u 2 ] F[e,\phi]=\int d{\mathbf{r}}\left[K|{\mathbf{\nabla}}\phi|^{2}+h_{0}f(\phi)+e_% {0}u^{2}\right]
  4. ϕ {\phi}
  5. u = e / e 0 + h ( ϕ ) / 2 u=e/e_{0}+h(\phi)/2
  6. e e
  7. h h
  8. ϕ \phi
  9. e 0 = L 2 / T M c p e_{0}={L^{2}}/{T_{M}c_{p}}
  10. L L
  11. T M T_{M}
  12. c p c_{p}
  13. ϕ \nabla\phi
  14. f ( ϕ ) f(\phi)
  15. f ( ϕ ) f(\phi)
  16. K K
  17. h 0 h_{0}
  18. W = K / h 0 W=\sqrt{K/h_{0}}
  19. t ϕ = - 1 τ ( δ F δ ϕ ) + η ( 𝐫 , t ) \partial_{t}\phi=-\frac{1}{\tau}\left(\frac{\delta F}{\delta\phi}\right)+{\eta% }({\mathbf{r}},t)
  20. t e = D e 0 2 ( δ F δ e ) - 𝐪 e ( 𝐫 , t ) . \partial_{t}e=De_{0}\nabla^{2}\left(\frac{\delta F}{\delta e}\right)-{\mathbf{% \nabla}}\cdot{\mathbf{q}}_{e}({\mathbf{r}},t).
  21. e e
  22. η \eta
  23. 𝐪 e \mathbf{q}_{e}
  24. l l
  25. l 2 / D l^{2}/D
  26. α ε 2 t ϕ = ε 2 2 ϕ - f ( ϕ ) - e 0 h 0 h ( ϕ ) u + η ~ ( 𝐫 , t ) \alpha\varepsilon^{2}\partial_{t}\phi=\varepsilon^{2}\nabla^{2}\phi-f^{\prime}% (\phi)-\frac{e_{0}}{h_{0}}h^{\prime}(\phi)u+\tilde{\eta}({\mathbf{r}},t)
  27. t u = 2 u + 1 2 t h - 𝐪 u ( 𝐫 , t ) \partial_{t}u=\nabla^{2}u+\frac{1}{2}\partial_{t}h-{\mathbf{\nabla}}\cdot{% \mathbf{q}_{u}}({\mathbf{r}},t)
  28. ε = W / l \varepsilon=W/l
  29. α = D τ / W 2 h 0 \alpha={D\tau}/{W^{2}h_{0}}
  30. η ~ ( 𝐫 , t ) \tilde{\eta}({\mathbf{r}},t)
  31. 𝐪 u ( 𝐫 , t ) {\mathbf{q}_{u}}({\mathbf{r}},t)
  32. f ( ϕ ) f(\phi)
  33. ε \varepsilon
  34. ε \varepsilon

Phase_shift_module.html

  1. π \pi

Photoacoustic_Doppler_effect.html

  1. v \vec{v}
  2. f 0 f_{0}
  3. I = I 0 [ 1 + c o s ( 2 π f 0 t ) ] / 2 I={I}_{0}\left[1+cos\left(2\pi f_{0}t\right)\right]/2
  4. v \vec{v}
  5. f 0 f_{0}
  6. v \vec{v}
  7. α \alpha
  8. θ \theta
  9. f P A D = - f 0 v c 0 c o s α + f 0 v c a c o s θ f_{PAD}=-f_{0}\frac{v}{c_{0}}cos\alpha+f_{0}\frac{v}{c_{a}}cos\theta
  10. c 0 c_{0}
  11. c a c_{a}
  12. c 0 c a 10 5 \frac{c_{0}}{c_{a}}\sim 10^{5}
  13. v c a v\ll c_{a}
  14. f P A D = f 0 v c a c o s θ = v λ c o s θ f_{PAD}=f_{0}\frac{v}{c_{a}}cos\theta=\frac{v}{\lambda}cos\theta
  15. > 1 >1
  16. 100 μ 100\mu
  17. 5 × 5 × 15 μ 5\times 5\times 15\mu
  18. 3 {}^{3}
  19. < 1 <1

Photodetection.html

  1. E ( 𝐫 , t ) = E ( + ) ( 𝐫 , t ) + E ( - ) ( 𝐫 , t ) E(\mathbf{r},t)=E^{(+)}(\mathbf{r},t)+E^{(-)}(\mathbf{r},t)
  2. E ( - ) ( 𝐫 , t ) = E ( + ) ( 𝐫 , t ) E^{(-)}(\mathbf{r},t)=E^{(+)}(\mathbf{r},t)^{\dagger}
  3. E ( + ) ( 𝐫 , t ) E^{(+)}(\mathbf{r},t)
  4. E ( + ) ( 𝐫 , t ) = i j ( ω j 2 ) 1 / 2 a ^ j ε j e i ( 𝐤 j 𝐫 - ω j t ) E^{(+)}(\mathbf{r},t)=i\sum_{j}\left(\frac{\hbar\omega_{j}}{2}\right)^{1/2}% \hat{a}_{j}\mathbf{\varepsilon}_{j}e^{i(\mathbf{k}_{j}\cdot\mathbf{r}-\omega_{% j}t)}
  5. ε j \mathbf{\varepsilon}_{j}
  6. a ^ j \hat{a}_{j}
  7. 𝐫 \mathbf{r}
  8. t {t}
  9. t + d t {\it t}+d{\it t}
  10. W I ( 𝐫 , t ) d t W_{I}(\mathbf{r},t)d{\it t}
  11. W I ( 𝐫 , t ) = ψ E ( - ) ( 𝐫 , t ) E ( + ) ( 𝐫 , t ) ψ {W_{I}(\mathbf{r},t)}=\langle\psi\mid{E^{(-)}(\mathbf{r},t)}\cdot{E^{(+)}(% \mathbf{r},t)}\mid\psi\rangle
  12. | ψ |\psi\rangle
  13. a ^ j a ^ j \langle\hat{a}_{j}^{\dagger}\hat{a}_{j}\rangle
  14. a ^ j \hat{a}_{j}^{\dagger}
  15. a ^ j \hat{a}_{j}
  16. [ a ^ j , a ^ j ] = 1 [\hat{a}_{j},\hat{a}_{j}^{\dagger}]=1
  17. a ^ j a ^ j \hat{a}_{j}^{\dagger}\hat{a}_{j}
  18. a ^ j a ^ j \hat{a}_{j}\hat{a}_{j}^{\dagger}

Photometric_stereo.html

  1. I = n L I=n\cdot L
  2. I I
  3. m m
  4. n n
  5. L L
  6. 3 × m 3\times m
  7. k k
  8. I = k ( n L ) I=k(n\cdot L)
  9. L L
  10. L - 1 I = k n L^{-1}I=kn
  11. k k
  12. k n kn
  13. n n
  14. L L
  15. L T L^{T}
  16. L T I = L T k ( n L ) L^{T}I=L^{T}k(n\cdot L)
  17. ( L T L ) - 1 L T I = k n (L^{T}L)^{-1}L^{T}I=kn

Photon_antibunching.html

  1. V n = Δ n 2 = n 2 - n 2 = ( a a ) 2 - a a 2 . V_{n}=\langle\Delta n^{2}\rangle=\langle n^{2}\rangle-\langle n\rangle^{2}=% \left\langle\left(a^{\dagger}a\right)^{2}\right\rangle-\langle a^{\dagger}a% \rangle^{2}.
  2. V n = ( a ) 2 a 2 + a a - a a 2 . V_{n}=\langle{(a^{\dagger}})^{2}a^{2}\rangle+\langle a^{\dagger}a\rangle-% \langle a^{\dagger}a\rangle^{2}.
  3. V n - n = ( a ) 2 a 2 - a a 2 . V_{n}-\langle n\rangle=\langle(a^{\dagger})^{2}a^{2}\rangle-\langle a^{\dagger% }a\rangle^{2}.
  4. g ( 2 ) ( 0 ) = ( a ) 2 a 2 a a 2 . g^{(2)}(0)={{\langle(a^{\dagger})^{2}a^{2}\rangle}\over{\langle a^{\dagger}a% \rangle^{2}}}.
  5. 1 ( n ) 2 ( V n - n ) = g ( 2 ) ( 0 ) - 1. {{1}\over{(\langle n\rangle)^{2}}}(V_{n}-\langle n\rangle)=g^{(2)}(0)-1.
  6. g ( 2 ) ( 0 ) < 1 g^{(2)}(0)<1
  7. Q < 0 Q<0
  8. Q V n n - 1. Q\equiv\frac{V_{n}}{\langle n\rangle}-1.
  9. g ( 2 ) ( 0 ) g^{(2)}(0)
  10. g ( 2 ) ( 0 ) = 0.0 g^{(2)}(0)=0.0
  11. g ( 2 ) ( τ ) = a ( 0 ) a ( τ ) a ( τ ) a ( 0 ) a a 2 . g^{(2)}(\tau)={{\langle a^{\dagger}(0)a^{\dagger}(\tau)a(\tau)a(0)\rangle}% \over{\langle a^{\dagger}a\rangle^{2}}}.
  12. g ( 2 ) ( 0 ) g ( 2 ) ( τ ) g^{(2)}(0)\leq g^{(2)}(\tau)
  13. g ( 2 ) ( τ ) = a a C a a g^{(2)}(\tau)={{\langle a^{\dagger}a\rangle_{C}}\over{\langle a^{\dagger}a% \rangle}}
  14. O C Ψ C | O | Ψ C . \langle O\rangle_{C}\equiv\langle\Psi_{C}|O|\Psi_{C}\rangle.
  15. | Ψ C |\Psi_{C}\rangle
  16. τ = 0 \tau=0

Photon_diffusion_equation.html

  1. ( D ( r ) ) Φ ( r , t ) - v μ a ( r ) Φ ( r , t ) + v S ( r , t ) = Φ ( r , t ) t \nabla(D(r)\cdot\nabla)\Phi(\vec{r},t)-v\mu_{a}(\vec{r})\Phi(\vec{r},t)+vS(% \vec{r},t)=\frac{\partial\Phi(\vec{r},t)}{\partial t}
  2. Φ \Phi
  3. μ a \mu_{a}
  4. D D
  5. v v
  6. S S

Pickering_emulsion.html

  1. Δ E = π r 2 γ O W ( 1 - | c o s θ O W | ) 2 \Delta E\ =\pi r^{2}\gamma_{OW}(1-|cos{\theta_{OW}}|)^{2}
  2. γ O W \gamma_{OW}
  3. θ O W \theta_{OW}

Piezoelectric_coefficient.html

  1. d = P σ d=\dfrac{P}{\sigma}
  2. σ \sigma

Pinch_point_(mathematics).html

  1. f ( u , v , w ) = u 2 - v w 2 + [ 4 ] f(u,v,w)=u^{2}-vw^{2}+[4]\,
  2. v v
  3. 1 - 2 x + x 2 - y z 2 = 0 1-2x+x^{2}-yz^{2}=0
  4. ( 1 , 0 , 0 ) (1,0,0)
  5. u = 1 - x , v = y u=1-x,v=y
  6. w = z w=z
  7. u , v , w u,v,w
  8. ( 1 , 0 , 0 ) (1,0,0)
  9. 1 - 2 x + x 2 - y z 2 = ( 1 - x ) 2 - y z 2 = u 2 - v w 2 1-2x+x^{2}-yz^{2}=(1-x)^{2}-yz^{2}=u^{2}-vw^{2}
  10. u 2 - v w 2 = 0 u^{2}-vw^{2}=0
  11. v v
  12. v v

Ping-pong_lemma.html

  1. H 1 , , H k = H 1 H k . \langle H_{1},\dots,H_{k}\rangle=H_{1}\ast\dots\ast H_{k}.
  2. w = i = 1 m w α i , β i . w=\prod_{i=1}^{m}w_{\alpha_{i},\beta_{i}}.
  3. w ( X 2 ) i = 1 m - 1 w α i , β i ( X 1 ) i = 1 m - 2 w α i , β i ( X α m - 1 ) w 1 , β 1 w α 2 , β 2 ( X α 3 ) w(X_{2})\subseteq\prod_{i=1}^{m-1}w_{\alpha_{i},\beta_{i}}(X_{1})\subseteq% \prod_{i=1}^{m-2}w_{\alpha_{i},\beta_{i}}(X_{\alpha_{m-1}})\subseteq\dots% \subseteq w_{1,\beta_{1}}w_{\alpha_{2},\beta_{2}}(X_{\alpha_{3}})\subseteq
  4. w 1 , β 1 ( X α 2 ) X 1 \subseteq w_{1,\beta_{1}}(X_{\alpha_{2}})\subseteq X_{1}
  5. α 1 = 1 ; α m 1 \alpha_{1}=1;\alpha_{m}\neq 1
  6. h H 1 { w 1 , β 1 - 1 , 1 } h\in H_{1}\setminus\{w_{1,\beta_{1}}^{-1},1\}
  7. α 1 1 ; α m = 1 \alpha_{1}\neq 1;\alpha_{m}=1
  8. h H 1 { w 1 , β m , 1 } h\in H_{1}\setminus\{w_{1,\beta_{m}},1\}
  9. α 1 1 ; α m 1 \alpha_{1}\neq 1;\alpha_{m}\neq 1
  10. h H 1 { 1 } h\in H_{1}\setminus\{1\}
  11. A = ( 1 2 0 1 ) \scriptstyle A=\begin{pmatrix}1&2\\ 0&1\end{pmatrix}
  12. B = ( 1 0 2 1 ) \scriptstyle B=\begin{pmatrix}1&0\\ 2&1\end{pmatrix}
  13. H 1 = { A n | n } = { ( 1 2 n 0 1 ) : n } H_{1}=\{A^{n}|n\in\mathbb{Z}\}=\left\{\begin{pmatrix}1&2n\\ 0&1\end{pmatrix}:n\in\mathbb{Z}\right\}
  14. H 2 = { B n | n } = { ( 1 0 2 n 1 ) : n } . H_{2}=\{B^{n}|n\in\mathbb{Z}\}=\left\{\begin{pmatrix}1&0\\ 2n&1\end{pmatrix}:n\in\mathbb{Z}\right\}.
  15. X 1 = { ( x y ) 2 : | x | > | y | } X_{1}=\left\{\begin{pmatrix}x\\ y\end{pmatrix}\in\mathbb{R}^{2}:|x|>|y|\right\}
  16. X 2 = { ( x y ) 2 : | x | < | y | } . X_{2}=\left\{\begin{pmatrix}x\\ y\end{pmatrix}\in\mathbb{R}^{2}:|x|<|y|\right\}.

Pinsker's_inequality.html

  1. δ ( P , Q ) 1 2 D KL ( P Q ) \delta(P,Q)\leq\sqrt{\frac{1}{2}D_{\mathrm{KL}}(P\|Q)}
  2. δ ( P , Q ) = sup { | P ( A ) - Q ( A ) | : A is an event to which probabilities are assigned. } \delta(P,Q)=\sup\{|P(A)-Q(A)|:A\,\text{ is an event to which probabilities are% assigned.}\}
  3. D KL ( P Q ) = i ln ( P ( i ) Q ( i ) ) P ( i ) D_{\mathrm{KL}}(P\|Q)=\sum_{i}\ln\left(\frac{P(i)}{Q(i)}\right)P(i)\!
  4. ϵ > 0 \epsilon>0
  5. δ ( P , Q ) ϵ \delta(P,Q)\leq\epsilon
  6. D KL ( P Q ) = D_{\mathrm{KL}}(P\|Q)=\infty

Piper_diagram.html

  1. [ x y 1 ] = [ cos ( π 2 ) sin ( π 2 ) 0 - 2 sin ( π 2 ) 2 cos ( π 2 ) 0 0 0 1 ] [ x y 1 ] \begin{bmatrix}x^{\prime}\\ y^{\prime}\\ 1\end{bmatrix}=\begin{bmatrix}\cos(\frac{\pi}{2})&\sin(\frac{\pi}{2})&0\\ -2\sin(\frac{\pi}{2})&2\cos(\frac{\pi}{2})&0\\ 0&0&1\end{bmatrix}\begin{bmatrix}x\\ y\\ 1\end{bmatrix}

PKCS_1.html

  1. n n
  2. p p
  3. q q
  4. n = p q n=pq
  5. r i r_{i}
  6. i i
  7. n = r 1 r 2 r u , n=r_{1}\cdot r_{2}\cdot...\cdot r_{u},
  8. u 2 u\geq 2
  9. p = r 1 p=r_{1}
  10. q = r 2 q=r_{2}
  11. ( n , e ) (n,e)
  12. e e
  13. ( n , d ) (n,d)
  14. d d

Planar_projection.html

  1. 𝐚 x , y , z \mathbf{a}_{x,y,z}
  2. 𝐛 u , v \mathbf{b}_{u,v}

Planar_separator_theorem.html

  1. 2 π 3 ( 1 + 3 2 2 + o ( 1 ) ) n 1.84 n . \sqrt{\frac{2\pi}{\sqrt{3}}}\left(\frac{1+\sqrt{3}}{2\sqrt{2}}+o(1)\right)% \sqrt{n}\approx 1.84\sqrt{n}.

Plancherel_theorem_for_spherical_functions.html

  1. 𝔄 \mathfrak{A}
  2. 𝔄 \mathfrak{A}
  3. 𝔄 \mathfrak{A}
  4. 𝔄 \mathfrak{A}^{\prime}
  5. 𝔄 \mathfrak{A}
  6. 𝔄 \mathfrak{A}
  7. 𝔄 \mathfrak{A}
  8. χ λ ( π ( f ) ) = G f ( g ) φ λ ( g ) d g . \chi_{\lambda}(\pi(f))=\int_{G}f(g)\cdot\varphi_{\lambda}(g)\,dg.
  9. 𝔄 \mathfrak{A}
  10. φ λ ( g ) = K λ ( g k ) - 1 d k . \varphi_{\lambda}(g)=\int_{K}\lambda^{\prime}(gk)^{-1}\,dk.
  11. λ ( k x ) = Δ A N ( x ) 1 / 2 λ ( x ) \lambda^{\prime}(kx)=\Delta_{AN}(x)^{1/2}\lambda(x)
  12. W = N K ( A ) / C K ( A ) , W=N_{K}(A)/C_{K}(A),
  13. f ( g b ) = Δ ( b ) 1 / 2 λ ( b ) f ( g ) f(gb)=\Delta(b)^{1/2}\lambda(b)f(g)
  14. π ( g ) f ( x ) = f ( g - 1 x ) , \pi(g)f(x)=f(g^{-1}x),
  15. f 2 = K | f ( k ) | 2 d k < . \|f\|^{2}=\int_{K}|f(k)|^{2}\,dk<\infty.
  16. χ λ ( g ) = ( π ( g ) 1 , 1 ) . \chi_{\lambda}(g)=(\pi(g)1,1).
  17. 3 = { x + y i + t j | t > 0 } \mathfrak{H}^{3}=\{x+yi+tj|t>0\}
  18. g = ( a b c d ) g=\begin{pmatrix}a&b\\ c&d\end{pmatrix}
  19. g ( w ) = ( a w + b ) ( c w + d ) - 1 . g(w)=(aw+b)(cw+d)^{-1}.
  20. 3 \mathfrak{H}^{3}
  21. d s 2 = r - 2 ( d x 2 + d y 2 + d r 2 ) ds^{2}=r^{-2}(dx^{2}+dy^{2}+dr^{2})
  22. d V = r - 3 d x d y d r dV=r^{-3}\,dx\,dy\,dr
  23. Δ = - r 2 ( x 2 + y 2 + r 2 ) + r r . \Delta=-r^{2}(\partial_{x}^{2}+\partial_{y}^{2}+\partial_{r}^{2})+r\partial_{r}.
  24. 3 \mathfrak{H}^{3}
  25. Δ = - t 2 - 2 coth t t . \Delta=-\partial_{t}^{2}-2\coth t\partial_{t}.
  26. f d V = - f ( t ) sinh 2 t d t . \int f\,dV=\int_{-\infty}^{\infty}f(t)\,\sinh^{2}t\,dt.
  27. U * Δ U = - d 2 d t 2 + 1. U^{*}\Delta U=-{d^{2}\over dt^{2}}+1.
  28. Φ λ ( t ) = sin λ t λ sinh t , \Phi_{\lambda}(t)={\sin\lambda t\over\lambda\sinh t},
  29. f ~ ( λ ) = f Φ - λ d V \tilde{f}(\lambda)=\int f\Phi_{-\lambda}\,dV
  30. f ( x ) = f ~ ( λ ) Φ λ ( x ) λ 2 d λ . f(x)=\int\tilde{f}(\lambda)\Phi_{\lambda}(x)\lambda^{2}\,d\lambda.
  31. f = f 2 * f 1 f=f_{2}^{*}\star f_{1}
  32. f * ( g ) = f ( g - 1 ) ¯ f^{*}(g)=\overline{f(g^{-1})}
  33. G f 1 f 2 ¯ d g = f ~ 1 ( λ ) f ~ 2 ( λ ) ¯ λ 2 d λ . \int_{G}f_{1}\overline{f_{2}}\,dg=\int\tilde{f}_{1}(\lambda)\overline{\tilde{f% }_{2}(\lambda)}\,\lambda^{2}\,d\lambda.
  34. U : f f ~ , L 2 ( K \ G / K ) L 2 ( , λ 2 d λ ) U:f\mapsto\tilde{f},\,\,L^{2}(K\backslash G/K)\rightarrow L^{2}({\mathbb{R}},% \lambda^{2}\,d\lambda)
  35. f f\in
  36. f ~ \tilde{f}
  37. Δ Φ λ = ( λ 2 + 1 ) Φ λ . \displaystyle{\Delta\Phi_{\lambda}=(\lambda^{2}+1)\Phi_{\lambda}.}
  38. 𝒮 = { f | sup t | ( 1 + t 2 ) N ( I + Δ ) M f ( t ) sinh ( t ) | < } . {\mathcal{S}}=\{f|\sup_{t}|(1+t^{2})^{N}(I+\Delta)^{M}f(t)\sinh(t)|<\infty\}.
  39. | F ( λ ) | C e R | Im λ | . |F(\lambda)|\leq Ce^{R\cdot|{\rm Im}\,\lambda|}.
  40. 3 \mathfrak{H}^{3}
  41. π λ ( g - 1 ) ξ ( z ) = | c z + d | - 2 - i λ ξ ( g ( z ) ) . \pi_{\lambda}(g^{-1})\xi(z)=|cz+d|^{-2-i\lambda}\xi(g(z)).
  42. ξ 0 ( z ) = π - 1 ( 1 + | z | 2 ) - 2 \xi_{0}(z)=\pi^{-1}(1+|z|^{2})^{-2}
  43. Φ λ ( g ) = ( π λ ( g ) ξ 0 , ξ 0 ) . \Phi_{\lambda}(g)=(\pi_{\lambda}(g)\xi_{0},\xi_{0}).
  44. L 2 ( 3 ) L^{2}({\mathfrak{H}}^{3})
  45. W f ( λ , z ) = G / K f ( g ) π λ ( g ) ξ 0 ( z ) d g Wf(\lambda,z)=\int_{G/K}f(g)\pi_{\lambda}(g)\xi_{0}(z)\,dg
  46. L 2 ( 3 ) L^{2}({\mathfrak{H}}^{3})
  47. 2 = { x + r i | r > 0 } \mathfrak{H}^{2}=\{x+ri|r>0\}
  48. g = ( a b c d ) g=\begin{pmatrix}a&b\\ c&d\end{pmatrix}
  49. g ( w ) = ( a w + b ) ( c w + d ) - 1 . g(w)=(aw+b)(cw+d)^{-1}.
  50. 2 \mathfrak{H}^{2}
  51. d s 2 = r - 2 ( d x 2 + d r 2 ) ds^{2}=r^{-2}(dx^{2}+dr^{2})
  52. d A = r - 2 d x d r dA=r^{-2}\,dx\,dr
  53. Δ = - r 2 ( x 2 + r 2 ) . \Delta=-r^{2}(\partial_{x}^{2}+\partial_{r}^{2}).
  54. 2 \mathfrak{H}^{2}
  55. Δ = - t 2 - coth t t . \Delta=-\partial_{t}^{2}-\coth t\partial_{t}.
  56. f d A = - f ( t ) | sinh t | d t . \int f\,dA=\int_{-\infty}^{\infty}f(t)\,|\sinh t|\,dt.
  57. 2 \mathfrak{H}^{2}
  58. M 1 f ( x , y , r ) = r 1 / 2 f ( x , r ) M_{1}f(x,y,r)=r^{1/2}\cdot f(x,r)
  59. Δ 3 M 1 f = M 1 ( Δ 2 + 3 4 ) f , \Delta_{3}M_{1}f=M_{1}(\Delta_{2}+{3\over 4})f,
  60. n {\mathfrak{H}}^{n}
  61. Δ 3 M 0 = M 0 ( Δ 2 + 3 4 ) . \Delta_{3}M_{0}=M_{0}(\Delta_{2}+{3\over 4}).
  62. M 1 * F ( x , r ) = r 1 / 2 - F ( x , y , r ) d y M_{1}^{*}F(x,r)=r^{1/2}\int_{-\infty}^{\infty}F(x,y,r)\,dy
  63. 3 ( M 1 f ) F d V = 2 f ( M 1 * F ) d A . \int_{\mathfrak{H}^{3}}(M_{1}f)\cdot F\,dV=\int_{\mathfrak{H}^{2}}f\cdot(M_{1}% ^{*}F)\,dA.
  64. 3 ( M 0 f ) F d V = 2 f ( M 0 * F ) d A \int_{\mathfrak{H}^{3}}(M_{0}f)\cdot F\,dV=\int_{\mathfrak{H}^{2}}f\cdot(M_{0}% ^{*}F)\,dA
  65. M i * Δ 3 = ( Δ 2 + 3 4 ) M i * . M_{i}^{*}\Delta_{3}=(\Delta_{2}+{3\over 4})M_{i}^{*}.
  66. f λ = M 1 * Φ λ \displaystyle{f_{\lambda}=M_{1}^{*}\Phi_{\lambda}}
  67. Δ 2 f λ = ( λ 2 + 1 4 ) f λ . \Delta_{2}f_{\lambda}=(\lambda^{2}+{1\over 4})f_{\lambda}.
  68. b ( λ ) = f λ ( i ) = sin λ t λ sinh t d t = π λ tanh π λ 2 , b(\lambda)=f_{\lambda}(i)=\int{\sin\lambda t\over\lambda\sinh t}\,dt={\pi\over% \lambda}\tanh{\pi\lambda\over 2},
  69. e i λ t / sinh t e^{i\lambda t}/\sinh t
  70. ϕ λ = b ( λ ) - 1 M 1 Φ λ \displaystyle{\phi_{\lambda}=b(\lambda)^{-1}M_{1}\Phi_{\lambda}}
  71. φ λ ( e t i ) = 1 2 π 0 2 π ( cosh t - sinh t cos θ ) - 1 - i λ d θ . \varphi_{\lambda}(e^{t}i)={1\over 2\pi}\int_{0}^{2\pi}(\cosh t-\sinh t\cos% \theta)^{-1-i\lambda}\,d\theta.
  72. Φ λ = M 1 ϕ λ . \displaystyle{\Phi_{\lambda}=M_{1}\phi_{\lambda}.}
  73. 2 \mathfrak{H}^{2}
  74. f ~ ( λ ) = f φ - λ d A , \tilde{f}(\lambda)=\int f\varphi_{-\lambda}\,dA,
  75. ( M 1 * F ) ( λ ) = F ~ ( λ ) . {(M_{1}^{*}F)}^{\sim}(\lambda)=\tilde{F}(\lambda).
  76. f ( x ) = - φ λ ( x ) f ~ ( λ ) λ π 2 tanh ( π λ 2 ) d λ , f(x)=\int_{-\infty}^{\infty}\varphi_{\lambda}(x)\tilde{f}(\lambda)\,{\lambda% \pi\over 2}\tanh({\pi\lambda\over 2})\,d\lambda,
  77. 2 \mathfrak{H}^{2}
  78. 2 f 1 f 2 ¯ d A = - f ~ 1 f ~ 2 ¯ λ π 2 tanh ( π λ 2 ) d λ . \int_{\mathfrak{H}^{2}}f_{1}\overline{f_{2}}\,dA=\int_{-\infty}^{\infty}\tilde% {f}_{1}\overline{\tilde{f}_{2}}\,{\lambda\pi\over 2}\tanh({\pi\lambda\over 2})% \,d\lambda.
  79. Δ 2 φ λ = ( λ 2 + 1 4 ) φ λ . \displaystyle{\Delta_{2}\varphi_{\lambda}=(\lambda^{2}+{1\over 4})\varphi_{% \lambda}.}
  80. 𝒮 = { f | sup t | ( 1 + t 2 ) N ( I + Δ ) M f ( t ) φ 0 ( t ) | < } . {\mathcal{S}}=\{f|\sup_{t}|(1+t^{2})^{N}(I+\Delta)^{M}f(t)\varphi_{0}(t)|<% \infty\}.
  81. | F ( λ ) | C e R | Im λ | . |F(\lambda)|\leq Ce^{R\cdot|{\rm Im}\,\lambda|}.
  82. ( M 1 * F ) = F ~ {(M_{1}^{*}F)}^{\sim}=\tilde{F}
  83. 2 \mathfrak{H}^{2}
  84. π λ ( g - 1 ) ξ ( x ) = | c x + d | - 1 - i λ ξ ( g ( x ) ) . \pi_{\lambda}(g^{-1})\xi(x)=|cx+d|^{-1-i\lambda}\xi(g(x)).
  85. ξ 0 ( x ) = π - 1 ( 1 + | x | 2 ) - 1 \xi_{0}(x)=\pi^{-1}(1+|x|^{2})^{-1}
  86. Φ λ ( g ) = ( π λ ( g ) ξ 0 , ξ 0 ) . \Phi_{\lambda}(g)=(\pi_{\lambda}(g)\xi_{0},\xi_{0}).
  87. L 2 ( 2 ) L^{2}({\mathfrak{H}}^{2})
  88. π λ / 2 tanh ( π λ / 2 ) d λ {\pi\lambda/2}\cdot\tanh(\pi\lambda/2)d\lambda
  89. W f ( λ , x ) = G / K f ( g ) π λ ( g ) ξ 0 ( x ) d g Wf(\lambda,x)=\int_{G/K}f(g)\pi_{\lambda}(g)\xi_{0}(x)\,dg
  90. L 2 ( 2 ) L^{2}({\mathfrak{H}}^{2})
  91. 3 \mathfrak{H}^{3}
  92. g t = ( cosh t i sinh t - i sinh t cosh t ) . g_{t}=\begin{pmatrix}\cosh t&i\sinh t\\ -i\sinh t&\cosh t\end{pmatrix}.
  93. A = ( a + b x + i y x - i y a - b ) A=\begin{pmatrix}a+b&x+iy\\ x-iy&a-b\end{pmatrix}
  94. g A = g A g * . g\cdot A=gAg^{*}.\,
  95. g t A = ( a cosh 2 t + y sinh 2 t + b x + i ( y cosh 2 t + a sinh 2 t ) x - i ( y cosh 2 t + a sinh 2 t ) a cosh 2 t + y sinh 2 t - b ) . g_{t}\cdot A=\begin{pmatrix}a\cosh 2t+y\sinh 2t+b&x+i(y\cosh 2t+a\sinh 2t)\\ x-i(y\cosh 2t+a\sinh 2t)&a\cosh 2t+y\sinh 2t-b\end{pmatrix}.
  96. a 2 = 1 + b 2 + x 2 + y 2 a^{2}=1+b^{2}+x^{2}+y^{2}
  97. d V = ( 1 + r 2 ) - 1 / 2 d b d x d y , d A = ( 1 + r 2 ) - 1 / 2 d b d x , dV=(1+r^{2})^{-1/2}\,db\,dx\,dy,\,\,\,dA=(1+r^{2})^{-1/2}\,db\,dx,
  98. r = sinh t r=\sinh t
  99. Δ n = - L n - R n 2 - ( n - 1 ) R n , \Delta_{n}=-L_{n}-R_{n}^{2}-(n-1)R_{n},\,
  100. L 2 = b 2 + x 2 , R 2 = b b + x x L_{2}=\partial_{b}^{2}+\partial_{x}^{2},\,\,\,R_{2}=b\partial_{b}+x\partial_{x}
  101. L 3 = b 2 + x 2 + y 2 , R 3 = b b + x x + y y . L_{3}=\partial_{b}^{2}+\partial_{x}^{2}+\partial_{y}^{2},\,\,\,R_{3}=b\partial% _{b}+x\partial_{x}+y\partial_{y}.\,
  102. H 3 F d V = 4 π - F ( t ) sinh 2 t d t , H 2 f d V = 2 π - f ( t ) sinh t d t . \int_{H^{3}}F\,dV=4\pi\int_{-\infty}^{\infty}F(t)\sinh^{2}t\,dt,\,\,\,\int_{H^% {2}}f\,dV=2\pi\int_{-\infty}^{\infty}f(t)\sinh t\,dt.
  103. E f ( b , x , y ) = f ( b , x ) . Ef(b,x,y)=f(b,x).\,
  104. Δ 3 E f = E ( Δ 2 - R 2 ) f . \Delta_{3}Ef=E(\Delta_{2}-R_{2})f.\,
  105. ( - Δ 2 + R 2 ) f = t 2 f + coth t t f + r r f = t 2 f + ( coth t + tanh t ) t f . (-\Delta_{2}+R_{2})f=\partial_{t}^{2}f+\coth t\partial_{t}f+r\partial_{r}f=% \partial_{t}^{2}f+(\coth t+\tanh t)\partial_{t}f.
  106. t 2 + ( coth t + tanh t ) t = t 2 + 2 coth ( 2 t ) t . \partial_{t}^{2}+(\coth t+\tanh t)\partial_{t}=\partial_{t}^{2}+2\coth(2t)% \partial_{t}.
  107. ( Δ 2 - R 2 ) S f = 4 S Δ 2 f , \displaystyle{(\Delta_{2}-R_{2})Sf=4S\Delta_{2}f},
  108. Δ 3 M 0 f = 4 M 0 Δ 2 f . \displaystyle{\Delta_{3}M_{0}f=4M_{0}\Delta_{2}f.}
  109. Q F = K 1 F g s d s QF=\int_{K_{1}}F\circ g_{s}\,ds
  110. H 3 F d V = H 2 ( 1 + b 2 + x 2 ) 1 / 2 Q F d A . \int_{H^{3}}F\,dV=\int_{H^{2}}(1+b^{2}+x^{2})^{1/2}QF\,dA.
  111. M * F ( t ) = Q F ( t / 2 ) . M^{*}F(t)=QF(t/2).
  112. H 3 F d V = H 2 M * F d A \int_{H^{3}}F\,dV=\int_{H^{2}}M^{*}F\,dA
  113. M * ( ( M f ) F ) = f ( M * F ) , M^{*}((Mf)\cdot F)=f\cdot(M^{*}F),
  114. H 3 ( M f ) F d V = H 2 f ( M * F ) d V . \int_{H^{3}}(Mf)\cdot F\,dV=\int_{H^{2}}f\cdot(M*F)\,dV.
  115. M * Δ 3 = 4 Δ 2 M * . M^{*}\Delta_{3}=4\Delta_{2}M^{*}.
  116. M * Φ 2 λ = b ( λ ) φ λ M^{*}\Phi_{2\lambda}=b(\lambda)\varphi_{\lambda}
  117. b ( λ ) = M * Φ 2 λ ( 0 ) = π tanh π λ \displaystyle{b(\lambda)=M^{*}\Phi_{2\lambda}(0)=\pi\tanh\pi\lambda}
  118. Φ 2 λ = M φ λ . \Phi_{2\lambda}=M\varphi_{\lambda}.
  119. ( M * F ) ( λ ) = F ~ ( 2 λ ) . {(M^{*}F)}^{\sim}(\lambda)=\tilde{F}(2\lambda).
  120. f ( x ) = - φ λ ( x ) f ~ ( λ ) λ π 2 tanh ( π λ 2 ) d λ , f(x)=\int_{-\infty}^{\infty}\varphi_{\lambda}(x)\tilde{f}(\lambda)\,{\lambda% \pi\over 2}\tanh({\pi\lambda\over 2})\,d\lambda,
  121. φ λ ( g ) = K α ( k g ) d k , \varphi_{\lambda}(g)=\int_{K}\alpha^{\prime}(kg)\,dk,
  122. f ~ ( λ ) = S f ( s ) α ( s ) d s , \tilde{f}(\lambda)=\int_{S}f(s)\alpha^{\prime}(s)\,ds,
  123. f ~ ( λ ) = - 0 f ( ( a 2 + a - 2 + b 2 ) / 2 ) a - i λ / 2 d a d b , \tilde{f}(\lambda)=\int_{-\infty}^{\infty}\int_{0}^{\infty}f((a^{2}+a^{-2}+b^{% 2})/2)a^{-i\lambda/2}\,da\,db,
  124. F ( u ) = - f ( u + t 2 2 ) d t , F(u)=\int_{-\infty}^{\infty}f(u+{t^{2}\over 2})\,dt,
  125. f ~ ( λ ) = 0 F ( a 2 + a - 2 2 ) a - i λ d a = 0 F ( cosh t ) e - i t λ d t . \tilde{f}(\lambda)=\int_{0}^{\infty}F({a^{2}+a^{-2}\over 2})a^{-i\lambda}\,da=% \int_{0}^{\infty}F(\cosh t)e^{-it\lambda}\,dt.
  126. f ( x ) = - 1 2 π - F ( x + t 2 2 ) d t . f(x)={-1\over 2\pi}\int_{-\infty}^{\infty}F^{\prime}(x+{t^{2}\over 2})\,dt.
  127. - F ( x + t 2 2 ) d t = - - f ( x + t 2 + u 2 2 ) d t d u = 2 π 0 f ( x + r 2 2 ) r d r = 2 π f ( x ) . \int_{-\infty}^{\infty}F^{\prime}(x+{t^{2}\over 2})\,dt=\int_{-\infty}^{\infty% }\int_{-\infty}^{\infty}f^{\prime}(x+{t^{2}+u^{2}\over 2})\,dt\,du={2\pi}\int_% {0}^{\infty}f^{\prime}(x+{r^{2}\over 2})r\,dr=2\pi f(x).
  128. f ~ \tilde{f}
  129. F ( cosh t ) = 2 π 0 f ~ ( i λ ) cos ( λ t ) d λ . F(\cosh t)={2\over\pi}\int_{0}^{\infty}\tilde{f}(i\lambda)\cos(\lambda t)\,d\lambda.
  130. f ( i ) = 1 2 π 2 0 f ~ ( λ ) λ d λ - sin λ t / 2 sinh t cosh t 2 d t = 1 2 π 2 - f ~ ( λ ) λ π 2 tanh ( π λ 2 ) d λ . f(i)={1\over 2\pi^{2}}\int_{0}^{\infty}\tilde{f}(\lambda)\lambda\,d\lambda\int% _{-\infty}^{\infty}{\sin\lambda t/2\over\sinh t}\cosh{t\over 2}\,dt={1\over 2% \pi^{2}}\int_{-\infty}^{\infty}\tilde{f}(\lambda){\lambda\pi\over 2}\tanh({\pi% \lambda\over 2})\,d\lambda.
  131. f 1 ( w ) = K f ( g k w ) d k , f_{1}(w)=\int_{K}f(gkw)\,dk,
  132. 2 \mathfrak{H}^{2}
  133. π λ ( f ) ξ 0 = f ~ ( λ ) ξ 0 \pi_{\lambda}(f)\xi_{0}=\tilde{f}(\lambda)\xi_{0}
  134. f ~ 1 ( λ ) = f ~ ( λ ) φ λ ( w ) , \tilde{f}_{1}(\lambda)=\tilde{f}(\lambda)\cdot\varphi_{\lambda}(w),
  135. f ( w ) = 1 π 2 0 f ~ ( λ ) φ λ ( w ) λ π 2 tanh ( π λ 2 ) d λ . f(w)={1\over\pi^{2}}\int_{0}^{\infty}\tilde{f}(\lambda)\varphi_{\lambda}(w){% \lambda\pi\over 2}\tanh({\pi\lambda\over 2})\,d\lambda.
  136. 𝔞 \mathfrak{a}
  137. 𝔤 {\mathfrak{g}}
  138. 𝔲 \mathfrak{u}
  139. 𝔤 = 𝔲 i 𝔲 . \mathfrak{g}=\mathfrak{u}\oplus i\mathfrak{u}.
  140. 𝔱 \mathfrak{t}
  141. A = exp i 𝔱 , P = exp i 𝔲 , A=\exp i\mathfrak{t},\,\,P=\exp i\mathfrak{u},
  142. G = P U = U A U . G=P\cdot U=UAU.
  143. 𝔱 * \mathfrak{t}^{*}
  144. χ λ ( e X ) = Tr π λ ( e X ) , ( X 𝔱 ) ) , d ( λ ) = dim π λ . \chi_{\lambda}(e^{X})={\rm Tr}\,\pi_{\lambda}(e^{X}),(X\in\mathfrak{t})),\,\,% \,d(\lambda)={\rm dim}\,\pi_{\lambda}.
  145. 𝔱 * × 𝔱 \mathfrak{t}^{*}\times\mathfrak{t}
  146. 𝔱 * \mathfrak{t}^{*}
  147. χ λ ( e X ) = σ W sign ( σ ) e i λ ( σ X ) δ ( e X ) , \chi_{\lambda}(e^{X})={\sum_{\sigma\in W}{\rm sign}(\sigma)e^{i\lambda(\sigma X% )}\over\delta(e^{X})},
  148. W = N U ( T ) / T W=N_{U}(T)/T
  149. 𝔱 \mathfrak{t}
  150. G F ( g ) d g = 1 | W | 𝔞 F ( e X ) | δ ( e X ) | 2 d X . \int_{G}F(g)\,dg={1\over|W|}\int_{\mathfrak{a}}F(e^{X})\,|\delta(e^{X})|^{2}\,dX.
  151. 𝔞 = i 𝔱 \mathfrak{a}=i\mathfrak{t}
  152. 𝔞 = i 𝔱 * \mathfrak{a}=i\mathfrak{t}^{*}
  153. Φ λ ( e X ) = χ λ ( e X ) d ( λ ) . \Phi_{\lambda}(e^{X})={\chi_{\lambda}(e^{X})\over d(\lambda)}.
  154. F ~ ( λ ) = G F ( g ) Φ - λ ( g ) d g \tilde{F}(\lambda)=\int_{G}F(g)\Phi_{-\lambda}(g)\,dg
  155. F ( g ) = 1 | W | 𝔞 * F ~ ( λ ) Φ λ ( g ) | d ( λ ) | 2 d λ = 𝔞 + * F ~ ( λ ) Φ λ ( g ) | d ( λ ) | 2 d λ , F(g)={1\over|W|}\int_{{\mathfrak{a}}^{*}}\tilde{F}(\lambda)\Phi_{\lambda}(g)|d% (\lambda)|^{2}\,d\,\lambda=\int_{{\mathfrak{a}}^{*}_{+}}\tilde{F}(\lambda)\Phi% _{\lambda}(g)|d(\lambda)|^{2}\,d\,\lambda,
  156. 𝔞 + * {\mathfrak{a}}^{*}_{+}
  157. 𝔞 \mathfrak{a}
  158. d ( λ ) δ ( e X ) Φ λ ( e X ) = σ W sign ( σ ) e i λ ( X ) , d(\lambda)\delta(e^{X})\Phi_{\lambda}(e^{X})=\sum_{\sigma\in W}{\rm sign}(% \sigma)e^{i\lambda(X)},
  159. d ( λ ) ¯ F ~ ( λ ) \displaystyle\overline{d(\lambda)}\tilde{F}(\lambda)
  160. F ( e X ) δ ( e X ) \displaystyle F(e^{X})\delta(e^{X})
  161. 𝔞 + * {\mathfrak{a}}^{*}_{+}
  162. f ~ ( λ ) = U f ( u ) χ λ ( u ) ¯ d ( λ ) d u \tilde{f}(\lambda)=\int_{U}f(u){\overline{\chi_{\lambda}(u)}\over d(\lambda)}% \,du
  163. f ( u ) = λ f ~ ( λ ) χ λ ( u ) d ( λ ) d ( λ ) 2 . f(u)=\sum_{\lambda}\tilde{f}(\lambda){\chi_{\lambda}(u)\over d(\lambda)}d(% \lambda)^{2}.
  164. G = K A + U , \displaystyle G=KA_{+}U,
  165. 𝔞 \mathfrak{a}
  166. K \ G / U = A + . K\backslash G/U=A_{+}.
  167. K 0 \ G 0 / K 0 = A + K_{0}\backslash G_{0}/K_{0}=A_{+}
  168. C c ( K 0 \ G 0 / K 0 ) C^{\infty}_{c}(K_{0}\backslash G_{0}/K_{0})
  169. C c ( U \ G / U ) C^{\infty}_{c}(U\backslash G/U)
  170. M f ( a ) = U f ( u a 2 ) d u . \displaystyle Mf(a)=\int_{U}f(ua^{2})\,du.
  171. 4 M Δ = Δ c M . \displaystyle 4M\Delta=\Delta_{c}M.
  172. C c ( U \ G / U ) C^{\infty}_{c}(U\backslash G/U)
  173. C c ( K 0 \ G 0 / K 0 ) C^{\infty}_{c}(K_{0}\backslash G_{0}/K_{0})
  174. M * F ( a 2 ) = K F ( g a ) d g . \displaystyle M^{*}F(a^{2})=\int_{K}F(ga)\,dg.
  175. G / U ( M f ) F = G 0 / K 0 f ( M * F ) . \displaystyle\int_{G/U}(Mf)\cdot F=\int_{G_{0}/K_{0}}f\cdot(M^{*}F).
  176. M * Δ c = 4 Δ M * . \displaystyle M^{*}\Delta_{c}=4\Delta M^{*}.
  177. M * Φ 2 λ M^{*}\Phi_{2\lambda}
  178. b ( λ ) = M * Φ 2 λ ( 1 ) = K Φ 2 λ ( k ) d k . b(\lambda)=M^{*}\Phi_{2\lambda}(1)=\int_{K}\Phi_{2\lambda}(k)\,dk.
  179. ( M * F ) ( λ ) = F ~ ( 2 λ ) . \displaystyle(M^{*}F)^{\sim}(\lambda)=\tilde{F}(2\lambda).
  180. f ( g ) = 𝔞 + * f ~ ( λ ) φ λ ( g ) 2 dim A | b ( λ ) | | d ( 2 λ ) | 2 d λ , f(g)=\int_{\mathfrak{a}^{*}_{+}}\tilde{f}(\lambda)\varphi_{\lambda}(g)\,\,2^{{% \rm dim}\,A}\cdot|b(\lambda)|\cdot|d(2\lambda)|^{2}\,d\lambda,
  181. f ( g ) = M * F ( g ) = 𝔞 + * F ~ ( 2 λ ) M * Φ 2 λ ( g ) 2 dim A | d ( 2 λ ) | 2 d λ = 𝔞 + * f ~ ( λ ) φ λ ( g ) b ( λ ) 2 dim A | d ( 2 λ ) | 2 d λ . f(g)=M^{*}F(g)=\int_{\mathfrak{a}_{+}^{*}}\tilde{F}(2\lambda)M^{*}\Phi_{2% \lambda}(g)2^{{\rm dim}\,A}|d(2\lambda)|^{2}\,d\lambda=\int_{\mathfrak{a}_{+}^% {*}}\tilde{f}(\lambda)\varphi_{\lambda}(g)\,\,b(\lambda)2^{{\rm dim}\,A}|d(2% \lambda)|^{2}\,d\lambda.
  182. b ( λ ) = C d ( 2 λ ) - 1 α > 0 tanh π ( α , λ ) ( α , α ) , b(\lambda)=C\cdot d(2\lambda)^{-1}\cdot\prod_{\alpha>0}\tanh{\pi(\alpha,% \lambda)\over(\alpha,\alpha)},
  183. 𝔞 \mathfrak{a}
  184. 𝔤 \mathfrak{g}
  185. 𝔤 ± \mathfrak{g}_{\pm}
  186. 𝔤 \mathfrak{g}
  187. 𝔨 = 𝔤 + \mathfrak{k}=\mathfrak{g}_{+}
  188. 𝔭 = 𝔤 - \mathfrak{p}=\mathfrak{g}_{-}
  189. 𝔤 = 𝔨 + 𝔭 , G = exp 𝔭 K . \mathfrak{g}=\mathfrak{k}+\mathfrak{p},\,\,G=\exp\mathfrak{p}\cdot K.
  190. 𝔞 \mathfrak{a}
  191. 𝔭 \mathfrak{p}
  192. 𝔞 * \mathfrak{a}^{*}
  193. 𝔤 α = { X 𝔤 : [ H , X ] = α ( H ) X ( H 𝔞 ) } . \mathfrak{g}_{\alpha}=\{X\in\mathfrak{g}:[H,X]=\alpha(H)X\,\,(H\in\mathfrak{a}% )\}.
  194. 𝔤 α ( 0 ) \mathfrak{g}_{\alpha}\neq(0)
  195. 𝔤 α \mathfrak{g}_{\alpha}
  196. 𝔞 \mathfrak{a}
  197. 𝔭 \mathfrak{p}
  198. 𝔞 \mathfrak{a}
  199. 𝔞 * \mathfrak{a}^{*}
  200. 𝔞 \mathfrak{a}
  201. W = N K ( A ) / C K ( A ) W=N_{K}(A)/C_{K}(A)
  202. 𝔞 + * \mathfrak{a}_{+}^{*}
  203. 𝔞 * \mathfrak{a}^{*}
  204. f ~ ( λ ) = G f ( g ) φ - λ ( g ) d g . \tilde{f}(\lambda)=\int_{G}f(g)\varphi_{-\lambda}(g)\,dg.
  205. f ( g ) = 𝔞 + * f ~ ( λ ) φ λ ( g ) | c ( λ ) | - 2 d λ , f(g)=\int_{\mathfrak{a}^{*}_{+}}\tilde{f}(\lambda)\varphi_{\lambda}(g)\,|c(% \lambda)|^{-2}\,d\lambda,
  206. c ( λ ) = c 0 α Σ 0 + 2 - i ( λ , α 0 ) Γ ( i ( λ , α 0 ) ) Γ ( 1 2 [ 1 2 m α + 1 + i ( λ , α 0 ) ] ) Γ ( 1 2 [ 1 2 m α + m 2 α + i ( λ , α 0 ) ] ) c(\lambda)=c_{0}\cdot\prod_{\alpha\in\Sigma_{0}^{+}}{2^{-i(\lambda,\alpha_{0})% }\Gamma(i(\lambda,\alpha_{0}))\over\Gamma({1\over 2}[{1\over 2}m_{\alpha}+1+i(% \lambda,\alpha_{0})])\Gamma({1\over 2}[{1\over 2}m_{\alpha}+m_{2\alpha}+i(% \lambda,\alpha_{0})])}
  207. α 0 = ( α , α ) - 1 α \alpha_{0}=(\alpha,\alpha)^{-1}\alpha
  208. ρ = 1 2 α Σ + m α α . \rho={1\over 2}\sum_{\alpha\in\Sigma^{+}}m_{\alpha}\alpha.
  209. W : f f ~ , L 2 ( K \ G / K ) L 2 ( 𝔞 + * , | c ( λ ) | - 2 d λ ) W:f\mapsto\tilde{f},\,\,\,\ L^{2}(K\backslash G/K)\rightarrow L^{2}(\mathfrak{% a}_{+}^{*},|c(\lambda)|^{-2}\,d\lambda)
  210. f L 1 ( K \ G / K ) f\in L^{1}(K\backslash G/K)
  211. f ~ \tilde{f}
  212. 𝔞 \mathfrak{a}
  213. 𝔞 \mathfrak{a}
  214. 𝔞 \mathfrak{a}
  215. X f ( y ) = d d t f ( y + t X ) | t = 0 . Xf(y)={d\over dt}f(y+tX)|_{t=0}.
  216. L = Δ 𝔞 - α > 0 m α coth α A α , \displaystyle L=\Delta_{\mathfrak{a}}-\sum_{\alpha>0}m_{\alpha}\,\coth\alpha\,% A_{\alpha},
  217. 𝔞 \mathfrak{a}
  218. ( A α , X ) = α ( X ) \displaystyle(A_{\alpha},X)=\alpha(X)
  219. Δ 𝔞 = - X i 2 \Delta_{\mathfrak{a}}=-\sum X_{i}^{2}
  220. 𝔞 \mathfrak{a}
  221. L = L 0 - α > 0 m α ( coth α - 1 ) A α , L=L_{0}-\sum_{\alpha>0}m_{\alpha}\,(\coth\alpha-1)A_{\alpha},
  222. L 0 = Δ 𝔞 - α > 0 A α , L_{0}=\Delta_{\mathfrak{a}}-\sum_{\alpha>0}A_{\alpha},
  223. Δ φ λ = ( λ 2 + ρ 2 ) φ λ \Delta\varphi_{\lambda}=(\|\lambda\|^{2}+\|\rho\|^{2})\varphi_{\lambda}
  224. 𝔞 \mathfrak{a}
  225. f λ = e i λ - ρ μ Λ a μ ( λ ) e - μ , f_{\lambda}=e^{i\lambda-\rho}\sum_{\mu\in\Lambda}a_{\mu}(\lambda)e^{-\mu},
  226. coth x - 1 = 2 m > 0 e - 2 m x , \displaystyle\coth x-1=2\sum_{m>0}e^{-2mx},
  227. 𝔞 + \mathfrak{a}_{+}
  228. 𝔞 \mathfrak{a}
  229. φ λ = s W c ( s λ ) f s λ . \varphi_{\lambda}=\sum_{s\in W}c(s\lambda)f_{s\lambda}.
  230. 𝔞 + \mathfrak{a}_{+}
  231. φ λ ( e t X ) c ( λ ) e ( i λ - ρ ) X t \varphi_{\lambda}(e^{t}X)\sim c(\lambda)e^{(i\lambda-\rho)Xt}
  232. 𝔞 + \mathfrak{a}_{+}
  233. G = s W B s B , G=\bigcup_{s\in W}BsB,
  234. 𝔞 + \mathfrak{a}_{+}
  235. - 𝔞 + -\mathfrak{a}_{+}
  236. φ λ ( g ) = K / M λ ( g k ) - 1 d k . \varphi_{\lambda}(g)=\int_{K/M}\lambda^{\prime}(gk)^{-1}\,dk.
  237. φ λ ( e X ) = e i λ - ρ σ ( N ) λ ( n ) ¯ λ ( e X n e - X ) d n , \varphi_{\lambda}(e^{X})=e^{i\lambda-\rho}\int_{\sigma(N)}{\overline{\lambda^{% \prime}(n)}\over\lambda^{\prime}(e^{X}ne^{-X})}\,dn,
  238. 𝔞 \mathfrak{a}
  239. lim t e t X n e - t X = 1 \lim_{t\rightarrow\infty}e^{tX}ne^{-tX}=1
  240. 𝔞 + \mathfrak{a}_{+}
  241. c ( λ ) = σ ( N ) λ ( n ) ¯ d n . c(\lambda)=\int_{\sigma(N)}\overline{\lambda^{\prime}(n)}\,dn.
  242. 𝔞 * \mathfrak{a}^{*}
  243. A ( s , λ ) F ( k ) = σ ( N ) s - 1 N s F ( k s n ) d n , \displaystyle A(s,\lambda)F(k)=\int_{\sigma(N)\cap s^{-1}Ns}F(ksn)\,dn,
  244. A ( s , λ ) π λ ( g ) = π s λ ( g ) A ( s , λ ) . \displaystyle A(s,\lambda)\pi_{\lambda}(g)=\pi_{s\lambda}(g)A(s,\lambda).
  245. A ( s 1 s 2 , λ ) = A ( s 1 , s 2 λ ) A ( s 2 , λ ) , \displaystyle A(s_{1}s_{2},\lambda)=A(s_{1},s_{2}\lambda)A(s_{2},\lambda),
  246. ( s 1 s 2 ) = ( s 1 ) + ( s 2 ) \ell(s_{1}s_{2})=\ell(s_{1})+\ell(s_{2})
  247. 𝔞 + * \mathfrak{a}_{+}^{*}
  248. - 𝔞 + * -\mathfrak{a}_{+}^{*}
  249. c ( λ ) = c s 0 ( λ ) . c(\lambda)=c_{s_{0}}(\lambda).
  250. A ( s , λ ) ξ 0 = c s ( λ ) ξ 0 , \displaystyle A(s,\lambda)\xi_{0}=c_{s}(\lambda)\xi_{0},
  251. c s 1 s 2 ( λ ) = c s 1 ( s 2 λ ) c s 2 ( λ ) c_{s_{1}s_{2}}(\lambda)=c_{s_{1}}(s_{2}\lambda)c_{s_{2}}(\lambda)
  252. ( s 1 s 2 ) = ( s 1 ) + ( s 2 ) . \ell(s_{1}s_{2})=\ell(s_{1})+\ell(s_{2}).
  253. 𝔤 ± α \mathfrak{g}_{\pm\alpha}
  254. c s α ( λ ) = c 0 2 - i ( λ , α 0 ) Γ ( i ( λ , α 0 ) ) Γ ( 1 2 ( 1 2 m α + 1 + i ( λ , α 0 ) ) Γ ( 1 2 ( 1 2 m α + m 2 α + i ( λ , α 0 ) ) , c_{s_{\alpha}}(\lambda)=c_{0}{2^{-i(\lambda,\alpha_{0})}\Gamma(i(\lambda,% \alpha_{0}))\over\Gamma({1\over 2}({1\over 2}m_{\alpha}+1+i(\lambda,\alpha_{0}% ))\Gamma({1\over 2}({1\over 2}m_{\alpha}+m_{2\alpha}+i(\lambda,\alpha_{0}))},
  255. c 0 = 2 m α / 2 + m 2 α Γ ( 1 2 ( m α + m 2 α + 1 ) ) . c_{0}=2^{m_{\alpha}/2+m_{2\alpha}}\Gamma({1\over 2}(m_{\alpha}+m_{2\alpha}+1)).
  256. | f ~ ( λ ) | C N ( 1 + | λ | ) - N e R | Im λ | . |\tilde{f}(\lambda)|\leq C_{N}(1+|\lambda|)^{-N}e^{R|{\rm Im}\,\lambda|}.
  257. 𝔞 * \mathfrak{a}^{*}
  258. 𝔞 * + i μ t \mathfrak{a}^{*}+i\mu t
  259. 𝔞 + * \mathfrak{a}^{*}_{+}
  260. T ( f ) = 𝔞 + * f ~ ( λ ) | c ( λ ) | - 2 d λ \displaystyle T(f)=\int_{\mathfrak{a}_{+}^{*}}\tilde{f}(\lambda)|c(\lambda)|^{% -2}\,d\lambda
  261. T ( f ) = C f ( o ) . \displaystyle T(f)=Cf(o).
  262. f 1 ( g ) = K f ( x - 1 k g ) d k , f_{1}(g)=\int_{K}f(x^{-1}kg)\,dk,
  263. C f = 𝔞 + * f ~ ( λ ) φ λ | c ( λ ) | - 2 d λ . Cf=\int_{\mathfrak{a}_{+}^{*}}\tilde{f}(\lambda)\varphi_{\lambda}|c(\lambda)|^% {-2}\,d\lambda.
  264. 𝔞 \mathfrak{a}
  265. 𝒮 ( K \ G / K ) = { f C ( G / K ) K : sup x | ( 1 + d ( x , o ) ) m ( Δ + I ) n f ( x ) | < } . \mathcal{S}(K\backslash G/K)=\{f\in C^{\infty}(G/K)^{K}:\sup_{x}|(1+d(x,o))^{m% }(\Delta+I)^{n}f(x)|<\infty\}.
  266. 𝒮 ( 𝔞 * ) W \mathcal{S}(\mathfrak{a}^{*})^{W}
  267. 𝔞 * \mathfrak{a}^{*}
  268. 𝔄 \mathfrak{A}
  269. χ λ d ( λ ) - 1 / 2 \chi_{\lambda}d(\lambda)^{-1/2}

Planck_constant.html

  1. h h
  2. E E
  3. ν ν
  4. E = h ν . E=h\nu.
  5. E E
  6. ν ν
  7. λ λ
  8. c c
  9. λ ν = c λν=c
  10. E = h c λ . E=\frac{hc}{\lambda}.
  11. p p
  12. λ λ
  13. λ = h p . \lambda=\frac{h}{p}.
  14. 2 π
  15. 2 π
  16. ħ ħ
  17. = h 2 π . \hbar=\frac{h}{2\pi}.
  18. ω ω
  19. ω = 2 π ν ω=2πν
  20. E = ω , E=\hbar\omega,
  21. p = k . p=\hbar k.
  22. h h
  23. h = 6.626 069 57 ( 29 ) × 10 - 34 J⋅s = 4.135 667 516 ( 91 ) × 10 - 15 eV⋅s . h=6.626\ 069\ 57(29)\times 10^{-34}\,\text{ J⋅s}=4.135\ 667\ 516(91)\times 10^% {-15}\,\text{ eV⋅s}.
  24. = h 2 π = 1.054 571 726 ( 47 ) × 10 - 34 J⋅s = 6.582 119 28 ( 15 ) × 10 - 16 eV⋅s . \hbar={{h}\over{2\pi}}=1.054\ 571\ 726(47)\times 10^{-34}\,\text{ J⋅s}=6.582\ % 119\ 28(15)\times 10^{-16}\,\text{ eV⋅s}.
  25. E = h ν . E=h\nu.
  26. E = h ν . E=h\nu.
  27. E n = - h c 0 R n 2 , E_{n}=-\frac{hc_{0}R_{\infty}}{n^{2}},
  28. h 2 π \frac{h}{2\pi}
  29. J 2 = j ( j + 1 ) 2 , j = 0 , 1 2 , 1 , 3 2 , , J z = m , m = - j , - j + 1 , , j . \begin{aligned}\displaystyle J^{2}=j(j+1)\hbar^{2},&\displaystyle j=0,\tfrac{1% }{2},1,\tfrac{3}{2},\ldots,\\ \displaystyle J_{z}=m\hbar,&\displaystyle m=-j,-j+1,\ldots,j.\end{aligned}
  30. Δ x Δ p 2 , \Delta x\,\Delta p\geq\frac{\hbar}{2},
  31. x ^ \hat{x}
  32. p ^ \hat{p}
  33. [ p ^ i , x ^ j ] = - i δ i j , [\hat{p}_{i},\hat{x}_{j}]=-i\hbar\delta_{ij},
  34. R = m e e 4 8 ϵ 0 2 h 3 c 0 = m e c 0 α 2 2 h . R_{\infty}=\frac{m_{\rm e}e^{4}}{8\epsilon_{0}^{2}h^{3}c_{0}}=\frac{m_{\rm e}c% _{0}\alpha^{2}}{2h}.
  35. m e = 2 R h c 0 α 2 , m_{\rm e}=\frac{2R_{\infty}h}{c_{0}\alpha^{2}},
  36. N A = M u A r ( e ) m e = M u A r ( e ) c 0 α 2 2 R h . N_{\rm A}=\frac{M_{\rm u}A_{\rm r}({\rm e})}{m_{\rm e}}=\frac{M_{\rm u}A_{\rm r% }({\rm e})c_{0}\alpha^{2}}{2R_{\infty}h}.
  37. α = e 2 c 0 4 π ϵ 0 = e 2 c 0 μ 0 2 h , \alpha\ =\ \frac{e^{2}}{\hbar c_{0}\ 4\pi\epsilon_{0}}\ =\ \frac{e^{2}c_{0}\mu% _{0}}{2h},
  38. e = 2 α h μ 0 c 0 = < m t p l > 2 α h ϵ 0 c 0 e=\sqrt{\frac{2\alpha h}{\mu_{0}c_{0}}}=\sqrt{<}mtpl>{{2\alpha h\epsilon_{0}c_% {0}}}
  39. μ B = e 2 m e = c 0 α 5 h 32 π 2 μ 0 R 2 \mu_{\rm B}=\frac{e\hbar}{2m_{\rm e}}=\sqrt{\frac{c_{0}\alpha^{5}h}{32\pi^{2}% \mu_{0}R_{\infty}^{2}}}
  40. μ N = μ B A r ( e ) A r ( p ) \mu_{\rm N}=\mu_{\rm B}\frac{A_{\rm r}({\rm e})}{A_{\rm r}({\rm p})}
  41. × 10 6 \times 10^{−}6
  42. × 10 8 \times 10^{−}8
  43. K J = ν U = 2 e h K_{\rm J}=\frac{\nu}{U}=\frac{2e}{h}\,
  44. h = 8 α μ 0 c 0 K J 2 . h=\frac{8\alpha}{\mu_{0}c_{0}K_{\rm J}^{2}}.
  45. h = 4 K J 2 R K . h=\frac{4}{K_{\rm J}^{2}R_{\rm K}}.
  46. γ p = μ p I = 2 μ p \gamma^{\prime}_{\rm p}=\frac{\mu^{\prime}_{\rm p}}{I\hbar}=\frac{2\mu^{\prime% }_{\rm p}}{\hbar}
  47. μ p = μ p μ e g e μ B 2 \mu^{\prime}_{\rm p}=\frac{\mu^{\prime}_{\rm p}}{\mu_{\rm e}}\frac{g_{\rm e}% \mu_{\rm B}}{2}
  48. γ p = μ p μ e g e μ B . \gamma^{\prime}_{\rm p}=\frac{\mu^{\prime}_{\rm p}}{\mu_{\rm e}}\frac{g_{\rm e% }\mu_{\rm B}}{\hbar}.
  49. γ p = K J - 90 R K - 90 K J R K Γ p - 90 ( hi ) = K J - 90 R K - 90 e 2 Γ p - 90 ( hi ) \gamma^{\prime}_{\rm p}=\frac{K_{\rm J-90}R_{\rm K-90}}{K_{\rm J}R_{\rm K}}% \Gamma^{\prime}_{\rm p-90}({\rm hi})=\frac{K_{\rm J-90}R_{\rm K-90}e}{2}\Gamma% ^{\prime}_{\rm p-90}({\rm hi})
  50. h = c 0 α 2 g e 2 K J - 90 R K - 90 R Γ p - 90 ( hi ) μ p μ e . h=\frac{c_{0}\alpha^{2}g_{\rm e}}{2K_{\rm J-90}R_{\rm K-90}R_{\infty}\Gamma^{% \prime}_{\rm p-90}({\rm hi})}\frac{\mu_{\rm p}^{\prime}}{\mu_{\rm e}}.
  51. h = c 0 M u A r ( e ) α 2 R 1 K J - 90 R K - 90 F 90 h=\frac{c_{0}M_{\rm u}A_{\rm r}({\rm e})\alpha^{2}}{R_{\infty}}\frac{1}{K_{\rm J% -90}R_{\rm K-90}F_{90}}
  52. h = M u A r ( e ) c 0 α 2 R 2 d 220 3 V m ( Si ) . h=\frac{M_{\rm u}A_{\rm r}({\rm e})c_{0}\alpha^{2}}{R_{\infty}}\frac{\sqrt{2}d% ^{3}_{220}}{V_{\rm m}({\rm Si})}.
  53. h 90 = 4 K J - 90 2 R K - 90 h_{90}=\frac{4}{K_{J-90}^{2}R_{K-90}}

Plate_notation.html

  1. θ i \theta_{i}
  2. N i N_{i}
  3. z i j z_{ij}
  4. w i j w_{ij}
  5. N i N_{i}
  6. w i j w_{ij}
  7. w i j w_{ij}
  8. z i j z_{ij}

Plato's_number.html

  1. 6 3 = 216 6^{3}=216
  2. 3 3 + 4 3 + 5 3 = 6 3 . 3^{3}+4^{3}+5^{3}=6^{3}.\,
  3. 48 × 27 = 36 × 36 = 1296 48\times 27=36\times 36=1296
  4. 17500 = 100 × 100 + 4800 + 2700 17500=100\times 100+4800+2700
  5. 760000 = 750000 + 10000 = 19 × 4 × 10000 760000=750000+10000=19\times 4\times 10000
  6. ( 4 3 + 5 ) × 3 (\tfrac{4}{3}+5)\times 3
  7. 8128 = 2 6 ( 2 7 - 1 ) 8128=2^{6}\cdot(2^{7}-1)
  8. 1728 = 12 3 = 8 12 18 1728=12^{3}=8\cdot 12\cdot 18
  9. = 144 × 35 = ( 3 + 4 + 5 ) 2 ( 2 3 + 3 3 ) =144\times 35=(3+4+5)^{2}\cdot(2^{3}+3^{3})

Playfair's_law.html

  1. ε ˙ = k A m S n \dot{\varepsilon}=kA^{m}S^{n}
  2. ε \ \varepsilon
  3. ε ˙ \ \dot{\varepsilon}
  4. k \ k
  5. A \ A
  6. S \ S
  7. m , n \ m,n
  8. k \ k
  9. m , n m,n

Plot_(graphics).html

  1. ln ( k ) \ln(k)
  2. 1 / T 1/T
  3. x ( i ) x ( j ) , \vec{x}(i)\approx\vec{x}(j),\,
  4. i i
  5. j j
  6. x \vec{x}

Pocket_set_theory.html

  1. 0 \scriptstyle{\aleph_{0}}
  2. 2 0 \scriptstyle{2^{\aleph_{0}}}
  3. \scriptstyle{\in}
  4. X Y \scriptstyle{X\in Y}
  5. z ( z X z Y ) X = Y \forall z\,(z\in X\leftrightarrow z\in Y)\rightarrow X=Y
  6. ϕ ( x ) \scriptstyle{\phi(x)}
  7. ϕ ( x ) \scriptstyle{\phi(x)}
  8. Y x ( x Y ϕ ( x ) ) \exists Y\forall x\,(x\in Y\leftrightarrow\phi(x))
  9. x ( inf ( x ) y ( inf ( y ) x y ) ) \exists x\,(\mathrm{inf}(x)\land\forall y\,(\mathrm{inf}(y)\rightarrow x% \approx y))
  10. x y \scriptstyle{x\approx y}
  11. X Y ( ( pr ( X ) pr ( Y ) ) ( pr ( X ) X Y ) ) \forall X\forall Y\,((\mathrm{pr}(X)\land\mathrm{pr}(Y))\leftrightarrow(% \mathrm{pr}(X)\land X\approx Y))
  12. x ϕ ( x ) def X ( set ( X ) ϕ ( X ) ) \forall x\,\phi(x)\Leftrightarrow_{\mathrm{def}}\forall X\,(\mathrm{set}(X)% \rightarrow\phi(X))
  13. ϕ ( x ) \scriptstyle{\phi(x)}
  14. R \scriptstyle{\mathrm{R}}
  15. R = def { x | x x } \scriptstyle{\mathrm{R}=_{\mathrm{def}}\{x\,|\,x\notin x\}}
  16. R \scriptstyle{\mathrm{R}}
  17. \scriptstyle{\emptyset}
  18. = def { x | x x } \scriptstyle{\emptyset=_{\mathrm{def}}\{x\,|\,x\neq x\}}
  19. \scriptstyle{\emptyset}
  20. \scriptstyle{\emptyset}
  21. R \scriptstyle{\mathrm{R}}
  22. R \scriptstyle{\mathrm{R}}
  23. { i } \scriptstyle{\{i\}}
  24. \scriptstyle{\emptyset}
  25. R \scriptstyle{\mathrm{R}}
  26. R \scriptstyle{\mathrm{R}}
  27. { } \scriptstyle{\{\emptyset\}}
  28. { } \scriptstyle{\{\emptyset\}}
  29. { , i } \scriptstyle{\{\emptyset,i\}}
  30. { , i } R \scriptstyle{\{\emptyset,i\}\in R}
  31. R \scriptstyle{\mathrm{R}}
  32. \scriptstyle{\emptyset}
  33. { , i } \scriptstyle{\{\emptyset,i\}}
  34. R \scriptstyle{\mathrm{R}}
  35. R - = def R { } \scriptstyle{\mathrm{R}^{-}=_{\mathrm{def}}\mathrm{R}\setminus\{\emptyset\}}
  36. R - R - \scriptstyle{\mathrm{R}^{-}\in\mathrm{R}^{-}}
  37. R - R - \scriptstyle{\mathrm{R}^{-}\notin\mathrm{R}^{-}}
  38. R - \scriptstyle{\mathrm{R}^{-}}
  39. R - R \scriptstyle{\mathrm{R}^{-}\in\mathrm{R}}
  40. R - R - \scriptstyle{\mathrm{R}^{-}\notin\mathrm{R}^{-}}
  41. R - \scriptstyle{\mathrm{R}^{-}}
  42. R - R \scriptstyle{\mathrm{R}^{-}\notin\mathrm{R}}
  43. R - R - \scriptstyle{\mathrm{R}^{-}\in\mathrm{R}^{-}}
  44. R - = \scriptstyle{\mathrm{R}^{-}=\emptyset}
  45. R - \scriptstyle{\mathrm{R}^{-}}
  46. { } \scriptstyle{\{\emptyset\}}
  47. F : X R \scriptstyle{F:X\longrightarrow\mathrm{R}}
  48. x 0 , \scriptstyle{\langle x_{0},\emptyset\rangle}
  49. R - \scriptstyle{\mathrm{R}^{-}}
  50. x , r \scriptstyle{\langle x,r\rangle}
  51. X - = X { x 0 } \scriptstyle{X^{-}=X\setminus\{x_{0}\}}
  52. F - = F { x 0 , } \scriptstyle{F^{-}=F\setminus\{\langle x_{0},\emptyset\rangle\}}
  53. F - : X - R - \scriptstyle{F^{-}:X^{-}\longrightarrow\mathrm{R}^{-}}
  54. X - \scriptstyle{X^{-}}
  55. X - X \scriptstyle{X^{-}\subseteq X}
  56. X - X \scriptstyle{X^{-}\neq X}
  57. G : X X - \scriptstyle{G:X\longrightarrow X^{-}}
  58. V = def { x | set ( x ) } \scriptstyle{\mathrm{V}=_{\mathrm{def}}\{x\,|\mathrm{set}(x)\}}
  59. 𝔠 {\mathfrak{c}}
  60. 𝒫 ( i ) \scriptstyle{\mathcal{P}(i)}
  61. 2 0 \scriptstyle{2^{\aleph_{0}}}
  62. 2 0 \scriptstyle{2^{\aleph_{0}}}
  63. x y ( inf ( x ) inf ( y ) | 𝒫 ( x ) | | 𝒫 ( y ) | z ( inf ( z ) ( | z | = | x | | z | = | y | ) ) ) \exists x\exists y\,(\mathrm{inf}(x)\land\mathrm{inf}(y)\land|\mathcal{P}(x)|% \neq|\mathcal{P}(y)|\land\forall z(\mathrm{inf}(z)\rightarrow(|z|=|x|\lor|z|=|% y|)))
  64. 0 \aleph_{0}
  65. 2 0 2^{\aleph_{0}}
  66. 2 2 0 2^{2^{\aleph_{0}}}

Poincaré_space.html

  1. [ M ] . [M].

Poisson_limit_theorem.html

  1. n , p 0 n\rightarrow\infty,p\rightarrow 0
  2. n p λ np\rightarrow\lambda
  3. n ! ( n - k ) ! k ! p k ( 1 - p ) n - k e - λ λ k k ! . \frac{n!}{(n-k)!k!}p^{k}(1-p)^{n-k}\rightarrow e^{-\lambda}\frac{\lambda^{k}}{% k!}.
  4. p n ( k ) p_{n}(k)
  5. p = 10 / 1000 = 0.01 p=10/1000=0.01
  6. n = 500 n=500
  7. n p = 5 np=5
  8. k k
  9. p n ( k ) = n ! ( n - k ) ! k ! p k ( 1 - p ) n - k . p_{n}(k)=\frac{n!}{(n-k)!k!}p^{k}(1-p)^{n-k}.
  10. p p
  11. n ! / ( k ! ( n - k ) ! ) n!/(k!\cdot(n-k)!)
  12. k k
  13. p k p^{k}
  14. k k
  15. ( 1 - p ) n - k (1-p)^{n-k}
  16. n - k {n-k}
  17. e - λ λ k k ! = e - 5 5 k k ! . e^{-\lambda}\frac{\lambda^{k}}{k!}=e^{-5}\frac{5^{k}}{k!}.
  18. n ! ( n - k ) ! k ! p k ( 1 - p ) n - k 2 π n ( n e ) n 2 π ( n - k ) ( n - k e ) n - k k ! p k ( 1 - p ) n - k . \frac{n!}{(n-k)!k!}p^{k}(1-p)^{n-k}\rightarrow\frac{\sqrt{2\pi n}\left(\frac{n% }{e}\right)^{n}}{\sqrt{2\pi\left(n-k\right)}\left(\frac{n-k}{e}\right)^{n-k}k!% }p^{k}(1-p)^{n-k}.
  19. 2 π n ( n e ) n 2 π ( n - k ) ( n - k e ) n - k k ! p k ( 1 - p ) n - k n n n p k ( 1 - p ) n - k n - k ( n - k ) n - k e k k ! n n p k ( 1 - p ) n - k ( n - k ) n - k e k k ! . \frac{\sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}}{\sqrt{2\pi\left(n-k\right)}% \left(\frac{n-k}{e}\right)^{n-k}k!}p^{k}(1-p)^{n-k}\rightarrow\frac{\sqrt{n}n^% {n}p^{k}(1-p)^{n-k}}{\sqrt{n-k}\left(n-k\right)^{n-k}e^{k}k!}\rightarrow\frac{% n^{n}p^{k}(1-p)^{n-k}}{\left(n-k\right)^{n-k}e^{k}k!}.
  20. n p λ np\rightarrow\lambda
  21. n n p k ( 1 - p ) n - k ( n - k ) n - k e k k ! n k ( λ n ) k ( 1 - λ n ) n - k ( 1 - k n ) n - k e k k ! = λ k ( 1 - λ n ) n - k ( 1 - k n ) n - k e k k ! λ k ( 1 - λ n ) n ( 1 - k n ) n e k k ! \frac{n^{n}p^{k}(1-p)^{n-k}}{\left(n-k\right)^{n-k}e^{k}k!}\rightarrow\frac{n^% {k}\left(\frac{\lambda}{n}\right)^{k}(1-\frac{\lambda}{n})^{n-k}}{\left(1-% \frac{k}{n}\right)^{n-k}e^{k}k!}=\frac{\lambda^{k}\left(1-\frac{\lambda}{n}% \right)^{n-k}}{\left(1-\frac{k}{n}\right)^{n-k}e^{k}k!}\rightarrow\frac{% \lambda^{k}\left(1-\frac{\lambda}{n}\right)^{n}}{\left(1-\frac{k}{n}\right)^{n% }e^{k}k!}
  22. n n\rightarrow\infty
  23. ( 1 + x n ) n e x \left(1+\frac{x}{n}\right)^{n}\rightarrow e^{x}
  24. λ k ( 1 - λ n ) n ( 1 - k n ) n e k k ! λ k e - λ e - k e k k ! = λ k e - λ k ! \frac{\lambda^{k}\left(1-\frac{\lambda}{n}\right)^{n}}{\left(1-\frac{k}{n}% \right)^{n}e^{k}k!}\rightarrow\frac{\lambda^{k}e^{-\lambda}}{e^{-k}e^{k}k!}=% \frac{\lambda^{k}e^{-\lambda}}{k!}
  25. n p = λ np=\lambda
  26. n p λ np\rightarrow\lambda
  27. n p = λ np=\lambda
  28. p = λ / n p=\lambda/n
  29. lim n n ! ( n - k ) ! k ! ( λ n ) k ( 1 - λ n ) n - k = lim n n ( n - 1 ) ( n - 2 ) ( n - k + 1 ) k ! λ k n k ( 1 - λ n ) n - k \lim_{n\to\infty}\frac{n!}{(n-k)!k!}\left(\frac{\lambda}{n}\right)^{k}\left(1-% \frac{\lambda}{n}\right)^{n-k}=\lim_{n\to\infty}\frac{n(n-1)(n-2)\dots(n-k+1)}% {k!}\frac{\lambda^{k}}{n^{k}}\left(1-\frac{\lambda}{n}\right)^{n-k}
  30. n ( n - 1 ) ( n - 2 ) ( n - k + 1 ) = n k + O ( n k - 1 ) n(n-1)(n-2)\dots(n-k+1)=n^{k}+O\left(n^{k-1}\right)
  31. lim n n ( n - 1 ) ( n - 2 ) ( n - k + 1 ) n k k ! = 1 k ! \lim_{n\to\infty}\frac{n(n-1)(n-2)\dots(n-k+1)}{n^{k}k!}=\frac{1}{k!}
  32. ( 1 - λ n ) n - k = ( 1 - λ n ) n ( 1 - λ n ) - k \left(1-\frac{\lambda}{n}\right)^{n-k}=\left(1-\frac{\lambda}{n}\right)^{n}% \left(1-\frac{\lambda}{n}\right)^{-k}
  33. e - λ e^{-\lambda}
  34. lim n ( 1 - λ n ) - k = lim n ( 1 - 0 ) - k = 1 \lim_{n\to\infty}\left(1-\frac{\lambda}{n}\right)^{-k}=\lim_{n\to\infty}\left(% 1-0\right)^{-k}=1
  35. 1 k ! λ k e - λ \frac{1}{k!}\lambda^{k}e^{-\lambda}
  36. G bin ( x ; p , N ) k = 0 N [ ( N k ) p k ( 1 - p ) N - k ] x k = [ 1 + ( x - 1 ) p ] N G_{\mathrm{bin}}(x;p,N)\equiv\sum_{k=0}^{N}\left[{\left({{N}\atop{k}}\right)}p% ^{k}(1-p)^{N-k}\right]x^{k}=\Big[1+(x-1)p\Big]^{N}
  37. N N\rightarrow\infty
  38. p N λ pN\equiv\lambda
  39. lim N G bin ( x ; p , N ) = lim N [ 1 + λ ( x - 1 ) N ] N = e λ ( x - 1 ) = k = 0 [ e - λ λ k k ! ] x k \lim_{N\rightarrow\infty}G_{\mathrm{bin}}(x;p,N)=\lim_{N\rightarrow\infty}\Big% [1+\frac{\lambda(x-1)}{N}\Big]^{N}=\mathrm{e}^{\lambda(x-1)}=\sum_{k=0}^{% \infty}\left[\frac{\mathrm{e}^{-\lambda}\lambda^{k}}{k!}\right]x^{k}

Polar_sine.html

  1. psin ( v 1 , , v n ) = Ω Π , \operatorname{psin}({v}_{1},\dots,{v}_{n})=\frac{\Omega}{\Pi},
  2. Ω = det [ 𝐯 1 𝐯 2 𝐯 n ] = | v 11 v 21 v n 1 v 12 v 22 v n 2 v 1 n v 2 n v n n | \begin{aligned}\displaystyle\Omega&\displaystyle=\det\begin{bmatrix}\mathbf{v}% _{1}&\mathbf{v}_{2}&\cdots&\mathbf{v}_{n}\end{bmatrix}=\begin{vmatrix}v_{11}&v% _{21}&\cdots&v_{n1}\\ v_{12}&v_{22}&\cdots&v_{n2}\\ \vdots&\vdots&\ddots&\vdots\\ v_{1n}&v_{2n}&\cdots&v_{nn}\\ \end{vmatrix}\end{aligned}
  3. 𝐯 1 = ( v 11 , v 12 , v 1 n ) T 𝐯 2 = ( v 21 , v 22 , v 2 n ) T 𝐯 n = ( v n 1 , v n 2 , v n n ) T \begin{aligned}\displaystyle\mathbf{v}_{1}&\displaystyle=(v_{11},v_{12},\cdots v% _{1n})^{T}\\ \displaystyle\mathbf{v}_{2}&\displaystyle=(v_{21},v_{22},\cdots v_{2n})^{T}\\ \displaystyle\vdots\\ \displaystyle\mathbf{v}_{n}&\displaystyle=(v_{n1},v_{n2},\cdots v_{nn})^{T}\\ \end{aligned}
  4. Π = i = 1 n v i \Pi=\prod_{i=1}^{n}\|{v}_{i}\|
  5. Ω Π Ω Π 1 \Omega\leq\Pi\Rightarrow\frac{\Omega}{\Pi}\leq 1
  6. - 1 psin ( v 1 , , v n ) 1 , -1\leq\operatorname{psin}({v}_{1},\dots,{v}_{n})\leq 1,\,
  7. Ω = det ( [ 𝐯 1 𝐯 2 𝐯 n ] T [ 𝐯 1 𝐯 2 𝐯 n ] ) , \Omega=\sqrt{\det\left(\begin{bmatrix}\mathbf{v}_{1}&\mathbf{v}_{2}&\cdots&% \mathbf{v}_{n}\end{bmatrix}^{T}\begin{bmatrix}\mathbf{v}_{1}&\mathbf{v}_{2}&% \cdots&\mathbf{v}_{n}\end{bmatrix}\right)}\,,
  8. Ω \displaystyle\Omega
  9. psin ( c 1 v 1 , , c n v n ) \displaystyle\operatorname{psin}(c_{1}{v}_{1},\dots,c_{n}{v}_{n})

Pole_and_polar.html

  1. A x x x 2 + 2 A x y x y + A y y y 2 + 2 B x x + 2 B y y + C = 0 A_{xx}x^{2}+2A_{xy}xy+A_{yy}y^{2}+2B_{x}x+2B_{y}y+C=0\,
  2. D x + E y + F = 0 Dx+Ey+F=0\,
  3. D = A x x ξ + A x y η + B x D=A_{xx}\xi+A_{xy}\eta+B_{x}\,
  4. E = A x y ξ + A y y η + B y E=A_{xy}\xi+A_{yy}\eta+B_{y}\,
  5. F = B x ξ + B y η + C F=B_{x}\xi+B_{y}\eta+C\,
  6. D x + E y + F = 0 Dx+Ey+F=0
  7. A x x x 2 + 2 A x y x y + A y y y 2 + 2 B x x + 2 B y y + C = 0 A_{xx}x^{2}+2A_{xy}xy+A_{yy}y^{2}+2B_{x}x+2B_{y}y+C=0\,
  8. [ x y z ] = [ A x x A x y B x A x y A y y B y B x B y C ] - 1 [ D E F ] \begin{bmatrix}x\\ y\\ z\end{bmatrix}=\begin{bmatrix}A_{xx}&A_{xy}&B_{x}\\ A_{xy}&A_{yy}&B_{y}\\ B_{x}&B_{y}&C\end{bmatrix}^{-1}\cdot\begin{bmatrix}D\\ E\\ F\end{bmatrix}
  9. ( x z , y z ) (\frac{x}{z},\frac{y}{z})

Poloidal–toroidal_decomposition.html

  1. 𝐅 = 0 , \nabla\cdot\mathbf{F}=0,
  2. 𝐅 = 𝐓 + 𝐏 = × Ψ 𝐫 + × ( × Φ 𝐫 ) , \mathbf{F}=\mathbf{T}+\mathbf{P}=\nabla\times\Psi\mathbf{r}+\nabla\times(% \nabla\times\Phi\mathbf{r}),
  3. 𝐫 \mathbf{r}
  4. ( r , θ , ϕ ) (r,\theta,\phi)
  5. 𝐓 \mathbf{T}
  6. 𝐓 = × Ψ 𝐫 \mathbf{T}=\nabla\times\Psi\mathbf{r}
  7. Ψ ( r , θ , ϕ ) \Psi(r,\theta,\phi)
  8. 𝐏 \mathbf{P}
  9. 𝐏 = × × Φ 𝐫 \mathbf{P}=\nabla\times\nabla\times\Phi\mathbf{r}
  10. Φ ( r , θ , ϕ ) \Phi(r,\theta,\phi)
  11. 𝐫 𝐓 = 0 \mathbf{r}\cdot\mathbf{T}=0
  12. 𝐫 ( × 𝐏 ) = 0 \mathbf{r}\cdot(\nabla\times\mathbf{P})=0
  13. Ψ \Psi
  14. Φ \Phi
  15. r r
  16. 𝐅 ( x , y , z ) = × g ( x , y , z ) 𝐳 ^ + × ( × h ( x , y , z ) 𝐳 ^ ) + b x ( z ) 𝐱 ^ + b y ( z ) 𝐲 ^ , \mathbf{F}(x,y,z)=\nabla\times g(x,y,z)\hat{\mathbf{z}}+\nabla\times(\nabla% \times h(x,y,z)\hat{\mathbf{z}})+b_{x}(z)\hat{\mathbf{x}}+b_{y}(z)\hat{\mathbf% {y}},
  17. 𝐱 ^ , 𝐲 ^ , 𝐳 ^ \hat{\mathbf{x}},\hat{\mathbf{y}},\hat{\mathbf{z}}

Poly(N-isopropylacrylamide).html

  1. Δ H \Delta H
  2. Δ G = Δ H - T Δ S \Delta G=\Delta H-T\Delta S\,
  3. Δ S \Delta S

Polyhedral_combinatorics.html

  1. f k - 1 i = 0 d / 2 ( ( d - i k - i ) + ( i k - d + i ) ) * ( n - d - 1 + i i ) , f_{k-1}\leq\sum_{i=0}^{d/2}{}^{*}\left({\left({{d-i}\atop{k-i}}\right)}+{\left% ({{i}\atop{k-d+i}}\right)}\right){\left({{n-d-1+i}\atop{i}}\right)},
  2. O ( n d / 2 ) \scriptstyle O(n^{\lfloor d/2\rfloor})

Polyhedral_space.html

  1. R 3 R^{3}

Polymer_field_theory.html

  1. n n
  2. β = 1 / k B T \beta=1/k_{B}T
  3. V V
  4. Z ( n , V , β ) = 1 n ! ( λ T 3 ) n N j = 1 n D 𝐫 j exp ( - β Φ 0 [ 𝐫 ] - β Φ ¯ [ 𝐫 ] ) , ( 1 ) Z(n,V,\beta)=\frac{1}{n!(\lambda_{T}^{3})^{nN}}\prod_{j=1}^{n}\int D\mathbf{r}% _{j}\exp\left(-\beta\Phi_{0}\left[\mathbf{r}\right]-\beta\bar{\Phi}\left[% \mathbf{r}\right]\right),\qquad(1)
  5. Φ ¯ [ 𝐫 ] \bar{\Phi}\left[\mathbf{r}\right]
  6. Φ ¯ [ 𝐫 ] = N 2 2 j = 1 n k = 1 n 0 1 d s 0 1 d s Φ ¯ ( | 𝐫 j ( s ) - 𝐫 k ( s ) | ) - 1 2 n N Φ ¯ ( 0 ) , ( 2 ) \bar{\Phi}\left[\mathbf{r}\right]=\frac{N^{2}}{2}\sum_{j=1}^{n}\sum_{k=1}^{n}% \int_{0}^{1}ds\int_{0}^{1}ds^{\prime}\bar{\Phi}\left(\left|\mathbf{r}_{j}(s)-% \mathbf{r}_{k}(s^{\prime})\right|\right)-\frac{1}{2}nN\bar{\Phi}(0),\qquad(2)
  7. Φ 0 [ 𝐫 ] \Phi_{0}[\mathbf{r}]
  8. Φ 0 [ 𝐫 ] = 3 k B T 2 N b 2 l = 1 n 0 1 d s | d 𝐫 l ( s ) d s | 2 , \Phi_{0}[\mathbf{r}]=\frac{3k_{B}T}{2Nb^{2}}\sum_{l=1}^{n}\int_{0}^{1}ds\left|% \frac{d\mathbf{r}_{l}(s)}{ds}\right|^{2},
  9. b b
  10. N N
  11. ρ ^ ( 𝐫 ) = N j = 1 n 0 1 d s δ ( 𝐫 - 𝐫 j ( s ) ) . \hat{\rho}(\mathbf{r})=N\sum_{j=1}^{n}\int_{0}^{1}ds\delta\left(\mathbf{r}-% \mathbf{r}_{j}(s)\right).
  12. Φ ¯ [ 𝐫 ] = 1 2 d 𝐫 d 𝐫 ρ ^ ( 𝐫 ) Φ ¯ ( | 𝐫 - 𝐫 | ) ρ ^ ( 𝐫 ) - 1 2 n N Φ ¯ ( 0 ) . ( 3 ) \bar{\Phi}\left[\mathbf{r}\right]=\frac{1}{2}\int d\mathbf{r}\int d\mathbf{r}^% {\prime}\hat{\rho}(\mathbf{r})\bar{\Phi}(\left|\mathbf{r}-\mathbf{r}^{\prime}% \right|)\hat{\rho}(\mathbf{r}^{\prime})-\frac{1}{2}nN\bar{\Phi}(0).\qquad(3)
  13. D ρ δ [ ρ - ρ ^ ] F [ ρ ] = F [ ρ ^ ] , ( 4 ) \int D\rho\;\delta\left[\rho-\hat{\rho}\right]F\left[\rho\right]=F\left[\hat{% \rho}\right],\qquad(4)
  14. F [ ρ ^ ] F\left[\hat{\rho}\right]
  15. δ [ ρ - ρ ^ ] \delta\left[\rho-\hat{\rho}\right]
  16. δ [ ρ - ρ ^ ] = D w e i d 𝐫 w ( 𝐫 ) [ ρ ( 𝐫 ) - ρ ^ ( 𝐫 ) ] , ( 5 ) \delta\left[\rho-\hat{\rho}\right]=\int Dwe^{i\int d\mathbf{r}w(\mathbf{r})% \left[\rho(\mathbf{r})-\hat{\rho}(\mathbf{r})\right]},\qquad(5)
  17. w ( 𝐫 ) = 𝐆 w ( 𝐆 ) exp [ i 𝐆𝐫 ] w(\mathbf{r})=\sum\nolimits_{\mathbf{G}}w(\mathbf{G})\exp\left[i\mathbf{G}% \mathbf{r}\right]
  18. 𝐆 \mathbf{G}
  19. Z ( n , V , β ) = Z 0 D w exp [ - 1 2 β V 2 d 𝐫 d 𝐫 w ( 𝐫 ) Φ ¯ - 1 ( 𝐫 - 𝐫 ) w ( 𝐫 ) ] Q n [ i w ] , ( 6 ) Z(n,V,\beta)=Z_{0}\int Dw\exp\left[-\frac{1}{2\beta V^{2}}\int d\mathbf{r}d% \mathbf{r}^{\prime}w(\mathbf{r})\bar{\Phi}^{-1}(\mathbf{r}-\mathbf{r}^{\prime}% )w(\mathbf{r}^{\prime})\right]Q^{n}[iw],\qquad(6)
  20. Z 0 = 1 n ! ( exp ( β / 2 N Φ ¯ ( 0 ) ) Z λ 3 N ( T ) ) n Z_{0}=\frac{1}{n!}\left(\frac{\exp\left(\beta/2N\bar{\Phi}(0)\right)Z^{\prime}% }{\lambda^{3N}(T)}\right)^{n}
  21. Z = D 𝐑 exp [ - β U 0 ( 𝐑 ) ] ( 7 ) Z^{\prime}=\int D\mathbf{R}\exp\left[-\beta U_{0}(\mathbf{R})\right]\qquad(7)
  22. U 0 [ 𝐑 ] = k B T 4 R g 0 2 0 1 d s | d 𝐑 ( s ) d s | 2 . U_{0}[\mathbf{R}]=\frac{k_{B}T}{4R_{g0}^{2}}\int_{0}^{1}ds\left|\frac{d\mathbf% {R}(s)}{ds}\right|^{2}.
  23. R g 0 = N b 2 / ( 6 ) R_{g0}=\sqrt{Nb^{2}/(6)}
  24. w ( 𝐑 ) w(\mathbf{R})
  25. Q [ i w ] = D 𝐑 exp [ - β U 0 [ 𝐑 ] - i N 0 1 d s w ( 𝐑 ( s ) ) ] D 𝐑 exp [ - β U 0 [ 𝐑 ] ] . ( 8 ) Q[iw]=\frac{\int D\mathbf{R}\exp\left[-\beta U_{0}[\mathbf{R}]-iN\int_{0}^{1}% ds\;w(\mathbf{R}(s))\right]}{\int D\mathbf{R}\exp\left[-\beta U_{0}[\mathbf{R}% ]\right]}.\qquad(8)
  26. Ξ ( μ , V , β ) = n = 0 e β μ n Z ( n , V , β ) , \Xi(\mu,V,\beta)=\sum_{n=0}^{\infty}e^{\beta\mu n}Z(n,V,\beta),
  27. μ \mu
  28. Z ( n , V , β ) Z(n,V,\beta)
  29. Ξ ( ξ , V , β ) = γ Φ ¯ D w exp [ - S [ w ] ] , \Xi(\xi,V,\beta)=\gamma_{\bar{\Phi}}\int Dw\exp\left[-S[w]\right],
  30. S [ w ] = 1 2 β V 2 d 𝐫 d 𝐫 w ( 𝐫 ) Φ ¯ - 1 ( 𝐫 - 𝐫 ) w ( 𝐫 ) - ξ Q [ i w ] S[w]=\frac{1}{2\beta V^{2}}\int d\mathbf{r}d\mathbf{r}^{\prime}w(\mathbf{r})% \bar{\Phi}^{-1}(\mathbf{r}-\mathbf{r}^{\prime})w(\mathbf{r}^{\prime})-\xi Q[iw]
  31. Q [ i w ] Q[iw]
  32. γ Φ ¯ = 1 2 𝐆 ( 1 π β Φ ¯ ( 𝐆 ) ) 1 / 2 . \gamma_{\bar{\Phi}}=\frac{1}{\sqrt{2}}\prod_{\mathbf{G}}\left(\frac{1}{\pi% \beta\bar{\Phi}(\mathbf{G})}\right)^{1/2}.
  33. ξ = exp ( β μ + β / 2 N Φ ¯ ( 0 ) ) Z λ 3 N ( T ) , \xi=\frac{\exp\left(\beta\mu+\beta/2N\bar{\Phi}(0)\right)Z^{\prime}}{\lambda^{% 3N}(T)},
  34. Z Z^{\prime}
  35. Q Q
  36. r r
  37. Q ( r ) Q(r)
  38. Q Q
  39. 0 t h 0^{th}

Polynomial_arithmetic.html

  1. \mathbb{Z}
  2. f ( x ) = i = 0 n a i x i ; g ( x ) = i = 0 m b i x i f(x)=\sum_{i=0}^{n}a_{i}x^{i};g(x)=\sum_{i=0}^{m}b_{i}x^{i}
  3. f ( x ) + g ( x ) = i = 0 m ( a i + b i ) x i f(x)+g(x)=\sum_{i=0}^{m}(a_{i}+b_{i})x^{i}
  4. f ( x ) = i = 0 n a i x i ; g ( x ) = i = 0 m b i x i f(x)=\sum_{i=0}^{n}a_{i}x^{i};g(x)=\sum_{i=0}^{m}b_{i}x^{i}
  5. f ( x ) × g ( x ) = i = 0 n + m c i x i f(x)\times g(x)=\sum_{i=0}^{n+m}c_{i}x^{i}
  6. c k = a 0 b k + a 1 b k - 1 + + a k - 1 b 1 + a k b 0 c_{k}=a_{0}b_{k}+a_{1}b_{k-1}+\cdots+a_{k-1}b_{1}+a_{k}b_{0}
  7. a i a_{i}
  8. i > n i>n
  9. P P
  10. \mathbb{R}
  11. \mathbb{C}
  12. r r
  13. P P
  14. P ( r ) = 0 P(r)=0
  15. B B
  16. A A
  17. A = B C A=BC
  18. B B
  19. A A
  20. B | A B|A
  21. r r
  22. P P
  23. ( X - r ) | P (X-r)|P
  24. U P + V Q = D UP+VQ=D

Polynomial_code.html

  1. G F ( q ) GF(q)
  2. n n
  3. a n - 1 a 0 a_{n-1}\ldots a_{0}
  4. a n - 1 x n - 1 + + a 1 x + a 0 . a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}.\,
  5. m n m\leq n
  6. g ( x ) g(x)
  7. m m
  8. g ( x ) g(x)
  9. n n
  10. g ( x ) g(x)
  11. G F ( 2 ) = { 0 , 1 } GF(2)=\{0,1\}
  12. n = 5 n=5
  13. m = 2 m=2
  14. g ( x ) = x 2 + x + 1 g(x)=x^{2}+x+1
  15. 0 g ( x ) , 1 g ( x ) , x g ( x ) , ( x + 1 ) g ( x ) , 0\cdot g(x),\quad 1\cdot g(x),\quad x\cdot g(x),\quad(x+1)\cdot g(x),
  16. x 2 g ( x ) , ( x 2 + 1 ) g ( x ) , ( x 2 + x ) g ( x ) , ( x 2 + x + 1 ) g ( x ) . x^{2}\cdot g(x),\quad(x^{2}+1)\cdot g(x),\quad(x^{2}+x)\cdot g(x),\quad(x^{2}+% x+1)\cdot g(x).
  17. 0 , x 2 + x + 1 , x 3 + x 2 + x , x 3 + 2 x 2 + 2 x + 1 , 0,\quad x^{2}+x+1,\quad x^{3}+x^{2}+x,\quad x^{3}+2x^{2}+2x+1,
  18. x 4 + x 3 + x 2 , x 4 + x 3 + 2 x 2 + x + 1 , x 4 + 2 x 3 + 2 x 2 + x , x 4 + 2 x 3 + 3 x 2 + 2 x + 1. x^{4}+x^{3}+x^{2},\quad x^{4}+x^{3}+2x^{2}+x+1,\quad x^{4}+2x^{3}+2x^{2}+x,% \quad x^{4}+2x^{3}+3x^{2}+2x+1.
  19. G F ( 2 ) = { 0 , 1 } GF(2)=\{0,1\}
  20. 0 , x 2 + x + 1 , x 3 + x 2 + x , x 3 + 1 , 0,\quad x^{2}+x+1,\quad x^{3}+x^{2}+x,\quad x^{3}+1,
  21. x 4 + x 3 + x 2 , x 4 + x 3 + x + 1 , x 4 + x , x 4 + x 2 + 1. x^{4}+x^{3}+x^{2},\quad x^{4}+x^{3}+x+1,\quad x^{4}+x,\quad x^{4}+x^{2}+1.
  22. 00000 , 00111 , 01110 , 01001 , 00000,\quad 00111,\quad 01110,\quad 01001,
  23. 11100 , 11011 , 10010 , 10101. 11100,\quad 11011,\quad 10010,\quad 10101.
  24. G F ( q ) GF(q)
  25. n n
  26. g ( x ) g(x)
  27. m m
  28. q n - m q^{n-m}
  29. p ( x ) p(x)
  30. p ( x ) = g ( x ) q ( x ) p(x)=g(x)\cdot q(x)
  31. q ( x ) q(x)
  32. n - m n-m
  33. q n - m q^{n-m}
  34. n - m n-m
  35. q ( x ) g ( x ) q ( x ) q(x)\mapsto g(x)\cdot q(x)
  36. d ( x ) d(x)
  37. n - m n-m
  38. d ( x ) d(x)
  39. x m x^{m}
  40. d ( x ) d(x)
  41. m m
  42. x m d ( x ) x^{m}d(x)
  43. g ( x ) g(x)
  44. m m
  45. x m d ( x ) x^{m}d(x)
  46. x m d ( x ) x^{m}d(x)
  47. g ( x ) g(x)
  48. x m d ( x ) = g ( x ) q ( x ) + r ( x ) , x^{m}d(x)=g(x)\cdot q(x)+r(x),\,
  49. r ( x ) r(x)
  50. m m
  51. d ( x ) d(x)
  52. p ( x ) := x m d ( x ) - r ( x ) , p(x):=x^{m}d(x)-r(x),\,
  53. p ( x ) = g ( x ) q ( x ) p(x)=g(x)\cdot q(x)
  54. g ( x ) g(x)
  55. p ( x ) p(x)
  56. r ( x ) r(x)
  57. m m
  58. n - m n-m
  59. p ( x ) p(x)
  60. x m d ( x ) x^{m}d(x)
  61. n - m n-m
  62. m m
  63. n = 5 n=5
  64. m = 2 m=2
  65. g ( x ) = x 2 + x + 1 g(x)=x^{2}+x+1
  66. \mapsto
  67. \mapsto
  68. \mapsto
  69. \mapsto
  70. \mapsto
  71. \mapsto
  72. \mapsto
  73. \mapsto
  74. m m
  75. x n - 1 x^{n}-1
  76. n 2 m - 1 n\leq 2^{m}-1

Polynomial_conjoint_measurement.html

  1. Y = { y 1 , y 2 , , y n } Y=\big\{y_{1},y_{2},\ldots,y_{n}\big\}
  2. S ( Y ) S\left(Y\right)
  3. y i S ( Y ) , i = 1 , , n y_{i}\in S\left(Y\right),i=1,\ldots,n
  4. Y 1 , Y 2 Y Y_{1},Y_{2}\subset Y
  5. Y 1 Y 2 = , G 1 S ( Y 1 ) Y_{1}\cap Y_{2}=\varnothing,G_{1}\in S\left(Y_{1}\right)
  6. G 2 S ( Y 2 ) G_{2}\in S\left(Y_{2}\right)
  7. G 1 + G 2 G_{1}+G_{2}\,
  8. G 1 G 2 G_{1}G_{2}\,
  9. S ( Y ) S\left(Y\right)
  10. × \times
  11. A + P + U A+P+U\,
  12. ( A + P ) U \left(A+P\right)U\,
  13. A P + U AP+U\,
  14. A P U APU\,
  15. A = { a , b , c , } A=\big\{a,b,c,\ldots\big\}
  16. P = { p , q , r , } P=\big\{p,q,r,\ldots\big\}
  17. U = { u , v , w , } U=\big\{u,v,w,\ldots\big\}
  18. \succsim
  19. Z = A , P , U , Z=\langle A,P,U,\succsim\rangle
  20. \succsim
  21. ( a , p , u ) ( b , p , u ) \left(a,p,u\right)\succsim\left(b,p,u\right)
  22. ( a , q , v ) ( b , q , v ) \left(a,q,v\right)\succsim\left(b,q,v\right)
  23. a , b A ; p , q P a,b\in A;p,q\in P
  24. u , v U u,v\in U
  25. \succsim
  26. A × P A\times P
  27. a , b , c A a,b,c\in A
  28. p , q , r P p,q,r\in P
  29. ( a , q , u ) ( b , p , u ) \left(a,q,u\right)\succsim\left(b,p,u\right)
  30. ( b , r , u ) ( c , q , u ) \left(b,r,u\right)\succsim\left(c,q,u\right)
  31. ( a , r , u ) ( c , p , u ) \left(a,r,u\right)\succsim\left(c,p,u\right)
  32. u U u\in U
  33. A × U A\times U
  34. U × P U\times P
  35. \succsim
  36. A × P A\times P
  37. ( a , p , u ) ( b , q , u ) \left(a,p,u\right)\succsim\left(b,q,u\right)
  38. ( a , p , v ) ( b , q , v ) \left(a,p,v\right)\succsim\left(b,q,v\right)
  39. a , b A ; p , q P a,b\in A;p,q\in P
  40. u , v U u,v\in U
  41. A × U A\times U
  42. U × P U\times P
  43. A × P × U A\times P\times U
  44. ( a , p , u ) ( c , r , v ) \left(a,p,u\right)\succsim\left(c,r,v\right)
  45. ( b , q , u ) ( d , s , v ) \left(b,q,u\right)\succsim\left(d,s,v\right)
  46. ( d , r , v ) ( b , p , u ) \left(d,r,v\right)\succsim\left(b,p,u\right)
  47. ( a , q , u ) ( c , s , v ) \left(a,q,u\right)\succsim\left(c,s,v\right)
  48. a , b , c , d A ; p , q , r , s P a,b,c,d\in A;p,q,r,s\in P
  49. u , v U u,v\in U
  50. A × P × U A\times P\times U
  51. ( a , r , w ) ( c , s , v ) \left(a,r,w\right)\succsim\left(c,s,v\right)
  52. ( d , p , u ) ( b , t , x ) \left(d,p,u\right)\succsim\left(b,t,x\right)
  53. ( d , r , x ) ( e , s , u ) \left(d,r,x\right)\succsim\left(e,s,u\right)
  54. ( c , t , y ) ( d , q , y ) \left(c,t,y\right)\succsim\left(d,q,y\right)
  55. ( a , p , v ) ( b , q , w ) \left(a,p,v\right)\succsim\left(b,q,w\right)
  56. a , b , c , d , e A ; p , q , r , s , t P a,b,c,d,e\in A;p,q,r,s,t\in P
  57. u , v , w , x , y U u,v,w,x,y\in U
  58. \succsim
  59. A × P × U A\times P\times U
  60. a , b A ; p , q P a,b\in A;p,q\in P
  61. u , v U u,v\in U
  62. c A ; r P c\in A;r\in P
  63. w U w\in U
  64. a ( b , q , w ) ( b , r , v ) ( c , q , v ) a\sim\left(b,q,w\right)\sim\left(b,r,v\right)\sim\left(c,q,v\right)
  65. Z = A , P , U , Z=\langle A,P,U,\succsim\rangle

Polynomial_SOS.html

  1. g 1 ( x ) , , g k ( x ) g_{1}(x),\ldots,g_{k}(x)
  2. h ( x ) = i = 1 k g i ( x ) 2 . h(x)=\sum_{i=1}^{k}g_{i}(x)^{2}.
  3. l 1 l_{1}
  4. { f ϵ } \{f_{\epsilon}\}
  5. h ( x ) = x { m } ( H + L ( α ) ) x { m } h(x)=x^{\{m\}^{\prime}}\left(H+L(\alpha)\right)x^{\{m\}}
  6. x { m } x^{\{m\}}
  7. h ( x ) = x { m } H x { m } h(x)=x^{\left\{m\right\}^{\prime}}Hx^{\{m\}}
  8. L ( α ) L(\alpha)
  9. = { L = L : x { m } L x { m } = 0 } . \mathcal{L}=\left\{L=L^{\prime}:~{}x^{\{m\}^{\prime}}Lx^{\{m\}}=0\right\}.
  10. x { m } x^{\{m\}}
  11. σ ( n , m ) = ( n + m - 1 m ) \sigma(n,m)={\left({{n+m-1}\atop{m}}\right)}
  12. α \alpha
  13. ω ( n , 2 m ) = 1 2 σ ( n , m ) ( 1 + σ ( n , m ) ) - σ ( n , 2 m ) . \omega(n,2m)=\frac{1}{2}\sigma(n,m)\left(1+\sigma(n,m)\right)-\sigma(n,2m).
  14. α \alpha
  15. H + L ( α ) 0 , H+L(\alpha)\geq 0,
  16. H + L ( α ) H+L(\alpha)
  17. h ( x ) = x { m } ( H + L ( α ) ) x { m } h(x)=x^{\{m\}^{\prime}}\left(H+L(\alpha)\right)x^{\{m\}}
  18. h ( x ) = x 1 4 - x 1 2 x 2 2 + x 2 4 h(x)=x_{1}^{4}-x_{1}^{2}x_{2}^{2}+x_{2}^{4}
  19. m = 2 , x { m } = ( x 1 2 x 1 x 2 x 2 2 ) , H + L ( α ) = ( 1 0 - α 1 0 - 1 + 2 α 1 0 - α 1 0 1 ) . m=2,~{}x^{\{m\}}=\left(\begin{array}[]{c}x_{1}^{2}\\ x_{1}x_{2}\\ x_{2}^{2}\end{array}\right),~{}H+L(\alpha)=\left(\begin{array}[]{ccc}1&0&-% \alpha_{1}\\ 0&-1+2\alpha_{1}&0\\ -\alpha_{1}&0&1\end{array}\right).
  20. H + L ( α ) 0 H+L(\alpha)\geq 0
  21. α = 1 \alpha=1
  22. h ( x ) = 2 x 1 4 - 2.5 x 1 3 x 2 + x 1 2 x 2 x 3 - 2 x 1 x 3 3 + 5 x 2 4 + x 3 4 h(x)=2x_{1}^{4}-2.5x_{1}^{3}x_{2}+x_{1}^{2}x_{2}x_{3}-2x_{1}x_{3}^{3}+5x_{2}^{% 4}+x_{3}^{4}
  23. m = 2 , x { m } = ( x 1 2 x 1 x 2 x 1 x 3 x 2 2 x 2 x 3 x 3 2 ) , H + L ( α ) = ( 2 - 1.25 0 - α 1 - α 2 - α 3 - 1.25 2 α 1 0.5 + α 2 0 - α 4 - α 5 0 0.5 + α 2 2 α 3 α 4 α 5 - 1 - α 1 0 α 4 5 0 - α 6 - α 2 - α 4 α 5 0 2 α 6 0 - α 3 - α 5 - 1 - α 6 0 1 ) . m=2,~{}x^{\{m\}}=\left(\begin{array}[]{c}x_{1}^{2}\\ x_{1}x_{2}\\ x_{1}x_{3}\\ x_{2}^{2}\\ x_{2}x_{3}\\ x_{3}^{2}\end{array}\right),~{}H+L(\alpha)=\left(\begin{array}[]{cccccc}2&-1.2% 5&0&-\alpha_{1}&-\alpha_{2}&-\alpha_{3}\\ -1.25&2\alpha_{1}&0.5+\alpha_{2}&0&-\alpha_{4}&-\alpha_{5}\\ 0&0.5+\alpha_{2}&2\alpha_{3}&\alpha_{4}&\alpha_{5}&-1\\ -\alpha_{1}&0&\alpha_{4}&5&0&-\alpha_{6}\\ -\alpha_{2}&-\alpha_{4}&\alpha_{5}&0&2\alpha_{6}&0\\ -\alpha_{3}&-\alpha_{5}&-1&-\alpha_{6}&0&1\end{array}\right).
  24. H + L ( α ) 0 H+L(\alpha)\geq 0
  25. α = ( 1.18 , - 0.43 , 0.73 , 1.13 , - 0.37 , 0.57 ) \alpha=(1.18,-0.43,0.73,1.13,-0.37,0.57)
  26. G 1 ( x ) , , G k ( x ) G_{1}(x),\ldots,G_{k}(x)
  27. F ( x ) = i = 1 k G i ( x ) G i ( x ) . F(x)=\sum_{i=1}^{k}G_{i}(x)^{\prime}G_{i}(x).
  28. F ( x ) = ( x { m } I r ) ( H + L ( α ) ) ( x { m } I r ) F(x)=\left(x^{\{m\}}\otimes I_{r}\right)^{\prime}\left(H+L(\alpha)\right)\left% (x^{\{m\}}\otimes I_{r}\right)
  29. \otimes
  30. F ( x ) = ( x { m } I r ) H ( x { m } I r ) F(x)=\left(x^{\{m\}}\otimes I_{r}\right)^{\prime}H\left(x^{\{m\}}\otimes I_{r}\right)
  31. L ( α ) L(\alpha)
  32. = { L = L : ( x { m } I r ) L ( x { m } I r ) = 0 } . \mathcal{L}=\left\{L=L^{\prime}:~{}\left(x^{\{m\}}\otimes I_{r}\right)^{\prime% }L\left(x^{\{m\}}\otimes I_{r}\right)=0\right\}.
  33. α \alpha
  34. ω ( n , 2 m , r ) = 1 2 r ( σ ( n , m ) ( r σ ( n , m ) + 1 ) - ( r + 1 ) σ ( n , 2 m ) ) . \omega(n,2m,r)=\frac{1}{2}r\left(\sigma(n,m)\left(r\sigma(n,m)+1\right)-(r+1)% \sigma(n,2m)\right).
  35. α \alpha
  36. H + L ( α ) 0. H+L(\alpha)\geq 0.
  37. F ( x ) = ( x { m } I r ) ( H + L ( α ) ) ( x { m } I r ) F(x)=\left(x^{\{m\}}\otimes I_{r}\right)^{\prime}\left(H+L(\alpha)\right)\left% (x^{\{m\}}\otimes I_{r}\right)

Polyphase_merge_sort.html

  1. N N
  2. N - 1 N-1

Ponderal_index.html

  1. PI = m a s s h e i g h t 3 \mathrm{PI}=\frac{mass}{height^{3}}
  2. PI = b i r t h w e i g h t C r o w n - h e e l - l e n g t h 3 \mathrm{PI}=\frac{birthweight}{Crown-heel-length^{3}}
  3. m a s s mass
  4. h e i g h t height
  5. PI = m a s s h e i g h t 3 \,\text{PI}=\frac{mass}{height^{3}}
  6. PI = 100 m a s s h e i g h t 3 \,\text{PI}=100\frac{mass}{height^{3}}
  7. PI = 1000 × m a s s 3 h e i g h t \,\text{PI}=1000\times\frac{\sqrt[3]{mass}}{height}
  8. PI = 100 × m a s s 3 h e i g h t \,\text{PI}=100\times\frac{\sqrt[3]{mass}}{height}
  9. PI = h e i g h t m a s s 3 \,\text{PI}=\frac{height}{\sqrt[3]{mass}}

Pool_factor.html

  1. O u t s t a n d i n g P r i n c i p a l B a l a n c e O r i g i n a l P r i n c i p a l B a l a n c e = P o o l F a c t o r {OutstandingPrincipalBalance\over OriginalPrincipalBalance}={PoolFactor}

Poole–Frenkel_effect.html

  1. J E exp ( - q ( ϕ B - q E / ( π ϵ ) ) k B T ) J\propto E\exp\left(\frac{-q\left(\phi_{B}-\sqrt{qE/(\pi\epsilon)}\ \right)}{k% _{B}T}\right)
  2. ϕ B \phi_{B}
  3. ϵ \epsilon
  4. k B k_{B}

Population_balance_equation.html

  1. d d t Ω x ( t ) d V x Ω r ( t ) d V r f ( x , r , t ) = Ω x ( t ) d V x Ω r ( t ) d V r h ( x , r , Y , t ) \frac{d}{dt}\int_{\Omega_{x}(t)}dV_{x}\int_{\Omega_{r}(t)}dV_{r}\,f({x},{r},t)% =\int_{\Omega_{x}(t)}dV_{x}\int_{\Omega_{r}(t)}dV_{r}\,h({x},{r},{Y},t)

Population_dynamics_of_fisheries.html

  1. d N d t = r N ( 1 - N K ) \frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)
  2. L ( t ) = r B ( L - L ( t ) ) L^{\prime}(t)=r_{B}\left(L_{\infty}-L(t)\right)
  3. H ( E , X ) = q E X H(E,X)=qEX\!
  4. X ˙ = 0 \dot{X}=0
  5. H ( E ) = q K E ( 1 - q E r ) H(E)=qKE(1-\frac{qE}{r})
  6. C = F F + M ( 1 - e - ( F + M ) T ) N 0 C=\frac{F}{F+M}(1-e^{-(F+M)T})N_{0}
  7. N t + 1 = N t e r ( 1 - N t k ) N_{t+1}=N_{t}e^{r(1-\frac{N_{t}}{k})}
  8. n t + 1 = R 0 n t 1 + n t / M . n_{t+1}=\frac{R_{0}n_{t}}{1+n_{t}/M}.
  9. d N d t = a N 2 - b N {dN\over dt}=aN^{2}-bN

Population_stratification.html

  1. Y 2 = N ( N ( r 1 + 2 r 2 ) - R ( n 1 + 2 n 2 ) ) 2 R ( N - R ) ( N ( n 1 + 4 n 2 ) - ( n 1 + 2 n 2 ) 2 ) Y^{2}=\frac{N(N(r_{1}+2r_{2})-R(n_{1}+2n_{2}))^{2}}{R(N-R)(N(n_{1}+4n_{2})-(n_% {1}+2n_{2})^{2})}
  2. χ 2 \chi^{2}
  3. χ 2 X A 2 = 2 N ( 2 N ( r 1 + 2 r 2 ) - R ( n 1 + 2 n 2 ) ) 2 4 R ( N - R ) ( 2 N ( n 1 + 2 n 2 ) - ( n 1 + 2 n 2 ) 2 ) \chi^{2}\sim X_{A}^{2}=\frac{2N(2N(r_{1}+2r_{2})-R(n_{1}+2n_{2}))^{2}}{4R(N-R)% (2N(n_{1}+2n_{2})-(n_{1}+2n_{2})^{2})}
  4. χ 2 \chi^{2}
  5. λ \lambda
  6. Y 2 λ χ 1 2 Y^{2}\sim\lambda\chi_{1}^{2}
  7. λ \lambda
  8. λ \lambda
  9. λ \lambda
  10. λ ^ = m e d i a n ( Y 1 2 , Y 2 2 , Y L 2 ) / 0.456 \hat{\lambda}=median(Y_{1}^{2},Y_{2}^{2},\ldots Y_{L}^{2})/0.456
  11. λ \lambda
  12. λ \lambda
  13. Y 2 Y^{2}
  14. χ 1 2 \chi^{2}_{1}
  15. α \alpha
  16. α = 0.05 \alpha=0.05

Porosity.html

  1. ϕ = V V V T \phi=\frac{V_{\mathrm{V}}}{V_{\mathrm{T}}}
  2. ϕ \phi
  3. n n
  4. ϕ ( z ) = ϕ 0 e - k z \phi(z)=\phi_{0}e^{-kz}\,
  5. ϕ 0 \phi_{0}
  6. k k
  7. z z
  8. ρ bulk \rho_{\,\text{bulk}}
  9. ρ particle \rho_{\,\text{particle}}
  10. ϕ = 1 - ρ bulk ρ particle \phi=1-\frac{\rho_{\,\text{bulk}}}{\rho_{\,\text{particle}}}
  11. V V = V T - V a - V b P 2 P 2 - P 1 V_{V}=V_{T}-V_{a}-V_{b}{P_{2}\over{P_{2}-P_{1}}}
  12. ϕ = V V V T \phi=\frac{V_{V}}{V_{T}}

Portal:Algebra::Selected_article::3.html

  1. 1 0 = \frac{1}{0}=\infty

Portal:Algebra::Selected_article::4.html

  1. a x 2 + b x + c = 0 , ax^{2}+bx+c=0,\,\!
  2. x = - b ± b 2 - 4 a c 2 a , x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a},
  3. f ( x ) = a x 2 + b x + c . f(x)=ax^{2}+bx+c.\,

Portal:Algebra::Selected_article::6.html

  1. R n R^{n}

Portal:Analysis::Selected_picture.html

  1. f f\,

Portal:Arctic.html

  1. 𝕋 𝔼 \mathbb{THE}
  2. 𝔸 𝕋 𝕀 \mathbb{ARCTIC}
  3. 𝕆 𝕋 𝔸 𝕃 \mathbb{PORTAL}

Portal:Arctic::Statistics.html

  1. 𝕋 𝔼 \mathbb{THE}
  2. 𝔸 𝕋 𝕀 \mathbb{ARCTIC}
  3. 𝕆 𝕋 𝔸 𝕃 \mathbb{PORTAL}

Portal:Arctic::WikiProjects.html

  1. 𝕋 𝔼 \mathbb{THE}
  2. 𝔸 𝕋 𝕀 \mathbb{ARCTIC}
  3. 𝕆 𝕋 𝔸 𝕃 \mathbb{PORTAL}

Portal:Atlas::Selected_article::16.html

  1. r r
  2. t t

Portal:Geometry::Selected_article::6.html

  1. a + b a = a b = φ . \frac{a+b}{a}=\frac{a}{b}=\varphi\,.
  2. φ = 1 + 5 2 , \varphi=\frac{1+\sqrt{5}}{2},\,

Portal:Mathematics::Featured_picture::2008_05.html

  1. ( ζ ( i t + 1 / 2 ) ) \Re(\zeta(it+1/2))
  2. ( ζ ( i t + 1 / 2 ) ) \Im(\zeta(it+1/2))

Portal:Mathematics::Featured_picture::2008_08.html

  1. e x e^{x}

Portal:Serbia::Selected_bio::August.html

  1. B B\,

Post-wall_waveguide.html

  1. a R W G = a S I W - 1.08 ( 2 r ) 2 / p + 0.1 ( 2 r ) 2 / a S I W a_{RWG}=a_{SIW}-1.08(2r)^{2}/p+0.1(2r)^{2}/a_{SIW}
  2. p p
  3. r r
  4. a R W G a_{RWG}
  5. a S I W a_{SIW}

Potential_of_mean_force.html

  1. - j w ( n ) = e - β V ( - j V ) d q n + 1 d q N e - β V d q n + 1 . d q N , j = 1 , 2 , . , n -\nabla_{j}w^{(n)}\,=\,\frac{\int e^{-\beta V}(-\nabla_{j}V)dq_{n+1}...dq_{N}}% {\int e^{-\beta V}dq_{n+1}....dq_{N}},~{}j=1,2,....,n
  2. - j w ( n ) -\nabla_{j}w^{(n)}
  3. w ( n ) w^{(n)}
  4. n = 2 n=2
  5. w ( 2 ) ( r ) w^{(2)}(r)
  6. r r
  7. g ( r ) g(r)
  8. g ( r ) = e - β w ( 2 ) ( r ) g(r)=e^{-\beta w^{(2)}(r)}
  9. w ( 2 ) w^{(2)}

Potentiometer_(measuring_instrument).html

  1. R 2 R 1 = cell voltage V S {R_{2}\over R_{1}}={\mbox{cell voltage}~{}\over V_{\mathrm{S}}}
  2. V U = V S A X A B V_{U}=V_{S}{AX\over AB}

Pound–Drever–Hall_technique.html

  1. E i = E 0 e i ( ω t + β sin ( ω m t ) ) E 0 e i ω t [ 1 + i β sin ( ω m t ) ] = E 0 e i ω t [ 1 + β 2 e i ω m t - β 2 e - i ω m t ] . \begin{aligned}\displaystyle E_{\,\text{i}}&\displaystyle=E_{0}e^{i(\omega t+% \beta\sin(\omega_{\mathrm{m}}t))}\\ &\displaystyle\approx E_{0}e^{i\omega t}[1+i\beta\sin(\omega_{\mathrm{m}}t)]\\ &\displaystyle=E_{0}e^{i\omega t}\left[1+\frac{\beta}{2}e^{i\omega_{\mathrm{m}% }t}-\frac{\beta}{2}e^{-i\omega_{\mathrm{m}}t}\right].\end{aligned}
  2. R ( ω ) = E r E i = - r 1 + ( r 1 2 + t 1 2 ) r 2 e i 2 α 1 - r 1 r 2 e i 2 α , R(\omega)=\frac{E_{\,\text{r}}}{E_{\,\text{i}}}=\frac{-r_{1}+(r_{1}^{2}+t_{1}^% {2})r_{2}e^{i2\alpha}}{1-r_{1}r_{2}e^{i2\alpha}},
  3. E r = E 0 [ R ( ω ) e i ω t + R ( ω + ω m ) β 2 e i ( ω + ω m ) t - R ( ω - ω m ) β 2 e i ( ω - ω m ) t ] . E_{\,\text{r}}=E_{0}\left[R(\omega)e^{i\omega t}+R(\omega+\omega_{\mathrm{m}})% \frac{\beta}{2}e^{i(\omega+\omega_{\mathrm{m}})t}-R(\omega-\omega_{\mathrm{m}}% )\frac{\beta}{2}e^{i(\omega-\omega_{\mathrm{m}})t}\right].
  4. P r = P 0 | R ( ω ) | 2 + P 0 β 2 4 { | R ( ω + ω m ) | 2 + | R ( ω - ω m ) | 2 } + P 0 β { Re [ χ ( ω ) ] cos ω m t + Im [ χ ( ω ) ] sin ω m t } + ( terms in 2 ω m ) . \begin{aligned}\displaystyle P_{\,\text{r}}=&\displaystyle\ P_{0}\left|R(% \omega)\right|^{2}+P_{0}\frac{\beta^{2}}{4}\Big\{\left|R(\omega+\omega_{% \mathrm{m}})\right|^{2}+\left|R(\omega-\omega_{\mathrm{m}})\right|^{2}\Big\}\\ &\displaystyle+P_{0}\beta\Big\{\textrm{Re}[\chi(\omega)]\cos{\omega_{\mathrm{m% }}t}+\textrm{Im}[\chi(\omega)]\sin{\omega_{\mathrm{m}}t}\Big\}+(\,\text{terms % in }2\omega_{\mathrm{m}}).\end{aligned}
  5. χ ( ω ) = R ( ω ) R * ( ω + ω m ) - R * ( ω ) R ( ω - ω m ) . \chi(\omega)=R(\omega)R^{*}(\omega+\omega_{\mathrm{m}})-R^{*}(\omega)R(\omega-% \omega_{\mathrm{m}}).
  6. V r \displaystyle V_{\,\text{r}}^{\prime}
  7. V ( ω ) Re [ χ ( ω ) ] cos φ + Im [ χ ( ω ) ] sin φ . V(\omega)\propto\textrm{Re}[\chi(\omega)]\cos\varphi+\textrm{Im}[\chi(\omega)]% \sin\varphi.

Power_graph_analysis.html

  1. G = ( V , E ) G=\bigl({V,E}\bigr)
  2. V = { v 0 , , v n } V=\bigl\{v_{0},\dots,v_{n}\bigr\}
  3. E V × V E\subseteq V\times V
  4. G = ( V , E ) G^{\prime}=\bigl({V^{\prime},E^{\prime}}\bigr)
  5. V 𝒫 ( V ) V^{\prime}\subseteq\mathcal{P}\bigl(V\bigr)
  6. E V × V E^{\prime}\subseteq V^{\prime}\times V^{\prime}
  7. G G
  8. t x t\mapsto x

Power_reverse_dual-currency_note.html

  1. t = 1 n M A X ( N F X t F X 0 r 1 t - r 2 t ( N - 1 ) , 0 ) \sum_{t=1}^{n}MAX(N\frac{FX_{t}}{FX_{0}}r_{1t}-r_{2t}(N-1),0)
  2. N = notional N=\text{notional}
  3. t = time of a cash flow t=\text{time of a cash flow}
  4. 0 = time at the start of the deal 0=\text{time at the start of the deal}
  5. r 1 = fixed rate at t of currency1. A set of rates for every t are fixed at time 0. r_{1}=\text{fixed rate at t of currency1. A set of rates for every t are fixed% at time 0. }
  6. r 2 = fixed rate at t of currency2. A set of rates for every t are fixed at time 0. r_{2}=\text{fixed rate at t of currency2. A set of rates for every t are fixed% at time 0. }
  7. F X = exchange rate between currency1 and currency2 FX=\text{exchange rate between currency1 and currency2}

Prandtl–Glauert_transformation.html

  1. 1 / β 1/\beta
  2. ϕ x x + ϕ y y + ϕ z z = M 2 ϕ x x (in flowfield) \phi_{xx}\,+\,\phi_{yy}\,+\,\phi_{zz}\;=\;M_{\infty}^{2}\phi_{xx}\;\;\;\;\mbox% {(in flowfield)}~{}
  3. V n x + ϕ y n y + ϕ z n z = 0 (on body surface) V_{\infty}\,n_{x}\,+\,\phi_{y}\,n_{y}\,+\,\phi_{z}\,n_{z}\;=\;0\;\;\;\;\mbox{(% on body surface)}~{}
  4. M M_{\infty}
  5. n x , n y , n z n_{x},n_{y},n_{z}
  6. ϕ ( x , y , z ) \phi(x,y,z)
  7. V V_{\infty}
  8. x x
  9. V = ϕ + V x ^ = ( V + ϕ x ) x ^ + ϕ y y ^ + ϕ z z ^ \vec{V}\;=\;\nabla\phi+V_{\infty}\hat{x}\;=\;(V_{\infty}+\phi_{x})\,\hat{x}\,+% \,\phi_{y}\,\hat{y}\,+\,\phi_{z}\,\hat{z}
  10. | ϕ | V |\nabla\phi|\ll V_{\infty}
  11. [ 1 + ( γ + 1 ) ϕ x V ] M 2 < 1 \left[1+(\gamma+1)\frac{\phi_{x}}{V_{\infty}}\right]M_{\infty}^{2}\;<\;1
  12. β 1 - M 2 \beta\equiv\sqrt{1-M_{\infty}^{2}}
  13. β \beta
  14. β 2 \beta^{2}
  15. x ¯ = x y ¯ = β y z ¯ = β z α ¯ = β α ϕ ¯ = β 2 ϕ \begin{array}[]{rcl}\bar{x}&=&x\\ \bar{y}&=&\beta y\\ \bar{z}&=&\beta z\\ \bar{\alpha}&=&\beta\alpha\\ \bar{\phi}&=&\beta^{2}\phi\end{array}
  16. ϕ ¯ x ¯ x ¯ + ϕ ¯ y ¯ y ¯ + ϕ ¯ z ¯ z ¯ = 0 (in flowfield) \bar{\phi}_{\bar{x}\bar{x}}\,+\,\bar{\phi}_{\bar{y}\bar{y}}\,+\,\bar{\phi}_{% \bar{z}\bar{z}}\;=\;0\;\;\;\;\mbox{(in flowfield)}~{}
  17. V n ¯ x ¯ + ϕ ¯ y ¯ n ¯ y ¯ + ϕ ¯ z ¯ n ¯ z ¯ = 0 (on body surface) V_{\infty}\,\bar{n}_{\bar{x}}\,+\,\bar{\phi}_{\bar{y}}\,\bar{n}_{\bar{y}}\,+\,% \bar{\phi}_{\bar{z}}\,\bar{n}_{\bar{z}}\;=\;0\;\;\;\;\mbox{(on body surface)}~{}
  18. n ¯ x ¯ , n ¯ y ¯ , n ¯ z ¯ \bar{n}_{\bar{x}},\bar{n}_{\bar{y}},\bar{n}_{\bar{z}}
  19. ϕ ¯ \bar{\phi}
  20. ϕ ¯ x ¯ , ϕ ¯ y ¯ , ϕ ¯ z ¯ \bar{\phi}_{\bar{x}},\bar{\phi}_{\bar{y}},\bar{\phi}_{\bar{z}}
  21. C p = - 2 ϕ x V = - 2 β 2 ϕ ¯ x ¯ V C_{p}\;=\;-2\frac{\phi_{x}}{V_{\infty}}\;=\;-\frac{2}{\beta^{2}}\frac{\bar{% \phi}_{\bar{x}}}{V_{\infty}}
  22. C p C_{p}
  23. c l , c m c_{l},c_{m}
  24. 1 / β 1/\beta
  25. C p = C p 0 β c l = c l 0 β c m = c m 0 β \begin{array}[]{rcl}C_{p}&=\displaystyle\frac{C_{p0}}{\beta}\\ c_{l}&=\displaystyle\frac{c_{l0}}{\beta}\\ c_{m}&=\displaystyle\frac{c_{m0}}{\beta}\end{array}
  26. C p 0 , c l 0 , c m 0 C_{p0},c_{l0},c_{m0}
  27. 1 / β 1/\beta
  28. C p C_{p}
  29. C L = 2 π α β + 2 / A R C_{L}=\frac{2\pi\alpha}{\beta+2/AR}
  30. c l 0 = 2 π α c_{l0}=2\pi\alpha
  31. M 1 M_{\infty}\simeq 1
  32. M 1 M_{\infty}\simeq 1

Predictor–corrector_method.html

  1. y = f ( t , y ) , y ( t 0 ) = y 0 , y^{\prime}=f(t,y),\quad y(t_{0})=y_{0},
  2. h h
  3. y i y_{i}
  4. y ~ i + 1 \tilde{y}_{i+1}
  5. y ~ i + 1 = y i + h f ( t i , y i ) . \tilde{y}_{i+1}=y_{i}+hf(t_{i},y_{i}).
  6. y i + 1 = y i + 1 2 h ( f ( t i , y i ) + f ( t i + 1 , y ~ i + 1 ) ) . y_{i+1}=y_{i}+\tfrac{1}{2}h\bigl(f(t_{i},y_{i})+f(t_{i+1},\tilde{y}_{i+1})% \bigr).
  7. y ~ i + 1 \displaystyle\tilde{y}_{i+1}
  8. y ~ i + 1 \displaystyle\tilde{y}_{i+1}
  9. y ~ i + 1 \displaystyle\tilde{y}_{i+1}

Preparata_code.html

  1. n = 2 m - 1 n=2^{m}-1
  2. 2 n + 2 = 2 m + 1 2n+2=2^{m+1}
  3. x X x = y Y y ; \sum_{x\in X}x=\sum_{y\in Y}y;
  4. x X x 3 + ( x X x ) 3 = y Y y 3 . \sum_{x\in X}x^{3}+\left(\sum_{x\in X}x\right)^{3}=\sum_{y\in Y}y^{3}.

Pressure-volume_loop_experiments.html

  1. V o l u m e = 4 3 × π × r 3 Volume=\frac{4}{3}\times\pi\times r^{3}
  2. V o l u m e = π 6 × L 1 × L 2 2 Volume=\frac{\pi}{6}\times L_{1}\times L_{2}^{2}
  3. V o l u m e = π 6 × L 1 × L 2 × L 3 Volume=\frac{\pi}{6}\times L_{1}\times L_{2}\times L_{3}
  4. V = 1 α × ρ × L 2 × ( G - G P ) V=\frac{1}{\alpha}\times\rho\times L^{2}\times(G-G^{P})
  5. d P d t m a x {\operatorname{d}P\over\operatorname{d}t_{max}}
  6. d P d t m a x {\operatorname{d}P\over\operatorname{d}t_{max}}

Pressure_exchanger.html

  1. η = Σ energy out Σ energy in = ( Q G - L ) × ( P G - H D P ) + ( Q B + L ) × ( P B - L D P ) Q G × P G + Q B × P B ( 1 ) \eta=\frac{\Sigma\,\text{ energy out}}{\Sigma\,\text{ energy in}}=\frac{(Q_{G}% -L)\times(P_{G}-HDP)+(Q_{B}+L)\times(P_{B}-LDP)}{Q_{G}\times P_{G}+Q_{B}\times P% _{B}}\qquad\qquad(1)

Price_of_anarchy.html

  1. G = ( N , S , u ) G=(N,S,u)
  2. N N
  3. S i S_{i}
  4. u i : S u_{i}:S\rightarrow\mathbb{R}
  5. S = S 1 × × S n S=S_{1}\times...\times S_{n}
  6. W : S W:S\rightarrow\mathbb{R}
  7. W ( s ) = i N u i ( s ) , W(s)=\sum_{i\in N}u_{i}(s),
  8. W ( s ) = min i N u i ( s ) , W(s)=\min_{i\in N}u_{i}(s),
  9. E S E\subseteq S
  10. P o A = max s S W ( s ) min s E W ( s ) PoA=\frac{\max_{s\in S}W(s)}{\min_{s\in E}W(s)}
  11. C : S C:S\rightarrow\mathbb{R}
  12. P o A = max s E C ( s ) min s S C ( s ) PoA=\frac{\max_{s\in E}C(s)}{\min_{s\in S}C(s)}
  13. P o S = max s S W ( s ) max s E W ( s ) PoS=\frac{\max_{s\in S}W(s)}{\max_{s\in E}W(s)}
  14. P o S = min s E C ( s ) min s S C ( s ) PoS=\frac{\min_{s\in E}C(s)}{\min_{s\in S}C(s)}
  15. 1 P o S P o A 1\leq PoS\leq PoA
  16. C ( s 1 , s 2 ) = u 1 ( s 1 , s 2 ) + u 2 ( s 1 , s 2 ) . C(s_{1},s_{2})=u_{1}(s_{1},s_{2})+u_{2}(s_{1},s_{2}).
  17. 1 + 1 = 2 1+1=2
  18. 5 + 5 = 10 , 5+5=10,
  19. 10 / 2 = 5 10/2=5
  20. N N
  21. M M
  22. s 1 , , s M > 0. s_{1},\ldots,s_{M}>0.
  23. w 1 , , w N > 0. w_{1},\ldots,w_{N}>0.
  24. A i = { 1 , 2 , , M } . A_{i}=\{1,2,\ldots,M\}.
  25. j j
  26. L j ( a ) = i : a i = j w i s j . L_{j}(a)=\frac{\sum_{i:a_{i}=j}w_{i}}{s_{j}}.
  27. i i
  28. c i ( a ) = L a i ( a ) , c_{i}(a)=L_{a_{i}}(a),
  29. MS ( a ) = max j L j ( a ) \mbox{MS}~{}(a)=\max_{j}L_{j}(a)
  30. N = 2 N=2
  31. w 1 = w 2 = 1 w_{1}=w_{2}=1
  32. M = 2 M=2
  33. s 1 = s 2 = 1 s_{1}=s_{2}=1
  34. σ 1 = σ 2 = ( 1 / 2 , 1 / 2 ) \sigma_{1}=\sigma_{2}=(1/2,1/2)
  35. 4 / 3 \leq 4/3
  36. a * a^{*}
  37. M M
  38. i i
  39. j j
  40. k k
  41. k k
  42. j j
  43. j j
  44. j j
  45. a a
  46. M M
  47. σ \sigma
  48. w ( σ ) i w i max j s j . w(\sigma)\leq\frac{\sum_{i}{w_{i}}}{\max_{j}{s_{j}}}.
  49. a a
  50. w ( a ) i w i j s j i w i M max j s j . w(a)\geq\frac{\sum_{i}{w_{i}}}{\sum_{j}{s_{j}}}\geq\frac{\sum_{i}{w_{i}}}{M% \cdot\max_{j}{s_{j}}}.
  51. w ( σ ) w(\sigma)
  52. w ( a ) w(a)
  53. G = ( V , E ) G=(V,E)
  54. s V s\in V
  55. t V t\in V
  56. f : E f:E\mapsto\Re
  57. L = { l e ( f e ) = a f e + b | e E , a 0 , b 0 } L=\{l_{e}(f_{e})=a\cdot f_{e}+b\;|\;e\in E,\;a\geq 0,\;b\geq 0\}
  58. f f
  59. w ( f ) = e f e l e ( f e ) w(f)=\sum_{e}{f_{e}\cdot l_{e}(f_{e})}
  60. s s
  61. t t
  62. s s
  63. t t
  64. G = ( V , E ) G=(V,E)
  65. L L
  66. w w
  67. R = { r 1 , r 2 , , r k , | r i > 0 } R=\{r_{1},r_{2},\dots,r_{k},\;|\;r_{i}>0\}
  68. Γ = { ( s 1 , t 1 ) , ( s 2 , t 2 ) , , ( s k , t k ) } ( V × V ) \Gamma=\{(s_{1},t_{1}),(s_{2},t_{2}),\dots,(s_{k},t_{k})\}\subseteq(V\times V)
  69. f Γ , R f_{\Gamma,R}
  70. p p\mapsto\Re
  71. p p
  72. s i s_{i}
  73. t i t_{i}
  74. Γ \in\Gamma
  75. p : s i t i f p = r i ( s i , t i ) Γ . \sum_{p:\,s_{i}\rightarrow t_{i}}{f_{p}}=r_{i}\;\;\forall(s_{i},t_{i})\in\Gamma.
  76. G G
  77. f e , Γ , R = p : e p f p . f_{e,\Gamma,R}=\sum_{p:\,e\in p}{f_{p}}.
  78. f e f_{e}
  79. Γ , R \Gamma,R
  80. f Γ , R f_{\Gamma,R}
  81. ( s i , t i ) Γ \forall(s_{i},t_{i})\in\Gamma
  82. p , q \forall p,q
  83. s i s_{i}
  84. t i t_{i}
  85. f p > 0 e p l e ( f e ) e q l e ( f e ) . f_{p}>0\Rightarrow\sum_{e\in p}{l_{e}(f_{e})}\leq\sum_{e\in q}{l_{e}(f_{e})}.
  86. f Γ , R f_{\Gamma,R}
  87. f Γ , R * f_{\Gamma,R}^{*}
  88. G G
  89. Γ \Gamma
  90. R R
  91. f f
  92. f * f^{*}
  93. f f
  94. w f ( f * ) = e E f e * l e ( f e ) w^{f}(f^{*})=\sum_{e\in E}{f^{*}_{e}\cdot l_{e}(f_{e})}
  95. f f
  96. f * f^{*}
  97. w ( f ) = w f ( f ) w f ( f * ) w(f)=w^{f}(f)\leq w^{f}(f^{*})
  98. w f ( f * ) < w f ( f ) w^{f}(f^{*})<w^{f}(f)
  99. i = 1 k p : s i t i f p * e p l e ( f e ) < i = 1 k p : s i t i f p e p l e ( f e ) \sum_{i=1}^{k}\sum_{p:s_{i}\rightarrow t_{i}}f_{p}^{*}\cdot\sum_{e\in p}l_{e}(% f_{e})<\sum_{i=1}^{k}\sum_{p:s_{i}\rightarrow t_{i}}f_{p}\cdot\sum_{e\in p}l_{% e}(f_{e})
  100. f f
  101. f * f^{*}
  102. Γ , R \Gamma,R
  103. p : s i t i f p = p : s i t i f p * = r i i . \sum_{p:s_{i}\rightarrow t_{i}}f_{p}=\sum_{p:s_{i}\rightarrow t_{i}}f_{p}^{*}=% r_{i}\;\;\forall i.
  104. ( s i , t i ) (s_{i},t_{i})
  105. p , q p,q
  106. s i s_{i}
  107. t i t_{i}
  108. f p * > f p f_{p}^{*}>f_{p}
  109. f q * < f q f_{q}^{*}<f_{q}
  110. e p l e ( f e ) < e q l e ( f e ) . \sum_{e\in p}l_{e}(f_{e})<\sum_{e\in q}l_{e}(f_{e}).
  111. f * f^{*}
  112. f f
  113. s i s_{i}
  114. t i t_{i}
  115. f * f^{*}
  116. f f
  117. f f
  118. L L
  119. x x
  120. y y
  121. x y x 2 + y 2 / 4 x\cdot y\leq x^{2}+y^{2}/4
  122. ( x - y / 2 ) 2 0 (x-y/2)^{2}\geq 0
  123. ( G , L ) (G,L)
  124. 4 / 3 \leq 4/3
  125. f f
  126. w ( f ) ( 4 / 3 ) min f * { w ( f * ) } w(f)\leq(4/3)\cdot\min_{f^{*}}\{w(f^{*})\}
  127. f * f^{*}
  128. w f ( f * ) = e E f e * ( a e f e + b e ) w^{f}(f^{*})=\sum_{e\in E}f_{e}^{*}(a_{e}\cdot f_{e}+b_{e})
  129. = e ( a e f e f e * ) + e E f e * b e . =\sum_{e}(a_{e}f_{e}f_{e}^{*})+\sum_{e\in E}f_{e}^{*}b_{e}.
  130. w f ( f * ) e E ( a e ( ( f e * ) 2 + ( f e ) 2 / 4 ) ) + e E f e * b e w^{f}(f^{*})\leq\sum_{e\in E}\left(a_{e}\cdot\left((f_{e}^{*})^{2}+(f_{e})^{2}% /4\right)\right)+\sum_{e\in E}f_{e}^{*}\cdot b_{e}
  131. = ( e E a e ( f e * ) 2 + f e * b e ) + e E a e ( f e ) 2 / 4 =\left(\sum_{e\in E}a_{e}(f_{e}^{*})^{2}+f_{e}^{*}b_{e}\right)+\sum_{e\in E}a_% {e}(f_{e})^{2}/4
  132. w ( f * ) + w ( f ) 4 , \leq w(f^{*})+\frac{w(f)}{4},
  133. ( 1 / 4 ) w ( f ) = ( 1 / 4 ) e E f e ( a e f e + b e ) (1/4)\cdot w(f)=(1/4)\cdot\sum_{e\in E}f_{e}(a_{e}f_{e}+b_{e})
  134. = ( 1 / 4 ) e E ( f e ) 2 + ( 1 / 4 ) e E f e b e 0 . =(1/4)\cdot\sum_{e\in E}(f_{e})^{2}+\underbrace{(1/4)\cdot\sum_{e\in E}f_{e}b_% {e}}_{\geq 0}.
  135. w f ( f * ) w ( f * ) + w ( f ) / 4 w^{f}(f^{*})\leq w(f^{*})+w(f)/4
  136. L L
  137. G G
  138. d d
  139. d + 1 \leq d+1
  140. d d
  141. x = 1 - 1 / d + 1 x=1-1/{\sqrt{d+1}}
  142. w = ( 1 - 1 d + 1 ) d ( 1 - 1 d + 1 ) + 1 1 d + 1 w=\left(1-\frac{1}{\sqrt{d+1}}\right)^{d}\cdot\left(1-\frac{1}{\sqrt{d+1}}% \right)+1\cdot\frac{1}{\sqrt{d+1}}
  143. = ( ( 1 - 1 d + 1 ) d + 1 ) d + 1 + 1 d + 1 =\left(\left(1-\frac{1}{\sqrt{d+1}}\right)^{\sqrt{d+1}}\right)^{\sqrt{d+1}}+% \frac{1}{\sqrt{d+1}}
  144. e - d + 1 + 1 d + 1 . \leq e^{-\sqrt{d+1}}+\frac{1}{\sqrt{d+1}}.
  145. d d

Pricing_science.html

  1. Q t + n i = Q t + n I * s I t + n i , i I ; Q_{t+n}^{i}=Q_{t+n}^{I}*sI_{t+n}^{i},i\in I;
  2. i i
  3. I I
  4. Q t + n I Q_{t+n}^{I}
  5. s I t + n i sI_{t+n}^{i}
  6. i i
  7. I I
  8. Q t + n I Q_{t+n}^{I}
  9. s I t + n i sI_{t+n}^{i}

Prime_manifold.html

  1. M M
  2. S S
  3. D D
  4. S = D S=\partial D
  5. D 3 = { x \R 3 | | x | 1 } . D^{3}=\{x\in\R^{3}\ |\ |x|\leq 1\}.
  6. M M
  7. M M
  8. N 1 # N 2 N_{1}\#N_{2}
  9. S 3 S^{3}
  10. M M
  11. \R 3 \R^{3}
  12. \R 3 \R^{3}
  13. S 3 S^{3}
  14. S 2 × S 1 S^{2}\times S^{1}
  15. S 2 × { p t } S^{2}\times\{pt\}
  16. S 1 S^{1}
  17. L ( p , q ) L(p,q)
  18. p 0 p\neq 0
  19. S 2 × S 1 S^{2}\times S^{1}
  20. S 2 × S 1 S^{2}\times S^{1}
  21. S 1 S^{1}
  22. M M
  23. M M
  24. M = N 1 # N 2 , M=N_{1}\#N_{2},
  25. M M
  26. N 1 N_{1}
  27. N 2 N_{2}
  28. M M
  29. M M
  30. N 1 N_{1}
  31. N 2 N_{2}
  32. N 1 N_{1}
  33. N 2 N_{2}
  34. M M
  35. M M
  36. S S
  37. S S
  38. N N
  39. M 1 M_{1}
  40. M 2 M_{2}
  41. N 1 N_{1}
  42. N 2 N_{2}
  43. M = N 1 # N 2 . M=N_{1}\#N_{2}.
  44. M M
  45. N 1 N_{1}
  46. S 3 S^{3}
  47. M 1 M_{1}
  48. S 3 S^{3}
  49. S S
  50. M M
  51. M M
  52. S S
  53. N N
  54. γ \gamma
  55. M M
  56. S S
  57. R R
  58. S S
  59. γ \gamma
  60. R \partial R
  61. M M
  62. R R
  63. R R
  64. M M
  65. R R
  66. M M
  67. S 2 × S 1 S^{2}\times S^{1}
  68. S 2 S^{2}
  69. S 1 S^{1}

Primitive_element_(finite_field).html

  1. α GF ( q ) \alpha\in\mathrm{GF}(q)
  2. GF ( q ) \mathrm{GF}(q)
  3. α i \alpha^{i}
  4. i i

Princeton_Papyri.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}

Principal_component_regression.html

  1. \;\;
  2. \;\;
  3. \;\;
  4. 𝐘 n × 1 = ( y 1 , , y n ) T \mathbf{Y}_{n\times 1}=\left(y_{1},\ldots,y_{n}\right)^{T}
  5. 𝐗 n × p = [ 𝐱 1 , , 𝐱 p ] T \mathbf{X}_{n\times p}=\left[\mathbf{x}_{1},\ldots,\mathbf{x}_{p}\right]^{T}
  6. n n
  7. p p
  8. n p n\geq p
  9. n n
  10. 𝐗 \mathbf{X}
  11. p p
  12. 𝐘 \mathbf{Y}
  13. 𝐘 \mathbf{Y}
  14. p p
  15. 𝐗 \mathbf{X}
  16. 𝐗 \mathbf{X}
  17. 𝐗 \mathbf{X}
  18. 𝐘 \mathbf{Y}
  19. 𝐗 \mathbf{X}
  20. 𝐘 = 𝐗 s y m b o l β + s y m b o l ε \mathbf{Y}=\mathbf{X}symbol{\beta}+symbol{\varepsilon}\;
  21. s y m b o l β p symbol{\beta}\in\mathbb{R}^{p}
  22. s y m b o l ε symbol{\varepsilon}
  23. E ( s y m b o l ε ) = 𝟎 \operatorname{E}\left(symbol{\varepsilon}\right)=\mathbf{0}\;
  24. Var ( s y m b o l ε ) = σ 2 I n × n \;\operatorname{Var}\left(symbol{\varepsilon}\right)=\sigma^{2}I_{n\times n}
  25. σ 2 > 0 \sigma^{2}>0\;\;
  26. s y m b o l β ^ \widehat{symbol\beta}
  27. s y m b o l β symbol\beta
  28. 𝐗 \mathbf{X}
  29. s y m b o l β ^ ols = ( 𝐗 T 𝐗 ) - 1 𝐗 T 𝐘 \widehat{symbol\beta}_{\mathrm{ols}}=(\mathbf{X}^{T}\mathbf{X})^{-1}\mathbf{X}% ^{T}\mathbf{Y}
  30. s y m b o l β symbol{\beta}
  31. s y m b o l β symbol{\beta}
  32. 𝐗 \mathbf{X}
  33. 𝐗 = U Δ V T \mathbf{X}=U\Delta V^{T}
  34. 𝐗 \mathbf{X}
  35. Δ p × p = diag [ δ 1 , , δ p ] \Delta_{p\times p}=\operatorname{diag}\left[\delta_{1},\ldots,\delta_{p}\right]
  36. δ 1 δ p 0 \delta_{1}\geq\cdots\geq\delta_{p}\geq 0
  37. 𝐗 \mathbf{X}
  38. U n × p = [ 𝐮 1 , , 𝐮 p ] U_{n\times p}=[\mathbf{u}_{1},\ldots,\mathbf{u}_{p}]
  39. V p × p = [ 𝐯 1 , , 𝐯 p ] V_{p\times p}=[\mathbf{v}_{1},\ldots,\mathbf{v}_{p}]
  40. 𝐗 \mathbf{X}
  41. V Λ V T V\Lambda V^{T}
  42. 𝐗 T 𝐗 \mathbf{X}^{T}\mathbf{X}
  43. Λ p × p = diag [ λ 1 , , λ p ] = diag [ δ 1 2 , , δ p 2 ] = Δ 2 \Lambda_{p\times p}=\operatorname{diag}\left[\lambda_{1},\ldots,\lambda_{p}% \right]=\operatorname{diag}\left[\delta_{1}^{2},\ldots,\delta_{p}^{2}\right]=% \Delta^{2}
  44. λ 1 λ p 0 \lambda_{1}\geq\cdots\geq\lambda_{p}\geq 0
  45. 𝐗 T 𝐗 \mathbf{X}^{T}\mathbf{X}
  46. V V
  47. 𝐗𝐯 j \mathbf{X}\mathbf{v}_{j}
  48. 𝐯 j \mathbf{v}_{j}
  49. j t h j^{th}
  50. j t h j^{th}
  51. j t h j^{th}
  52. λ j \lambda_{j}
  53. j { 1 , , p } j\in\{1,\ldots,p\}
  54. k { 1 , , p } k\in\{1,\ldots,p\}
  55. V k V_{k}
  56. p × k p\times k
  57. k k
  58. V V
  59. W k = 𝐗 V k W_{k}=\mathbf{X}V_{k}
  60. = [ 𝐗𝐯 1 , , 𝐗𝐯 k ] =[\mathbf{X}\mathbf{v}_{1},\ldots,\mathbf{X}\mathbf{v}_{k}]
  61. n × k n\times k
  62. k k
  63. W W
  64. 𝐱 i k = V k T 𝐱 i k \mathbf{x}_{i}^{k}=V_{k}^{T}\mathbf{x}_{i}\in\mathbb{R}^{k}
  65. 𝐱 i p 1 i n \mathbf{x}_{i}\in\mathbb{R}^{p}\;\;\forall\;\;1\leq i\leq n
  66. γ ^ k = ( W k T W k ) - 1 W k T 𝐘 k \widehat{\gamma}_{k}=\left(W_{k}^{T}W_{k}\right)^{-1}W_{k}^{T}\mathbf{Y}\in% \mathbb{R}^{k}
  67. 𝐘 \mathbf{Y}
  68. W k W_{k}
  69. k { 1 , , p } k\in\{1,\ldots,p\}
  70. s y m b o l β symbol{\beta}
  71. k k
  72. s y m b o l β ^ k = V k γ ^ k p \widehat{symbol{\beta}}_{k}=V_{k}\widehat{\gamma}_{k}\in\mathbb{R}^{p}
  73. W k W_{k}
  74. k { 1 , , p } k\in\{1,\ldots,p\}
  75. k k
  76. k k
  77. k k
  78. k = p k=p
  79. s y m b o l β ^ p = s y m b o l β ^ ols \widehat{symbol{\beta}}_{p}=\widehat{symbol{\beta}}_{\mathrm{ols}}
  80. W p = 𝐗 V p = 𝐗 V W_{p}=\mathbf{X}V_{p}=\mathbf{X}V
  81. V V
  82. k { 1 , , p } k\in\{1,\ldots,p\}
  83. s y m b o l β ^ k \widehat{symbol{\beta}}_{k}
  84. Var ( s y m b o l β ^ k ) = σ 2 V k ( W k T W k ) - 1 V k T = σ 2 V k diag ( λ 1 - 1 , , λ k - 1 ) V k T = σ 2 j = 1 k 𝐯 j 𝐯 j T λ j . \operatorname{Var}(\widehat{symbol{\beta}}_{k})=\sigma^{2}\;V_{k}(W_{k}^{T}W_{% k})^{-1}V_{k}^{T}=\sigma^{2}\;V_{k}\;\operatorname{diag}\left(\lambda_{1}^{-1}% ,\ldots,\lambda_{k}^{-1}\right)V_{k}^{T}=\sigma^{2}\sideset{}{}{\sum}_{j=1}^{k% }\frac{\mathbf{v}_{j}\mathbf{v}_{j}^{T}}{\lambda_{j}}.\;
  85. Var ( s y m b o l β ^ p ) = Var ( s y m b o l β ^ ols ) = σ 2 j = 1 p 𝐯 j 𝐯 j T λ j \operatorname{Var}(\widehat{symbol{\beta}}_{p})=\operatorname{Var}(\widehat{% symbol{\beta}}_{\mathrm{ols}})=\sigma^{2}\sideset{}{}{\sum}_{j=1}^{p}\frac{% \mathbf{v}_{j}\mathbf{v}_{j}^{T}}{\lambda_{j}}
  86. Var ( s y m b o l β ^ ols ) - Var ( s y m b o l β ^ k ) = σ 2 j = k + 1 p 𝐯 j 𝐯 j T λ j \operatorname{Var}(\widehat{symbol{\beta}}_{\mathrm{ols}})-\operatorname{Var}(% \widehat{symbol{\beta}}_{k})=\sigma^{2}\sideset{}{}{\sum}_{j=k+1}^{p}\frac{% \mathbf{v}_{j}\mathbf{v}_{j}^{T}}{\lambda_{j}}
  87. k { 1 , , ( p - 1 ) } k\in\{1,\ldots,(p-1)\}
  88. k { 1 , , p } , Var ( s y m b o l β ^ ols ) - Var ( s y m b o l β ^ k ) 0 k\in\{1,\ldots,p\},\operatorname{Var}(\widehat{symbol{\beta}}_{\mathrm{ols}})-% \operatorname{Var}(\widehat{symbol{\beta}}_{k})\succeq 0
  89. A 0 A\succeq 0
  90. A A
  91. 𝐗 \mathbf{X}
  92. 𝐗 \mathbf{X}
  93. 𝐗 T 𝐗 \mathbf{X}^{T}\mathbf{X}
  94. 0
  95. 0
  96. L k L_{k}
  97. p × k p\times k
  98. k { 1 , , p } k\in\{1,\ldots,p\}
  99. 𝐱 i \mathbf{x}_{i}
  100. k k
  101. L k 𝐳 i L_{k}\mathbf{z}_{i}
  102. 𝐳 i k ( 1 i n ) \mathbf{z}_{i}\in\mathbb{R}^{k}\;(1\leq i\leq n)
  103. i = 1 n 𝐱 i - L k 𝐳 i 2 \sum_{i=1}^{n}\|\mathbf{x}_{i}-L_{k}\mathbf{z}_{i}\|^{2}
  104. L k = V k L_{k}=V_{k}\rightarrow
  105. k k
  106. 𝐳 i = 𝐱 i k = V k T 𝐱 i \mathbf{z}_{i}=\mathbf{x}_{i}^{k}=V_{k}^{T}\mathbf{x}_{i}\rightarrow
  107. k k
  108. k k
  109. k k
  110. 𝐗 \mathbf{X}
  111. i = 1 n 𝐱 i - V k 𝐱 i k 2 = j = k + 1 n ( λ j ) \sum_{i=1}^{n}\|\mathbf{x}_{i}-V_{k}\mathbf{x}_{i}^{k}\|^{2}=\sum_{j=k+1}^{n}% \left(\lambda_{j}\right)\;
  112. 1 k < p 1\leq k<p
  113. i = 1 n 𝐱 i - V k 𝐱 i k 2 = 0 \sum_{i=1}^{n}\|\mathbf{x}_{i}-V_{k}\mathbf{x}_{i}^{k}\|^{2}=0
  114. k = p k=p
  115. k k
  116. 𝐗 T 𝐗 \mathbf{X}^{T}\mathbf{X}
  117. 1 k < p 1\leq k<p
  118. s y m b o l β ^ k \widehat{symbol{\beta}}_{k}
  119. min s y m b o l β * p 𝐘 - 𝐗 s y m b o l β * 2 \underset{symbol{\beta}_{*}\in\mathbb{R}^{p}}{\min}\;\|\mathbf{Y}-\mathbf{X}% symbol{\beta}_{*}\|^{2}
  120. s y m b o l β * { 𝐯 k + 1 , , 𝐯 p } symbol{\beta}_{*}\perp\{\mathbf{v}_{k+1},\ldots,\mathbf{v}_{p}\}
  121. V ( p - k ) T s y m b o l β * = 𝟎 V_{\left(p-k\right)}^{T}symbol{\beta}_{*}=\mathbf{0}
  122. V ( p - k ) = [ 𝐯 k + 1 , , 𝐯 p ] p × ( p - k ) . V_{\left(p-k\right)}=\left[\mathbf{v}_{k+1},\ldots,\mathbf{v}_{p}\right]_{p% \times\left(p-k\right)}.
  123. min s y m b o l β * p 𝐘 - 𝐗 s y m b o l β * 2 \underset{symbol{\beta}_{*}\in\mathbb{R}^{p}}{\min}\;\|\mathbf{Y}-\mathbf{X}% symbol{\beta}_{*}\|^{2}
  124. L ( p - k ) T s y m b o l β * = 𝟎 L_{\left(p-k\right)}^{T}symbol{\beta}_{*}=\mathbf{0}
  125. L ( p - k ) L_{\left(p-k\right)}
  126. p × ( p - k ) p\times\left(p-k\right)
  127. 1 k < p 1\leq k<p
  128. s y m b o l β ^ L \widehat{symbol{\beta}}_{L}
  129. s y m b o l β ^ L = arg min s y m b o l β * p 𝐘 - 𝐗 s y m b o l β * 2 \widehat{symbol{\beta}}_{L}=\underset{symbol{\beta}_{*}\in\mathbb{R}^{p}}{% \operatorname{arg}\;\min}\;\|\mathbf{Y}-\mathbf{X}symbol{\beta}_{*}\|^{2}
  130. L ( p - k ) T s y m b o l β * = 𝟎 L_{\left(p-k\right)}^{T}symbol{\beta}_{*}=\mathbf{0}
  131. L ( p - k ) L_{\left(p-k\right)}
  132. s y m b o l β ^ L \widehat{symbol{\beta}}_{L}
  133. L ( p - k ) * = V ( p - k ) Λ ( p - k ) 1 / 2 L^{*}_{\left(p-k\right)}=V_{\left(p-k\right)}\Lambda_{\left(p-k\right)}^{1/2}
  134. Λ ( p - k ) 1 / 2 = diag ( λ k + 1 1 / 2 , , λ p 1 / 2 ) . \Lambda_{\left(p-k\right)}^{1/2}=\operatorname{diag}\left(\lambda_{k+1}^{1/2},% \ldots,\lambda_{p}^{1/2}\right).
  135. s y m b o l β ^ L * \widehat{symbol{\beta}}_{L^{*}}
  136. s y m b o l β ^ k \widehat{symbol{\beta}}_{k}
  137. k k
  138. s y m b o l β symbol{\beta}
  139. Var ( s y m b o l β ^ ols ) = MSE ( s y m b o l β ^ ols ) \operatorname{Var}(\widehat{symbol{\beta}}_{\mathrm{ols}})=\operatorname{MSE}(% \widehat{symbol{\beta}}_{\mathrm{ols}})
  140. k { 1 , , p } k\in\{1,\ldots,p\}
  141. V ( p - k ) T s y m b o l β = 𝟎 V_{\left(p-k\right)}^{T}symbol{\beta}=\mathbf{0}
  142. s y m b o l β ^ k \widehat{symbol{\beta}}_{k}
  143. s y m b o l β symbol{\beta}
  144. Var ( s y m b o l β ^ k ) = MSE ( s y m b o l β ^ k ) \operatorname{Var}(\widehat{symbol{\beta}}_{k})=\operatorname{MSE}(\widehat{% symbol{\beta}}_{k})
  145. Var ( s y m b o l β ^ ols ) - Var ( s y m b o l β ^ j ) \operatorname{Var}(\widehat{symbol{\beta}}_{\mathrm{ols}})-\operatorname{Var}(% \widehat{symbol{\beta}}_{j})
  146. 0 1 j p \succeq 0\;\forall\;1\leq j\leq p
  147. MSE ( s y m b o l β ^ ols ) - MSE ( s y m b o l β ^ k ) 0 \operatorname{MSE}(\widehat{symbol{\beta}}_{\mathrm{ols}})-\operatorname{MSE}(% \widehat{symbol{\beta}}_{k})\succeq 0
  148. k k
  149. s y m b o l β ^ k \widehat{symbol{\beta}}_{k}
  150. s y m b o l β symbol{\beta}
  151. s y m b o l β ^ ols \widehat{symbol{\beta}}_{\mathrm{ols}}
  152. s y m b o l β ^ k \widehat{symbol{\beta}}_{k}
  153. s y m b o l β ^ ols \widehat{symbol{\beta}}_{\mathrm{ols}}
  154. k { 1 , , p } k\in\{1,\ldots,p\}
  155. V ( p - k ) T s y m b o l β 𝟎 V_{\left(p-k\right)}^{T}symbol{\beta}\neq\mathbf{0}
  156. s y m b o l β ^ k \widehat{symbol{\beta}}_{k}
  157. s y m b o l β symbol{\beta}
  158. Var ( s y m b o l β ^ ols ) - Var ( s y m b o l β ^ k ) \operatorname{Var}(\widehat{symbol{\beta}}_{\mathrm{ols}})-\operatorname{Var}(% \widehat{symbol{\beta}}_{k})
  159. 0 1 k p \succeq 0\;\forall\;1\leq k\leq p
  160. MSE ( s y m b o l β ^ ols ) - MSE ( s y m b o l β ^ k ) 0 \operatorname{MSE}(\widehat{symbol{\beta}}_{\mathrm{ols}})-\operatorname{MSE}(% \widehat{symbol{\beta}}_{k})\succeq 0
  161. k k
  162. s y m b o l β symbol{\beta}
  163. j t h j^{th}
  164. λ j < ( p σ 2 ) / s y m b o l β T s y m b o l β \lambda_{j}<(p\sigma^{2})/symbol{\beta}^{T}symbol{\beta}
  165. σ 2 \sigma^{2}
  166. s y m b o l β symbol{\beta}
  167. 𝐗 T 𝐗 \mathbf{X}^{T}\mathbf{X}
  168. 𝐗 T 𝐗 \mathbf{X}^{T}\mathbf{X}
  169. 𝐗 T 𝐗 \mathbf{X}^{T}\mathbf{X}
  170. 𝐗 \mathbf{X}
  171. p p
  172. p p
  173. m { 1 , , p } m\in\{1,\ldots,p\}
  174. m m
  175. n × m n\times m
  176. m { 1 , , p } m\in\{1,\ldots,p\}
  177. k { 1 , , m } k\in\{1,\ldots,m\}
  178. n × n n\times n
  179. n × n n\times n
  180. 𝐗𝐗 T \mathbf{X}\mathbf{X}^{T}
  181. 𝐗𝐗 T \mathbf{X}\mathbf{X}^{T}

Product-form_solution.html

  1. P ( x 1 , x 2 , x 3 , , x n ) = B i = 1 n P ( x i ) \,\text{P}(x_{1},x_{2},x_{3},\ldots,x_{n})=B\prod_{i=1}^{n}\,\text{P}(x_{i})

Proebsting's_paradox.html

  1. f * = b - 1 2 b f^{*}=\frac{b-1}{2b}\!
  2. 0.5 ln ( 1.5 + 5 f * ) + 0.5 ln ( 0.75 - f * ) 0.5\ln(1.5+5f^{*})+0.5\ln(0.75-f^{*})\!
  3. 5 ( 0.75 - f * ) = 1.5 + 5 f * 5(0.75-f^{*})=1.5+5f^{*}\!
  4. 2.25 = 10 f * 2.25=10f^{*}\!
  5. 3 n - 1 4 n \frac{3^{n-1}}{4^{n}}\!
  6. f * = b 2 - 1 2 b 2 + b 1 - 1 4 ( 1 f 1 - 1 f 2 ) . f^{*}=\frac{b_{2}-1}{2b_{2}}+\frac{b_{1}-1}{4}\left(\frac{1}{f_{1}}-\frac{1}{f% _{2}}\right).\!
  7. 0.25 ln ( 1 + 2 f 1 ) + 0.25 ln ( 1 - f 1 ) + 0.25 ln ( 1 + 2 f 1 + 5 f 2 ) + 0.25 ln ( 1 - f 1 - f 2 ) 0.25\ln(1+2f_{1})+0.25\ln(1-f_{1})+0.25\ln(1+2f_{1}+5f_{2})+0.25\ln(1-f_{1}-f_% {2})\!
  8. f * = b 2 - 1 2 b 2 - b 1 - 1 4 ( 1 f 1 - 1 f 2 ) f 2 - 1 f 2 + 1 f^{*}=\frac{b_{2}-1}{2b_{2}}-\frac{b_{1}-1}{4}\left(\frac{1}{f_{1}}-\frac{1}{f% _{2}}\right)\frac{f_{2}-1}{f_{2}+1}\!

Projected_area.html

  1. A p r o j e c t e d = A cos β d A A_{projected}=\int_{A}\cos{\beta}\,dA
  2. β \beta\,
  3. A = L × W A=L\times W
  4. A p r o j = L × W cos β A_{proj}=L\times W\cos{\beta}
  5. A = π r 2 A=\pi r^{2}
  6. A p r o j = π r 2 cos β A_{proj}=\pi r^{2}\cos{\beta}
  7. A = 4 π r 2 A=4\pi r^{2}
  8. A p r o j = A 4 = π r 2 A_{proj}=\frac{A}{4}=\pi r^{2}

Projection_method_(fluid_dynamics).html

  1. 𝐮 \mathbf{u}
  2. 𝐮 sol \mathbf{u}_{\,\text{sol}}
  3. 𝐮 irrot \mathbf{u}_{\,\text{irrot}}
  4. 𝐮 = 𝐮 sol + 𝐮 irrot = 𝐮 sol + ϕ \mathbf{u}=\mathbf{u}_{\,\text{sol}}+\mathbf{u}_{\,\text{irrot}}=\mathbf{u}_{% \,\text{sol}}+\nabla\phi
  5. × ϕ = 0 \nabla\times\nabla\phi=0
  6. ϕ \,\phi
  7. 𝐮 = 2 ϕ ( since, 𝐮 sol = 0 ) \nabla\cdot\mathbf{u}=\nabla^{2}\phi\qquad(\,\text{since,}\;\nabla\cdot\mathbf% {u}_{\,\text{sol}}=0)
  8. ϕ \,\phi
  9. 𝐮 \mathbf{u}
  10. ϕ \,\phi
  11. 𝐮 \mathbf{u}
  12. 𝐮 sol = 𝐮 - ϕ \mathbf{u}_{\,\text{sol}}=\mathbf{u}-\nabla\phi
  13. 𝐮 t + ( 𝐮 ) 𝐮 = - 1 ρ p + ν 2 𝐮 \frac{\partial\mathbf{u}}{\partial t}+(\mathbf{u}\cdot\nabla)\mathbf{u}=-\frac% {1}{\rho}\nabla p+\nu\nabla^{2}\mathbf{u}
  14. 𝐮 * \mathbf{u}^{*}
  15. ( 1 ) 𝐮 * - 𝐮 n Δ t = - ( 𝐮 n ) 𝐮 n + ν 2 𝐮 n \quad(1)\qquad\frac{\mathbf{u}^{*}-\mathbf{u}^{n}}{\Delta t}=-(\mathbf{u}^{n}% \cdot\nabla)\mathbf{u}^{n}+\nu\nabla^{2}\mathbf{u}^{n}
  16. 𝐮 n \mathbf{u}^{n}
  17. n \,n
  18. 𝐮 n + 1 \mathbf{u}^{n+1}
  19. ( 2 ) 𝐮 n + 1 = 𝐮 * - Δ t ρ p n + 1 \quad(2)\qquad\mathbf{u}^{n+1}=\mathbf{u}^{*}-\frac{\Delta t}{\rho}\,\nabla p^% {n+1}
  20. 𝐮 n + 1 - 𝐮 * Δ t = - 1 ρ p n + 1 \frac{\mathbf{u}^{n+1}-\mathbf{u}^{*}}{\Delta t}=-\frac{1}{\rho}\,\nabla p^{n+1}
  21. p \,p
  22. ( n + 1 ) \,(n+1)
  23. 𝐮 n + 1 = 0 \nabla\cdot\mathbf{u}^{n+1}=0
  24. p n + 1 \,p^{n+1}
  25. 2 p n + 1 = ρ Δ t 𝐮 * \nabla^{2}p^{n+1}=\frac{\rho}{\Delta t}\,\nabla\cdot\mathbf{u}^{*}
  26. 𝐮 * = 𝐮 n + 1 + Δ t ρ p n + 1 \mathbf{u}^{*}=\mathbf{u}^{n+1}+\frac{\Delta t}{\rho}\,\nabla p^{n+1}
  27. p \,p
  28. Ω \partial\Omega
  29. p n + 1 𝐧 = 0 \nabla p^{n+1}\cdot\mathbf{n}=0
  30. 𝐮 * \mathbf{u}^{*}
  31. 𝐮 𝐧 = 0 \mathbf{u}\cdot\mathbf{n}=0
  32. Ω \partial\Omega
  33. p n + 1 n = 0 on Ω \frac{\partial p^{n+1}}{\partial n}=0\qquad\,\text{on}\quad\partial\Omega
  34. 𝐮 n + 1 \nabla\cdot\mathbf{u}^{n+1}

Projective_harmonic_conjugate.html

  1. A C : B C = A D : D B {AC}:{BC}={AD}:{DB}\,
  2. ( A , B ; C , D ) = A C A D / B C - D B , (A,B;C,D)=\frac{AC}{AD}/\frac{BC}{-DB},
  3. ( A , B ; C , D ) = A C A D . B D B C = - 1. (A,B;C,D)=\frac{AC}{AD}.\frac{BD}{BC}=-1.\,
  4. t ( x ) = x - a x - b . t(x)=\frac{x-a}{x-b}.
  5. t ( x ) = x - a x - b = - 1. t(x)=\frac{x-a}{x-b}=-1.
  6. lim y t ( y ) = 1 , \lim_{y\to\infty}t(y)=1,

Proof_without_words.html

  1. a 2 + b 2 = c 2 a^{2}+b^{2}=c^{2}\,

Propensity_score_matching.html

  1. p ( x ) = def Pr ( T = 1 | X = x ) . p(x)\ \stackrel{\mathrm{def}}{=}\ \Pr(T=1|X=x).
  2. Y ( 0 ) , Y ( 1 ) T | X Y(0),Y(1)\perp T\,|\,X
  3. \perp
  4. Y ( 0 ) , Y ( 1 ) T | p ( X ) . Y(0),Y(1)\perp T\,|\,p(X).

Property_testing.html

  1. ϵ ( n 2 ) \epsilon{\textstyle\left({{n}\atop{2}}\right)}
  2. Ω ( n ) \Omega(\sqrt{n})

Proportionality_for_Solid_Coalitions.html

  1. V V
  2. C C
  3. V V
  4. C C
  5. C C
  6. n n
  7. k k
  8. j j
  9. k k
  10. k k
  11. n / k n/k
  12. V V
  13. C C
  14. V V
  15. j j
  16. j j
  17. C C
  18. C C
  19. j j
  20. k + 1 k+1
  21. k + 1 k+1
  22. k k
  23. n / ( k + 1 ) n/(k+1)
  24. V V
  25. j j
  26. n / ( k + 1 ) n/(k+1)
  27. k + 1 k+1

PROPT.html

  1. f f
  2. c c

Proton-to-electron_mass_ratio.html

  1. α s = - 2 π β 0 ln ( E / Λ Q C D ) \alpha_{s}=-\frac{2\pi}{\beta_{0}\ln(E/\Lambda_{QCD})}
  2. λ i = λ 0 [ 1 + K i ( Δ μ / μ ) ] , \ \lambda_{i}=\lambda_{0}[1+K_{i}(\Delta\mu/\mu)],
  3. × 10 - 5 \times 10^{-}5
  4. × 10 - 6 \times 10^{-}6

Proton-transfer-reaction_mass_spectrometry.html

  1. R R
  2. H 3 O + + R R H + + H 2 O H_{3}O^{+}+R\longrightarrow RH^{+}+H_{2}O
  3. R R
  4. [ R H + ] = [ H 3 O + ] 0 ( 1 - e - k [ R ] t ) [ H 3 O + ] 0 [ R ] k t [RH^{+}]=[H_{3}O^{+}]_{0}\left(1-e^{-k[R]t}\right)\approx[H_{3}O^{+}]_{0}[R]kt
  5. [ R H + ] [RH^{+}]
  6. [ H 3 O + ] 0 [H_{3}O^{+}]_{0}
  7. k k
  8. t t
  9. [ R ] [R]
  10. e - + H 2 O H 2 O + + 2 e - e^{-}+H_{2}O\longrightarrow H_{2}O^{+}+2e^{-}
  11. e - + H 2 O H 2 + + O + 2 e - e^{-}+H_{2}O\longrightarrow H_{2}^{+}+O+2e^{-}
  12. e - + H 2 O H + + O H + 2 e - e^{-}+H_{2}O\longrightarrow H^{+}+OH+2e^{-}
  13. e - + H 2 O O + + H 2 + 2 e - e^{-}+H_{2}O\longrightarrow O^{+}+H_{2}+2e^{-}
  14. H 2 + + H 2 O H 2 O + + H 2 H_{2}^{+}+H_{2}O\rightarrow H_{2}O^{+}+H_{2}
  15. H + + H 2 O H 2 O + + H H^{+}+H_{2}O\rightarrow H_{2}O^{+}+H
  16. O + + H 2 O H 2 O + + O O^{+}+H_{2}O\rightarrow H_{2}O^{+}+O
  17. H 2 O + + H 2 O H 3 O + + O H H_{2}O^{+}+H_{2}O\longrightarrow H_{3}O^{+}+OH

Proton_ATPase.html

  1. \rightleftharpoons

Prototype_filter.html

  1. i ω ( ω c ω c ) i ω i\omega\to\left(\frac{\omega_{c}^{\prime}}{\omega_{c}}\right)i\omega
  2. A ( i ω ) A ( i ω ω c ) A(i\omega)\to A\left(i\frac{\omega}{\omega_{c}}\right)
  3. L ω c ω c L L\to\frac{\omega_{c}^{\prime}}{\omega_{c}}\,L
  4. C ω c ω c C C\to\frac{\omega_{c}^{\prime}}{\omega_{c}}\,C
  5. Z R R Z Z\to\frac{R}{R^{\prime}}\,Z
  6. Y R R Y Y\to\frac{R^{\prime}}{R}\,Y
  7. L R R L L\to\frac{R}{R^{\prime}}\,L
  8. C R R C C\to\frac{R^{\prime}}{R}\,C
  9. L ω c ω c R R L L\to\,\frac{\omega_{c}^{\prime}}{\omega_{c}}\,\frac{R}{R^{\prime}}\,L
  10. C ω c ω c R R C C\to\,\frac{\omega_{c}^{\prime}}{\omega_{c}}\,\frac{R^{\prime}}{R}\,C
  11. i ω ω c ω c i ω \frac{i\omega}{\omega_{c}^{\prime}}\to\frac{\omega_{c}}{i\omega}
  12. A ( i ω ) A ( ω c ω c i ω ) A(i\omega)\to A\left(\frac{\omega_{c}\,\omega_{c}^{\prime}}{i\omega}\right)
  13. L C = 1 ω c ω c L L^{\prime}\to C=\frac{1}{\omega_{c}\,\omega_{c}^{\prime}\,L^{\prime}}
  14. C L = 1 ω c ω c C C^{\prime}\to L=\frac{1}{\omega_{c}\,\omega_{c}^{\prime}\,C^{\prime}}
  15. i ω ω c Q ( i ω ω 0 + ω 0 i ω ) \frac{i\omega}{\omega_{c}^{\prime}}\to Q\left(\frac{i\omega}{\omega_{0}}+\frac% {\omega_{0}}{i\omega}\right)
  16. Q = ω 0 Δ ω Q=\frac{\omega_{0}}{\Delta\omega}
  17. Δ ω = ω 2 - ω 1 \Delta\omega=\omega_{2}-\omega_{1}\,
  18. ω 0 = ω 1 ω 2 \omega_{0}=\sqrt{\omega_{1}\omega_{2}}
  19. A ( i ω ) A ( ω c Q [ i ω ω 0 + ω 0 i ω ] ) A(i\omega)\to A\left(\omega_{c}^{\prime}Q\left[\frac{i\omega}{\omega_{0}}+% \frac{\omega_{0}}{i\omega}\right]\right)
  20. L L = ω c Q ω 0 L , C = 1 ω 0 ω c Q 1 L L^{\prime}\to L=\frac{\omega_{c}^{\prime}Q}{\omega_{0}}L^{\prime}\,,\,C=\frac{% 1}{\omega_{0}\omega_{c}^{\prime}Q}\frac{1}{L^{\prime}}
  21. C C = ω c Q ω 0 C L = 1 ω 0 ω c Q 1 C C^{\prime}\to C=\frac{\omega_{c}^{\prime}Q}{\omega_{0}}C^{\prime}\,\lVert\,L=% \frac{1}{\omega_{0}\omega_{c}^{\prime}Q}\frac{1}{C^{\prime}}
  22. ω c i ω Q ( i ω ω 0 + ω 0 i ω ) \frac{\omega_{c}^{\prime}}{i\omega}\to Q\left(\frac{i\omega}{\omega_{0}}+% \dfrac{\omega_{0}}{i\omega}\right)
  23. L L = ω c ω 0 Q L C = Q ω 0 ω c 1 L L^{\prime}\to L=\frac{\omega_{c}^{\prime}}{\omega_{0}Q}L^{\prime}\,\lVert\,C=% \frac{Q}{\omega_{0}\omega_{c}^{\prime}}\frac{1}{L^{\prime}}
  24. C C = ω c ω 0 Q C , L = Q ω 0 ω c 1 C C^{\prime}\to C=\frac{\omega_{c}^{\prime}}{\omega_{0}Q}C^{\prime}\,,\,L=\frac{% Q}{\omega_{0}\omega_{c}^{\prime}}\frac{1}{C^{\prime}}
  25. ω c i ω 1 Q 1 ( i ω ω 01 + ω 01 i ω ) + 1 Q 2 ( i ω ω 02 + ω 02 i ω ) + \frac{\omega_{c}^{\prime}}{i\omega}\to\dfrac{1}{Q_{1}\left(\dfrac{i\omega}{% \omega_{01}}+\dfrac{\omega_{01}}{i\omega}\right)}+\dfrac{1}{Q_{2}\left(\dfrac{% i\omega}{\omega_{02}}+\dfrac{\omega_{02}}{i\omega}\right)}+\cdots
  26. i ω ω c 1 Q 1 ( i ω ω 01 + ω 01 i ω ) + 1 Q 2 ( i ω ω 02 + ω 02 i ω ) + \frac{i\omega}{\omega_{c}^{\prime}}\to\dfrac{1}{Q_{1}\left(\dfrac{i\omega}{% \omega_{01}}+\dfrac{\omega_{01}}{i\omega}\right)}+\dfrac{1}{Q_{2}\left(\dfrac{% i\omega}{\omega_{02}}+\dfrac{\omega_{02}}{i\omega}\right)}+\cdots
  27. Z Y = U + i V ZY=U+iV\,\!
  28. Z Y = U ( ω ) + i V ( ω ) ZY=U(\omega)+iV(\omega)\,\!
  29. Z Y = U k ( ω ) + i V k ( ω ) ZY=U_{k}(\omega)+iV_{k}(\omega)\,\!
  30. R 0 = 1 , ω c = 1 R_{0}=1\,,\,\omega_{c}=1
  31. U k ( ω ) = - ω 2 U_{k}(\omega)=-\omega^{2}\,\!
  32. U k ( ω ) ( i ω ω c ) 2 U_{k}(\omega)\to\left(\frac{i\omega}{\omega_{c}}\right)^{2}
  33. U k ( ω ) ( ω c i ω ) 2 U_{k}(\omega)\to\left(\frac{\omega_{c}}{i\omega}\right)^{2}
  34. U k ( ω ) Q 2 ( i ω ω 0 + ω 0 i ω ) 2 U_{k}(\omega)\to Q^{2}\left(\frac{i\omega}{\omega_{0}}+\frac{\omega_{0}}{i% \omega}\right)^{2}

Prouhet–Tarry–Escott_problem.html

  1. a A a i = b B b i \sum_{a\in A}a^{i}=\sum_{b\in B}b^{i}
  2. n , k n,k\in\mathbb{N}
  3. { ( x 1 , y 1 ) , , ( x n , y n ) } \{(x_{1},y_{1}),\dots,(x_{n},y_{n})\}
  4. { ( x 1 , y 1 ) , , ( x n , y n ) } \{(x_{1}^{\prime},y_{1}^{\prime}),\dots,(x_{n}^{\prime},y_{n}^{\prime})\}
  5. 2 \mathbb{Z}^{2}
  6. i = 1 n x i j y i d - j = i = 1 n x i j y i d - j \sum_{i=1}^{n}x_{i}^{j}y_{i}^{d-j}=\sum_{i=1}^{n}{x^{\prime}}_{i}^{j}{y^{% \prime}}_{i}^{d-j}
  7. d , j { 0 , , k } d,j\in\{0,\dots,k\}
  8. j d . j\leq d.
  9. n = 6 n=6
  10. k = 5 k=5
  11. { ( x 1 , y 1 ) , , ( x 6 , y 6 ) } = { ( 2 , 1 ) , ( 1 , 3 ) , ( 3 , 6 ) , ( 6 , 7 ) , ( 7 , 5 ) , ( 5 , 2 ) } \{(x_{1},y_{1}),\dots,(x_{6},y_{6})\}=\{(2,1),(1,3),(3,6),(6,7),(7,5),(5,2)\}
  12. { ( x 1 , y 1 ) , , ( x 6 , y 6 ) } = { ( 1 , 2 ) , ( 2 , 5 ) , ( 5 , 7 ) , ( 7 , 6 ) , ( 6 , 3 ) , ( 3 , 1 ) } \{(x^{\prime}_{1},y^{\prime}_{1}),\dots,(x^{\prime}_{6},y^{\prime}_{6})\}=\{(1% ,2),(2,5),(5,7),(7,6),(6,3),(3,1)\}
  13. n = k + 1 n=k+1
  14. k 6 k\geq 6

Pseudo-abelian_category.html

  1. p p
  2. p p = p p\circ p=p
  3. C C
  4. k a r ( C ) kar(C)
  5. s : C k a r ( C ) s:C\rightarrow kar(C)
  6. s ( p ) s(p)
  7. p p
  8. C C
  9. k a r ( C ) kar(C)
  10. C C
  11. k a r ( C ) kar(C)
  12. C C
  13. C k a r ( C ) C\rightarrow kar(C)
  14. C C
  15. k a r ( C ) kar(C)
  16. k a r ( C ) kar(C)
  17. ( X , p ) (X,p)
  18. X X
  19. C C
  20. p p
  21. X X
  22. f : ( X , p ) ( Y , q ) f:(X,p)\rightarrow(Y,q)
  23. k a r ( C ) kar(C)
  24. f : X Y f:X\rightarrow Y
  25. f = q f p f=q\circ f\circ p
  26. C C
  27. C k a r ( C ) C\rightarrow kar(C)
  28. X X
  29. ( X , i d X ) (X,id_{X})

Pseudo-order.html

  1. x , y : ¬ ( x < y y < x ) \forall x,y:\neg\;(x<y\;\wedge\;y<x)
  2. x x
  3. y y
  4. z z
  5. x , y , z : x < y ( x < z z < y ) \forall x,y,z:x<y\;\to\;(x<z\;\vee\;z<y)
  6. x , y : ¬ ( x < y y < x ) x = y \forall x,y:\neg\;(x<y\;\vee\;y<x)\;\to\;x=y
  7. x # y x < y y < x x\#y\;\leftrightarrow\;x<y\;\vee\;y<x

Pseudo_amino_acid_composition.html

  1. L L
  2. 𝐏 = [ R 1 R 2 R 3 R 4 R 5 R 6 R 7 R L ] (1) \mathbf{P}={\begin{bmatrix}\mathrm{R}_{1}\mathrm{R}_{2}\mathrm{R}_{3}\mathrm{R% }_{4}\mathrm{R}_{5}\mathrm{R}_{6}\mathrm{R}_{7}\cdots\mathrm{R}_{L}\end{% bmatrix}}\qquad\,\text{(1)}
  3. 𝐏 = [ f 1 f 2 f 20 ] 𝐓 (2) \mathbf{P}={\begin{bmatrix}f_{1}&f_{2}&\cdots&f_{20}\end{bmatrix}}^{\mathbf{T}% }\qquad\,\text{(2)}
  4. f u ( u = 1 , 2 , , 20 ) \,f_{u}\,(u=1,2,\cdots,20)
  5. J i , j J_{i,j}
  6. 𝐏 = [ p 1 , p 2 , , p 20 , p 20 + 1 , , p 20 + λ ] 𝐓 , ( λ < L ) (3) \mathbf{P}={\begin{bmatrix}p_{1},\,p_{2},\,\ldots,\,p_{20},\,p_{20+1},\,\ldots% ,\,p_{20+\lambda}\end{bmatrix}}^{\mathbf{T}},\,\,\,(\lambda<L)\qquad\,\text{(3)}
  7. 20 + λ 20+\lambda
  8. p u = { f u i = 1 20 f i + w k = 1 λ τ k , ( 1 u 20 ) w τ u - 20 i = 1 20 f i + w k = 1 λ τ k , ( 20 + 1 u 20 + λ ) (4) p_{u}=\begin{cases}\dfrac{f_{u}}{\sum_{i=1}^{20}f_{i}\,+\,w\sum_{k=1}^{\lambda% }\tau_{k}},&(1\leq u\leq 20)\\ \dfrac{w\tau_{u-20}}{\sum_{i=1}^{20}f_{i}\,+\,w\sum_{k=1}^{\lambda}\tau_{k}},&% (20+1\leq u\leq 20+\lambda)\end{cases}\qquad\,\text{(4)}
  9. w w
  10. τ k \tau_{k}
  11. k k
  12. k k
  13. τ k = 1 L - k i = 1 L - k J i , i + k , ( k < L ) (5) \tau_{k}=\frac{1}{L-k}\sum_{i=1}^{L-k}\,\mathrm{J}_{i,i+k},\,\,\,(k<L)\qquad\,% \text{(5)}
  14. J i , i + k = 1 Γ q = 1 Γ [ Φ q ( R i + k ) - Φ q ( R i ) ] 2 (6) \mathrm{J}_{i,i+k}=\frac{1}{\Gamma}\sum_{q=1}^{\Gamma}\left[\Phi_{q}\left(% \mathrm{R}_{i+k}\right)-\Phi_{q}\left(\mathrm{R}_{i}\right)\right]^{2}\qquad\,% \text{(6)}
  15. Φ q ( R i ) \Phi_{q}\left(\mathrm{R}_{i}\right)
  16. q {q}
  17. R i \mathrm{R}_{i}\,
  18. Γ \Gamma\,
  19. Φ 1 ( R i ) \Phi_{1}\left(\mathrm{R}_{i}\right)
  20. Ψ 2 ( R i ) \Psi_{2}\left(\mathrm{R}_{i}\right)
  21. Ψ 3 ( R i ) \Psi_{3}\left(\mathrm{R}_{i}\right)
  22. R i \mathrm{R}_{i}\,
  23. Φ 1 ( R i + 1 ) \Phi_{1}\left(\mathrm{R}_{i+1}\right)
  24. Φ 2 ( R i + 1 ) \Phi_{2}\left(\mathrm{R}_{i+1}\right)
  25. Φ 3 ( R i + 1 ) \Phi_{3}\left(\mathrm{R}_{i+1}\right)
  26. R i + 1 \mathrm{R}_{i+1}\,
  27. Γ = 3 \Gamma=3\,
  28. p 1 , p 2 , , p 20 p_{1},\,p_{2},\,\cdots,\,p_{20}
  29. p 20 + 1 , , p 20 + λ p_{20+1},\,\cdots,\,p_{20+\lambda}
  30. λ \lambda\,
  31. λ \lambda\,
  32. λ \lambda\,
  33. λ \lambda\,

Pseudorandom_generators_for_polynomials.html

  1. G : 𝔽 𝔽 n G:\mathbb{F}^{\ell}\rightarrow\mathbb{F}^{n}
  2. d d
  3. 𝔽 \mathbb{F}
  4. \ell
  5. n n
  6. n n
  7. 𝔽 \mathbb{F}
  8. d d
  9. G G
  10. p ( x 1 , , x n ) p(x_{1},\dots,x_{n})
  11. p ( U n ) p(U_{n})
  12. p ( G ( U ) ) p(G(U_{\ell}))
  13. ϵ \epsilon
  14. U k U_{k}
  15. 𝔽 k \mathbb{F}^{k}
  16. d = 1 d=1
  17. = log n + O ( log ( ϵ - 1 ) ) \ell=\log n+O(\log(\epsilon^{-1}))
  18. 2 d 2^{d}
  19. d d
  20. d d
  21. d d
  22. = d log n + O ( 2 d log ( ϵ - 1 ) ) \ell=d\cdot\log n+O(2^{d}\cdot\log(\epsilon^{-1}))

Pseudoreflection.html

  1. g : V V g:V\to V
  2. V g = { v V : g v = v } V^{g}=\{v\in V:\ gv=v\}
  3. [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 r ] \begin{bmatrix}1&0&0&\cdots&0\\ 0&1&0&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&1&0\\ 0&0&0&\cdots&r\\ \end{bmatrix}
  4. [ 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1 ] \begin{bmatrix}1&0&0&\cdots&0\\ 0&1&0&\cdots&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&\cdots&1&1\\ 0&0&0&\cdots&1\\ \end{bmatrix}

Pseudospectral_optimal_control.html

  1. \infty
  2. 2 N - 1 2N-1

PSRK.html

  1. P = R T v - b - a α ( T ) v ( v + b ) P={{R\;T}\over{v-b}}-{{a\;\alpha(T)}\over{v(v+b)}}
  2. α ( T r ) = [ 1 + c 1 ( 1 - T r ) + c 2 ( 1 - T r ) 2 + c 3 ( 1 - T r ) 3 ] 2 \alpha(T_{r})=\left[1+c_{1}\left(1-\sqrt{T}_{r}\right)+c_{2}\left(1-\sqrt{T}_{% r}\right)^{2}+c_{3}\left(1-\sqrt{T}_{r}\right)^{3}\right]^{2}
  3. c 1 = 0.48 + 1.574 ω - 0.176 ω 2 c_{1}=0.48+1.574\;\omega-0.176\;\omega^{2}
  4. a b R T = i x i a i i b i R T - g 0 E R T + x i l n b b i 0.64663 {a\over{bRT}}=\sum_{i}{{x_{i}}{a_{ii}\over{b_{i}RT}}-{{{g^{E}_{0}}\over{RT}}+% \sum{x_{i}ln{b\over{b_{i}}}}\over{0.64663}}}
  5. b = i x i b i b=\sum_{i}x_{i}\;b_{i}

Pushforward_(homology).html

  1. f f
  2. X Y X\rightarrow Y
  3. f * : H n ( X ) H n ( Y ) f_{*}:H_{n}\left(X\right)\rightarrow H_{n}\left(Y\right)
  4. n 0 n\geq 0
  5. X X
  6. H n ( X ) H_{n}\left(X\right)
  7. H * ( X ) H_{*}\left(X\right)
  8. C n ( X ) C_{n}\left(X\right)
  9. C n ( Y ) C_{n}\left(Y\right)
  10. σ X \sigma_{X}
  11. Δ n X \Delta^{n}\rightarrow X
  12. f f
  13. Y Y
  14. f # ( σ X ) = f σ X f_{\#}\left(\sigma_{X}\right)=f\sigma_{X}
  15. Δ n Y \Delta^{n}\rightarrow Y
  16. f # f_{\#}
  17. f # ( t n t σ t ) = t n t f # ( σ t ) f_{\#}\left(\sum_{t}n_{t}\sigma_{t}\right)=\sum_{t}n_{t}f_{\#}\left(\sigma_{t}\right)
  18. f # f_{\#}
  19. C n ( X ) C n ( Y ) C_{n}\left(X\right)\rightarrow C_{n}\left(Y\right)
  20. f # = f # f_{\#}\partial=\partial f_{\#}
  21. \partial
  22. f # \partial f_{\#}
  23. f # f_{\#}
  24. α = 0 \partial\alpha=0
  25. f # ( α ) = f # ( α ) = 0 \partial f_{\#}\left(\alpha\right)=f_{\#}\left(\partial\alpha\right)=0
  26. f # f_{\#}
  27. f # ( β ) = f # ( β ) f_{\#}\left(\partial\beta\right)=\partial f_{\#}\left(\beta\right)
  28. f # f_{\#}
  29. f * : H n ( X ) H n ( Y ) f_{*}:H_{n}\left(X\right)\rightarrow H_{n}\left(Y\right)
  30. n 0 n\geq 0
  31. ( f g ) * = f * g * \left(f\circ g\right)_{*}=f_{*}\circ g_{*}
  32. X 𝑓 Y 𝑔 Z X\overset{f}{\rightarrow}Y\overset{g}{\rightarrow}Z
  33. ( i d X ) * = i d \left(id_{X}\right)_{*}=id
  34. i d X id_{X}
  35. X X X\rightarrow X
  36. X X
  37. i d : H n ( X ) H n ( X ) id:H_{n}\left(X\right)\rightarrow H_{n}\left(X\right)
  38. f , g : X Y f,g:X\rightarrow Y
  39. f * = g * : H n ( X ) H n ( Y ) f_{*}=g_{*}:H_{n}\left(X\right)\rightarrow H_{n}\left(Y\right)
  40. f * : H n ( X ) H n ( Y ) f_{*}:H_{n}\left(X\right)\rightarrow H_{n}\left(Y\right)
  41. f f
  42. X Y X\rightarrow Y
  43. n n

Pyraminx_Crystal.html

  1. 30 ! × 2 27 × 20 ! × 3 19 60 1.68 × 10 66 \frac{30!\times 2^{27}\times 20!\times 3^{19}}{60}\approx 1.68\times 10^{66}

Q-Charlier_polynomials.html

  1. c n ( q - x ; a ; q ) = ϕ 1 2 ( q - n , q - x ; 0 ; q , - q n + 1 / a ) \displaystyle c_{n}(q^{-x};a;q)={}_{2}\phi_{1}(q^{-n},q^{-x};0;q,-q^{n+1}/a)

Q-Racah_polynomials.html

  1. p n ( q - x + q x + 1 c d ; a , b , c , d ; q ) = ϕ 3 4 [ q - n a b q n + 1 q - x q x + 1 c d a q b d q c q ; q ; q ] p_{n}(q^{-x}+q^{x+1}cd;a,b,c,d;q)={}_{4}\phi_{3}\left[\begin{matrix}q^{-n}&abq% ^{n+1}&q^{-x}&q^{x+1}cd\\ aq&bdq&cq\\ \end{matrix};q;q\right]
  2. W n ( x ; a , b , c , N ; q ) = ϕ 3 4 [ q - n a b q n + 1 q - x c q x - n a q b c q q - N ; q ; q ] W_{n}(x;a,b,c,N;q)={}_{4}\phi_{3}\left[\begin{matrix}q^{-n}&abq^{n+1}&q^{-x}&% cq^{x-n}\\ aq&bcq&q^{-N}\\ \end{matrix};q;q\right]

Q-ratio.html

  1. Q = c u r r e n t C h i p s s t a r t i n g C h i p s × c u r r e n t N u m P l a y e r s s t a r t i n g N u m P l a y e r s Q=\frac{currentChips}{startingChips}\times\frac{currentNumPlayers}{startingNumPlayers}
  2. Q = 20 , 000 10 , 000 × 20 50 = 0.8 Q=\frac{20,000}{10,000}\times\frac{20}{50}=0.8
  3. Q = c u r r e n t C h i p s s t a r t i n g C h i p s × c u r r e n t N u m P l a y e r s s t a r t i n g N u m P l a y e r s + n u m T o t a l R e b u y s + n u m T o t a l A d d O n s Q=\frac{currentChips}{startingChips}\times\frac{currentNumPlayers}{% startingNumPlayers+numTotalRebuys+numTotalAddOns}

QCD_sum_rules.html

  1. 0 | T { 𝒪 1 ( x ) 𝒪 2 ( 0 ) } | 0 \left\langle 0|T\left\{\mathcal{O}_{1}(x)\mathcal{O}_{2}(0)\right\}|0\right\rangle

QI_(F_series).html

  1. W = π t 2 ( 3 / 2 ) ( n - 1 ) . W=\pi t2^{(3/2)\left(n-1\right)}.
  2. L = π t 6 ( 2 n + 4 ) ( 2 n - 1 ) . L=\frac{\pi t}{6}\left(2^{n}+4\right)\left(2^{n}-1\right).
  3. d = 1.5 h d=\sqrt{1.5h}

Quad-Ominos.html

  1. ( 6 + 4 - 1 4 ) = 126 {\textstyle\left({{6+4-1}\atop{4}}\right)}=126

Quadratic_unconstrained_binary_optimization.html

  1. E ( X 1 , X 2 , , X N ) = i < j = 1 N Q i j × X i × X j E(X_{1},X_{2},...,X_{N})=\sum_{i<j=1}^{N}Q_{ij}\times X_{i}\times X_{j}

Quadrature_modulation.html

  1. y ( t ) = I ( t ) cos ( ω c t ) y(t)=I(t)\cdot\cos(\omega_{c}t)
  2. I ( t ) I(t)
  3. cos ( ω c t ) \cos(\omega_{c}t)
  4. ω c \omega_{c}
  5. z ( t ) = I ( t ) cos ( ω c t ) - Q ( t ) sin ( ω c t ) z(t)=I(t)\cdot\cos(\omega_{c}t)-Q(t)\cdot\sin(\omega_{c}t)
  6. I ( t ) I(t)
  7. Q ( t ) Q(t)
  8. e i t = cos t + i sin t , e^{it}=\cos t+i\sin t,
  9. ( I , Q ) (I,Q)

Quantitative_models_of_the_action_potential.html

  1. C d V d t = I tot = I ext + I Na + I K + I L C\frac{dV}{dt}=I_{\mathrm{tot}}=I_{\mathrm{ext}}+I_{\mathrm{Na}}+I_{\mathrm{K}% }+I_{\mathrm{L}}
  2. I K = g K ( V - E K ) p open , K I_{\mathrm{K}}=g_{\mathrm{K}}\left(V-E_{\mathrm{K}}\right)p_{\mathrm{open,K}}
  3. d m d t = - m - m eq τ m \frac{dm}{dt}=-\frac{m-m_{\mathrm{eq}}}{\tau_{m}}
  4. 1 τ h = 0.07 e - V / 20 + 1 1 + e 3 - V / 10 . \frac{1}{\tau_{h}}=0.07e^{-V/20}+\frac{1}{1+e^{3-V/10}}.
  5. C d V d t = I - g ( V ) , C\frac{dV}{dt}=I-g(V),
  6. L d I d t = E - V - R I L\frac{dI}{dt}=E-V-RI
  7. C d V d t = I - ϵ ( V 3 3 - V ) , C\frac{dV}{dt}=I-\epsilon\left(\frac{V^{3}}{3}-V\right),
  8. L d I d t = - V L\frac{dI}{dt}=-V
  9. C d 2 V d t 2 + ϵ ( V 2 - 1 ) d V d t + V L = 0. C\frac{d^{2}V}{dt^{2}}+\epsilon\left(V^{2}-1\right)\frac{dV}{dt}+\frac{V}{L}=0.
  10. 𝐣 = σ 𝐄 \mathbf{j}=\sigma\mathbf{E}
  11. ϕ ( 𝐱 ) = 1 4 π σ outside membrane n 1 | 𝐱 - s y m b o l ξ | [ σ outside ϕ outside ( s y m b o l ξ ) - σ inside ϕ inside ( s y m b o l ξ ) ] d S \phi(\mathbf{x})=\frac{1}{4\pi\sigma_{\mathrm{outside}}}\oint_{\mathrm{% membrane}}\frac{\partial}{\partial n}\frac{1}{\left|\mathbf{x}-symbol\xi\right% |}\left[\sigma_{\mathrm{outside}}\phi_{\mathrm{outside}}(symbol\xi)-\sigma_{% \mathrm{inside}}\phi_{\mathrm{inside}}(symbol\xi)\right]dS
  12. s y m b o l ξ symbol\xi

Quantum_Byzantine_agreement.html

  1. 1 3 \frac{1}{3}
  2. 1 3 \frac{1}{3}
  3. c A , c B [ 0 , 1 ] c_{A},c_{B}\in[0,1]
  4. c A = c B c_{A}=c_{B}
  5. P r ( c A = c B = b ) = 1 2 Pr(c_{A}=c_{B}=b)=\frac{1}{2}
  6. a , b { 0 , 1 } a,b\in\{0,1\}
  7. 1 2 + ϵ \frac{1}{2}+\epsilon
  8. P r ( c A = c B = 1 ) 1 2 + ϵ Pr(c_{A}=c_{B}=1)\leq\frac{1}{2}+\epsilon
  9. P r ( c A = c B = 0 ) 1 2 + ϵ Pr(c_{A}=c_{B}=0)\leq\frac{1}{2}+\epsilon
  10. ϵ \epsilon
  11. P i P_{i}
  12. | C o i n i = 1 2 | 0 , 0 , , 0 + 1 2 | 1 , 1 , , 1 |Coin_{i}\rangle=\frac{1}{\sqrt{2}}|0,0,\ldots,0\rangle+\frac{1}{\sqrt{2}}|1,1% ,\ldots,1\rangle
  13. | L e a d e r i = 1 n 3 / 2 a = 1 n 3 | a , a , , a |Leader_{i}\rangle=\frac{1}{n^{3/2}}\sum_{a=1}^{n^{3}}|a,a,\ldots,a\rangle
  14. n 3 n^{3}
  15. L e a d e r j Leader_{j}
  16. v i v_{i}
  17. P k P_{k}
  18. s k i s{{}_{k}^{i}}
  19. P i P_{i}
  20. P i P_{i}
  21. s i = s k i for all secrets properly shared mod n s_{i}=\sum\,{s_{k}^{i}}{\,\text{for all secrets properly shared}}\mod n
  22. | ϕ = 1 n a = 0 n - 1 | a |\phi\rangle=\frac{1}{\sqrt{n}}\sum_{a=0}^{n-1}|a\rangle
  23. | ϕ , ϕ , ϕ |\phi,\phi,\ldots\phi\rangle
  24. P i P_{i}
  25. ( v a l u e i , c o n f i d e n c e i ) (value_{i},confidence_{i})
  26. ( i , c o n f i d e n c e i { 0 , 1 , 2 } ) (\forall i,confidence_{i}\in\{0,1,2\})
  27. v a l u e i value_{i}
  28. c o n f i d e n c e i confidence_{i}
  29. P i P_{i}
  30. P i P_{i}
  31. P j , P_{j},
  32. | c o n f i d e n c e i - c o n f i d e n c e j | 1 |confidence_{i}-confidence_{j}|\leq 1
  33. P i P_{i}
  34. P j P_{j}
  35. c o n f i d e n c e i > 0 confidence_{i}>0
  36. c o n f i d e n c e j > 0 confidence_{j}>0
  37. v a l u e i = v a l u e j value_{i}=value_{j}
  38. t < n 4 t<\frac{n}{4}
  39. t < n 4 t<\frac{n}{4}

Quantum_digital_signature.html

  1. x f ( x ) x\mapsto f(x)
  2. f ( x ) x f(x)\mapsto x
  3. k | f k k\mapsto|f_{k}\rangle
  4. | k | f k |k\rangle\mapsto|f_{k}\rangle
  5. | f k | f k | < δ for k k and 0 δ 1 |\langle f_{k}|f_{k}^{\prime}\rangle|<\delta\qquad\,\text{ for }k\neq k^{% \prime}\and 0\leq\delta\leq 1
  6. 2 n 2^{n}
  7. | 0 |0\rangle
  8. | 1 |1\rangle
  9. 1 2 ( | 0 + | 1 ) \frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)
  10. 3 m = 2 n 3^{m}=2^{n}
  11. δ \delta
  12. n - T m n-Tm
  13. | f k |f_{k}\rangle
  14. n - T m n-Tm
  15. | f k = | f k |f_{k}^{\prime}\rangle=|f_{k}\rangle
  16. | a = 1 2 ( | 0 + | 1 ) |a\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)
  17. | f k |f_{k}\rangle
  18. | f k \ |f_{k}^{\prime}\rangle
  19. | 0 |0\rangle
  20. | 0 |0\rangle
  21. | 1 \ |1\rangle
  22. | ψ 0 = | a | f k | f k |\psi_{0}\rangle=|a\rangle|f_{k}\rangle|f_{k}^{\prime}\rangle
  23. | ψ 0 = 1 2 ( | 0 + | 1 ) | f k | f k |\psi_{0}\rangle=\frac{1}{\sqrt{2}}\bigg(|0\rangle+|1\rangle\bigg)|f_{k}% \rangle|f_{k}^{\prime}\rangle
  24. | ψ 0 = 1 2 ( | 0 | f k | f k + | 1 | f k | f k ) |\psi_{0}\rangle=\frac{1}{\sqrt{2}}\bigg(|0\rangle|f_{k}\rangle|f_{k}^{\prime}% \rangle+|1\rangle|f_{k}\rangle|f_{k}^{\prime}\rangle\bigg)
  25. 1 2 ( | 0 | f k | f k + | 1 | 𝐟 𝐤 | 𝐟 𝐤 ) \Rightarrow\frac{1}{\sqrt{2}}\bigg(|0\rangle|f_{k}\rangle|f_{k}^{\prime}% \rangle+|1\rangle\mathbf{|f_{k}^{\prime}\rangle|f_{k}\rangle}\bigg)
  26. 1 2 [ ( | 0 + | 1 ) | f k | f k + ( | 0 - | 1 ) | f k | f k ] \Rightarrow\frac{1}{2}\bigg[\bigg(|0\rangle+|1\rangle\bigg)|f_{k}\rangle|f_{k}% ^{\prime}\rangle+\bigg(|0\rangle-|1\rangle\bigg)|f_{k}^{\prime}\rangle|f_{k}% \rangle\bigg]
  27. | 0 and | 1 |0\rangle\,\text{ and }|1\rangle
  28. | ψ = 1 2 [ | 0 ( | f k | f k + | f k | f k ) + | 1 ( | f k | f k - | f k | f k ) ] \Rightarrow|\psi\rangle=\frac{1}{2}\bigg[|0\rangle\bigg(|f_{k}\rangle|f_{k}^{% \prime}\rangle+|f_{k}^{\prime}\rangle|f_{k}\rangle\bigg)+|1\rangle\bigg(|f_{k}% \rangle|f_{k}^{\prime}\rangle-|f_{k}^{\prime}\rangle|f_{k}\rangle\bigg)\bigg]
  29. | f k = | f k |f_{k}\rangle=|f_{k}^{\prime}\rangle
  30. | ψ = | 0 | f k | f k \ |\psi\rangle=|0\rangle|f_{k}\rangle|f_{k}\rangle
  31. { 0 , 1 } \in\{0,1\}
  32. { k 0 i , k 1 i } 1 i M \{k_{0}^{i},k_{1}^{i}\}\quad 1\leq i\leq M
  33. k 0 k_{0}
  34. k 1 k_{1}
  35. k | f k k\mapsto|f_{k}\rangle
  36. { | f k 0 i , | f k 1 i } \{|f_{k_{0}}^{i}\rangle,|f_{k_{1}}^{i}\rangle\}
  37. ( n T m has to hold ) \left(n\gg Tm\,\text{ has to hold }\right)
  38. k 0 k_{0}
  39. k 1 k_{1}
  40. k 0 or k 1 k_{0}\,\text{ or }k_{1}
  41. { | f k 0 , | f k 1 } \{|f_{k_{0}}\rangle,|f_{k_{1}}\rangle\}
  42. | f k b |f_{k_{b}}\rangle
  43. k 0 or k 1 k_{0}\,\text{ or }k_{1}
  44. T a = c 1 M T_{a}=c_{1}M
  45. T r = c 2 M T_{r}=c_{2}M
  46. T a T_{a}
  47. T r T_{r}
  48. T a T_{a}

Quantum_dissipation.html

  1. H = P 2 2 M + V ( X ) + i ( p i 2 2 m i + 1 2 m i ω i 2 q i 2 ) + X i C i q i + X 2 i C i 2 2 m i ω i 2 H=\frac{P^{2}}{2M}+V(X)+\sum_{i}\left(\frac{p_{i}^{2}}{2m_{i}}+\frac{1}{2}m_{i% }\omega_{i}^{2}q_{i}^{2}\right)+X\sum_{i}{C_{i}q_{i}}+X^{2}\sum_{i}\frac{C_{i}% ^{2}}{2m_{i}\omega_{i}^{2}}
  2. M M
  3. P P
  4. V V
  5. X X
  6. m i m_{i}
  7. p i p_{i}
  8. q i q_{i}
  9. ω i \omega_{i}
  10. C i C_{i}
  11. J ( ω ) = π 2 i C i 2 m i ω i δ ( ω - ω i ) J(\omega)=\frac{\pi}{2}\sum_{i}\frac{C_{i}^{2}}{m_{i}\omega_{i}}\delta(\omega-% \omega_{i})
  12. C i C_{i}
  13. J ( ω ) = η ω J(\omega)=\eta\omega
  14. J ( ω ) ω s J(\omega)\propto\omega^{s}
  15. s > 1 s>1
  16. s < 1 s<1
  17. 0 \hbar\rightarrow 0
  18. M d 2 d t 2 X ( t ) = - V ( X ) X - 0 T d t α ( t - t ) ( X ( t ) - X ( t ) ) M\frac{d^{2}}{dt^{2}}X(t)=-\frac{\partial V(X)}{\partial X}-\int_{0}^{T}dt^{% \prime}\alpha(t-t^{\prime})(X(t)-X(t^{\prime}))
  19. α ( t - t ) = 1 2 π 0 J ( ω ) e - ω | t - t | d ω \alpha(t-t^{\prime})=\frac{1}{2\pi}\int_{0}^{\infty}J(\omega)e^{-\omega|t-t^{% \prime}|}d\omega
  20. M d 2 d t 2 X ( t ) = - V ( X ) X - η d X ( t ) d t M\frac{d^{2}}{dt^{2}}X(t)=-\frac{\partial V(X)}{\partial X}-\eta\frac{dX(t)}{dt}
  21. 1 / 2 1/2
  22. H = Δ S x + i ( p i 2 2 m i + 1 2 m i ω i 2 q i 2 ) + S z i C i q i H=\Delta S_{x}+\sum_{i}\left(\frac{p_{i}^{2}}{2m_{i}}+\frac{1}{2}m_{i}\omega_{% i}^{2}q_{i}^{2}\right)+S_{z}\sum_{i}{C_{i}q_{i}}
  23. S i = σ i 2 , i = x , y , z S_{i}=\frac{\sigma_{i}}{2},i=x,y,z
  24. σ i \sigma_{i}
  25. Δ \Delta
  26. S z S_{z}

Quantum_no-deleting_theorem.html

  1. U | ψ A | ψ B | A C = | ψ A | 0 B | A C U|\psi\rangle_{A}|\psi\rangle_{B}|A\rangle_{C}=|\psi\rangle_{A}|0\rangle_{B}|A% ^{\prime}\rangle_{C}
  2. U U
  3. | ψ A |\psi\rangle_{A}
  4. | 0 B |0\rangle_{B}
  5. | A C |A\rangle_{C}
  6. | A C |A^{\prime}\rangle_{C}
  7. | 00 |00\rangle
  8. | 11 |11\rangle
  9. | 00 |00\rangle
  10. | 10 |10\rangle
  11. U U
  12. | ψ |\psi\rangle
  13. | ψ |\psi\rangle
  14. | ψ A | ψ B | A C | ψ A | 0 B | A C |\psi\rangle_{A}|\psi\rangle_{B}|A\rangle_{C}\rightarrow|\psi\rangle_{A}|0% \rangle_{B}|A^{\prime}\rangle_{C}
  15. | ψ |\psi\rangle
  16. | 0 A | 0 B | A C | 0 A | 0 B | A 0 C |0\rangle_{A}|0\rangle_{B}|A\rangle_{C}\rightarrow|0\rangle_{A}|0\rangle_{B}|A% _{0}\rangle_{C}
  17. | 1 A | 1 B | A C | 1 A | 0 B | A 1 C |1\rangle_{A}|1\rangle_{B}|A\rangle_{C}\rightarrow|1\rangle_{A}|0\rangle_{B}|A% _{1}\rangle_{C}
  18. | ψ = α | 0 + β | 1 |\psi\rangle=\alpha|0\rangle+\beta|1\rangle
  19. | ψ A | ψ B | A C = [ α 2 | 0 A | 0 B + β 2 | 1 A | 1 B + α β ( | 0 A | 1 B + | 1 A | 0 B ) ] | A C |\psi\rangle_{A}|\psi\rangle_{B}|A\rangle_{C}=[\alpha^{2}|0\rangle_{A}|0% \rangle_{B}+\beta^{2}|1\rangle_{A}|1\rangle_{B}+\alpha\beta(|0\rangle_{A}|1% \rangle_{B}+|1\rangle_{A}|0\rangle_{B})]|A\rangle_{C}
  20. α 2 | 0 A | 0 B | A 0 C + β 2 | 1 A | 0 B | A 1 C + 2 α β | Φ A B C . \qquad\rightarrow\alpha^{2}|0\rangle_{A}|0\rangle_{B}|A_{0}\rangle_{C}+\beta^{% 2}|1\rangle_{A}|0\rangle_{B}|A_{1}\rangle_{C}+{\sqrt{2}}\alpha\beta|\Phi% \rangle_{ABC}.
  21. 1 / 2 ( | 0 A | 1 B + | 1 A | 0 B ) | A C | Φ A B C . 1/{\sqrt{2}}(|0\rangle_{A}|1\rangle_{B}+|1\rangle_{A}|0\rangle_{B})|A\rangle_{% C}\rightarrow|\Phi\rangle_{ABC}.
  22. | ψ A | 0 B | A C = ( α | 0 A | 0 B + β | 1 A | 0 B ) | A C |\psi\rangle_{A}|0\rangle_{B}|A^{\prime}\rangle_{C}=(\alpha|0\rangle_{A}|0% \rangle_{B}+\beta|1\rangle_{A}|0\rangle_{B})|A^{\prime}\rangle_{C}
  23. | Φ = 1 / 2 ( | 0 A | 0 B | A 1 C + | 1 A | 0 B | A 0 C ) , |\Phi\rangle=1/{\sqrt{2}}(|0\rangle_{A}|0\rangle_{B}|A_{1}\rangle_{C}+|1% \rangle_{A}|0\rangle_{B}|A_{0}\rangle_{C}),
  24. | A = α | A 0 C + β | A 1 C . |A^{\prime}\rangle=\alpha|A_{0}\rangle_{C}+\beta|A_{1}\rangle_{C}.
  25. α , β \alpha,\beta
  26. | A 0 |A_{0}\rangle
  27. | A 1 |A_{1}\rangle

Quantum_pseudo-telepathy.html

  1. | ϕ = 1 2 ( | + a | + b + | - a | - b ) 1 2 ( | + c | + d + | - c | - d ) \left|\phi\right\rangle=\frac{1}{\sqrt{2}}\bigg(\left|+\right\rangle_{a}% \otimes\left|+\right\rangle_{b}+\left|-\right\rangle_{a}\otimes\left|-\right% \rangle_{b}\bigg)\otimes\frac{1}{\sqrt{2}}\bigg(\left|+\right\rangle_{c}% \otimes\left|+\right\rangle_{d}+\left|-\right\rangle_{c}\otimes\left|-\right% \rangle_{d}\bigg)
  2. S x = [ 0 1 1 0 ] , S y = [ 0 - i i 0 ] , S z = [ 1 0 0 - 1 ] S_{x}=\begin{bmatrix}0&1\\ 1&0\end{bmatrix},S_{y}=\begin{bmatrix}0&-i\\ i&0\end{bmatrix},S_{z}=\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}
  3. + S x I +S_{x}\otimes I
  4. + S x S x +S_{x}\otimes S_{x}
  5. + I S x +I\otimes S_{x}
  6. - S x S z -S_{x}\otimes S_{z}
  7. + S y S y +S_{y}\otimes S_{y}
  8. - S z S x -S_{z}\otimes S_{x}
  9. + I S z +I\otimes S_{z}
  10. + S z S z +S_{z}\otimes S_{z}
  11. + S z I +S_{z}\otimes I

Quantum_reference_frame.html

  1. 10 - 5 10^{-5}
  2. | z |\uparrow z\rangle
  3. V ( r ) = - Z e 2 r V(r)=\frac{-Ze^{2}}{r}
  4. - 1 2 m 2 ψ ( r ) + - Z e 2 r ψ ( r ) = E ψ ( r ) -\frac{1}{2m}\nabla^{2}\psi(\vec{r})+\frac{-Ze^{2}}{r}\psi(\vec{r})=E\psi(\vec% {r})
  5. Φ ( r , θ , ϕ ) = R n l ( r ) Y l m ( θ , ϕ ) \Phi(r,\theta,\phi)=R_{nl}(r)Y_{lm}(\theta,\phi)
  6. l , l,
  7. m m
  8. n n
  9. t Ψ ( x 1 , y 1 , z 1 , x 2 , y 2 , z 2 , t ) = i H Ψ ( x 1 , y 1 , z 1 , x 2 , y 2 , z 2 , t ) \frac{\partial}{\partial t}\Psi(x_{1},y_{1},z_{1},x_{2},y_{2},z_{2},t)=iH\Psi(% x_{1},y_{1},z_{1},x_{2},y_{2},z_{2},t)
  10. Ψ ( x , y , z , X , Y , Z ) t = i [ 1 2 M c . o . m . 2 + 1 2 μ r e l 2 + V ( x , y , z ) ] Ψ \frac{\partial\Psi(x,y,z,X,Y,Z)}{\partial t}=i[\frac{1}{2M}\nabla_{c.o.m.}^{2}% +\frac{1}{2\mu}\nabla_{rel}^{2}+V(x,y,z)]\Psi
  11. M M
  12. μ \mu
  13. Φ ( r , θ , ϕ ) \Phi(r,\theta,\phi)
  14. Φ ( r , θ , ϕ ) \Phi(r,\theta,\phi)
  15. r = ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 + ( z 1 - z 2 ) 2 r=\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}+(z_{1}-z_{2})^{2}}
  16. θ = tan - 1 ( ( x 1 - x 2 ) 2 + ( y 1 - y 2 ) 2 z 1 - z 2 ) \theta=\tan^{-1}\left(\frac{\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}}{z_{1}-% z_{2}}\right)
  17. ϕ = tan - 1 ( y 1 - y 2 x 1 - x 2 ) \phi=\tan^{-1}\left(\frac{y_{1}-y_{2}}{x_{1}-x_{2}}\right)
  18. X = m 1 x 1 + m 2 x 2 m 1 + m 2 X=\frac{m_{1}x_{1}+m_{2}x_{2}}{m_{1}+m_{2}}
  19. Y = m 1 y 1 + m 2 y 2 m 1 + m 2 Y=\frac{m_{1}y_{1}+m_{2}y_{2}}{m_{1}+m_{2}}
  20. Z = m 1 z 1 + m 2 z 2 m 1 + m 2 Z=\frac{m_{1}z_{1}+m_{2}z_{2}}{m_{1}+m_{2}}
  21. j j
  22. ϵ \epsilon
  23. n m a x ϵ j 2 n_{max}\simeq\epsilon j^{2}
  24. j j

Quantum_t-design.html

  1. 1 2 \tfrac{1}{2}
  2. d 2 d^{2}
  3. N \mathbb{R}^{N}
  4. N / 2 \mathbb{C}^{N/2}
  5. ( p i , | ϕ i ) (p_{i},|\phi_{i}\rangle)
  6. i p i ( | ϕ i ϕ i | ) t = ψ ( | ψ ψ | ) t d ψ \sum_{i}p_{i}(|\phi_{i}\rangle\langle\phi_{i}|)^{\otimes t}=\int_{\psi}(|\psi% \rangle\langle\psi|)^{\otimes t}d\psi
  7. C N C^{N}
  8. | ϕ |\phi\rangle
  9. p i | ϕ i p_{i}|\phi_{i}\rangle
  10. O ( log c N ) O(\log^{c}N)
  11. N p i | ϕ i ϕ i | Np_{i}|\phi_{i}\rangle\langle\phi_{i}|
  12. O ( log c N ) O(\log^{c}N)
  13. i p i | ϕ i ϕ i | = ψ | ψ ψ | d ψ \sum_{i}p_{i}|\phi_{i}\rangle\langle\phi_{i}|=\int_{\psi}|\psi\rangle\langle% \psi|d\psi
  14. ( 1 - ϵ ) ψ ( | ψ ψ | ) t d ψ i p i ( | ϕ i ϕ i | ) t ( 1 + ϵ ) ψ ( | ψ ψ | ) t d ψ (1-\epsilon)\int_{\psi}(|\psi\rangle\langle\psi|)^{\otimes t}d\psi\leq\sum_{i}% p_{i}(|\phi_{i}\rangle\langle\phi_{i}|)^{\otimes t}\leq(1+\epsilon)\int_{\psi}% (|\psi\rangle\langle\psi|)^{\otimes t}d\psi
  15. ( p i , | ϕ i ) (p_{i},|\phi_{i}\rangle)
  16. ϵ \epsilon
  17. ϵ \epsilon
  18. N d N^{d}
  19. \rightarrow
  20. k 1 , , k d k_{1},...,k_{d}\in
  21. | ψ = i = 1 N α | i |\psi\rangle=\sum_{i=1}^{N}\alpha|i\rangle
  22. P n P_{n}
  23. α n \alpha_{n}
  24. P = lim N N P N P=\lim_{N\rightarrow\infty}\sqrt{N}P_{N}
  25. α \alpha
  26. X = | α | X=|\alpha|
  27. 1 2 \tfrac{1}{2}
  28. X = - | α | X=-|\alpha|
  29. 1 2 \tfrac{1}{2}
  30. E [ X j ] = 0 E[X^{j}]=0
  31. E [ X j ] = ( j 2 ) ! E[X^{j}]=(\tfrac{j}{2})!
  32. p f , g = i = 1 N a f , i 2 | S 1 | | S 2 | p_{f,g}=\frac{\sum_{i=1}^{N}a_{f,i}^{2}}{|S_{1}||S_{2}|}
  33. p f , g | ψ f , g p_{f,g}|\psi_{f,g}\rangle
  34. d × d d\times d
  35. U k {U_{k}}
  36. | ψ |\psi\rangle
  37. | ψ k = U k | ψ |\psi_{k}\rangle=U_{k}|\psi\rangle
  38. | ψ k {|\psi_{k}\rangle}
  39. 1 | X | U X U t ( U * ) t = U ( d ) U t ( U * ) t d U \frac{1}{|X|}\sum_{U\in X}U^{\otimes t}\otimes(U^{*})^{\otimes t}=\int_{U(d)}U% ^{\otimes t}\otimes(U^{*})^{\otimes t}dU
  40. U r ( U * ) s d U U^{\otimes r}\otimes(U^{*})^{\otimes s}dU
  41. U X U\in X
  42. r + s = t r+s=t
  43. X U ( d ) X\subseteq U(d)
  44. 1 | X | 2 U , V X | t r ( U * V ) | 2 t U ( d ) | t r ( U * V ) | 2 t d U \frac{1}{|X|^{2}}\sum_{U,V\in X}|tr(U*V)|^{2t}\geq\int_{U(d)}|tr(U*V)|^{2t}dU
  45. H o m ( U ( d ) , t , t ) Hom(U(d),t,t)
  46. U U
  47. U * U^{*}
  48. f H o m ( U ( d ) , t , t ) f\in Hom(U(d),t,t)
  49. 1 | X | U X f ( U ) = U ( d ) f ( U ) d U \frac{1}{|X|}\sum_{U\in X}f(U)=\int_{U(d)}f(U)dU
  50. f f
  51. g g
  52. U ( d ) U(d)
  53. f ¯ g \bar{f}g
  54. f , g := U ( d ) f ( U ) ¯ g ( U ) d X \langle f,g\rangle:=\int_{U(d)}\bar{f(U)}g(U)dX
  55. f , g X \langle f,g\rangle_{X}
  56. f ¯ g \bar{f}g
  57. X U ( d ) X\subset U(d)
  58. 1 , f X = 1 , f f \langle 1,f\rangle_{X}=\langle 1,f\rangle\quad\forall f
  59. | X | d i m ( H o m ( U ( d ) , t 2 , t 2 ) ) |X|\geq dim(Hom(U(d),\left\lceil\tfrac{t}{2}\right\rceil,\left\lfloor\tfrac{t}% {2}\right\rfloor))
  60. X U ( d ) X\subset U(d)
  61. U M U\neq M
  62. | t r ( U * M ) | 2 |tr(U^{*}M)|^{2}
  63. | X | d i m ( H o m ( U ( d ) , s , s ) ) |X|\leq dim(Hom(U(d),s,s))
  64. | X | d i m ( H o m ( U ( d ) , s , s - 1 ) ) |X|\leq dim(Hom(U(d),s,s-1))

Quarter-wave_impedance_transformer.html

  1. Z in Z 0 = Z 0 Z L \frac{Z_{\mathrm{in}}}{Z_{0}}=\frac{Z_{0}}{Z_{L}}
  2. Z in = Z 0 Z L + Z 0 tanh ( γ l ) Z 0 + Z L tanh ( γ l ) Z_{\mathrm{in}}=Z_{0}\frac{Z_{L}+Z_{0}\tanh(\gamma l)}{Z_{0}+Z_{L}\tanh(\gamma l)}
  3. Z in = Z 0 Z L + i Z 0 tan ( β l ) Z 0 + i Z L tan ( β l ) Z_{\mathrm{in}}=Z_{0}\frac{Z_{L}+iZ_{0}\tan(\beta l)}{Z_{0}+iZ_{L}\tan(\beta l)}
  4. β = 2 π λ , \beta=\frac{2\pi}{\lambda}\ ,
  5. l = λ 4 , l=\frac{\lambda}{4}\ ,
  6. β l = π 2 , \beta l={\pi\over 2}\ ,
  7. π 2 \pi\over 2
  8. Z in = lim β l π / 2 Z 0 Z L + i Z 0 tan ( β l ) Z 0 + i Z L tan ( β l ) = Z 0 i Z 0 i Z L = Z 0 2 Z L Z_{\mathrm{in}}=\lim_{\beta l\rightarrow\pi/2}{Z_{0}\frac{Z_{L}+iZ_{0}\tan({% \beta l})}{Z_{0}+iZ_{L}\tan({\beta l})}}=Z_{0}\frac{iZ_{0}}{iZ_{L}}=\frac{{Z_{% 0}}^{2}}{Z_{L}}
  9. Z in Z 0 = Z 0 Z L \frac{Z_{\mathrm{in}}}{Z_{0}}=\frac{Z_{0}}{Z_{L}}

Quartic_reciprocity.html

  1. a p - 1 2 1 ( mod p ) , a^{\frac{p-1}{2}}\equiv 1\;\;(\mathop{{\rm mod}}p),
  2. a p - 1 4 ± 1 ( mod p ) . a^{\frac{p-1}{4}}\equiv\pm 1\;\;(\mathop{{\rm mod}}p).
  3. ( a p ) 4 = ± 1 a p - 1 4 ( mod p ) . \Bigg(\frac{a}{p}\Bigg)_{4}=\pm 1\equiv a^{\frac{p-1}{4}}\;\;(\mathop{{\rm mod% }}p).
  4. ( a p ) 4 = 1. \Bigg(\frac{a}{p}\Bigg)_{4}=1.
  5. ( 2 p ) 4 i a b 2 ( mod p ) . \Bigg(\frac{2}{p}\Bigg)_{4}\equiv i^{\frac{ab}{2}}\;\;(\mathop{{\rm mod}}p).
  6. ( 2 p ) 4 = ( - 1 ) b 4 = ( 2 c ) = ( - 1 ) n + d 2 = ( - 2 e ) , \Bigg(\frac{2}{p}\Bigg)_{4}=\left(-1\right)^{\frac{b}{4}}=\Bigg(\frac{2}{c}% \Bigg)=\left(-1\right)^{n+\frac{d}{2}}=\Bigg(\frac{-2}{e}\Bigg),
  7. ( x q ) (\tfrac{x}{q})
  8. ( p q ) = 1. (\tfrac{p}{q})=1.
  9. ( q * p ) = 1 , (\tfrac{q^{*}}{p})=1,
  10. q * = ( - 1 ) q - 1 2 q . q^{*}=(-1)^{\frac{q-1}{2}}q.
  11. ( q * p ) 4 = ( σ ( b + σ ) q ) . \Bigg(\frac{q^{*}}{p}\Bigg)_{4}=\Bigg(\frac{\sigma(b+\sigma)}{q}\Bigg).
  12. ( q * p ) 4 = 1 if and only if { b 0 ( mod q ) ; or a 0 ( mod q ) and ( 2 q ) = 1 ; or a μ b , μ 2 + 1 λ 2 ( mod q ) , and ( λ ( λ + 1 ) q ) = 1. \Bigg(\frac{q^{*}}{p}\Bigg)_{4}=1\mbox{ if and only if }~{}\begin{cases}b% \equiv 0\;\;(\mathop{{\rm mod}}q);&\mbox{ or }\\ a\equiv 0\;\;(\mathop{{\rm mod}}q)\mbox{ and }~{}\left(\frac{2}{q}\right)=1;&% \mbox{ or }\\ a\equiv\mu b,\;\;\mu^{2}+1\equiv\lambda^{2}\;\;(\mathop{{\rm mod}}q)\mbox{, % and }~{}\left(\frac{\lambda(\lambda+1)}{q}\right)=1.\end{cases}
  13. ( - 3 p ) 4 = 1 if and only if b 0 ( mod 3 ) ( 5 p ) 4 = 1 if and only if b 0 ( mod 5 ) ( - 7 p ) 4 = 1 if and only if a b 0 ( mod 7 ) ( - 11 p ) 4 = 1 if and only if b ( b 2 - 3 a 2 ) 0 ( mod 11 ) ( 13 p ) 4 = 1 if and only if b ( b 2 - 3 a 2 ) 0 ( mod 13 ) ( 17 p ) 4 = 1 if and only if a b ( b 2 - a 2 ) 0 ( mod 17 ) . \begin{aligned}\displaystyle\left(\frac{-3}{p}\right)_{4}=1&\displaystyle\mbox% { if and only if }&\displaystyle b&\displaystyle\equiv 0\;\;(\mathop{{\rm mod}% }3)\\ \displaystyle\left(\frac{5}{p}\right)_{4}=1&\displaystyle\mbox{ if and only if% }&\displaystyle b&\displaystyle\equiv 0\;\;(\mathop{{\rm mod}}5)\\ \displaystyle\left(\frac{-7}{p}\right)_{4}=1&\displaystyle\mbox{ if and only % if }&\displaystyle ab&\displaystyle\equiv 0\;\;(\mathop{{\rm mod}}7)\\ \displaystyle\left(\frac{-11}{p}\right)_{4}=1&\displaystyle\mbox{ if and only % if }&\displaystyle b(b^{2}-3a^{2})&\displaystyle\equiv 0\;\;(\mathop{{\rm mod}% }11)\\ \displaystyle\left(\frac{13}{p}\right)_{4}=1&\displaystyle\mbox{ if and only % if }&\displaystyle b(b^{2}-3a^{2})&\displaystyle\equiv 0\;\;(\mathop{{\rm mod}% }13)\\ \displaystyle\left(\frac{17}{p}\right)_{4}=1&\displaystyle\mbox{ if and only % if }&\displaystyle ab(b^{2}-a^{2})&\displaystyle\equiv 0\;\;(\mathop{{\rm mod}% }17).\\ \end{aligned}
  14. ( 17 p ) = 1 (\tfrac{17}{p})=1
  15. ( 17 p ) 4 ( p 17 ) 4 = { + 1 if and only if p = x 2 + 17 y 2 - 1 if and only if 2 p = x 2 + 17 y 2 \Bigg(\frac{17}{p}\Bigg)_{4}\Bigg(\frac{p}{17}\Bigg)_{4}=\begin{cases}+1\mbox{% if and only if }~{}\;\;p=x^{2}+17y^{2}\\ -1\mbox{ if and only if }~{}2p=x^{2}+17y^{2}\end{cases}
  16. ( p q ) = 1. (\tfrac{p}{q})=1.
  17. ( q p ) 4 ( a / b - c / d a / b + c / d ) q - 1 4 ( mod q ) . \Bigg(\frac{q}{p}\Bigg)_{4}\equiv\Bigg(\frac{a/b-c/d}{a/b+c/d}\Bigg)^{\frac{q-% 1}{4}}\;\;(\mathop{{\rm mod}}q).
  18. ( q p ) 4 ( p q ) 4 = ( a + b j q ) = ( c + d i p ) . \Bigg(\frac{q}{p}\Bigg)_{4}\Bigg(\frac{p}{q}\Bigg)_{4}=\Bigg(\frac{a+bj}{q}% \Bigg)=\Bigg(\frac{c+di}{p}\Bigg).
  19. ( q p ) 4 ( p q ) 4 = ( a c - b d q ) . \Bigg(\frac{q}{p}\Bigg)_{4}\Bigg(\frac{p}{q}\Bigg)_{4}=\Bigg(\frac{ac-bd}{q}% \Bigg).
  20. ( a c + b d p ) = ( p q ) ( a c - b d p ) . \Bigg(\frac{ac+bd}{p}\Bigg)=\Bigg(\frac{p}{q}\Bigg)\Bigg(\frac{ac-bd}{p}\Bigg).
  21. ( p q ) = 1 (\tfrac{p}{q})=1
  22. ( p q ) 4 ( q p ) 4 = ( - 1 ) f g 2 ( - 1 e ) . \Bigg(\frac{p}{q}\Bigg)_{4}\Bigg(\frac{q}{p}\Bigg)_{4}=\left(-1\right)^{\frac{% fg}{2}}\left(\frac{-1}{e}\right).
  23. ( p q ) 4 ( q p ) 4 = ( 2 q ) s . \Bigg(\frac{p}{q}\Bigg)_{4}\Bigg(\frac{q}{p}\Bigg)_{4}=\left(\frac{2}{q}\right% )^{s}.
  24. a * = ( - 1 ) a - 1 2 a . a^{*}=\left(-1\right)^{\frac{a-1}{2}}a.
  25. λ = i μ ( 1 + i ) ν π 1 α 1 π 2 α 2 π 3 α 3 \lambda=i^{\mu}(1+i)^{\nu}\pi_{1}^{\alpha_{1}}\pi_{2}^{\alpha_{2}}\pi_{3}^{% \alpha_{3}}\dots
  26. α N π - 1 1 ( mod π ) \alpha^{N\pi-1}\equiv 1\;\;(\mathop{{\rm mod}}\pi)
  27. α N π - 1 4 \alpha^{\frac{N\pi-1}{4}}
  28. α N π - 1 4 i k ( mod π ) \alpha^{\frac{N\pi-1}{4}}\equiv i^{k}\;\;(\mathop{{\rm mod}}\pi)
  29. [ α π ] = i k α N π - 1 4 ( mod π ) . \left[\frac{\alpha}{\pi}\right]=i^{k}\equiv\alpha^{\frac{N\pi-1}{4}}\;\;(% \mathop{{\rm mod}}\pi).
  30. x 4 α ( mod π ) x^{4}\equiv\alpha\;\;(\mathop{{\rm mod}}\pi)
  31. [ α π ] = 1. \left[\frac{\alpha}{\pi}\right]=1.
  32. [ α β π ] = [ α π ] [ β π ] \Bigg[\frac{\alpha\beta}{\pi}\Bigg]=\Bigg[\frac{\alpha}{\pi}\Bigg]\Bigg[\frac{% \beta}{\pi}\Bigg]
  33. [ α π ] ¯ = [ α ¯ π ¯ ] \overline{\Bigg[\frac{\alpha}{\pi}\Bigg]}=\Bigg[\frac{\overline{\alpha}}{% \overline{\pi}}\Bigg]
  34. [ α π ] = [ α θ ] \Bigg[\frac{\alpha}{\pi}\Bigg]=\Bigg[\frac{\alpha}{\theta}\Bigg]
  35. [ α π ] = [ β π ] \Bigg[\frac{\alpha}{\pi}\Bigg]=\Bigg[\frac{\beta}{\pi}\Bigg]
  36. [ α λ ] = [ α π 1 ] α 1 [ α π 2 ] α 2 \left[\frac{\alpha}{\lambda}\right]=\left[\frac{\alpha}{\pi_{1}}\right]^{% \alpha_{1}}\left[\frac{\alpha}{\pi_{2}}\right]^{\alpha_{2}}\dots
  37. λ = π 1 α 1 π 2 α 2 π 3 α 3 \lambda=\pi_{1}^{\alpha_{1}}\pi_{2}^{\alpha_{2}}\pi_{3}^{\alpha_{3}}\dots
  38. [ a b ] = 1. \left[\frac{a}{b}\right]=1.
  39. [ π θ ] = [ θ π ] , \Bigg[\frac{\pi}{\theta}\Bigg]=\left[\frac{\theta}{\pi}\right],
  40. [ π θ ] = - [ θ π ] . \Bigg[\frac{\pi}{\theta}\Bigg]=-\left[\frac{\theta}{\pi}\right].
  41. [ π θ ] [ θ π ] - 1 = ( - 1 ) N π - 1 4 N θ - 1 4 . \Bigg[\frac{\pi}{\theta}\Bigg]\left[\frac{\theta}{\pi}\right]^{-1}=(-1)^{\frac% {N\pi-1}{4}\frac{N\theta-1}{4}}.
  42. [ i π ] = i - a - 1 2 , [ 1 + i π ] = i a - b - 1 - b 2 4 , \Bigg[\frac{i}{\pi}\Bigg]=i^{-\frac{a-1}{2}},\;\;\;\Bigg[\frac{1+i}{\pi}\Bigg]% =i^{\frac{a-b-1-b^{2}}{4}},
  43. [ - 1 π ] = ( - 1 ) a - 1 2 , [ 2 π ] = i - b 2 . \Bigg[\frac{-1}{\pi}\Bigg]=(-1)^{\frac{a-1}{2}},\;\;\;\Bigg[\frac{2}{\pi}\Bigg% ]=i^{-\frac{b}{2}}.
  44. [ π ¯ π ] = [ - 2 π ] ( - 1 ) a 2 - 1 8 \Bigg[\frac{\overline{\pi}}{\pi}\Bigg]=\Bigg[\frac{-2}{\pi}\Bigg](-1)^{\frac{a% ^{2}-1}{8}}
  45. [ α β ] [ β α ] - 1 = ( - 1 ) b d 4 \left[\frac{\alpha}{\beta}\right]\left[\frac{\beta}{\alpha}\right]^{-1}=(-1)^{% \frac{bd}{4}}
  46. [ α β ] [ β α ] - 1 = ( - 1 ) b d + a - 1 2 d + c - 1 2 b , [ 1 + i α ] = i b ( a - 3 b ) 2 - a 2 - 1 8 \left[\frac{\alpha}{\beta}\right]\left[\frac{\beta}{\alpha}\right]^{-1}=(-1)^{% bd+\frac{a-1}{2}d+\frac{c-1}{2}b},\;\;\;\;\left[\frac{1+i}{\alpha}\right]=i^{% \frac{b(a-3b)}{2}-\frac{a^{2}-1}{8}}
  47. [ α β ] [ β α ] - 1 = ( - 1 ) N α - 1 4 N β - 1 4 ϵ ( α ) N β - 1 4 ϵ ( β ) N α - 1 4 \left[\frac{\alpha}{\beta}\right]\left[\frac{\beta}{\alpha}\right]^{-1}=(-1)^{% \frac{N\alpha-1}{4}\frac{N\beta-1}{4}}\epsilon(\alpha)^{\frac{N\beta-1}{4}}% \epsilon(\beta)^{\frac{N\alpha-1}{4}}
  48. λ * = ( - 1 ) N λ - 1 4 λ . \lambda^{*}=(-1)^{\frac{N\lambda-1}{4}}\lambda.
  49. [ λ μ ] = [ μ * λ ] . \left[\frac{\lambda}{\mu}\right]=\Bigg[\frac{\mu^{*}}{\lambda}\Bigg].

Quartic_surface.html

  1. f ( x , y , z ) = 0 f(x,y,z)=0

Quartz_clock.html

  1. f = 1.875 2 2 π a l 2 E 12 ρ f=\frac{1.875^{2}}{2\pi}\frac{a}{l^{2}}\sqrt{\frac{E}{12\rho}}

Quasi_Fermi_level.html

  1. E > E g E>E_{g}
  2. f c ( k , r ) f 0 ( E , E F c , T c ) f_{c}(k,r)\approx f_{0}(E,E_{Fc},T_{c})
  3. f v ( k , r ) f 0 ( E , E F v , T v ) f_{v}(k,r)\approx f_{0}(E,E_{Fv},T_{v})
  4. f 0 ( E , E F , T ) = 1 1 + e ( E - E F ) / ( k B T ) f_{0}(E,E_{F},T)=\frac{1}{1+e^{(E-E_{F})/(k_{B}T)}}
  5. E F c E_{Fc}
  6. E F v E_{Fv}
  7. T c T_{c}
  8. T v T_{v}
  9. f c ( k , r ) f_{c}(k,r)
  10. f v ( k , r ) f_{v}(k,r)
  11. E E
  12. k B k_{B}
  13. n n
  14. p p
  15. n = n ( E F c ) n=n(E_{F_{c}})
  16. p = p ( E F v ) p=p(E_{F_{v}})
  17. n ( E ) n(E)
  18. E E
  19. p ( E ) p(E)
  20. E E
  21. μ \mu
  22. E F ( 𝐫 ) E_{F}(\mathbf{r})
  23. 𝐫 \mathbf{r}
  24. 𝐉 n ( 𝐫 ) = - μ n n ( E F c ) \mathbf{J}_{n}(\mathbf{r})=-\mu_{n}n\cdot(\nabla E_{F_{c}})
  25. 𝐉 p ( 𝐫 ) = μ p p ( E F v ) . \mathbf{J}_{p}(\mathbf{r})=\mu_{p}p\cdot(\nabla E_{F_{v}}).

Quasideterminant.html

  1. | a 11 a 12 a 21 a 22 | 11 = a 11 - a 12 a 22 - 1 a 21 | a 11 a 12 a 21 a 22 | 12 = a 12 - a 11 a 21 - 1 a 22 . \left|\begin{array}[]{cc}a_{11}&a_{12}\\ a_{21}&a_{22}\end{array}\right|_{11}=a_{11}-a_{12}{a_{22}}^{-1}a_{21}\qquad% \left|\begin{array}[]{cc}a_{11}&a_{12}\\ a_{21}&a_{22}\end{array}\right|_{12}=a_{12}-a_{11}{a_{21}}^{-1}a_{22}.
  2. | A | i j = ( - 1 ) i + j det A det A i j , \left|A\right|_{ij}=(-1)^{i+j}\frac{\det A}{\det A^{ij}},
  3. A i j A^{ij}
  4. 2 × 2 2\times 2
  5. A A
  6. B B
  7. A A
  8. i i
  9. ρ \left.\rho\right.
  10. | B | i j = ρ | A | i j \left|B\right|_{ij}=\rho\left|A\right|_{ij}
  11. B B
  12. A A
  13. k k
  14. | B | i j = | A | i j ( j ; k i ) \left|B\right|_{ij}=\left|A\right|_{ij}\,\,(\forall j;\forall k\neq i)
  15. A A
  16. n × n n\times n
  17. R R
  18. 1 i , j n 1\leq i,j\leq n
  19. a i j a_{ij}
  20. i , j i,j
  21. A A
  22. r i j r_{i}^{j}
  23. i i
  24. A A
  25. j j
  26. c j i c_{j}^{i}
  27. j j
  28. A A
  29. i i
  30. i , j i,j
  31. A A
  32. A i j A^{ij}
  33. R R
  34. | A | i j = a i j - r i j ( A i j ) - 1 c j i . \left|A\right|_{ij}=a_{ij}-r_{i}^{j}\,\bigl(A^{ij}\bigr)^{-1}\,c_{j}^{i}.
  35. A - 1 A^{-1}
  36. ( A - 1 ) j i = ( - 1 ) i + j det A i j det A (A^{-1})_{ji}=(-1)^{i+j}\frac{\det A^{ij}}{\det A}
  37. ( A - 1 ) j i = | A | i j - 1 \bigl(A^{-1}\bigr)_{\!ji}=\left|A\right|_{ij}^{\,-1}
  38. ( A 11 A 12 A 21 A 22 ) \left(\begin{array}[]{cc}A_{11}&A_{12}\\ A_{21}&A_{22}\end{array}\right)
  39. n × n n\times n
  40. A A
  41. A 11 A_{11}
  42. k × k k\times k
  43. i , j i,j
  44. A A
  45. A 11 A_{11}
  46. | A | i j = | A 11 - A 12 A 22 - 1 A 21 | i j . \left|A\right|_{ij}=\left|A_{11}-A_{12}\,{A_{22}}^{-1}\,A_{21}\right|_{ij}.
  47. | A | i j | A i l | k j - 1 = - | A | i l | A i j | k l - 1 \left|A\right|_{ij}|A^{il}|_{kj}^{\,-1}=-\left|A\right|_{il}|A^{ij}|_{kl}^{\,-1}
  48. | A k j | i l - 1 | A | i j = - | A i j | k l - 1 | A | k j , |A^{kj}|_{il}^{\,-1}\left|A\right|_{ij}=-|A^{ij}|_{kl}^{\,-1}\left|A\right|_{% kj},
  49. i k i\neq k
  50. j l j\neq l
  51. A 0 A_{0}
  52. k × k k\times k
  53. n × n n\times n
  54. A A
  55. i , j i,j
  56. A 0 A_{0}
  57. B = ( b p q ) B=(b_{pq})
  58. ( n - k ) × ( n - k ) (n-k)\times(n-k)
  59. b p q b_{pq}
  60. p , q p,q
  61. ( k + 1 ) × ( k + 1 ) (k+1)\times(k+1)
  62. A 0 A_{0}
  63. k k
  64. p p
  65. k k
  66. q q
  67. a a
  68. p q pq
  69. | B | i j = | A | i j . \left|B\right|_{ij}=\left|A\right|_{ij}.
  70. i i
  71. det A = l ( - 1 ) i + l a i l det A i l \det A=\sum_{l}(-1)^{i+l}a_{il}\cdot\det A^{il}
  72. | A | i j = a i j - l j a i l | A i j | k l - 1 | A i l | k j \left|A\right|_{ij}=a_{ij}-\sum_{l\neq j}a_{il}\cdot|A^{ij}|_{kl}^{\,-1}|A^{il% }|_{kj}
  73. j j
  74. | A | i j = a i j - k i | A k j | i l | A i j | k l - 1 a k j \left|A\right|_{ij}=a_{ij}-\sum_{k\neq i}|A^{kj}|_{il}|A^{ij}|_{kl}^{\,-1}% \cdot a_{kj}
  75. i i
  76. det q A = | A | 11 | A 11 | 22 | A 12 , 12 | 33 | a n n | n n , {\det}_{q}A=\bigl|A\bigr|_{11}\,\left|A^{11}\right|_{22}\,\left|A^{12,12}% \right|_{33}\,\cdots\,|a_{nn}|_{nn},

Quasilinear_utility.html

  1. u ( x 1 , x 2 , , x n ) = x 1 + θ ( x 2 , , x n ) u(x_{1},x_{2},\ldots,x_{n})=x_{1}+\theta(x_{2},\ldots,x_{n})
  2. θ \theta
  3. \preceq
  4. x y x\sim y
  5. ( x + α e 1 ) ( y + α e 1 ) (x+\alpha e_{1})\sim(y+\alpha e_{1})
  6. e 1 = ( 1 , 0 , 0 , , 0 ) e_{1}=(1,0,0,\ldots,0)
  7. α \alpha
  8. u ( x ) = x 1 + θ ( x 2 , , x L ) u\left(x\right)=x_{1}+\theta\left(x_{2},...,x_{L}\right)
  9. θ \theta
  10. u ( x ) = x 1 + x 2 u\left(x\right)=x_{1}+\sqrt{x_{2}}
  11. \succsim
  12. X = ( - , ) × + L - 1 X=\left(-\infty,\infty\right)\times\mathbb{R}^{L-1}_{+}
  13. ( x + α e 1 ) ( y + α e 1 ) , α , e 1 = ( 1 , 0 , , 0 ) \left(x+\alpha e_{1}\right)\sim\left(y+\alpha e_{1}\right),\forall\alpha\in% \mathbb{R},e_{1}=\left(1,0,...,0\right)
  14. ( x + α e 1 ) ( x ) , α > 0 \left(x+\alpha e_{1}\right)\succ\left(x\right),\forall\alpha>0

Quaternionic_vector_space.html

  1. q ( q 1 , q 2 , q n ) = ( q q 1 , q q 2 , q q n ) q(q_{1},q_{2},\ldots q_{n})=(qq_{1},qq_{2},\ldots qq_{n})
  2. ( q 1 , q 2 , q n ) q = ( q 1 q , q 2 q , q n q ) (q_{1},q_{2},\ldots q_{n})q=(q_{1}q,q_{2}q,\ldots q_{n}q)

QUIET.html

  1. 10 16 10^{16}

Quincunx_matrix.html

  1. ( 1 - 1 1 1 ) \begin{pmatrix}1&-1\\ 1&1\end{pmatrix}

Quintom_scenario.html

  1. w = - 1 w=-1

R_(cross_section_ratio).html

  1. R = σ ( 0 ) ( e + e - hadrons ) σ ( 0 ) ( e + e - μ + μ - ) , R=\frac{\sigma^{(0)}(e^{+}e^{-}\rightarrow\mathrm{hadrons})}{\sigma^{(0)}(e^{+% }e^{-}\rightarrow\mu^{+}\mu^{-})},
  2. R = 3 q e q 2 , R=3\sum_{q}e_{q}^{2},

Racah_polynomials.html

  1. p n ( x ( x + γ + δ + 1 ) ) = F 3 4 [ - n n + α + β + 1 - x x + γ + δ + 1 α + 1 γ + 1 β + δ + 1 ; 1 ] . p_{n}(x(x+\gamma+\delta+1))={}_{4}F_{3}\left[\begin{matrix}-n&n+\alpha+\beta+1% &-x&x+\gamma+\delta+1\\ \alpha+1&\gamma+1&\beta+\delta+1\\ \end{matrix};1\right].
  2. p n ( q - x + q x + 1 c d ; a , b , c , d ; q ) = ϕ 3 4 [ q - n a b q n + 1 q - x q x + 1 c d a q b d q c q ; q ; q ] . p_{n}(q^{-x}+q^{x+1}cd;a,b,c,d;q)={}_{4}\phi_{3}\left[\begin{matrix}q^{-n}&abq% ^{n+1}&q^{-x}&q^{x+1}cd\\ aq&bdq&cq\\ \end{matrix};q;q\right].
  3. W n ( x ; a , b , c , N ; q ) = ϕ 3 4 [ q - n a b q n + 1 q - x c q x - n a q b c q q - N ; q ; q ] . W_{n}(x;a,b,c,N;q)={}_{4}\phi_{3}\left[\begin{matrix}q^{-n}&abq^{n+1}&q^{-x}&% cq^{x-n}\\ aq&bcq&q^{-N}\\ \end{matrix};q;q\right].

Racetrack_principle.html

  1. f ( x ) > g ( x ) f^{\prime}(x)>g^{\prime}(x)
  2. x > 0 x>0
  3. f ( 0 ) = g ( 0 ) f(0)=g(0)
  4. f ( x ) > g ( x ) f(x)>g(x)
  5. x > 0 x>0
  6. f ( x ) g ( x ) f^{\prime}(x)\geq g^{\prime}(x)
  7. x > 0 x>0
  8. f ( 0 ) = g ( 0 ) f(0)=g(0)
  9. f ( x ) g ( x ) f(x)\geq g(x)
  10. x > 0 x>0
  11. h = f - g > 0. h^{\prime}=f^{\prime}-g^{\prime}>0.
  12. h ( x 0 ) = h ( x ) - h ( 0 ) x - 0 = f ( x ) - g ( x ) x > 0. h^{\prime}(x_{0})=\frac{h(x)-h(0)}{x-0}=\frac{f(x)-g(x)}{x}>0.
  13. f ( x ) > g ( x ) f^{\prime}(x)>g^{\prime}(x)
  14. x > a x>a
  15. f ( a ) = g ( a ) f(a)=g(a)
  16. f ( x ) > g ( x ) f(x)>g(x)
  17. x > a x>a
  18. f ( x ) g ( x ) f^{\prime}(x)\geq g^{\prime}(x)
  19. x > a x>a
  20. f ( a ) = g ( a ) f(a)=g(a)
  21. f ( x ) g ( x ) f(x)\geq g(x)
  22. x > a x>a
  23. f ( x ) > g ( x ) f^{\prime}(x)>g^{\prime}(x)
  24. x > a x>a
  25. f ( a ) = g ( a ) f(a)=g(a)
  26. f 2 ( x ) = f ( x - a ) f_{2}(x)=f(x-a)
  27. g 2 ( x ) = g ( x - a ) g_{2}(x)=g(x-a)
  28. f 2 ( x ) > g 2 ( x ) f_{2}^{\prime}(x)>g_{2}^{\prime}(x)
  29. x > 0 x>0
  30. f 2 ( 0 ) = g 2 ( 0 ) f_{2}(0)=g_{2}(0)
  31. f 2 ( x ) > g 2 ( x ) f_{2}(x)>g_{2}(x)
  32. x > 0 x>0
  33. f ( x ) > g ( x ) f(x)>g(x)
  34. x > a x>a
  35. e x > x e^{x}>x
  36. f ( x ) = e x f(x)=e^{x}
  37. g ( x ) = x + 1. g(x)=x+1.
  38. e x > 1 e^{x}>1
  39. f ( x ) > g ( x ) f^{\prime}(x)>g^{\prime}(x)
  40. e x > x + 1 > x e^{x}>x+1>x

Radical_(chemistry).html

  1. Cl 2 U V Cl + Cl \mathrm{Cl}_{2}\;\xrightarrow{UV}\;{\mathrm{Cl}\cdot}+{\mathrm{Cl}\cdot}
  2. 1. NO 2 h ν NO + O 1.\;\;\mathrm{NO}_{2}\;\xrightarrow{h\nu}\;\mathrm{NO+O}
  3. 2. O + O 2 O 3 2.\;\;\mathrm{O}+\mathrm{O}_{2}\;\xrightarrow{}\;\mathrm{O}_{3}
  4. 3. NO 2 + O 2 h ν NO + O 3 3.\;\;\mathrm{NO}_{2}+\mathrm{O}_{2}\;\xrightarrow{h\nu}\;\mathrm{NO}+\mathrm{% O}_{3}\;
  5. 4. NO + O 3 NO 2 + O 2 4.\;\;\mathrm{NO}+\mathrm{O}_{3}\;\xrightarrow{}\;\mathrm{NO}_{2}+\mathrm{O}_{% 2}\;
  6. 1. CFCS h ν Cl 1.\;\;\mathrm{CFCS}\;\xrightarrow{h\nu}\;{\mathrm{Cl}\cdot}
  7. 2. Cl + O 3 ClO + O 2 2.\;\;{\mathrm{Cl}\cdot}+\mathrm{O}_{3}\;\xrightarrow{}\;{\mathrm{ClO}\cdot}+% \mathrm{O}_{2}\;
  8. 3. O 3 h ν O + O 2 3.\;\;\mathrm{O}_{3}\;\xrightarrow{h\nu}\;\mathrm{O}+\mathrm{O}_{2}
  9. 4. O + ClO Cl + O 2 4.\;\;\mathrm{O}+{\mathrm{ClO}\cdot}\;\xrightarrow{\;}{\mathrm{Cl}\cdot}+% \mathrm{O}_{2}
  10. 5.  2 O 3 h ν 3 O 2 5.\;\;2\mathrm{O}_{3}\;\xrightarrow{h\nu}\;3\mathrm{O}_{2}

Radical_clock.html

  1. U + A B k R U A + B k r R + A B R A + B \begin{array}[]{lcl}U\cdot+AB&\xrightarrow{k_{R}}&UA+B\cdot\\ \bigg\downarrow{k_{r}}\\ R\cdot+AB&\xrightarrow{}&RA+B\cdot\end{array}
  2. k R = k r [ U A ] [ A B ] [ R A ] k_{R}=\frac{k_{r}[UA]}{[AB][RA]}

Radius_of_curvature_(mathematics).html

  1. d s d φ = 1 κ , \frac{ds}{d\varphi}=\frac{1}{\kappa},
  2. κ \scriptstyle\kappa
  3. R = | ( 1 + y 2 ) 3 / 2 y ′′ | , where y = d y d x , y ′′ = d 2 y d x 2 , R=\left|\frac{\left(1+y^{\prime\,2}\right)^{3/2}}{y^{\prime\prime}}\right|,% \qquad\mbox{where}~{}\quad y^{\prime}=\frac{dy}{dx},\quad y^{\prime\prime}=% \frac{d^{2}y}{dx^{2}},
  4. R = | d s d φ | = | ( x ˙ 2 + y ˙ 2 ) 3 / 2 x ˙ y ¨ - y ˙ x ¨ | , where x ˙ = d x d t , x ¨ = d 2 x d t 2 , y ˙ = d y d t , y ¨ = d 2 y d t 2 . R=\;\left|\frac{ds}{d\varphi}\right|\;=\;\left|\frac{\big({\dot{x}^{2}+\dot{y}% ^{2}}\big)^{3/2}}{\dot{x}\ddot{y}-\dot{y}\ddot{x}}\right|,\qquad\mbox{where}~{% }\quad\dot{x}=\frac{dx}{dt},\quad\ddot{x}=\frac{d^{2}x}{dt^{2}},\quad\dot{y}=% \frac{dy}{dt},\quad\ddot{y}=\frac{d^{2}y}{dt^{2}}.
  5. R = | 𝐯 | 3 | 𝐯 × 𝐯 ˙ | , where | 𝐯 | = | ( x ˙ , y ˙ ) | = R d φ d t . R=\frac{\left|\mathbf{v}\right|^{3}}{\left|\mathbf{v}\times\mathbf{\dot{v}}% \right|},\qquad\mbox{where}~{}\quad\left|\mathbf{v}\right|=\left|(\dot{x},\dot% {y})\right|=R\frac{d\varphi}{dt}.
  6. y = a 2 - x 2 , y = - x a 2 - x 2 , y ′′ = - a 2 ( a 2 - x 2 ) 3 / 2 , R = | - a | = a . y=\sqrt{a^{2}-x^{2}},\quad y^{\prime}=\frac{-x}{\sqrt{a^{2}-x^{2}}},\quad y^{% \prime\prime}=\frac{-a^{2}}{(a^{2}-x^{2})^{3/2}},\quad R=|-a|=a.
  7. y = - a 2 - x 2 , R = | a | = a . y=-\sqrt{a^{2}-x^{2}},\quad R=|a|=a.
  8. ( R = b 2 a ) \left(R=\frac{b^{2}}{a}\right)
  9. ( R = a 2 b ) \left(R=\frac{a^{2}}{b}\right)

Radon–Riesz_property.html

  1. ( x n ) (x_{n})
  2. x x
  3. ( x n ) (x_{n})
  4. x x
  5. lim n x n = x \lim_{n\to\infty}\|x_{n}\|=\|x\|
  6. ( x n ) (x_{n})
  7. x x
  8. lim n x n - x = 0 \lim_{n\to\infty}\|x_{n}-x\|=0
  9. ( x n ) (x_{n})
  10. x x
  11. x n - x , x n - x = x n , x n - x n , x - x , x n + x , x , \langle x_{n}-x,x_{n}-x\rangle=\langle x_{n},x_{n}\rangle-\langle x_{n},x% \rangle-\langle x,x_{n}\rangle+\langle x,x\rangle,
  12. lim n x n - x , x n - x = 0. \lim_{n\to\infty}{\langle x_{n}-x,x_{n}-x\rangle}=0.

Rail_speed_limits_in_the_United_States.html

  1. V m a x = E a + 3 0.0007 d V_{max}=\sqrt{\frac{E_{a}+3}{0.0007d}}
  2. E a E_{a}
  3. d d
  4. V m a x V_{max}

Rain.html

  1. d d
  2. D + d D D+dD
  3. n ( d ) = n 0 e - d / d d D n(d)=n_{0}e^{-d/\langle d\rangle}dD
  4. d - 1 = 41 R - 0.21 \langle d\rangle^{-1}=41R^{-0.21}
  5. Z = A R b Z=AR^{b}

Randomness_extractor.html

  1. X X
  2. H ( X ) H_{\infty}(X)
  3. k k
  4. Pr [ X = x ] 2 - k \Pr[X=x]\leq 2^{-k}
  5. x x
  6. X X
  7. X X
  8. X X
  9. U U_{\ell}
  10. { 0 , 1 } \{0,1\}^{\ell}
  11. H ( U ) = H_{\infty}(U_{\ell})=\ell
  12. X X
  13. X X
  14. ( n , k ) (n,k)
  15. Ext : { 0 , 1 } n × { 0 , 1 } d { 0 , 1 } m \,\text{Ext}:\{0,1\}^{n}\times\{0,1\}^{d}\to\{0,1\}^{m}
  16. ( n , k ) (n,k)
  17. X X
  18. U d U_{d}
  19. Ext \,\text{Ext}
  20. ( n , k ) (n,k)
  21. X X
  22. Ext \,\text{Ext}
  23. U m U_{m}
  24. d d
  25. m m
  26. ( k , ϵ ) (k,\epsilon)
  27. Ext : { 0 , 1 } n × { 0 , 1 } d { 0 , 1 } m \,\text{Ext}:\{0,1\}^{n}\times\{0,1\}^{d}\rightarrow\{0,1\}^{m}\,
  28. ( n , k ) (n,k)
  29. X X
  30. U d Ext ( X , U d ) U_{d}\circ\,\text{Ext}(X,U_{d})
  31. U d U_{d}
  32. ϵ \epsilon
  33. { 0 , 1 } m + d \{0,1\}^{m+d}
  34. Ext n : { 0 , 1 } n × { 0 , 1 } d ( n ) { 0 , 1 } m ( n ) \,\text{Ext}_{n}:\{0,1\}^{n}\times\{0,1\}^{d(n)}\rightarrow\{0,1\}^{m(n)}
  35. d = log ( n - k ) + 2 log ( 1 ε ) + O ( 1 ) d=\log{(n-k)}+2\log\left(\frac{1}{\varepsilon}\right)+O(1)
  36. m = k + d - 2 log ( 1 ε ) - O ( 1 ) m=k+d-2\log\left(\frac{1}{\varepsilon}\right)-O(1)
  37. f f
  38. r r
  39. A A
  40. r r
  41. k k
  42. | Pr { A r ( f ( r ) ) = 1 } - Pr { A r ( R ) = 1 } | ϵ ( n ) |\Pr\{A^{r}(f(r))=1\}-\Pr\{A^{r}(R)=1\}|\leq\epsilon(n)
  43. ϵ ( n ) \epsilon(n)
  44. k = k ( n ) k=k(n)
  45. f f
  46. n - k n-k
  47. r r
  48. p p
  49. f ( r ) f(r)
  50. k k
  51. | p i - 2 - m | < 2 - m ϵ ( n ) |p_{i}-2^{-m}|<2^{-m}\epsilon(n)
  52. i i
  53. ϵ ( n ) \epsilon(n)
  54. f f
  55. f f
  56. 2 - m ϵ ( n ) 2^{-m}\epsilon(n)
  57. f : { 0 , 1 } n { 0 , 1 } m f:\{0,1\}^{n}\rightarrow\{0,1\}^{m}
  58. ( n , k ) (n,k)
  59. ϵ ( n ) \epsilon(n)
  60. 2 - m ϵ ( n ) 2^{-m}\epsilon(n)
  61. 2 - m ϵ ( n ) 2^{-m}\epsilon(n)
  62. γ 1 2 \gamma\leq\frac{1}{2}
  63. f : { 0 , 1 } n { 0 , 1 } m f:\{0,1\}^{n}\rightarrow\{0,1\}^{m}
  64. m m
  65. m = Ω ( n 2 γ ) m=\Omega(n^{2\gamma})
  66. k = n 1 2 + γ k=n^{\frac{1}{2}+\gamma}
  67. ϵ ( n ) \epsilon(n)
  68. ϵ ( n ) = O ( 1 n c ) \epsilon(n)=O(\frac{1}{n^{c}})
  69. c c
  70. ϵ \epsilon
  71. f f
  72. ( n , δ n ) (n,\delta n)
  73. f : { 0 , 1 } n { 0 , 1 } m f:\{0,1\}^{n}\rightarrow\{0,1\}^{m}
  74. f f
  75. m = Ω ( δ 2 n ) m=\Omega(\delta^{2}n)
  76. ϵ = 2 - c m \epsilon=2^{-cm}
  77. c > 1 c>1
  78. δ 1 \delta\leq 1
  79. m m
  80. ϵ = 2 - c m \epsilon=2^{-cm}
  81. m m
  82. m = Ω ( δ 2 n ) = Ω ( n ) Ω ( n 2 γ ) m=\Omega(\delta^{2}n)=\Omega(n)\geq\Omega(n^{2\gamma})
  83. δ 1 \delta\leq 1
  84. m m
  85. n n
  86. γ 1 2 \gamma\leq\frac{1}{2}
  87. n n
  88. 1 1
  89. n n
  90. n 2 γ n^{2\gamma}
  91. n n
  92. k k
  93. ( n , k ) = ( n , δ n ) k = δ n (n,k)=(n,\delta n)\Rightarrow k=\delta n
  94. m m
  95. m = δ 2 n = n 2 γ m=\delta^{2}n=n^{2\gamma}
  96. m m
  97. k k
  98. δ 2 n = n 2 γ \delta^{2}n=n^{2\gamma}
  99. δ 2 = n 2 γ - 1 \Rightarrow\delta^{2}=n^{2\gamma-1}
  100. δ = n γ - 1 2 \Rightarrow\delta=n^{\gamma-\frac{1}{2}}
  101. k k
  102. k = δ n = n γ - 1 2 n = n γ + 1 2 k=\delta n=n^{\gamma-\frac{1}{2}}n=n^{\gamma+\frac{1}{2}}
  103. \Box
  104. p = 1 / 2. p=1/2.
  105. p q = q p p\cdot q=q\cdot p

Range_accrual.html

  1. P × i = 1 N 1 index ( i ) Range × 1 N P\times\sum_{i=1}^{N}1_{\,\text{index}(i)\in\,\text{Range}}\times\frac{1}{N}
  2. P × n N P\times\frac{n}{N}

Rank_of_a_group.html

  1. rank ( G ) = min { | X | : X G , X = G } . \operatorname{rank}(G)=\min\{|X|:X\subseteq G,\langle X\rangle=G\}.
  2. sr ( G ) = max H G min { | X | : X H , X = H } . \operatorname{sr}(G)=\max_{H\leq G}\min\{|X|:X\subseteq H,\langle X\rangle=H\}.
  3. n \mathbb{Z}^{n}
  4. rank ( n ) = n . {\rm rank}(\mathbb{Z}^{n})=n.
  5. L = H K L=H\cap K
  6. \ast
  7. G = x 1 , , x n | r = 1 G=\langle x_{1},\dots,x_{n}|r=1\rangle

Rarefaction_(ecology).html

  1. i = 1 K N i = N \sum_{i=1}^{K}N_{i}=N
  2. j = 1 M j = K \sum_{j=1}^{\infty}M_{j}=K
  3. j = 1 j M j = N \sum_{j=1}^{\infty}jM_{j}=N
  4. X n = X_{n}=
  5. X n X_{n}
  6. f n f_{n}
  7. f n = E [ X n ] = K - ( N n ) - 1 i = 1 K ( N - N i n ) f_{n}=E[X_{n}]=K-{\left({{N}\atop{n}}\right)}^{-1}\sum_{i=1}^{K}{{\left({{N-N_% {i}}\atop{n}}\right)}}
  8. f ( 0 ) = 0 , f ( 1 ) = 1 , f ( N ) = K f(0)=0,f(1)=1,f(N)=K

Rate_of_infusion.html

  1. K i n = C s s C L \textstyle K_{in}=C_{ss}\cdot CL

Rational_dependence.html

  1. independent 3 , 8 , 1 + 2 dependent \begin{matrix}\mbox{independent}\\ \underbrace{\overbrace{3,\quad\sqrt{8}\quad},1+\sqrt{2}}\\ \mbox{dependent}\\ \end{matrix}
  2. k 1 ω 1 + k 2 ω 2 + + k n ω n = 0. k_{1}\omega_{1}+k_{2}\omega_{2}+\cdots+k_{n}\omega_{n}=0.
  3. k 1 ω 1 + k 2 ω 2 + + k n ω n = 0 k_{1}\omega_{1}+k_{2}\omega_{2}+\cdots+k_{n}\omega_{n}=0

Rational_motion.html

  1. 𝐪 ^ = 𝐪 + ε 𝐪 0 \hat{\,\textbf{q}}=\,\textbf{q}+\varepsilon\,\textbf{q}^{0}
  2. 𝐐 ^ = 𝐐 + ε 𝐐 0 \hat{\,\textbf{Q}}=\,\textbf{Q}+\varepsilon\,\textbf{Q}^{0}
  3. 𝐐 = w 𝐪 , 𝐐 0 = w 𝐪 0 + w 0 𝐪 \,\textbf{Q}=w\,\textbf{q},\,\textbf{Q}^{0}=w\,\textbf{q}^{0}+w^{0}\,\textbf{q}
  4. 𝐐 ^ = w ^ 𝐪 ^ \hat{\,\textbf{Q}}=\hat{w}\hat{\,\textbf{q}}
  5. w ^ = w + ε w 0 \hat{w}=w+\varepsilon w^{0}
  6. 𝐏 : ( P 1 , P 2 , P 3 , P 4 ) \,\textbf{P}:(P_{1},P_{2},P_{3},P_{4})
  7. 𝐏 ~ = 𝐐 𝐏 𝐐 + P 4 [ ( 𝐐 0 ) 𝐐 - 𝐐 ( 𝐐 0 ) ] , \tilde{\,\textbf{P}}=\,\textbf{Q}\,\textbf{P}\,\textbf{Q}^{\ast}+P_{4}[(\,% \textbf{Q}^{0})\,\textbf{Q}^{\ast}-\,\textbf{Q}(\,\textbf{Q}^{0})^{\ast}],
  8. 𝐐 \,\textbf{Q}^{\ast}
  9. ( 𝐐 0 ) (\,\textbf{Q}^{0})^{\ast}
  10. 𝐐 \,\textbf{Q}
  11. 𝐐 0 \,\textbf{Q}^{0}
  12. 𝐏 ~ \tilde{\,\textbf{P}}
  13. 𝐪 ^ i , w ^ i ; i = 0... n \hat{\,\textbf{q}}_{i},\hat{w}_{i};i=0...n
  14. 𝐐 ^ ( t ) = i = 0 n B i n ( t ) 𝐐 ^ i = i = 0 n B i n ( t ) w ^ i 𝐪 ^ i \hat{\,\textbf{Q}}(t)=\sum\limits_{i=0}^{n}{B_{i}^{n}(t)\hat{\,\textbf{Q}}_{i}% }=\sum\limits_{i=0}^{n}{B_{i}^{n}(t)\hat{w}_{i}\hat{\,\textbf{q}}_{i}}
  15. B i n ( t ) B_{i}^{n}(t)
  16. 2 n 2n
  17. 𝐐 ^ ( t ) = i = 0 n N i , p ( t ) 𝐐 ^ i = i = 0 n N i , p ( t ) w ^ i 𝐪 ^ i \hat{\,\textbf{Q}}(t)=\sum\limits_{i=0}^{n}{N_{i,p}(t)\hat{\,\textbf{Q}}_{i}}=% \sum\limits_{i=0}^{n}{N_{i,p}(t)\hat{w}_{i}\hat{\,\textbf{q}}_{i}}
  18. N i , p ( t ) N_{i,p}(t)
  19. 𝐐 ^ ( t ) \hat{\,\textbf{Q}}(t)
  20. 𝐏 ~ 2 n ( t ) = [ H 2 n ( t ) ] 𝐏 , \tilde{\,\textbf{P}}^{2n}(t)=[H^{2n}(t)]\,\textbf{P},
  21. H 2 n ( t ) ] = k = 0 2 n B k 2 n ( t ) [ H k ] , H^{2n}(t)]=\sum\limits_{k=0}^{2n}{B_{k}^{2n}(t)[H_{k}]},
  22. [ H 2 n ( t ) ] [H^{2n}(t)]
  23. 2 n 2n
  24. [ H k ] [H_{k}]
  25. [ H k ] = 1 C k 2 n i + j = k C i n C j n w i w j [ H i j ] , [H_{k}]=\frac{1}{C_{k}^{2n}}\sum\limits_{i+j=k}{C_{i}^{n}C_{j}^{n}w_{i}w_{j}[H% _{ij}^{\ast}]},
  26. [ H i j ] = [ H i + ] [ H j - ] + [ H j - ] [ H i 0 + ] - [ H i + ] [ H j 0 - ] + ( α i - α j ) [ H j - ] [ Q i + ] [H_{ij}^{\ast}]=[H_{i}^{+}][H_{j}^{-}]+[H_{j}^{-}][H_{i}^{0+}]-[H_{i}^{+}][H_{% j}^{0-}]+(\alpha_{i}-\alpha_{j})[H_{j}^{-}][Q_{i}^{+}]
  27. C i n C_{i}^{n}
  28. C j n C_{j}^{n}
  29. α i = w i 0 / w i , α j = w j 0 / w j \alpha_{i}=w_{i}^{0}/w_{i},\alpha_{j}=w_{j}^{0}/w_{j}
  30. [ H j - ] = [ q j , 4 - q j , 3 q j , 2 - q j , 1 q j , 3 q j , 4 - q j , 1 - q j , 2 - q j , 2 q j , 1 q j , 4 - q j , 3 q j , 1 q j , 2 q j , 3 q j , 4 ] , [H_{j}^{-}]=\left[\begin{array}[]{rrrr}q_{j,4}&-q_{j,3}&q_{j,2}&-q_{j,1}\\ q_{j,3}&q_{j,4}&-q_{j,1}&-q_{j,2}\\ -q_{j,2}&q_{j,1}&q_{j,4}&-q_{j,3}\\ q_{j,1}&q_{j,2}&q_{j,3}&q_{j,4}\\ \end{array}\right],
  31. [ Q i + ] = [ 0 0 0 q i , 1 0 0 0 q i , 2 0 0 0 q i , 3 0 0 0 q i , 4 ] , [Q_{i}^{+}]=\left[\begin{array}[]{rrrr}0&0&0&q_{i,1}\\ 0&0&0&q_{i,2}\\ 0&0&0&q_{i,3}\\ 0&0&0&q_{i,4}\\ \end{array}\right],
  32. [ H i 0 + ] = [ 0 0 0 q i , 1 0 0 0 0 q i , 2 0 0 0 0 q i , 3 0 0 0 0 q i , 4 0 ] , [H_{i}^{0+}]=\left[\begin{array}[]{rrrr}0&0&0&q_{i,1}^{0}\\ 0&0&0&q_{i,2}^{0}\\ 0&0&0&q_{i,3}^{0}\\ 0&0&0&q_{i,4}^{0}\\ \end{array}\right],
  33. [ H j 0 - ] = [ 0 0 0 - q j , 1 0 0 0 0 - q j , 2 0 0 0 0 - q j , 3 0 0 0 0 q j , 4 0 ] , [H_{j}^{0-}]=\left[\begin{array}[]{rrrr}0&0&0&-q_{j,1}^{0}\\ 0&0&0&-q_{j,2}^{0}\\ 0&0&0&-q_{j,3}^{0}\\ 0&0&0&q_{j,4}^{0}\\ \end{array}\right],
  34. [ H i + ] = [ q i , 4 - q i , 3 q i , 2 q i , 1 q i , 3 q i , 4 - q i , 1 q i , 2 - q i , 2 q i , 1 q i , 4 q i , 3 - q i , 1 - q i , 2 - q i , 3 q i , 4 ] . [H_{i}^{+}]=\left[\begin{array}[]{rrrr}q_{i,4}&-q_{i,3}&q_{i,2}&q_{i,1}\\ q_{i,3}&q_{i,4}&-q_{i,1}&q_{i,2}\\ -q_{i,2}&q_{i,1}&q_{i,4}&q_{i,3}\\ -q_{i,1}&-q_{i,2}&-q_{i,3}&q_{i,4}\\ \end{array}\right].
  35. ( q i , 1 , q i , 2 , q i , 3 , q i , 4 ) (q_{i,1},q_{i,2},q_{i,3},q_{i,4})
  36. ( 𝐪 i ) (\,\textbf{q}_{i})
  37. ( q i , 1 0 , q i , 2 0 , q i , 3 0 , q i , 4 0 ) (q_{i,1}^{0},q_{i,2}^{0},q_{i,3}^{0},q_{i,4}^{0})
  38. ( 𝐪 i 0 ) (\,\textbf{q}_{i}^{0})
  39. ( 𝐪 ^ i ) (\hat{\,\textbf{q}}_{i})
  40. w ^ i = 1 + ϵ 0 ; i = 0..3 \hat{w}_{i}=1+\epsilon 0;i=0..3
  41. w ^ i = 1 + ϵ 0 ; i = 0 , 3 \hat{w}_{i}=1+\epsilon 0;i=0,3
  42. w ^ i = 4 + ϵ 0 ; i = 1 , 2 \hat{w}_{i}=4+\epsilon 0;i=1,2