wpmath0000007_12

Segal's_law.html

  1. S = (WW, WB, BW, BB) S=\text{(WW, WB, BW, BB)}
  2. p 2 + p q + q p + q 2 = p 2 + 2 p q + q 2 = 1 p^{2}+pq+qp+q^{2}=p^{2}+2pq+q^{2}=1
  3. P = p 2 + p q P=p^{2}+pq
  4. P = p 2 + p ( 1 - p ) = p P=p^{2}+p(1-p)=p
  5. p 3 + 3 p 2 q + 3 p q 2 + q 3 = 1 p^{3}+3p^{2}q+3pq^{2}+q^{3}=1
  6. P = p 3 + 3 p 2 q + p q 2 = p + p 2 ( 1 - p ) \begin{aligned}\displaystyle P&\displaystyle=p^{3}+3p^{2}q+pq^{2}\\ &\displaystyle=p+p^{2}(1-p)\end{aligned}

Segal_conjecture.html

  1. lim π S 0 ( B G + ( k ) ) A ^ ( G ) . \underleftarrow{\lim}\pi_{S}^{0}(BG^{(k)}_{+})\to\hat{A}(G).
  2. K U 0 ( B G ) R ^ [ G ] KU^{0}(BG)\to\hat{R}[G]

Seiberg–Witten_theory.html

  1. V ( x ) = 1 g 2 Tr [ ϕ , ϕ ¯ ] 2 V(x)=\frac{1}{g^{2}}\operatorname{Tr}[\phi,\bar{\phi}]^{2}\,
  2. 1 4 π Im [ d 4 θ d F d A A ¯ + d 2 θ 1 2 d 2 F d A 2 W α W α ] \frac{1}{4\pi}\operatorname{Im}\Bigl[\int d^{4}\theta\frac{dF}{dA}\bar{A}+\int d% ^{2}\theta\frac{1}{2}\frac{d^{2}F}{dA^{2}}W_{\alpha}W^{\alpha}\Bigr]\,
  3. F = i 2 π 𝒜 2 ln 𝒜 2 Λ 2 + k = 1 F k Λ 4 k 𝒜 4 k 𝒜 2 F=\frac{i}{2\pi}\mathcal{A}^{2}\operatorname{\ln}\frac{\mathcal{A}^{2}}{% \Lambda^{2}}+\sum_{k=1}^{\infty}F_{k}\frac{\Lambda^{4k}}{\mathcal{A}^{4k}}% \mathcal{A}^{2}\,
  4. a D = d F d a a_{D}=\frac{dF}{da}\,
  5. Z ( a ; ε 1 , ε 2 , Λ ) = exp ( - 1 ε 1 ε 2 ( ( a ; Λ ) + O ( ε 1 , ε 2 ) ) Z(a;\varepsilon_{1},\varepsilon_{2},\Lambda)=\exp(-\frac{1}{\varepsilon_{1}% \varepsilon_{2}}(\mathcal{F}(a;\Lambda)+O(\varepsilon_{1},\varepsilon_{2}))\,

Seismic_anisotropy.html

  1. σ i j = C i j k l e k l i , j , k , l = 1 , 2 , 3 \sigma_{ij}=C_{ijkl}e_{kl}\quad i,j,k,l=1,2,3
  2. 𝖢 ¯ ¯ = [ C 11 C 11 - 2 C 66 C 13 0 0 0 C 11 - 2 C 66 C 11 C 13 0 0 0 C 13 C 13 C 33 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 66 ] \underline{\underline{\mathsf{C}}}=\begin{bmatrix}C_{11}&C_{11}-2C_{66}&C_{13}% &0&0&0\\ C_{11}-2C_{66}&C_{11}&C_{13}&0&0&0\\ C_{13}&C_{13}&C_{33}&0&0&0\\ 0&0&0&C_{44}&0&0\\ 0&0&0&0&C_{44}&0\\ 0&0&0&0&0&C_{66}\end{bmatrix}
  3. 𝖢 ¯ ¯ = [ C 11 C 13 C 13 0 0 0 C 13 C 33 C 33 - 2 C 44 0 0 0 C 13 C 33 - 2 C 44 C 33 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 66 0 0 0 0 0 0 C 66 ] \underline{\underline{\mathsf{C}}}=\begin{bmatrix}C_{11}&C_{13}&C_{13}&0&0&0\\ C_{13}&C_{33}&C_{33}-2C_{44}&0&0&0\\ C_{13}&C_{33}-2C_{44}&C_{33}&0&0&0\\ 0&0&0&C_{44}&0&0\\ 0&0&0&0&C_{66}&0\\ 0&0&0&0&0&C_{66}\end{bmatrix}
  4. V q P ( θ ) = C 11 sin 2 ( θ ) + C 33 cos 2 ( θ ) + C 44 + M ( θ ) 2 ρ V q S ( θ ) = C 11 sin 2 ( θ ) + C 33 cos 2 ( θ ) + C 44 - M ( θ ) 2 ρ V S = C 66 sin 2 ( θ ) + C 44 cos 2 ( θ ) ρ M ( θ ) = [ ( C 11 - C 44 ) sin 2 ( θ ) - ( C 33 - C 44 ) cos 2 ( θ ) ] 2 + ( C 13 + C 44 ) 2 sin 2 ( 2 θ ) \begin{aligned}\displaystyle V_{qP}(\theta)&\displaystyle=\sqrt{\frac{C_{11}% \sin^{2}(\theta)+C_{33}\cos^{2}(\theta)+C_{44}+\sqrt{M(\theta)}}{2\rho}}\\ \displaystyle V_{qS}(\theta)&\displaystyle=\sqrt{\frac{C_{11}\sin^{2}(\theta)+% C_{33}\cos^{2}(\theta)+C_{44}-\sqrt{M(\theta)}}{2\rho}}\\ \displaystyle V_{S}&\displaystyle=\sqrt{\frac{C_{66}\sin^{2}(\theta)+C_{44}% \cos^{2}(\theta)}{\rho}}\\ \displaystyle M(\theta)&\displaystyle=\left[\left(C_{11}-C_{44}\right)\sin^{2}% (\theta)-\left(C_{33}-C_{44}\right)\cos^{2}(\theta)\right]^{2}+\left(C_{13}+C_% {44}\right)^{2}\sin^{2}(2\theta)\\ \end{aligned}
  5. θ \begin{aligned}\displaystyle\theta\end{aligned}
  6. ρ \rho
  7. C i j C_{ij}
  8. V q P ( θ ) V P 0 ( 1 + δ sin 2 θ cos 2 θ + ϵ sin 4 θ ) V q S ( θ ) V S 0 [ 1 + ( V P 0 V S 0 ) 2 ( ϵ - δ ) sin 2 θ cos 2 θ ] V S ( θ ) V S 0 ( 1 + γ sin 2 θ ) \begin{aligned}\displaystyle V_{qP}(\theta)&\displaystyle\approx V_{P0}(1+% \delta\sin^{2}\theta\cos^{2}\theta+\epsilon\sin^{4}\theta)\\ \displaystyle V_{qS}(\theta)&\displaystyle\approx V_{S0}\left[1+\left(\frac{V_% {P0}}{V_{S0}}\right)^{2}(\epsilon-\delta)\sin^{2}\theta\cos^{2}\theta\right]\\ \displaystyle V_{S}(\theta)&\displaystyle\approx V_{S0}(1+\gamma\sin^{2}\theta% )\end{aligned}
  9. V P 0 = C 33 / ρ ; V S 0 = C 44 / ρ V_{P0}=\sqrt{C_{33}/\rho}~{};~{}~{}V_{S0}=\sqrt{C_{44}/\rho}
  10. 𝐞 3 \mathbf{e}_{3}
  11. δ \delta
  12. ϵ = C 11 - C 33 2 C 33 δ = ( C 13 + C 44 ) 2 - ( C 33 - C 44 ) 2 2 C 33 ( C 33 - C 44 ) γ = C 66 - C 44 2 C 44 \begin{aligned}\displaystyle\epsilon&\displaystyle=\frac{C_{11}-C_{33}}{2C_{33% }}\\ \displaystyle\delta&\displaystyle=\frac{(C_{13}+C_{44})^{2}-(C_{33}-C_{44})^{2% }}{2C_{33}(C_{33}-C_{44})}\\ \displaystyle\gamma&\displaystyle=\frac{C_{66}-C_{44}}{2C_{44}}\end{aligned}
  13. η \eta
  14. t 2 ( x ) = t 0 2 + x 2 V n m o 2 - 2 η x 4 V n m o 2 ( [ t 0 2 V ] n m o 2 + ( 1 + 2 η ) x 2 ) t^{2}(x)=t_{0}^{2}+\frac{x^{2}}{V_{nmo}^{2}}-\frac{2\eta x^{4}}{V_{nmo}^{2}([t% _{0}^{2}V]_{nmo}^{2}+(1+2\eta)x^{2})}
  15. t ( x ) t(x)
  16. x x
  17. t 0 t_{0}
  18. V n m o V_{nmo}
  19. η \eta
  20. C 33 = ρ V P 2 ( 0 ) \displaystyle C_{33}=\rho V_{P}^{2}(0^{\circ})
  21. ϵ , δ \epsilon,\delta
  22. ϵ \epsilon
  23. δ \delta
  24. R P P = A + ( b 11 cos 2 ϕ + 2 b 12 cos ϕ sin ϕ + b 22 sin 2 ϕ ) R_{PP}=A+(b_{11}\cos^{2}\phi+2b_{12}\cos\phi\sin\phi+b_{22}\sin^{2}\phi)
  25. ϕ \phi
  26. b i j b_{ij}
  27. V P ( ϕ ) = V P ( 0 ) ( 1 + δ sin 2 ϕ cos 2 ϕ + ϵ sin 4 ϕ ) V_{P}(\phi)=V_{P}(0)(1+\delta\sin^{2}\phi\cos^{2}\phi+\epsilon\sin^{4}\phi)
  28. ϕ \phi
  29. δ \delta
  30. ϵ \epsilon
  31. ϕ \phi
  32. δ \delta
  33. ϵ \epsilon
  34. δ \delta
  35. ϵ \epsilon
  36. V P ( 0 ) V_{P}(0)
  37. V P ( 0 ) = V P ( ϕ ) 1 + δ sin 2 ϕ cos 2 ϕ + ϵ sin 4 ϕ V_{P}(0)=\frac{V_{P}(\phi)}{1+\delta\sin^{2}\phi\cos^{2}\phi+\epsilon\sin^{4}\phi}
  38. V P ( 0 ) V_{P}(0)

Selberg_zeta_function.html

  1. ζ ( s ) = p 1 1 - p - s \zeta(s)=\prod_{p\in\mathbb{P}}\frac{1}{1-p^{-s}}
  2. \mathbb{P}
  3. Γ \Gamma
  4. ζ Γ ( s ) = p ( 1 - N ( p ) - s ) - 1 , \zeta_{\Gamma}(s)=\prod_{p}(1-N(p)^{-s})^{-1},
  5. Z Γ ( s ) = p n = 0 ( 1 - N ( p ) - s - n ) , Z_{\Gamma}(s)=\prod_{p}\prod^{\infty}_{n=0}(1-N(p)^{-s-n}),
  6. s 0 ( 1 - s 0 ) s_{0}(1-s_{0})
  7. s 0 s_{0}
  8. ϕ ( s ) \phi(s)
  9. 1 / 2 - 1/2-\mathbb{N}
  10. - -\mathbb{N}
  11. Γ \ 2 \Gamma\backslash\mathbb{H}^{2}
  12. Γ \Gamma
  13. φ ( s ) = π 1 / 2 Γ ( s - 1 / 2 ) ζ ( 2 s - 1 ) Γ ( s ) ζ ( 2 s ) . \varphi(s)=\pi^{1/2}\frac{\Gamma(s-1/2)\zeta(2s-1)}{\Gamma(s)\zeta(2s)}.
  14. s 0 s_{0}
  15. s 0 / 2 s_{0}/2
  16. s 0 / 2 s_{0}/2

Selective_yellow.html

  1. y 0.138 + 0.580 x y\geq 0.138+0.580x
  2. y 1.290 x - 0.100 y\leq 1.290x-0.100
  3. y 0.966 - x y\geq 0.966-x
  4. y 0.992 - x y\leq 0.992-x
  5. y 0.940 - x y\geq 0.940-x
  6. y 0.440 y\geq 0.440

Self-creation_cosmology.html

  1. T ϕ μ ν T_{\phi\,\mu\nu}
  2. R μ ν - 1 2 g μ ν R = 8 π ϕ [ T M μ ν + T ϕ μ ν ] R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\frac{8\pi}{\phi}\left[T_{M\mu\nu}+T_{\phi\,% \mu\nu}\right]
  3. T M μ ν T_{M\mu\nu}
  4. ϕ 1 G N \phi\approx\frac{1}{G_{N}}
  5. G N G_{N}
  6. ϕ = 4 π λ T M \Box\phi=4\pi\lambda T_{M}
  7. T M T_{M\;}
  8. ( T M σ σ ) (T_{M\;\sigma}^{\;\;\sigma})
  9. λ \lambda
  10. μ T M ν μ 0 \nabla_{\mu}T_{M\nu}^{\;\mu}\neq 0
  11. μ T M ν μ = 4 π f ν ( ϕ ) T M = f ν ( ϕ ) ϕ \nabla_{\mu}T_{M\;\nu}^{\;\mu}=4\pi f_{\nu}\left(\phi\right)T_{M\;}=f_{\nu}% \left(\phi\right)\Box\phi
  12. μ T e m ν μ = 4 π f ν ( ϕ ) T e m = 4 π f ν ( ϕ ) ( 3 p e m - ρ e m ) = 0 \nabla_{\mu}T_{em\ \nu}^{\mu}=4\pi f_{\nu}\left(\phi\right)T_{em}=4\pi f_{\nu}% \left(\phi\right)\left(3p_{em}-\rho_{em}\right)=0
  13. f ν ( ϕ ) = 1 8 π ϕ ν ϕ f_{\nu}\left(\phi\right)=\frac{1}{8\pi\phi}\nabla_{\nu}\phi
  14. μ T M ν . μ = 1 8 π ν ϕ ϕ ϕ = 1 2 ν ϕ ϕ T M \nabla_{\mu}T_{M\;\nu}^{.\;\mu}=\frac{1}{8\pi}\frac{\nabla_{\nu}\phi}{\phi}% \Box\phi=\frac{1}{2}\frac{\nabla_{\nu}\phi}{\phi}T_{M\;}
  15. Φ N ( x μ ) \Phi_{N}\left(x^{\mu}\right)
  16. d 2 r d t 2 = - Φ N ( r ) \frac{d^{2}r}{dt^{2}}=-\nabla\Phi_{N}\left(r\right)
  17. Φ N ( ) = 0 , \Phi_{N}\left(\infty\right)=0,
  18. m p ( x μ ) = m 0 exp [ Φ N ( x μ ) ] , m_{p}(x^{\mu})=m_{0}\exp[\Phi_{N}\left(x^{\mu}\right)],
  19. m p ( r ) m 0 m_{p}\left(r\right)\rightarrow m_{0}
  20. r r\rightarrow\infty
  21. G N G_{N}
  22. G M G_{M}
  23. G N = 2 3 G M . G_{N}=\frac{2}{3}G_{M}.
  24. G N G_{N}
  25. G M G_{M}
  26. p ρ c 2 p\ll\rho c^{2}
  27. 10 17 10^{17}
  28. g 00 \sqrt{g}_{00}
  29. g μ ν g ~ μ ν = Ω 2 g μ ν . g_{\mu\nu}\rightarrow\tilde{g}_{\mu\nu}=\Omega^{2}g_{\mu\nu}.
  30. m ( x μ ) = Ω m ~ 0 , m\left(x^{\mu}\right)=\Omega\tilde{m}_{0},
  31. m ( x μ ) m\left(x^{\mu}\right)
  32. m ~ 0 \tilde{m}_{0}
  33. Ω = exp [ Φ N ( x μ ) ] . \Omega=\exp\left[\Phi_{N}\left(x^{\mu}\right)\right].
  34. ω = - 3 2 \omega=-\frac{3}{2}
  35. L S C C [ g , ϕ ] = - g 16 π ( ϕ R + 3 2 ϕ g μ ν μ ϕ ν ϕ ) + L m a t t e r S C C [ g , ϕ ] , L^{SCC}[g,\phi]=\frac{\sqrt{-g}}{16\pi}\left(\phi R+\frac{3}{2\phi}g^{\mu\nu}% \nabla_{\mu}\phi\nabla_{\nu}\phi\right)+L_{matter}^{SCC}[g,\phi],
  36. G = 1 ϕ G=\frac{1}{\phi}
  37. m m
  38. G m = c o n s t a n t Gm=constant
  39. G m 2 Gm^{2}
  40. G m Gm
  41. G G
  42. ϕ ~ \tilde{\phi}
  43. L S C C [ g ~ , ϕ ~ ] = - g ~ 16 π G N R ~ + L ~ m a t t e r S C C [ g ~ ] + 3 - g ~ 8 π G N ~ Φ ~ N ( x ~ μ ) , L^{SCC}[\tilde{g},\tilde{\phi}]=\frac{\sqrt{-\tilde{g}}}{16\pi G_{N}}\tilde{R}% +\tilde{L}_{matter}^{SCC}[\tilde{g}]+\frac{3\sqrt{-\tilde{g}}}{8\pi G_{N}}% \tilde{\square}\tilde{\Phi}_{N}\left(\tilde{x}^{\mu}\right),
  44. ~ Φ ~ N ( x ~ μ ) = 0 \tilde{\square}\tilde{\Phi}_{N}\left(\tilde{x}^{\mu}\right)=0
  45. L S C C [ g ~ , ϕ ~ ] = - g ~ 16 π G N R ~ + L ~ m a t t e r S C C [ g ~ ] . L^{SCC}[\tilde{g},\tilde{\phi}]=\frac{\sqrt{-\tilde{g}}}{16\pi G_{N}}\tilde{R}% +\tilde{L}_{matter}^{SCC}[\tilde{g}].
  46. ϕ \phi
  47. ϕ \phi
  48. ϕ = 4 π T M , \Box\phi=4\pi T_{M},
  49. R μ ν - 1 2 g μ ν R = 8 π ϕ T M μ ν - 3 2 ϕ 2 ( μ ϕ ν ϕ - 1 2 g μ ν g α β α ϕ β ϕ ) R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\frac{8\pi}{\phi}T_{M\mu\nu}-\frac{3}{2\phi^% {2}}\left(\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{1}{2}g_{\mu\nu}g^{\alpha\beta% }\nabla_{\alpha}\phi\nabla_{\beta}\phi\right)
  50. + 1 ϕ ( μ ν ϕ - g μ ν ϕ ) , +\frac{1}{\phi}\left(\nabla_{\mu}\nabla_{\nu}\phi-g_{\mu\nu}\Box\phi\right),
  51. μ T M ν μ = 1 8 π ν ϕ ϕ ϕ , \nabla_{\mu}T_{M\;\nu}^{\;\mu}=\frac{1}{8\pi}\frac{\nabla_{\nu}\phi}{\phi}\Box\phi,
  52. ( 1 + γ 2 ) G M , (\frac{1+\gamma}{2})G_{M},
  53. G M G_{M}
  54. G N G_{N}
  55. γ = 1 \gamma=1
  56. G M = G N ; G_{M}=G_{N};
  57. γ = 1 3 \gamma=\frac{1}{3}
  58. G M = 3 2 G N G_{M}=\frac{3}{2}G_{N}
  59. γ \gamma
  60. γ \gamma
  61. ( 1 + 2 γ 3 ) G M , (\frac{1+2\gamma}{3})G_{M},
  62. δ = 1 4 g S u n ( l c ) 2 2 × 10 - 12 \delta=\frac{1}{4}g_{Sun}\left(\frac{l}{c}\right)^{2}\approx 2\times 10^{-12}
  63. g S u n g_{Sun}
  64. δ l 10 - 18 \delta_{l}\approx 10^{-18}
  65. R ( t ) = R 0 t t 0 . R(t)=R_{0}\frac{t}{t_{0}}.
  66. Ω m = 0.22 , \Omega_{m}=0.22,\,
  67. Ω Λ = 0.11 , \Omega_{\Lambda}=0.11,\,
  68. Ω b 0.2 , \Omega_{b}\approx 0.2,

Self-focusing_transducers.html

  1. D = d 2 4 λ D=\frac{d^{2}}{4\lambda}
  2. d = d=
  3. λ = \lambda=
  4. d 2 \frac{d}{2}

Semantic_holism.html

  1. p q p ( B ( x , p ) B ( x , q ) ) \forall p\exists q\neq p\Box(B(x,p)\rightarrow B(x,q))
  2. ( B ( x , p ) q p ( B ( x , q ) ) \Box(B(x,p)\rightarrow\exists q\neq p(B(x,q))
  3. ( B ( x , p ) B ( y , p ) q p ( B ( x , q ) B ( y , q ) ) \Box(B(x,p)\land B(y,p)\rightarrow\exists q\neq p(B(x,q)\land B(y,q))
  4. ( a ¬ b ) c (a\lor\lnot b)\rightarrow c
  5. a ¬ b a\lor\lnot b
  6. P Q P\lor Q
  7. ¬ Q \lnot Q
  8. L ( α , β , γ ) P ( α , β ) I ( β , γ ) L(\alpha,\beta,\gamma)\leftrightarrow P(\alpha,\beta)\land I(\beta,\gamma)
  9. R ( α , β , γ ) P ( α , γ ) I ( β , γ ) R(\alpha,\beta,\gamma)\leftrightarrow P(\alpha,\gamma)\land I(\beta,\gamma)
  10. G ( α ) = ( [ ( β , γ ) : L ( α , β , γ ) ] , [ ( β , γ ) : R ( α , β , γ ) ] ) G(\alpha)=([(\beta,\gamma):L(\alpha,\beta,\gamma)],[(\beta,\gamma):R(\alpha,% \beta,\gamma)])\;
  11. α \alpha
  12. β \beta
  13. γ \gamma
  14. α \alpha
  15. γ \gamma
  16. ( β , γ ) (\beta,\gamma)
  17. G ( β ) = [ G ( α 1 ) G ( α n ) ] G(\beta)=[G(\alpha_{1})...G(\alpha_{n})]\;
  18. h ( G ( β ) ) = h ( [ G ( α 1 ) G ( α n ) ] ) = F ( h ( G ( α 1 ) ) h ( G ( α n ) ) ) . h(G(\beta))=h([G(\alpha_{1})...G(\alpha_{n})])=F(h(G(\alpha_{1}))...h(G(\alpha% _{n}))).\;
  19. H 2 O H_{2}O

Semi-log_plot.html

  1. y = λ a γ x y=\lambda a^{\gamma x}
  2. log a y = γ x + log a λ . \log_{a}y=\gamma x+\log_{a}\lambda.
  3. γ \gamma
  4. log a λ \log_{a}\lambda
  5. log ( y ) = ( γ log ( a ) ) x + log ( λ ) . \log(y)=(\gamma\log(a))x+\log(\lambda).
  6. log 10 ( F ( x ) ) = m x + b \log_{10}(F(x))=mx+b
  7. F ( x ) = 10 m x + b = ( 10 m x ) ( 10 b ) . F(x)=10^{mx+b}=(10^{mx})(10^{b}).
  8. F ( x ) = m log 10 ( x ) + b . F(x)=m\log_{10}(x)+b.\,

Semicomma.html

  1. 3 3 5 7 : 2 21 3^{3}\cdot 5^{7}:2^{21}

Semidefinite_programming.html

  1. min x 1 , , x n n i , j [ n ] c i , j ( x i x j ) subject to i , j [ n ] a i , j , k ( x i x j ) b k k . \begin{array}[]{rl}{\displaystyle\min_{x^{1},\ldots,x^{n}\in\mathbb{R}^{n}}}&{% \displaystyle\sum_{i,j\in[n]}c_{i,j}(x^{i}\cdot x^{j})}\\ \,\text{subject to}&{\displaystyle\sum_{i,j\in[n]}a_{i,j,k}(x^{i}\cdot x^{j})% \leq b_{k}\qquad\forall k}.\\ \end{array}
  2. n × n n\times n
  3. M M
  4. x 1 , , x n x^{1},\ldots,x^{n}
  5. m i , j = x i x j m_{i,j}=x^{i}\cdot x^{j}
  6. i , j i,j
  7. M 0 M\succeq 0
  8. 𝕊 n \mathbb{S}^{n}
  9. n × n n\times n
  10. tr {\rm tr}
  11. A , B 𝕊 n = tr ( A T B ) = i = 1 , j = 1 n A i j B i j . \langle A,B\rangle_{\mathbb{S}^{n}}={\rm tr}(A^{T}B)=\sum_{i=1,j=1}^{n}A_{ij}B% _{ij}.
  12. min X 𝕊 n C , X 𝕊 n subject to A k , X 𝕊 n b k , k = 1 , , m X 0 \begin{array}[]{rl}{\displaystyle\min_{X\in\mathbb{S}^{n}}}&\langle C,X\rangle% _{\mathbb{S}^{n}}\\ \,\text{subject to}&\langle A_{k},X\rangle_{\mathbb{S}^{n}}\leq b_{k},\quad k=% 1,\ldots,m\\ &X\succeq 0\end{array}
  13. i , j i,j
  14. C C
  15. c i , j c_{i,j}
  16. A k A_{k}
  17. n × n n\times n
  18. i , j i,j
  19. a i , j , k a_{i,j,k}
  20. min X 𝕊 n C , X 𝕊 n subject to A i , X 𝕊 n = b i , i = 1 , , m X 0. \begin{array}[]{rl}{\displaystyle\min_{X\in\mathbb{S}^{n}}}&\langle C,X\rangle% _{\mathbb{S}^{n}}\\ \,\text{subject to}&\langle A_{i},X\rangle_{\mathbb{S}^{n}}=b_{i},\quad i=1,% \ldots,m\\ &X\succeq 0.\end{array}
  21. X X
  22. X i i X_{ii}
  23. i i
  24. X i i 0 X_{ii}\geq 0
  25. X i j = 0 X_{ij}=0
  26. j i j\neq i
  27. X X
  28. { v i } \{v_{i}\}
  29. i i
  30. j j
  31. X X
  32. X i j = ( v i , v j ) X_{ij}=(v_{i},v_{j})
  33. v i v_{i}
  34. v j v_{j}
  35. { v i } \{v_{i}\}
  36. O ( n 3 ) O(n^{3})
  37. min X 𝕊 n C , X 𝕊 n subject to A i , X 𝕊 n = b i , i = 1 , , m X 0 \begin{array}[]{rl}{\displaystyle\min_{X\in\mathbb{S}^{n}}}&\langle C,X\rangle% _{\mathbb{S}^{n}}\\ \,\text{subject to}&\langle A_{i},X\rangle_{\mathbb{S}^{n}}=b_{i},\quad i=1,% \ldots,m\\ &X\succeq 0\end{array}
  38. max y m b , y m subject to i = 1 m y i A i C \begin{array}[]{rl}{\displaystyle\max_{y\in\mathbb{R}^{m}}}&\langle b,y\rangle% _{\mathbb{R}^{m}}\\ \,\text{subject to}&{\displaystyle\sum_{i=1}^{m}}y_{i}A_{i}\preceq C\end{array}
  39. P P
  40. Q Q
  41. P Q P\succeq Q
  42. P - Q 0 P-Q\succeq 0
  43. C , X - b , y = C , X - i = 1 m y i b i = C , X - i = 1 m y i A i , X = C - i = 1 m y i A i , X 0 , \langle C,X\rangle-\langle b,y\rangle=\langle C,X\rangle-\sum_{i=1}^{m}y_{i}b_% {i}=\langle C,X\rangle-\sum_{i=1}^{m}y_{i}\langle A_{i},X\rangle=\langle C-% \sum_{i=1}^{m}y_{i}A_{i},X\rangle\geq 0,
  44. X 0 𝕊 n , X 0 0 X_{0}\in\mathbb{S}^{n},X_{0}\succ 0
  45. A i , X 0 𝕊 n = b i \langle A_{i},X_{0}\rangle_{\mathbb{S}^{n}}=b_{i}
  46. i = 1 , , m i=1,\ldots,m
  47. y * y^{*}
  48. C , X * 𝕊 n = b , y * \R m . \langle C,X^{*}\rangle_{\mathbb{S}^{n}}=\langle b,y^{*}\rangle_{\R^{m}}.
  49. i = 1 m ( y 0 ) i A i C \sum_{i=1}^{m}(y_{0})_{i}A_{i}\prec C
  50. y 0 \R m y_{0}\in\R^{m}
  51. X * X^{*}
  52. A A
  53. B B
  54. C C
  55. ρ A B , ρ A C , ρ B C \rho_{AB},\ \rho_{AC},\rho_{BC}
  56. ( 1 ρ A B ρ A C ρ A B 1 ρ B C ρ A C ρ B C 1 ) 0 \begin{pmatrix}1&\rho_{AB}&\rho_{AC}\\ \rho_{AB}&1&\rho_{BC}\\ \rho_{AC}&\rho_{BC}&1\end{pmatrix}\succeq 0
  57. - 0.2 ρ A B - 0.1 -0.2\leq\rho_{AB}\leq-0.1
  58. 0.4 ρ B C 0.5 0.4\leq\rho_{BC}\leq 0.5
  59. ρ A C \rho_{AC}
  60. x 13 x_{13}
  61. - 0.2 x 12 - 0.1 -0.2\leq x_{12}\leq-0.1
  62. 0.4 x 23 0.5 0.4\leq x_{23}\leq 0.5
  63. x 11 = x 22 = x 33 = 1 x_{11}=x_{22}=x_{33}=1
  64. ( 1 x 12 x 13 x 12 1 x 23 x 13 x 23 1 ) 0 \begin{pmatrix}1&x_{12}&x_{13}\\ x_{12}&1&x_{23}\\ x_{13}&x_{23}&1\end{pmatrix}\succeq 0
  65. ρ A B = x 12 , ρ A C = x 13 , ρ B C = x 23 \rho_{AB}=x_{12},\ \rho_{AC}=x_{13},\ \rho_{BC}=x_{23}
  66. tr ( ( 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) ( 1 x 12 x 13 0 0 0 x 12 1 x 23 0 0 0 x 13 x 23 1 0 0 0 0 0 0 s 1 0 0 0 0 0 0 s 2 0 0 0 0 0 0 s 3 ) ) = x 12 + s 1 = - 0.1 \mathrm{tr}\left(\left(\begin{array}[]{cccccc}0&1&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&1&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\end{array}\right)\cdot\left(\begin{array}[]{cccccc}1&x_{12}&x_{13}% &0&0&0\\ x_{12}&1&x_{23}&0&0&0\\ x_{13}&x_{23}&1&0&0&0\\ 0&0&0&s_{1}&0&0\\ 0&0&0&0&s_{2}&0\\ 0&0&0&0&0&s_{3}\end{array}\right)\right)=x_{12}+s_{1}=-0.1
  67. ρ A C = x 13 \rho_{AC}=x_{13}
  68. - 0.978 -0.978
  69. 0.872 0.872
  70. ( c T x ) 2 d T x \frac{(c^{T}x)^{2}}{d^{T}x}
  71. A x + b 0 Ax+b\geq 0
  72. d T x > 0 d^{T}x>0
  73. A x + b 0 Ax+b\geq 0
  74. t t
  75. t t
  76. A x + b 0 , ( c T x ) 2 d T x t Ax+b\geq 0,\,\frac{(c^{T}x)^{2}}{d^{T}x}\leq t
  77. x , t x,t
  78. 𝐝𝐢𝐚𝐠 ( A x + b ) 0 \,\textbf{diag}(Ax+b)\geq 0
  79. 𝐝𝐢𝐚𝐠 ( A x + b ) \,\textbf{diag}(Ax+b)
  80. A x + b Ax+b
  81. t d T x - ( c T x ) 2 0 td^{T}x-(c^{T}x)^{2}\geq 0
  82. D D
  83. D = [ t c T x c T x d T x ] D=\left[\begin{array}[]{cc}t&c^{T}x\\ c^{T}x&d^{T}x\end{array}\right]
  84. D 0 D\succeq 0
  85. t t
  86. [ 𝐝𝐢𝐚𝐠 ( A x + b ) 0 0 0 t c T x 0 c T x d T x ] 0 \left[\begin{array}[]{ccc}\,\textbf{diag}(Ax+b)&0&0\\ 0&t&c^{T}x\\ 0&c^{T}x&d^{T}x\end{array}\right]\succeq 0
  87. ( i , j ) E 1 - v i v j 2 , \sum_{(i,j)\in E}\frac{1-v_{i}v_{j}}{2},
  88. v i { 1 , - 1 } v_{i}\in\{1,-1\}
  89. ϵ \epsilon
  90. max ( i , j ) E 1 - v i , v j 2 , \max\sum_{(i,j)\in E}\frac{1-\langle v_{i},v_{j}\rangle}{2},
  91. v i 2 = 1 \lVert v_{i}\rVert^{2}=1
  92. { v i } \{v_{i}\}
  93. 𝐑 𝐧 \mathbf{R^{n}}
  94. cos - 1 v i , v j \cos^{-1}\langle v_{i},v_{j}\rangle
  95. π \pi
  96. ( 1 - v i , v j ) / 2 (1-\langle v_{i},v_{j}\rangle)/{2}
  97. ϵ \epsilon
  98. log ( 1 / ϵ ) \log(1/\epsilon)

Seminormal_subgroup.html

  1. A A
  2. G G
  3. B B
  4. A B = G AB=G
  5. C C
  6. B B
  7. A C AC
  8. G G

Semiotic_information_theory.html

  1. log 5 \log 5
  2. log 3. \log 3.
  3. log 5 \log 5
  4. log 2. \log 2.
  5. log 5 \log 5
  6. log 3 \log 3
  7. ( log 5 - log 3 ) . (\log 5-\log 3).
  8. log 5 \log 5
  9. log 2 \log 2
  10. ( log 5 - log 2 ) . (\log 5-\log 2).
  11. ( log 5 - log 3 ) (\log 5-\log 3)
  12. ( log 5 - log 2 ) (\log 5-\log 2)
  13. 1 2 + 3 ( 3 ( log 5 - log 3 ) + 2 ( log 5 - log 2 ) ) {1\over{2+3}}(3(\log 5-\log 3)+2(\log 5-\log 2))
  14. = 1 n ( k 1 ( log n - log k 1 ) + k 2 ( log n - log k 2 ) ) =\quad{1\over n}(k_{1}(\log n-\log k_{1})+k_{2}(\log n-\log k_{2}))
  15. = k 1 n ( log n - log k 1 ) + k 2 n ( log n - log k 2 ) =\quad{k_{1}\over n}(\log n-\log k_{1})+{k_{2}\over n}(\log n-\log k_{2})
  16. = - k 1 n ( log k 1 - log n ) - k 2 n ( log k 2 - log n ) =\quad-{k_{1}\over n}(\log k_{1}-\log n)-{k_{2}\over n}(\log k_{2}-\log n)
  17. = - k 1 n ( log k 1 n ) - k 2 n ( log k 2 n ) =\quad-{k_{1}\over n}(\log{k_{1}\over n})-{k_{2}\over n}(\log{k_{2}\over n})
  18. = - ( p 1 log p 1 + p 2 log p 2 ) =\quad-(p_{1}\log p_{1}+p_{2}\log p_{2})
  19. = - ( 0.6 log 0.6 + 0.4 log 0.4 ) =\quad-(0.6\log 0.6+0.4\log 0.4)
  20. = 0.971 =\quad 0.971

Semiparametric_model.html

  1. { P θ : θ Θ } \{P_{\theta}:\theta\in\Theta\}
  2. θ \theta
  3. k k
  4. k k
  5. θ \theta
  6. k \mathbb{R}^{k}
  7. Θ k \Theta\subset\mathbb{R}^{k}
  8. θ \theta
  9. θ \theta
  10. Θ 𝔽 \Theta\subset\mathbb{F}
  11. 𝔽 \mathbb{F}
  12. Θ \Theta
  13. Θ k × 𝔽 \Theta\subset\mathbb{R}^{k}\times\mathbb{F}
  14. 𝔽 \mathbb{F}
  15. θ \theta
  16. T T
  17. T T
  18. F ( t ) = 1 - exp ( - 0 t λ 0 ( u ) e β x d u ) , F(t)=1-\exp\left(-\int_{0}^{t}\lambda_{0}(u)e^{\beta^{\prime}x}du\right),
  19. x x
  20. β \beta
  21. λ 0 ( u ) \lambda_{0}(u)
  22. θ = ( β , λ 0 ( u ) ) \theta=(\beta,\lambda_{0}(u))
  23. β \beta
  24. λ 0 ( u ) \lambda_{0}(u)
  25. λ 0 ( u ) \lambda_{0}(u)

Semiparametric_regression.html

  1. Y i = X i β + g ( Z i ) + u i , i = 1 , , n , Y_{i}=X^{\prime}_{i}\beta+g\left(Z_{i}\right)+u_{i},\,\quad i=1,\ldots,n,\,
  2. Y i Y_{i}
  3. X i X_{i}
  4. Z i Z_{i}
  5. p × 1 p\times 1
  6. β \beta
  7. p × 1 p\times 1
  8. Z i R q Z_{i}\in\operatorname{R}^{q}
  9. β \beta
  10. g ( Z i ) g\left(Z_{i}\right)
  11. E ( u i | X i , Z i ) = 0 E\left(u_{i}|X_{i},Z_{i}\right)=0
  12. E ( u i 2 | x , z ) = σ 2 ( x , z ) E\left(u^{2}_{i}|x,z\right)=\sigma^{2}\left(x,z\right)
  13. n \sqrt{n}
  14. β \beta
  15. g ( Z i ) g\left(Z_{i}\right)
  16. Y i - X i β ^ Y_{i}-X^{\prime}_{i}\hat{\beta}
  17. z z
  18. Y = g ( X β 0 ) + u , Y=g\left(X^{\prime}\beta_{0}\right)+u,\,
  19. Y Y
  20. X X
  21. β 0 \beta_{0}
  22. u u
  23. E ( u | X ) = 0 E\left(u|X\right)=0
  24. x β x^{\prime}\beta
  25. g ( ) g\left(\cdot\right)
  26. y y
  27. g ( ) g\left(\cdot\right)
  28. β 0 \beta_{0}
  29. i = 1 ( Y i - g ( X i β ) ) 2 . \sum_{i=1}\left(Y_{i}-g\left(X^{\prime}_{i}\beta\right)\right)^{2}.
  30. g ( ) g\left(\cdot\right)
  31. β \beta
  32. G ( X i β ) = E ( Y i | X i β ) = E [ g ( X i β o ) | X i β ] G\left(X^{\prime}_{i}\beta\right)=E\left(Y_{i}|X^{\prime}_{i}\beta\right)=E% \left[g\left(X^{\prime}_{i}\beta_{o}\right)|X^{\prime}_{i}\beta\right]
  33. g ( X i β ) g\left(X^{\prime}_{i}\beta\right)
  34. G ^ - i ( X i β ) , \hat{G}_{-i}\left(X^{\prime}_{i}\beta\right),\,
  35. G ( X i β ) G\left(X^{\prime}_{i}\beta\right)
  36. y y
  37. X i X_{i}
  38. u i u_{i}
  39. β \beta
  40. L ( β ) = i ( 1 - Y i ) ln ( 1 - g ^ - i ( X i β ) ) + i Y i ln ( g ^ - i ( X i β ) ) , L\left(\beta\right)=\sum_{i}\left(1-Y_{i}\right)\ln\left(1-\hat{g}_{-i}\left(X% ^{\prime}_{i}\beta\right)\right)+\sum_{i}Y_{i}\ln\left(\hat{g}_{-i}\left(X^{% \prime}_{i}\beta\right)\right),
  41. g ^ - i ( X i β ) \hat{g}_{-i}\left(X^{\prime}_{i}\beta\right)
  42. Y i = α ( Z i ) + X i β ( Z i ) + u i = ( 1 + X i ) ( α ( Z i ) β ( Z i ) ) + u i = W i γ ( Z i ) + u i , Y_{i}=\alpha\left(Z_{i}\right)+X^{\prime}_{i}\beta\left(Z_{i}\right)+u_{i}=% \left(1+X^{\prime}_{i}\right)\left(\begin{array}[]{c}\alpha\left(Z_{i}\right)% \\ \beta\left(Z_{i}\right)\end{array}\right)+u_{i}=W^{\prime}_{i}\gamma\left(Z_{i% }\right)+u_{i},
  43. X i X_{i}
  44. k × 1 k\times 1
  45. β ( z ) \beta\left(z\right)
  46. z z
  47. γ ( ) \gamma\left(\cdot\right)
  48. γ ( Z i ) = ( E [ W i W i | Z i ] ) - 1 E [ W i Y i | Z i ] . \gamma\left(Z_{i}\right)=\left(E\left[W_{i}W^{\prime}_{i}|Z_{i}\right]\right)^% {-1}E\left[W_{i}Y_{i}|Z_{i}\right].

Semisimple_algebra.html

  1. { 0 } = J 0 J n A \{0\}=J_{0}\subset\cdots\subset J_{n}\subset A
  2. A J 1 × J 2 / J 1 × J 3 / J 2 × × J n / J n - 1 × A / J n A\simeq J_{1}\times J_{2}/J_{1}\times J_{3}/J_{2}\times...\times J_{n}/J_{n-1}% \times A/J_{n}
  3. J i + 1 / J i J_{i+1}/J_{i}\,
  4. J 2 J 1 × J 2 / J 1 . J_{2}\simeq J_{1}\times J_{2}/J_{1}.
  5. J 2 / J 1 J_{2}/J_{1}\,
  6. J 3 J 2 × J 3 / J 2 J 1 × J 2 / J 1 × J 3 / J 2 . J_{3}\simeq J_{2}\times J_{3}/J_{2}\simeq J_{1}\times J_{2}/J_{1}\times J_{3}/% J_{2}.
  7. k k
  8. M n i ( D i ) \prod M_{n_{i}}(D_{i})
  9. n i n_{i}
  10. D i D_{i}
  11. k k
  12. M n i ( D i ) M_{n_{i}}(D_{i})
  13. n i × n i n_{i}\times n_{i}
  14. D i D_{i}

Separation_logic.html

  1. s s
  2. h h
  3. h h
  4. h h^{\prime}
  5. h h h\,\bot\,h^{\prime}
  6. \ell
  7. h ( ) h(\ell)
  8. h ( ) h^{\prime}(\ell)
  9. s , h P s,h\models P
  10. s s
  11. h h
  12. P P
  13. P P
  14. Q Q
  15. R R
  16. 𝐞𝐦𝐩 \mathbf{e}\mathbf{m}\mathbf{p}
  17. e e e\mapsto e^{\prime}
  18. P Q P\ast Q
  19. P - Q P{-\!\!\ast}\,Q
  20. e e
  21. e e^{\prime}
  22. 𝐞𝐦𝐩 \mathbf{e}\mathbf{m}\mathbf{p}
  23. s , h 𝐞𝐦𝐩 s,h\models\mathbf{e}\mathbf{m}\mathbf{p}
  24. h h
  25. \mapsto
  26. s , h e e s,h\models e\mapsto e^{\prime}
  27. h ( [ [ e ] ] s ) = [ [ e ] ] s h([\![e]\!]_{s})=[\![e^{\prime}]\!]_{s}
  28. [ [ e ] ] s [\![e]\!]_{s}
  29. e e
  30. s s
  31. h h
  32. \ast
  33. s , h P Q s,h\models P\ast Q
  34. h 1 , h 2 h_{1},h_{2}
  35. h 1 h 2 h_{1}\,\bot\,h_{2}
  36. h = h 1 h 2 h=h_{1}\cup h_{2}
  37. s , h 1 P s,h_{1}\models P
  38. s , h 2 Q s,h_{2}\models Q
  39. - -\!\!\ast
  40. s , h P - Q s,h\models P-\!\!\ast\,Q
  41. h h h^{\prime}\,\bot\,h
  42. s , h P s,h^{\prime}\models P
  43. s , h h Q s,h\cup h^{\prime}\models Q
  44. \ast
  45. - -\!\!\ast
  46. s , h P ( P - Q ) s , h Q \frac{s,h\models P\ast(P-\!\!\ast\,Q)}{s,h\models Q}
  47. s , h h P Q R s,h\cup h^{\prime}\models P\ast Q\Rightarrow R
  48. s , h P Q - R s,h\models P\Rightarrow Q-\!\!\ast\,R
  49. h h h\,\bot\,h^{\prime}
  50. _ Q \_\ast Q
  51. Q - _ Q-\!\!\ast\,\_
  52. { P } C { Q } \{P\}\ C\ \{Q\}
  53. C C
  54. P P
  55. Q Q
  56. C C
  57. C C
  58. { P } C { Q } { P R } C { Q R } 𝗆𝗈𝖽 ( C ) 𝖿𝗏 ( R ) = \frac{\{P\}\ C\ \{Q\}}{\{P\ast R\}\ C\ \{Q\ast R\}}~{}\mathsf{mod}(C)\cap% \mathsf{fv}(R)=\emptyset
  59. P P
  60. P R P\ast R
  61. R R
  62. C C
  63. R R
  64. 𝖿𝗏 \mathsf{fv}
  65. R R

Septimal_meantone_temperament.html

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

Septimal_semicomma.html

  1. 2 * 3 2 * 5 - 3 * 7 2*3^{2}*5^{-3}*7
  2. ( 6 / 5 ) 3 * ( 7 / 6 ) * ( 2 / 1 ) - 1 (6/5)^{3}*(7/6)*(2/1)^{-1}

Sequential_space.html

  1. A X A\subset X
  2. X X
  3. [ A ] seq [A]_{\,\text{seq}}
  4. [ A ] seq = { x X : { a n } x , a n A } [A]_{\,\text{seq}}=\{x\in X:\exists\{a_{n}\}\to x,a_{n}\in A\}
  5. x X x\in X
  6. A A
  7. x x
  8. [ ] seq : A [ A ] seq [\;]_{\,\text{seq}}:A\mapsto[A]_{\,\text{seq}}
  9. [ ] seq = . [\varnothing]_{\,\text{seq}}=\varnothing.
  10. A [ A ] seq A ¯ A\subset[A]_{\,\text{seq}}\subset\overline{A}
  11. A X A\subset X
  12. A ¯ \overline{A}
  13. A A
  14. [ A B ] seq = [ A ] seq [ B ] seq [A\cup B]_{\,\text{seq}}=[A]_{\,\text{seq}}\cup[B]_{\,\text{seq}}
  15. A , B X A,B\subset X
  16. [ A ] seq [ [ A ] seq ] seq , [A]_{\,\text{seq}}\subsetneq\Big[[A]_{\,\text{seq}}\Big]_{\,\text{seq}},
  17. A X A\subset X
  18. X X
  19. A 0 A_{0}
  20. A A
  21. A α + 1 A_{\alpha+1}
  22. [ A α ] seq [A_{\alpha}]_{\,\text{seq}}
  23. α \alpha
  24. A α A_{\alpha}
  25. β < α A β \bigcup_{\beta<\alpha}A_{\beta}
  26. α \alpha
  27. A α = A α + 1 A_{\alpha}=A_{\alpha+1}
  28. α \alpha
  29. A α A_{\alpha}
  30. A A
  31. α ω 1 \alpha\leq\omega_{1}
  32. ω 1 \omega_{1}
  33. α \alpha
  34. A α = A ¯ A_{\alpha}=\overline{A}
  35. A X A\subseteq X
  36. [ A ] seq = A ¯ [A]\text{seq}=\overline{A}\,
  37. A X A\subset X

Set-theoretic_topology.html

  1. 0 \aleph_{0}
  2. 2 0 2^{\aleph_{0}}
  3. 2 0 2^{\aleph_{0}}
  4. ω 0 \aleph_{\omega}^{\aleph_{0}}
  5. ω + 1 \aleph_{\omega+1}
  6. + 0 \;\;+\;\aleph_{0}
  7. 0 \aleph_{0}
  8. π \pi
  9. π \pi
  10. χ ( X ) = sup { χ ( x , X ) : x X } . \chi(X)=\sup\;\{\chi(x,X):x\in X\}.
  11. χ ( X ) = 0 \chi(X)=\aleph_{0}
  12. d ( X ) = 0 \rm{d}(X)=\aleph_{0}
  13. L ( X ) = 0 \rm{L}(X)=\aleph_{0}
  14. c ( X ) = sup { | 𝒰 | : 𝒰 {\rm c}(X)=\sup\{|{\mathcal{U}}|:{\mathcal{U}}
  15. X } X\}
  16. s ( X ) = hc ( X ) = sup { c ( Y ) : Y X } s(X)={\rm hc}(X)=\sup\{{\rm c}(Y):Y\subseteq X\}
  17. s ( X ) = sup { | Y | : Y X s(X)=\sup\{|Y|:Y\subseteq X
  18. } \}
  19. x X x\in X
  20. α \alpha
  21. x cl X ( Y ) x\in{\rm cl}_{X}(Y)
  22. α \alpha
  23. x cl X ( Z ) x\in{\rm cl}_{X}(Z)
  24. t ( x , X ) = sup { min { | Z | : Z Y x cl X ( Z ) } : Y X x cl X ( Y ) } . t(x,X)=\sup\big\{\min\{|Z|:Z\subseteq Y\ \wedge\ x\in{\rm cl}_{X}(Z)\}:Y% \subseteq X\ \wedge\ x\in{\rm cl}_{X}(Y)\big\}.
  25. t ( X ) = sup { t ( x , X ) : x X } t(X)=\sup\{t(x,X):x\in X\}
  26. 0 \aleph_{0}
  27. t + ( X ) t^{+}(X)
  28. α \alpha
  29. Y X Y\subseteq X
  30. x cl X ( Y ) x\in{\rm cl}_{X}(Y)
  31. α \alpha
  32. x cl X ( Z ) x\in{\rm cl}_{X}(Z)
  33. 2 0 2^{\aleph_{0}}
  34. 2 0 2^{\aleph_{0}}
  35. V * = V × { 0 , 1 } , V^{*}=V\times\{0,1\},\,
  36. x V x\in V
  37. ( x , 0 ) (x,0)
  38. ( x , 1 ) (x,1)
  39. ω + 1 \aleph_{\omega+1}

Set_theory_of_the_real_line.html

  1. n o n ( 𝒩 ) non(\mathcal{N})
  2. 1 \aleph_{1}
  3. n o n ( 𝒩 ) non(\mathcal{N})
  4. 1 \aleph_{1}
  5. 𝔠 \mathfrak{c}
  6. 𝔠 > 1 \mathfrak{c}>\aleph_{1}
  7. κ \kappa
  8. 1 κ 𝔠 \aleph_{1}\leq\kappa\leq\mathfrak{c}
  9. n o n ( 𝒩 ) κ non(\mathcal{N})\geq\kappa
  10. κ \kappa
  11. κ \kappa
  12. \mathbb{R}
  13. 𝔡 < 𝔞 , \mathfrak{d}<\mathfrak{a},

SETAR_(model).html

  1. y t = γ 0 + γ 1 y t - 1 + γ 2 y t - 2 + + γ p y t - p + ϵ t . y_{t}=\gamma_{0}+\gamma_{1}y_{t-1}+\gamma_{2}y_{t-2}+...+\gamma_{p}y_{t-p}+% \epsilon_{t}.\,
  2. γ i \gamma_{i}\,
  3. ϵ t i i d W N ( 0 ; σ 2 ) \epsilon_{t}\sim^{iid}WN(0;\sigma^{2})\,
  4. y t = 𝐗 𝐭 γ + σ ϵ t . y_{t}=\mathbf{X_{t}\gamma}+\sigma\epsilon_{t}.\,
  5. 𝐗 𝐭 = ( 1 , y t - 1 , y t - 2 , , y t - p ) \mathbf{X_{t}}=(1,y_{t-1},y_{t-2},\ldots,y_{t-p})\,
  6. γ \gamma\,
  7. γ 0 , γ 1 , γ 2 , , γ p \gamma_{0},\gamma_{1},\gamma_{2},...,\gamma_{p}\,
  8. ϵ t i i d W N ( 0 ; 1 ) \epsilon_{t}\sim^{iid}WN(0;1)\,
  9. y t = 𝐗 𝐭 γ ( j ) + σ ( j ) ϵ t y_{t}=\mathbf{X_{t}}\gamma^{(j)}+\sigma^{(j)}\epsilon_{t}\,
  10. r j - 1 < z t < r j . r_{j-1}<z_{t}<r_{j}.\,
  11. X t = ( 1 , y t - 1 , y t - 2 , , y t - p ) X_{t}=(1,y_{t-1},y_{t-2},...,y_{t-p})\,
  12. - = r 0 < r 1 < < r k = + -\infty=r_{0}<r_{1}<\ldots<r_{k}=+\infty\,
  13. γ ( j ) \gamma^{(j)}\,
  14. γ ( j ) \gamma^{(j)}\,

Settling_time.html

  1. ζ 1 \zeta\ll 1
  2. T s = - ln ( tolerance fraction ) damping ratio × natural freq T_{s}=-\frac{\ln(\,\text{tolerance fraction})}{\,\text{damping ratio}\times\,% \text{natural freq}}
  3. T s = - ln ( 0.02 ) ζ ω n 3.9 ζ ω n T_{s}=-\frac{\ln(0.02)}{\zeta\omega_{n}}\approx\frac{3.9}{\zeta\omega_{n}}

Sfermion.html

  1. 1 / 2 {1}/{2}
  2. 1 / 2 {1}/{2}
  3. u ~ \tilde{u}
  4. u u
  5. d ~ \tilde{d}
  6. d d
  7. c ~ \tilde{c}
  8. c c
  9. s ~ \tilde{s}
  10. s s
  11. t ~ \tilde{t}
  12. t t
  13. b ~ \tilde{b}
  14. b b
  15. e ~ \tilde{e}
  16. e e
  17. ν ~ e \tilde{\nu}_{e}
  18. ν e \nu_{e}
  19. μ ~ \tilde{\mu}
  20. μ \mu
  21. ν ~ μ \tilde{\nu}_{\mu}
  22. ν μ \nu_{\mu}
  23. τ ~ \tilde{\tau}
  24. τ \tau
  25. ν ~ τ \tilde{\nu}_{\tau}
  26. ν τ \nu_{\tau}

Shadow_zone.html

  1. k k
  2. p p
  3. u u
  4. ( k + 4 3 u ) / p \sqrt{(k+\tfrac{4}{3}u)/p}
  5. ( u / p ) \sqrt{(u/p)}

Shanks'_square_forms_factorization.html

  1. x x
  2. y y
  3. x 2 - y 2 = N x^{2}-y^{2}=N
  4. N N
  5. x x
  6. y y
  7. x 2 y 2 ( mod N ) x^{2}\equiv y^{2}\;\;(\mathop{{\rm mod}}N)
  8. ( x , y ) (x,y)
  9. N N
  10. N N
  11. x 2 - y 2 = ( x - y ) ( x + y ) x^{2}-y^{2}=(x-y)(x+y)
  12. N N
  13. N N
  14. x - y x-y
  15. N N
  16. ( x , y ) (x,y)
  17. x 2 y 2 ( mod N ) x^{2}\equiv y^{2}\;\;(\mathop{{\rm mod}}N)
  18. N N
  19. k k
  20. N N
  21. P 0 = k N , Q 0 = 1 , Q 1 = k N - P 0 2 . P_{0}=\lfloor\sqrt{kN}\rfloor,Q_{0}=1,Q_{1}=kN-P_{0}^{2}.
  22. b i = k N + P i - 1 Q i , P i = b i Q i - P i - 1 , Q i + 1 = Q i - 1 + b i ( P i - 1 - P i ) b_{i}=\left\lfloor\frac{\lfloor\sqrt{kN}\rfloor+P_{i-1}}{Q_{i}}\right\rfloor,P% _{i}=b_{i}Q_{i}-P_{i-1},Q_{i+1}=Q_{i-1}+b_{i}(P_{i-1}-P_{i})
  23. Q i Q_{i}
  24. i i
  25. b 0 = k N - P i - 1 Q i , P 0 = b 0 Q i + P i - 1 , Q 0 = Q i , Q 1 = k N - P 0 2 Q 0 b_{0}=\left\lfloor\frac{\lfloor\sqrt{kN}\rfloor-P_{i-1}}{\sqrt{Q_{i}}}\right% \rfloor,P_{0}=b_{0}\sqrt{Q_{i}}+P_{i-1},Q_{0}=\sqrt{Q_{i}},Q_{1}=\frac{kN-P_{0% }^{2}}{Q_{0}}
  26. b i = k N + P i - 1 Q i , P i = b i Q i - P i - 1 , Q i + 1 = Q i - 1 + b i ( P i - 1 - P i ) b_{i}=\left\lfloor\frac{\lfloor\sqrt{kN}\rfloor+P_{i-1}}{Q_{i}}\right\rfloor,P% _{i}=b_{i}Q_{i}-P_{i-1},Q_{i+1}=Q_{i-1}+b_{i}(P_{i-1}-P_{i})
  27. P i = P i - 1 . P_{i}=P_{i-1}.
  28. f = gcd ( N , P i ) f=\gcd(N,P_{i})
  29. 1 1
  30. N N
  31. f f
  32. N N
  33. k k
  34. O ( N 4 ) O(\sqrt[4]{N})

Shannon_number.html

  1. 64 ! 32 ! 8 ! 2 2 ! 6 \scriptstyle\frac{64!}{32!{8!}^{2}{2!}^{6}}

Shannon–Weaver_model.html

  1. C = W log 2 ( 1 + S N ) , C=W\log_{2}(1+\tfrac{S}{N}),

Shear_flow.html

  1. q = V y Q x I x q=\frac{V_{y}Q_{x}}{I_{x}}

Shift_rule.html

  1. ( a n ) (a_{n})
  2. ( a n + N ) (a_{n+N})
  3. Σ n = 1 ( a n ) {\Sigma^{\infty}_{n=1}}(a_{n})
  4. Σ n = 1 ( a n + N ) {\Sigma^{\infty}_{n=1}}(a_{n+N})

Shock_diamond.html

  1. x = 0.67 D 0 P 0 P 1 x=0.67D_{0}\sqrt{\frac{P_{0}}{P_{1}}}

Shore_durometer.html

  1. E = 0.0981 ( 56 + 7.62336 S ) 0.137505 ( 254 - 2.54 S ) E=\cfrac{0.0981(56+7.62336S)}{0.137505(254-2.54S)}
  2. E E
  3. S S
  4. E = E=\infty
  5. S = 100 S=100
  6. S < 40 S<40
  7. S = 100 erf ( 3.186 × 10 - 4 E 1 / 2 ) S=100~{}\mathrm{erf}(3.186\times 10^{-4}~{}E^{1/2})
  8. erf \mathrm{erf}
  9. E E
  10. S D = 100 - 20 ( - 78.188 + 6113.36 + 781.88 E ) E S_{D}=100-\cfrac{20(-78.188+\sqrt{6113.36+781.88E})}{E}
  11. S D S_{D}
  12. E E
  13. log 10 ( E ) = 0.0235 S - 0.6403 ; S = { S A for 20 < S A < 80 S D + 50 for 30 < S D < 85 \log_{10}(E)=0.0235S-0.6403~{};~{}~{}S=\begin{cases}S_{A}&\mathrm{for}~{}20<S_% {A}<80\\ S_{D}+50&\mathrm{for}~{}30<S_{D}<85\end{cases}
  14. S A S_{A}
  15. S D S_{D}
  16. E E

Shot_transition_detection.html

  1. V = C C + M V={C\over C+M}
  2. P = C C + F P={C\over C+F}
  3. F 1 = 2 * P * V P + V F1={2*P*V\over P+V}

Sicherman_dice.html

  1. n n
  2. x + x 2 + x 3 + x 4 + x 5 + x 6 x+x^{2}+x^{3}+x^{4}+x^{5}+x^{6}
  3. x 2 + 2 x 3 + 3 x 4 + 4 x 5 + 5 x 6 + 6 x 7 + 5 x 8 + 4 x 9 + 3 x 10 + 2 x 11 + x 12 x^{2}+2x^{3}+3x^{4}+4x^{5}+5x^{6}+6x^{7}+5x^{8}+4x^{9}+3x^{10}+2x^{11}+x^{12}
  4. x n - 1 = d n n Φ d ( x ) . x^{n}-1=\prod_{d\,\mid\,n}^{n}\Phi_{d}(x).\;
  5. Φ d ( x ) \Phi_{d}(x)\,
  6. x n - 1 x - 1 = i = 0 n - 1 x i = 1 + x + + x n - 1 \frac{x^{n}-1}{x-1}=\sum_{i=0}^{n-1}x^{i}=1+x+\cdots+x^{n-1}
  7. x + x 2 + + x n = x x - 1 d n n Φ d ( x ) x+x^{2}+\cdots+x^{n}=\frac{x}{x-1}\prod_{d\,\mid\,n}^{n}\Phi_{d}(x)
  8. Φ 1 ( x ) = x - 1 \Phi_{1}(x)=x-1\,
  9. x Φ 2 ( x ) Φ 3 ( x ) Φ 6 ( x ) = x ( x + 1 ) ( x 2 + x + 1 ) ( x 2 - x + 1 ) x\,\Phi_{2}(x)\,\Phi_{3}(x)\,\Phi_{6}(x)=x\;(x+1)\;(x^{2}+x+1)\;(x^{2}-x+1)
  10. x ( x + 1 ) ( x 2 + x + 1 ) = x + 2 x 2 + 2 x 3 + x 4 x\;(x+1)\;(x^{2}+x+1)=x+2x^{2}+2x^{3}+x^{4}
  11. x ( x + 1 ) ( x 2 + x + 1 ) ( x 2 - x + 1 ) 2 = x + x 3 + x 4 + x 5 + x 6 + x 8 x\;(x+1)\;(x^{2}+x+1)\;(x^{2}-x+1)^{2}=x+x^{3}+x^{4}+x^{5}+x^{6}+x^{8}

Sidney_Lau_romanisation.html

  1. a 1 a^{1}
  2. a 2 a^{2}
  3. a 3 a^{3}
  4. a 1 0 a^{1^{0}}
  5. a 4 a^{4}
  6. a 5 a^{5}
  7. a 6 a^{6}

SIGCUM.html

  1. 5 ! = 120 5!=120
  2. 10 × 9 × 8 × 7 × 6 × 2 5 = 967 , 680 10\times 9\times 8\times 7\times 6\times 2^{5}=967,680

Signal-to-quantization-noise_ratio.html

  1. SNR = 3 × 2 2 n 1 + 4 P e × ( 2 2 n - 1 ) m m ( t ) 2 m p ( t ) 2 \mathrm{SNR}=\frac{3\times 2^{2n}}{1+4P_{e}\times(2^{2n}-1)}\frac{m_{m}(t)^{2}% }{m_{p}(t)^{2}}
  2. P e P_{e}
  3. m p ( t ) m_{p}(t)
  4. m m ( t ) m_{m}(t)
  5. m ( t ) m(t)
  6. x ( n ) x(n)
  7. N N
  8. x x
  9. ν = log 2 N \nu=\log_{2}N
  10. x x
  11. f ( x ) f(x)
  12. x x
  13. x m a x x_{max}
  14. SQNR = P s i g n a l P n o i s e = E [ x 2 ] E [ x ~ 2 ] \mathrm{SQNR}=\frac{P_{signal}}{P_{noise}}=\frac{E[x^{2}]}{E[\tilde{x}^{2}]}
  15. x 2 ¯ = E [ x 2 ] = P x ν = x 2 f ( x ) d x \overline{x^{2}}=E[x^{2}]=P_{x^{\nu}}=\int x^{2}f(x)dx
  16. E [ x ~ 2 ] = x m a x 2 3 × 4 ν E[\tilde{x}^{2}]=\frac{x_{max}^{2}}{3\times 4^{\nu}}
  17. SQNR = 3 × 4 ν × x 2 ¯ x m a x 2 \mathrm{SQNR}=\frac{3\times 4^{\nu}\times\overline{x^{2}}}{x_{max}^{2}}
  18. SQNR | d B = P x ν + 6 ν + 4.8 \mathrm{SQNR}|_{dB}=P_{x^{\nu}}+6\nu+4.8
  19. ν \nu
  20. P x ν P_{x^{\nu}}
  21. 20 × l o g 10 ( 2 ) 20\times log_{10}(2)

Simple_extension.html

  1. L = K ( θ ) . L=K(\theta).
  2. q = p d q=p^{d}
  3. 𝔽 q \mathbb{F}_{q}
  4. 𝔽 p . \mathbb{F}_{p}.
  5. 𝔽 q \mathbb{F}_{q}
  6. x q - 1 - 1 = 0 , x^{q-1}-1=0,
  7. φ : K [ X ] L p ( X ) p ( θ ) . \begin{aligned}\displaystyle\varphi:K[X]&\displaystyle\rightarrow L\\ \displaystyle p(X)&\displaystyle\mapsto p(\theta)\,.\end{aligned}
  8. φ \varphi
  9. φ \varphi
  10. φ \varphi
  11. φ \varphi
  12. K [ X ] / p K[X]/\langle p\rangle
  13. φ \varphi
  14. φ \varphi
  15. K [ X ] / p K[X]/\langle p\rangle
  16. 𝐐 ( 3 , 7 ) \mathbf{Q}(\sqrt{3},\sqrt{7})
  17. 3 + 7 \sqrt{3}+\sqrt{7}

Simple_rational_approximation.html

  1. f ( x ) = 0 \,f(x)=0
  2. h ( z ) = a z + b + c . h(z)=\frac{a}{z+b}+c.
  3. h ( i ) ( x ) = f ( i ) ( x ) , i = 0 , 1 , 2. h^{(i)}(x)=f^{(i)}(x),\qquad i=0,1,2.
  4. h ( z ) = 0 \,h(z)=0
  5. x n + 1 = x n - f ( x n ) f ( x n ) ( 1 1 - f ( x n ) f ′′ ( x n ) 2 ( f ( x n ) ) 2 ) . x_{n+1}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})}\left({\frac{1}{1-\frac{f(x_{n% })f^{\prime\prime}(x_{n})}{2(f^{\prime}(x_{n}))^{2}}}}\right).
  6. h ( z ) h(z)
  7. g ( x ) = f ( x ) f ( x ) = 0. g(x)=\frac{f(x)}{\sqrt{f^{\prime}(x)}}=0.
  8. g ( x ) g(x)
  9. f ( x ) = α ( 0 ) \,f(x)=\alpha(\neq 0)
  10. h ( z ) = a z + b . h(z)=\frac{a}{z+b}.
  11. h ( i ) ( x ) = f ( i ) ( x ) , i = 0 , 1. h^{(i)}(x)=f^{(i)}(x),\qquad i=0,1.
  12. x n + 1 = x n - f ( x n ) - α f ( x n ) ( f ( x n ) α ) . x_{n+1}=x_{n}-\frac{f(x_{n})-\alpha}{f^{\prime}(x_{n})}\left(\frac{f(x_{n})}{% \alpha}\right).
  13. g ( x ) = 1 - α < m t p l > f ( x ) = 0. g(x)=1-\frac{\alpha}{<}mtpl>{{f(x)}}=0.

Simson_line.html

  1. N M P + P M L = 180 \angle NMP+\angle PML=180^{\circ}
  2. P C A B PCAB
  3. P B N + A C P = P B A + A C P = 180 \angle PBN+\angle ACP=\angle PBA+\angle ACP=180^{\circ}
  4. P M N B PMNB
  5. P B N + N M P = 180 \angle PBN+\angle NMP=180^{\circ}
  6. N M P = A C P \angle NMP=\angle ACP
  7. P L C M PLCM
  8. P M L = P C L = 180 - A C P \angle PML=\angle PCL=180^{\circ}-\angle ACP
  9. N M P + P M L = A C P + ( 180 - A C P ) = 180 \angle NMP+\angle PML=\angle ACP+(180^{\circ}-\angle ACP)=180^{\circ}

Sine.html

  1. θ \theta
  2. sin θ = cos ( π 2 - θ ) = 1 csc θ \sin\theta=\cos\left(\frac{\pi}{2}-\theta\right)=\frac{1}{\csc\theta}
  3. csc A = 1 sin A = hypotenuse opposite = h a . \csc A=\frac{1}{\sin A}=\frac{\textrm{hypotenuse}}{\textrm{opposite}}=\frac{h}% {a}.
  4. 1 {}^{−1}
  5. θ = arcsin ( opposite hypotenuse ) = sin - 1 ( a h ) . \theta=\arcsin\left(\frac{\,\text{opposite}}{\,\text{hypotenuse}}\right)=\sin^% {-1}\left(\frac{a}{h}\right).
  6. sin y = x \displaystyle\sin y=x\ \Leftrightarrow
  7. sin y = x y = ( - 1 ) k arcsin x + k π \sin y=x\ \Leftrightarrow\ y=(-1)^{k}\arcsin x+k\pi
  8. sin ( arcsin x ) = x \sin(\arcsin x)=x\!
  9. arcsin ( sin θ ) = θ for - π / 2 θ π / 2. \arcsin(\sin\theta)=\theta\quad\,\text{for }-\pi/2\leq\theta\leq\pi/2.
  10. f ( x ) = sin x f(x)=\sin x\,
  11. f ( x ) = cos x f^{\prime}(x)=\cos x\,
  12. f ( x ) d x = - cos x + C \int f(x)\,dx=-\cos x+C
  13. sin θ \sin\theta
  14. = ± 1 - cos 2 θ =\pm\sqrt{1-\cos^{2}\theta}
  15. = sgn ( cos ( θ - π 2 ) ) 1 - cos 2 θ =\operatorname{sgn}\left(\cos\left(\theta-\frac{\pi}{2}\right)\right)\sqrt{1-% \cos^{2}\theta}
  16. cos θ \cos\theta
  17. = ± 1 - sin 2 θ =\pm\sqrt{1-\sin^{2}\theta}
  18. = sgn ( sin ( θ + π 2 ) ) 1 - sin 2 θ =\operatorname{sgn}\left(\sin\left(\theta+\frac{\pi}{2}\right)\right)\sqrt{1-% \sin^{2}\theta}
  19. sin θ \sin\theta
  20. = ± 1 1 + cot 2 θ =\pm\frac{1}{\sqrt{1+\cot^{2}\theta}}
  21. = sgn ( cot ( θ 2 ) ) 1 1 + cot 2 θ =\operatorname{sgn}\left(\cot\left(\frac{\theta}{2}\right)\right)\frac{1}{% \sqrt{1+\cot^{2}\theta}}
  22. cot θ \cot\theta
  23. = ± 1 - sin 2 θ sin θ =\pm\frac{\sqrt{1-\sin^{2}\theta}}{\sin\theta}
  24. = sgn ( sin ( θ + π 2 ) ) 1 - sin 2 θ sin θ =\operatorname{sgn}\left(\sin\left(\theta+\frac{\pi}{2}\right)\right)\frac{% \sqrt{1-\sin^{2}\theta}}{\sin\theta}
  25. sin θ \sin\theta
  26. = ± tan θ 1 + tan 2 θ =\pm\frac{\tan\theta}{\sqrt{1+\tan^{2}\theta}}
  27. = sgn ( tan ( 2 θ + π 4 ) ) tan θ 1 + tan 2 θ =\operatorname{sgn}\left(\tan\left(\frac{2\theta+\pi}{4}\right)\right)\frac{% \tan\theta}{\sqrt{1+\tan^{2}\theta}}
  28. tan θ \tan\theta
  29. = ± sin θ 1 - sin 2 θ =\pm\frac{\sin\theta}{\sqrt{1-\sin^{2}\theta}}
  30. = sgn ( sin ( θ + π 2 ) ) sin θ 1 - sin 2 θ =\operatorname{sgn}\left(\sin\left(\theta+\frac{\pi}{2}\right)\right)\frac{% \sin\theta}{\sqrt{1-\sin^{2}\theta}}
  31. sin θ \sin\theta
  32. = ± sec 2 θ - 1 sec θ =\pm\frac{\sqrt{\sec^{2}\theta-1}}{\sec\theta}
  33. = sgn ( sec ( 4 θ - π 2 ) ) sec 2 θ - 1 sec θ =\operatorname{sgn}\left(\sec\left(\frac{4\theta-\pi}{2}\right)\right)\frac{% \sqrt{\sec^{2}\theta-1}}{\sec\theta}
  34. sec θ \sec\theta
  35. = ± 1 1 - sin 2 θ =\pm\frac{1}{\sqrt{1-\sin^{2}\theta}}
  36. = sgn ( sin ( θ + π 2 ) ) 1 1 - sin 2 θ =\operatorname{sgn}\left(\sin\left(\theta+\frac{\pi}{2}\right)\right)\frac{1}{% \sqrt{1-\sin^{2}\theta}}
  37. cos 2 θ + sin 2 θ = 1 \cos^{2}\theta+\sin^{2}\theta=1\!
  38. 0 < x < 90 0^{\circ}<x<90^{\circ}
  39. 0 < x < π / 2 0<x<\pi/2
  40. 0 < sin x < 1 0<\sin x<1
  41. + +
  42. 90 < x < 180 90^{\circ}<x<180^{\circ}
  43. π / 2 < x < π \pi/2<x<\pi
  44. 0 < sin x < 1 0<\sin x<1
  45. + +
  46. 180 < x < 270 180^{\circ}<x<270^{\circ}
  47. π < x < 3 π / 2 \pi<x<3\pi/2
  48. - 1 < sin x < 0 -1<\sin x<0
  49. - -
  50. 270 < x < 360 270^{\circ}<x<360^{\circ}
  51. 3 π / 2 < x < 2 π 3\pi/2<x<2\pi
  52. - 1 < sin x < 0 -1<\sin x<0
  53. - -
  54. 2 π k 2\pi k
  55. π / 2 \pi/2
  56. 2 π k + π / 2 2\pi k+\pi/2
  57. π \pi
  58. 2 π k - π 2\pi k-\pi
  59. 3 π / 2 3\pi/2
  60. 2 π k - π / 2 2\pi k-\pi/2
  61. sin ( α + 360 ) = sin ( α ) \sin(\alpha+360^{\circ})=\sin(\alpha)
  62. sin ( α + 180 ) = - sin ( α ) \sin(\alpha+180^{\circ})=-\sin(\alpha)
  63. cos ( x ) = e i x + e - i x 2 \cos(x)=\frac{e^{ix}+e^{-ix}}{2}
  64. sin ( x ) = e i x - e - i x 2 i \sin(x)=\frac{e^{ix}-e^{-ix}}{2i}
  65. sin ( 180 - α ) = sin ( α ) \sin(180^{\circ}-\alpha)=\sin(\alpha)
  66. sin ( 4 n + k ) ( 0 ) = { 0 when k = 0 1 when k = 1 0 when k = 2 - 1 when k = 3 \sin^{(4n+k)}(0)=\begin{cases}0&\,\text{when }k=0\\ 1&\,\text{when }k=1\\ 0&\,\text{when }k=2\\ -1&\,\text{when }k=3\end{cases}
  67. sin x = x - x 3 3 ! + x 5 5 ! - x 7 7 ! + = n = 0 ( - 1 ) n ( 2 n + 1 ) ! x 2 n + 1 \begin{aligned}\displaystyle\sin x&\displaystyle=x-\frac{x^{3}}{3!}+\frac{x^{5% }}{5!}-\frac{x^{7}}{7!}+\cdots\\ &\displaystyle=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)!}x^{2n+1}\\ \end{aligned}
  68. sin x deg = sin y rad = π 180 x - ( π 180 ) 3 x 3 3 ! + ( π 180 ) 5 x 5 5 ! - ( π 180 ) 7 x 7 7 ! + . \begin{aligned}\displaystyle\sin x_{\mathrm{deg}}&\displaystyle=\sin y_{% \mathrm{rad}}\\ &\displaystyle=\frac{\pi}{180}x-\left(\frac{\pi}{180}\right)^{3}\ \frac{x^{3}}% {3!}+\left(\frac{\pi}{180}\right)^{5}\ \frac{x^{5}}{5!}-\left(\frac{\pi}{180}% \right)^{7}\ \frac{x^{7}}{7!}+\cdots.\end{aligned}
  69. sin 0 = 0 and sin 2 x = 2 sin x cos x cos 2 x + sin 2 x = 1 and cos 2 x = cos 2 x - sin 2 x \begin{aligned}\displaystyle\sin 0=0&\displaystyle\,\text{ and }\sin{2x}=2\sin x% \cos x\\ \displaystyle\cos^{2}x+\sin^{2}x=1&\displaystyle\,\text{ and }\cos{2x}=\cos^{2% }x-\sin^{2}x\\ \end{aligned}
  70. sin x x when x 0. \sin x\approx x\,\text{ when }x\approx 0.
  71. sin x = x 1 + x 2 2 3 - x 2 + 2 3 x 2 4 5 - x 2 + 4 5 x 2 6 7 - x 2 + . \sin x=\cfrac{x}{1+\cfrac{x^{2}}{2\cdot 3-x^{2}+\cfrac{2\cdot 3x^{2}}{4\cdot 5% -x^{2}+\cfrac{4\cdot 5x^{2}}{6\cdot 7-x^{2}+\ddots}}}}.
  72. a a
  73. b b
  74. a b 1 + cos ( x ) 2 d x \int_{a}^{b}\!\sqrt{1+\cos(x)^{2}}\,dx
  75. 4 2 π 3 / 2 Γ ( 1 / 4 ) 2 + Γ ( 1 / 4 ) 2 2 π = 7.640395578 \frac{4\sqrt{2}\,\pi^{3/2}}{\Gamma(1/4)^{2}}+\frac{\Gamma(1/4)^{2}}{\sqrt{2\pi% }}=7.640395578\ldots
  76. Γ \Gamma
  77. 2 π 2\pi
  78. π \pi
  79. 1.21600672 × x + 0.10317093 sin ( 2 x ) - 0.00220445 sin ( 4 x ) + 0.00012584 sin ( 6 x ) - 0.00001011 sin ( 8 x ) + 1.21600672\,\times\,x\,+\,0.10317093\,\sin(2x)-0.00220445\sin(4x)+0.00012584% \sin(6x)-0.00001011\sin(8x)+\cdots
  80. sin A a = sin B b = sin C c . \frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}.
  81. a sin A = b sin B = c sin C = 2 R , \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R,
  82. 0
  83. π \pi
  84. 1 12 π \tfrac{1}{12}\pi
  85. 2 / 3 {2}/{3}
  86. 6 - 2 4 \frac{\sqrt{6}-\sqrt{2}}{4}
  87. 11 12 π \tfrac{11}{12}\pi
  88. 1 / 3 {1}/{3}
  89. 1 6 π \tfrac{1}{6}\pi
  90. 1 / 3 {1}/{3}
  91. 1 2 \frac{1}{2}
  92. 5 6 π \tfrac{5}{6}\pi
  93. 2 / 3 {2}/{3}
  94. 1 4 π \tfrac{1}{4}\pi
  95. 2 2 \frac{\sqrt{2}}{2}
  96. 3 4 π \tfrac{3}{4}\pi
  97. 1 3 π \tfrac{1}{3}\pi
  98. 2 / 3 {2}/{3}
  99. 3 2 \frac{\sqrt{3}}{2}
  100. 2 3 π \tfrac{2}{3}\pi
  101. 1 / 3 {1}/{3}
  102. 5 12 π \tfrac{5}{12}\pi
  103. 1 / 3 {1}/{3}
  104. 6 + 2 4 \frac{\sqrt{6}+\sqrt{2}}{4}
  105. 7 12 π \tfrac{7}{12}\pi
  106. 2 / 3 {2}/{3}
  107. 1 2 π \tfrac{1}{2}\pi
  108. 1 1
  109. sin x \mathrm{sin}\,x
  110. 0 2 \frac{\sqrt{0}}{2}
  111. 1 2 \frac{\sqrt{1}}{2}
  112. 2 2 \frac{\sqrt{2}}{2}
  113. 3 2 \frac{\sqrt{3}}{2}
  114. 4 2 \frac{\sqrt{4}}{2}
  115. sin x \mathrm{sin}\,x
  116. sin π 60 = sin 3 = ( 2 - 12 ) 5 + 5 + ( 10 - 2 ) ( 3 + 1 ) 16 \sin\frac{\pi}{60}=\sin 3^{\circ}=\frac{(2-\sqrt{12})\sqrt{5+\sqrt{5}}+(\sqrt{% 10}-\sqrt{2})(\sqrt{3}+1)}{16}\,
  117. sin π 30 = sin 6 = 30 - 180 - 5 - 1 8 \sin\frac{\pi}{30}=\sin 6^{\circ}=\frac{\sqrt{30-\sqrt{180}}-\sqrt{5}-1}{8}\,
  118. sin π 20 = sin 9 = 10 + 2 - 20 - 80 8 \sin\frac{\pi}{20}=\sin 9^{\circ}=\frac{\sqrt{10}+\sqrt{2}-\sqrt{20-\sqrt{80}}% }{8}\,
  119. sin π 15 = sin 12 = 10 + 20 + 3 - 15 8 \sin\frac{\pi}{15}=\sin 12^{\circ}=\frac{\sqrt{10+\sqrt{20}}+\sqrt{3}-\sqrt{15% }}{8}\,
  120. sin π 10 = sin 18 = 5 - 1 4 = 1 2 φ - 1 \sin\frac{\pi}{10}=\sin 18^{\circ}=\frac{\sqrt{5}-1}{4}=\tfrac{1}{2}\varphi^{-% 1}\,
  121. sin 7 π 60 = sin 21 = ( 2 + 12 ) 5 - 5 - ( 10 + 2 ) ( 3 - 1 ) 16 \sin\frac{7\pi}{60}=\sin 21^{\circ}=\frac{(2+\sqrt{12})\sqrt{5-\sqrt{5}}-(% \sqrt{10}+\sqrt{2})(\sqrt{3}-1)}{16}\,
  122. sin π 8 = sin 22.5 = 2 - 2 2 , \sin\frac{\pi}{8}=\sin 22.5^{\circ}=\frac{\sqrt{2-\sqrt{2}}}{2},
  123. sin 2 π 15 = sin 24 = 3 + 15 - 10 - 20 8 \sin\frac{2\pi}{15}=\sin 24^{\circ}=\frac{\sqrt{3}+\sqrt{15}-\sqrt{10-\sqrt{20% }}}{8}\,
  124. sin 3 π 20 = sin 27 = 20 + 80 - 10 + 2 8 \sin\frac{3\pi}{20}=\sin 27^{\circ}=\frac{\sqrt{20+\sqrt{80}}-\sqrt{10}+\sqrt{% 2}}{8}\,
  125. sin 11 π 60 = sin 33 = ( 12 - 2 ) 5 + 5 + ( 10 - 2 ) ( 3 + 1 ) 16 \sin\frac{11\pi}{60}=\sin 33^{\circ}=\frac{(\sqrt{12}-2)\sqrt{5+\sqrt{5}}+(% \sqrt{10}-\sqrt{2})(\sqrt{3}+1)}{16}\,
  126. sin π 5 = sin 36 = 10 - 20 4 \sin\frac{\pi}{5}=\sin 36^{\circ}=\frac{\sqrt{10-\sqrt{20}}}{4}\,
  127. sin 13 π 60 = sin 39 = ( 2 - 12 ) 5 - 5 + ( 10 + 2 ) ( 3 + 1 ) 16 \sin\frac{13\pi}{60}=\sin 39^{\circ}=\frac{(2-\sqrt{12})\sqrt{5-\sqrt{5}}+(% \sqrt{10}+\sqrt{2})(\sqrt{3}+1)}{16}\,
  128. sin 7 π 30 = sin 42 = 30 + 180 - 5 + 1 8 \sin\frac{7\pi}{30}=\sin 42^{\circ}=\frac{\sqrt{30+\sqrt{180}}-\sqrt{5}+1}{8}\,
  129. sin θ = sin ( θ + 2 π k ) , \sin\theta=\sin\left(\theta+2\pi k\right),\,
  130. z = r ( cos φ + i sin φ ) z=r(\cos\varphi+i\sin\varphi)\,
  131. Im ( z ) = r sin φ \operatorname{Im}(z)=r\sin\varphi
  132. sin z \displaystyle\sin z
  133. sin x = Im ( e i x ) . \sin x=\operatorname{Im}(e^{ix}).\,
  134. sin i y = i sinh y . \sin iy=i\sinh y.\,
  135. sin ( x + i y ) \displaystyle\sin(x+iy)
  136. n = - ( - 1 ) n z - n = 1 z - 2 z n = 1 ( - 1 ) n n 2 - z 2 \displaystyle\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}}{z-n}=\frac{1}{z}-2z\sum_% {n=1}^{\infty}\frac{(-1)^{n}}{n^{2}-z^{2}}
  137. π sin π z \frac{\pi}{\sin\pi z}
  138. π 2 sin 2 π z = n = - 1 ( z - n ) 2 . \displaystyle\frac{\pi^{2}}{\sin^{2}\pi z}=\sum_{n=-\infty}^{\infty}\frac{1}{(% z-n)^{2}}.
  139. sin π z = π z n = 1 ( 1 - z 2 n 2 ) . \displaystyle\sin\pi z=\pi z\prod_{n=1}^{\infty}\Bigl(1-\frac{z^{2}}{n^{2}}% \Bigr).
  140. Γ ( s ) Γ ( 1 - s ) = π sin π s , \Gamma(s)\Gamma(1-s)={\pi\over\sin\pi s},
  141. ζ ( s ) = 2 ( 2 π ) s - 1 Γ ( 1 - s ) sin ( π s / 2 ) ζ ( 1 - s ) . \zeta(s)=2(2\pi)^{s-1}\Gamma(1-s)\sin(\pi s/2)\zeta(1-s).
  142. Δ u ( x 1 , x 2 ) = 0. \Delta u(x_{1},x_{2})=0.

Single-machine_scheduling.html

  1. m a x ( 0 , r e c e i p t d a t e - d u e d a t e ) max(0,receipt\;date-due\;date)
  2. m a x ( 0 , d u e d a t e - r e c e i p t d a t e ) max(0,due\;date-receipt\;date)
  3. r e c e i p t d a t e - d u e d a t e receipt\;date-due\;date
  4. e n d d a t e - s t a r t d a t e end\;date-start\;date

Single_crossing_condition.html

  1. x , x y F ( x ) G ( y ) \forall x,x\geq y\implies F(x)\geq G(y)
  2. x , x y F ( x ) G ( y ) \forall x,x\leq y\implies F(x)\leq G(y)
  3. h ( x ) = F ( x ) - G ( y ) h(x)=F(x)-G(y)
  4. F ( x , t ) F ( x , t ) F ( x , t ) F ( x , t ) F(x^{\prime},t)\geq F(x,t)\implies F(x^{\prime},t^{\prime})\geq F(x,t^{\prime})
  5. F ( x , t ) > F ( x , t ) F ( x , t ) > F ( x , t ) F(x^{\prime},t)>F(x,t)\implies F(x^{\prime},t^{\prime})>F(x,t^{\prime})

Single_loss_expectancy.html

  1. S i n g l e L o s s E x p e c t a n c y ( S L E ) = A s s e t V a l u e ( A V ) × E x p o s u r e F a c t o r ( E F ) {Single\ Loss\ Expectancy\ (SLE)}={Asset\ Value\ (AV)\ }\times{\ Exposure\ % Factor\ (EF)}

Singmaster's_conjecture.html

  1. N ( a ) = O ( 1 ) . N(a)=O(1).\,
  2. N ( a ) = O ( log a ) . N(a)=O(\log a).\,
  3. N ( a ) = O ( ( log a ) ( log log log a ) ( log log a ) 3 ) , N(a)=O\left(\frac{(\log a)(\log\log\log a)}{(\log\log a)^{3}}\right),\,
  4. N ( a ) = O ( log ( a ) 2 / 3 + ε ) N(a)=O\left(\log(a)^{2/3+\varepsilon}\right)
  5. ε > 0 \varepsilon>0
  6. ( n + 1 k + 1 ) = ( n k + 2 ) , {n+1\choose k+1}={n\choose k+2},
  7. n = F 2 i + 2 F 2 i + 3 - 1 , n=F_{2i+2}F_{2i+3}-1,\,
  8. k = F 2 i F 2 i + 3 - 1 , k=F_{2i}F_{2i+3}-1,\,
  9. ( 120 1 ) = ( 16 2 ) = ( 10 3 ) {120\choose 1}={16\choose 2}={10\choose 3}
  10. ( 210 1 ) = ( 21 2 ) = ( 10 4 ) {210\choose 1}={21\choose 2}={10\choose 4}
  11. ( 1540 1 ) = ( 56 2 ) = ( 22 3 ) {1540\choose 1}={56\choose 2}={22\choose 3}
  12. ( 7140 1 ) = ( 120 2 ) = ( 36 3 ) {7140\choose 1}={120\choose 2}={36\choose 3}
  13. ( 11628 1 ) = ( 153 2 ) = ( 19 5 ) {11628\choose 1}={153\choose 2}={19\choose 5}
  14. ( 24310 1 ) = ( 221 2 ) = ( 17 8 ) {24310\choose 1}={221\choose 2}={17\choose 8}
  15. ( 3003 1 ) = ( 78 2 ) = ( 15 5 ) = ( 14 6 ) {3003\choose 1}={78\choose 2}={15\choose 5}={14\choose 6}

Singular_cardinals_hypothesis.html

  1. κ \kappa
  2. c f ( κ ) = ω cf(\kappa)=\omega
  3. κ \kappa
  4. κ \kappa
  5. 𝒫 ( κ ) \mathcal{P}(\kappa)
  6. { α < κ : 2 α = α + } D \{\alpha<\kappa:2^{\alpha}=\alpha^{+}\}\in D
  7. 2 κ = κ + 2^{\kappa}=\kappa^{+}
  8. κ \kappa
  9. κ \kappa
  10. 2 κ > κ + 2^{\kappa}>\kappa^{+}
  11. κ \kappa
  12. κ \kappa
  13. 2 κ > κ + 2^{\kappa}>\kappa^{+}
  14. κ + + \kappa^{++}
  15. κ + + \kappa^{++}
  16. 2 0 = 2 2^{\aleph_{0}}=\aleph_{2}
  17. κ \kappa
  18. κ \kappa
  19. κ + + \kappa^{++}

Singular_measure.html

  1. μ ν . \mu\perp\nu.
  2. H ( x ) = def { 0 , x < 0 ; 1 , x 0 ; H(x)\ \stackrel{\mathrm{def}}{=}\begin{cases}0,&x<0;\\ 1,&x\geq 0;\end{cases}
  3. δ 0 \delta_{0}
  4. δ 0 \delta_{0}
  5. λ \lambda
  6. λ \lambda
  7. δ 0 \delta_{0}
  8. λ ( { 0 } ) = 0 \lambda(\{0\})=0
  9. δ 0 ( { 0 } ) = 1 \delta_{0}(\{0\})=1
  10. U U
  11. λ ( U ) > 0 \lambda(U)>0
  12. δ 0 ( U ) = 0 \delta_{0}(U)=0

Singular_point_of_a_curve.html

  1. f = a 0 + b 0 x + b 1 y + c 0 x 2 + 2 c 1 x y + c 2 y 2 + f=a_{0}+b_{0}x+b_{1}y+c_{0}x^{2}+2c_{1}xy+c_{2}y^{2}+\dots\,
  2. b 0 = f x , b 1 = f y , b_{0}={\partial f\over\partial x},\,b_{1}={\partial f\over\partial y},
  3. f ( x , y ) = f x = f y = 0. f(x,y)={\partial f\over\partial x}={\partial f\over\partial y}=0.
  4. f = ( b 0 + m b 1 ) x + ( c 0 + 2 m c 1 + c 2 m 2 ) x 2 + . f=(b_{0}+mb_{1})x+(c_{0}+2mc_{1}+c_{2}m^{2})x^{2}+\dots.\,
  5. f = ( c 0 + 2 m c 1 + c 2 m 2 ) x 2 + ( d 0 + 3 m d 1 + 3 m 2 d 2 + d 3 m 3 ) x 3 + . f=(c_{0}+2mc_{1}+c_{2}m^{2})x^{2}+(d_{0}+3md_{1}+3m^{2}d_{2}+d_{3}m^{3})x^{3}+% \dots.\,
  6. d g 1 d t = d g 2 d t = 0. {dg_{1}\over dt}={dg_{2}\over dt}=0.

Size_consistency_and_size_extensivity.html

  1. E ( A + B ) = E ( A ) + E ( B ) E(A+B)=E(A)+E(B)

Skew_flip_turnover.html

  1. T = 2 D A T=2\sqrt{D\over A}

Skew_heap.html

  1. s h 1 sh_{1}
  2. s h 2 sh_{2}

Sky_brightness.html

  1. Airglow / S 10 = 145 + 108 ( S - 0.8 ) {\rm Airglow}/{\rm S}_{10}=145+108(S-0.8)
  2. ZodiacalLight / S 10 = 140 - 90 sin ( | β | ) {\rm ZodiacalLight}/{\rm S}_{10}=140-90\sin(|\beta|)

Slice_knot.html

  1. S 3 = { 𝐱 4 | 𝐱 | = 1 } S^{3}=\{\mathbf{x}\in\mathbb{R}^{4}\mid|\mathbf{x}|=1\}
  2. B 4 = { 𝐱 4 | 𝐱 | 1 } . B^{4}=\{\mathbf{x}\in\mathbb{R}^{4}\mid|\mathbf{x}|\leq 1\}.
  3. K S 3 K\subset S^{3}
  4. f ( t ) f ( t - 1 ) f(t)f(t^{-1})
  5. f ( t ) f(t)
  6. 8 8 8_{8}
  7. 8 9 8_{9}
  8. 8 20 8_{20}
  9. 9 27 9_{27}
  10. 9 41 9_{41}
  11. 9 46 9_{46}
  12. 10 3 10_{3}
  13. 10 22 10_{22}
  14. 10 35 10_{35}
  15. 10 42 10_{42}
  16. 10 48 10_{48}
  17. 10 75 10_{75}
  18. 10 87 10_{87}
  19. 10 99 10_{99}
  20. 10 123 10_{123}
  21. 10 129 10_{129}
  22. 10 137 10_{137}
  23. 10 140 10_{140}
  24. 10 153 10_{153}
  25. 10 155 10_{155}

Slingshot_argument.html

  1. ι x \iota x
  2. S S
  3. ϕ ( a ) \phi(a)
  4. a = ι x ( ϕ ( x ) x = a ) a=\iota x(\phi(x)\land x=a)
  5. a = ι x ( π ( x , b ) x = a ) a=\iota x(\pi(x,b)\land x=a)
  6. π ( a , b ) \pi(a,b)
  7. b = ι x ( π ( a , x ) x = b ) b=\iota x(\pi(a,x)\land x=b)
  8. b = ι x ( ψ ( x ) x = b ) b=\iota x(\psi(x)\land x=b)
  9. ψ ( b ) \psi(b)
  10. T T
  11. x ( y ( F ( y ) y = x ) G ( x ) ) \exists x(\forall y(F(y)\leftrightarrow y=x)\land G(x))
  12. ι \iota
  13. x ( y ( ( ϕ ( y ) y = a ) y = x ) a = x ) \exists x(\forall y((\phi(y)\land y=a)\leftrightarrow y=x)\land a=x)
  14. x ( y ( ( π ( y , b ) y = a ) y = x ) a = x ) \exists x(\forall y((\pi(y,b)\land y=a)\leftrightarrow y=x)\land a=x)

Slip_(materials_science).html

  1. 1 ¯ \overline{1}
  2. | b | = a 2 | < 110 > | = a 2 |b|=\frac{a}{2}|<110>|=\frac{a}{\sqrt{2}}
  3. 1 ¯ \overline{1}
  4. | b | = a 2 | < 111 > | = 3 a 2 |b|=\frac{a}{2}|<111>|=\frac{\sqrt{3}a}{2}

Slope_One.html

  1. f ( x ) = a x + b f(x)=ax+b
  2. f ( x ) = x + b f(x)=x+b
  3. ( 1 , 0 , 0 ) ( 0 , 1 , 1 ) ( 1 , 0 , 0 ) ( 0 , 1 , 1 ) = 0 \frac{(1,0,0)\cdot(0,1,1)}{\|(1,0,0)\|\|(0,1,1)\|}=0
  4. ( 1 , 0 , 0 ) ( 1 , 1 , 0 ) ( 1 , 0 , 0 ) ( 1 , 1 , 0 ) = 1 2 \frac{(1,0,0)\cdot(1,1,0)}{\|(1,0,0)\|\|(1,1,0)\|}=\frac{1}{\sqrt{2}}
  5. ( 0 , 1 , 1 ) ( 1 , 1 , 0 ) ( 0 , 1 , 1 ) ( 1 , 1 , 0 ) = 1 2 \frac{(0,1,1)\cdot(1,1,0)}{\|(0,1,1)\|\|(1,1,0)\|}=\frac{1}{2}
  6. f ( x ) = a x + b f(x)=ax+b
  7. f ( x ) = x + b f(x)=x+b
  8. 2 × 2.5 + 1 × 8 2 + 1 = 13 3 = 4.33 \frac{2\times 2.5+1\times 8}{2+1}=\frac{13}{3}=4.33

Small_Latin_squares_and_quasigroups.html

  1. [ 1 2 3 4 5 2 4 1 5 3 3 5 4 2 1 4 1 5 3 2 5 3 2 1 4 ] . \begin{bmatrix}1&2&3&4&5\\ 2&4&1&5&3\\ 3&5&4&2&1\\ 4&1&5&3&2\\ 5&3&2&1&4\end{bmatrix}.

SMS4.html

  1. Z 2 e Z^{e}_{2}
  2. Z 2 32 Z^{32}_{2}
  3. Z 2 8 Z^{8}_{2}
  4. M K = ( M K 0 , M K 1 , M K 2 , M K 3 ) MK=(MK_{0},\ MK_{1},\ MK_{2},\ MK_{3})
  5. M K i ( i = 0 , 1 , 2 , 3 ) MK_{i}\ (i=0,\ 1,\ 2,\ 3)
  6. ( r k 0 , r k 1 , , r k 31 ) (rk_{0},\ rk_{1},\ \ldots,\ rk_{31})
  7. F K = ( F K 0 , F K 1 , F K 2 , F K 3 ) FK=(FK_{0},\ FK_{1},\ FK_{2},\ FK_{3})
  8. C K = ( C K 0 , C K 1 , , C K 31 ) CK=(CK_{0},\ CK_{1},\ \ldots,\ CK_{31})
  9. F K i FK_{i}
  10. C K i CK_{i}

SO(5).html

  1. \cong

So_What_chord.html

  1. G / D A 5 / E \frac{G/D}{A5/E}

Sobolev_conjugate.html

  1. 1 p < n 1\leq p<n
  2. p * = p n n - p > p p^{*}=\frac{pn}{n-p}>p
  3. W 1 , p ( n ) W^{1,p}(\mathbb{R}^{n})
  4. L q ( n ) L^{q}(\mathbb{R}^{n})
  5. D u L p ( n ) \|Du\|_{L^{p}(\mathbb{R}^{n})}
  6. u L q ( n ) \|u\|_{L^{q}(\mathbb{R}^{n})}
  7. u L q ( n ) C ( p , q ) D u L p ( n ) \|u\|_{L^{q}(\mathbb{R}^{n})}\leq C(p,q)\|Du\|_{L^{p}(\mathbb{R}^{n})}
  8. u ( x ) C c ( n ) u(x)\in C^{\infty}_{c}(\mathbb{R}^{n})
  9. u λ ( x ) := u ( λ x ) u_{\lambda}(x):=u(\lambda x)
  10. u λ L q ( n ) q = n | u ( λ x ) | q d x = 1 λ n n | u ( y ) | q d y = λ - n u L q ( n ) q \|u_{\lambda}\|_{L^{q}(\mathbb{R}^{n})}^{q}=\int_{\mathbb{R}^{n}}|u(\lambda x)% |^{q}dx=\frac{1}{\lambda^{n}}\int_{\mathbb{R}^{n}}|u(y)|^{q}dy=\lambda^{-n}\|u% \|_{L^{q}(\mathbb{R}^{n})}^{q}
  11. D u λ L p ( n ) p = n | λ D u ( λ x ) | p d x = λ p λ n n | D u ( y ) | p d y = λ p - n D u L p ( n ) p \|Du_{\lambda}\|_{L^{p}(\mathbb{R}^{n})}^{p}=\int_{\mathbb{R}^{n}}|\lambda Du(% \lambda x)|^{p}dx=\frac{\lambda^{p}}{\lambda^{n}}\int_{\mathbb{R}^{n}}|Du(y)|^% {p}dy=\lambda^{p-n}\|Du\|_{L^{p}(\mathbb{R}^{n})}^{p}
  12. u λ u_{\lambda}
  13. u u
  14. u L q ( n ) λ 1 - n / p + n / q C ( p , q ) D u L p ( n ) \|u\|_{L^{q}(\mathbb{R}^{n})}\leq\lambda^{1-n/p+n/q}C(p,q)\|Du\|_{L^{p}(% \mathbb{R}^{n})}
  15. 1 - n / p + n / q 0 1-n/p+n/q\not=0
  16. λ \lambda
  17. q = p n n - p q=\frac{pn}{n-p}

Sobolev_inequality.html

  1. k k
  2. k k
  3. k > k>ℓ
  4. 1 q = 1 p - k - n , \frac{1}{q}=\frac{1}{p}-\frac{k-\ell}{n},
  5. W k , p ( 𝐑 n ) W , q ( 𝐑 n ) W^{k,p}(\mathbf{R}^{n})\subseteq W^{\ell,q}(\mathbf{R}^{n})
  6. k = 1 k=1
  7. = 0 ℓ=0
  8. W 1 , p ( 𝐑 n ) L p * ( 𝐑 n ) W^{1,p}(\mathbf{R}^{n})\subseteq L^{p^{*}}(\mathbf{R}^{n})
  9. p p
  10. 1 p * = 1 p - 1 n . \frac{1}{p^{*}}=\frac{1}{p}-\frac{1}{n}.
  11. ( k r α ) / n = 1 / p (k−r−α)/n=1/p
  12. α ( 0 , 1 ) α∈(0,1)
  13. W k , p ( 𝐑 n ) C r , α ( 𝐑 n ) . W^{k,p}(\mathbf{R}^{n})\subset C^{r,\alpha}(\mathbf{R}^{n}).
  14. M M
  15. M M
  16. M M
  17. M M
  18. M M
  19. δ > 0 δ>0
  20. k > k>ℓ
  21. k n / p > n / q k−n/p>ℓ−n/q
  22. W k , p ( M ) W , q ( M ) W^{k,p}(M)\subset W^{\ell,q}(M)
  23. u u
  24. C C
  25. n n
  26. p p
  27. u L p * ( 𝐑 n ) C D u L p ( 𝐑 n ) . \|u\|_{L^{p^{*}}(\mathbf{R}^{n})}\leq C\|Du\|_{L^{p}(\mathbf{R}^{n})}.
  28. W 1 , p ( 𝐑 n ) \sub L p * ( 𝐑 n ) . W^{1,p}(\mathbf{R}^{n})\sub L^{p^{*}}(\mathbf{R}^{n}).
  29. q = p n n - α p q=\frac{pn}{n-\alpha p}
  30. C C
  31. p p
  32. I α f q C f p . \left\|I_{\alpha}f\right\|_{q}\leq C\|f\|_{p}.
  33. p = 1 p=1
  34. m { x : | I α f ( x ) | > λ } C ( f 1 λ ) q m\left\{x:\left|I_{\alpha}f(x)\right|>\lambda\right\}\leq C\left(\frac{\|f\|_{% 1}}{\lambda}\right)^{q}
  35. 1 / q = 1 α / n 1/q=1−α/n
  36. C C
  37. p p
  38. n n
  39. u C 0 , γ ( 𝐑 n ) C u W 1 , p ( 𝐑 n ) \|u\|_{C^{0,\gamma}(\mathbf{R}^{n})}\leq C\|u\|_{W^{1,p}(\mathbf{R}^{n})}
  40. γ = 1 - n p . \gamma=1-\frac{n}{p}.
  41. u u
  42. γ γ
  43. U U
  44. u C 0 , γ ( U ) C u W 1 , p ( U ) \|u\|_{C^{0,\gamma}(U)}\leq C\|u\|_{W^{1,p}(U)}
  45. C C
  46. n , p n,p
  47. U U
  48. U U
  49. U U
  50. 1 q = 1 p - k n . \frac{1}{q}=\frac{1}{p}-\frac{k}{n}.
  51. u L q ( U ) C u W k , p ( U ) \|u\|_{L^{q}(U)}\leq C\|u\|_{W^{k,p}(U)}
  52. C C
  53. k , p , n k,p,n
  54. U U
  55. k > n / p k>n/p
  56. u u
  57. u C k - [ n p ] - 1 , γ ( U ) , u\in C^{k-\left[\frac{n}{p}\right]-1,\gamma}(U),
  58. γ = { [ n p ] + 1 - n p n p 𝐙 any element in ( 0 , 1 ) n p 𝐙 \gamma=\begin{cases}\left[\frac{n}{p}\right]+1-\frac{n}{p}&\frac{n}{p}\notin% \mathbf{Z}\\ \,\text{any element in }(0,1)&\frac{n}{p}\in\mathbf{Z}\end{cases}
  59. u C k - [ n p ] - 1 , γ ( U ) C u W k , p ( U ) , \|u\|_{C^{k-\left[\frac{n}{p}\right]-1,\gamma}(U)}\leq C\|u\|_{W^{k,p}(U)},
  60. C C
  61. k , p , n , γ k,p,n,γ
  62. U U
  63. p = n , k = 1 p=n,k=1
  64. u W 1 , n ( 𝐑 n ) u\in W^{1,n}(\mathbf{R}^{n})
  65. u u
  66. u B M O C D u L n ( 𝐑 n ) , \|u\|_{BMO}\leq C\|Du\|_{L^{n}(\mathbf{R}^{n})},
  67. C C
  68. n n
  69. C > 0 C>0
  70. u L 2 ( 𝐑 n ) 1 + 2 / n C u L 1 ( 𝐑 n ) 2 / n D u L 2 ( 𝐑 n ) . \|u\|_{L^{2}(\mathbf{R}^{n})}^{1+2/n}\leq C\|u\|_{L^{1}(\mathbf{R}^{n})}^{2/n}% \|Du\|_{L^{2}(\mathbf{R}^{n})}.
  71. ρ ρ
  72. | x | ρ | u ^ ( x ) | 2 d x | x | ρ | x | 2 ρ 2 | u ^ ( x ) | 2 d x ρ - 2 𝐑 n | D u | 2 d x \int_{|x|\geq\rho}\left|\hat{u}(x)\right|^{2}\,dx\leq\int_{|x|\geq\rho}\frac{|% x|^{2}}{\rho^{2}}\left|\hat{u}(x)\right|^{2}\,dx\leq\rho^{-2}\int_{\mathbf{R}^% {n}}|Du|^{2}\,dx
  73. | u ^ | u L 1 |\hat{u}|\leq\|u\|_{L^{1}}
  74. ρ ρ
  75. | x | ρ | u ^ ( x ) | 2 d x ρ n ω n u L 1 2 \int_{|x|\leq\rho}|\hat{u}(x)|^{2}\,dx\leq\rho^{n}\omega_{n}\|u\|_{L^{1}}^{2}
  76. n n
  77. ρ ρ
  78. u ^ L 2 = u L 2 \|\hat{u}\|_{L^{2}}=\|u\|_{L^{2}}
  79. n = 1 n=1
  80. I I
  81. u L p ( I ) C u L q ( I ) 1 - a u W 1 , r ( I ) a , \|u\|_{L^{p}(I)}\leq C\|u\|^{1-a}_{L^{q}(I)}\|u\|^{a}_{W^{1,r}(I)},
  82. a ( 1 q - 1 r + 1 ) = 1 q - 1 p . a\left(\frac{1}{q}-\frac{1}{r}+1\right)=\frac{1}{q}-\frac{1}{p}.

Social_discount_rate.html

  1. ( 1 / ( 1 + r ) t ) \ (1/(1+r)^{t})
  2. ( 1 / r ) \ (1/r)

Socle_(mathematics).html

  1. soc ( M ) = { N N is a simple submodule of M } . \mathrm{soc}(M)=\sum\{N\mid N\,\text{ is a simple submodule of }M\}.\,
  2. soc ( M ) = { E E is an essential submodule of M } . \mathrm{soc}(M)=\bigcap\{E\mid E\,\text{ is an essential submodule of }M\}.\,

Solar_core.html

  1. × 10 3 8 \times 10^{3}8
  2. × 10 2 6 \times 10^{2}6
  3. { H 1 + 1 H D 2 + e + + v then D 2 + H 1 H 3 e + γ then H 3 e + H 3 e H 4 e + H 1 + H 1 \left\{\begin{aligned}&&\displaystyle{}^{1}\!H+^{1}\!H&\displaystyle% \rightarrow{}^{2}\!D+e^{+}+v\\ \displaystyle\,\text{then}&&\displaystyle{}^{2}\!D+{}^{1}\!H&\displaystyle% \rightarrow{}^{3}\!He+\gamma\\ \displaystyle\,\text{then}&&\displaystyle{}^{3}\!He+{}^{3}\!He&\displaystyle% \rightarrow{}^{4}\!He+{}^{1}\!H+{}^{1}\!H\\ \end{aligned}\right.
  4. { C 12 + H 1 N 13 + γ then N 13 C 13 + e + + ν then C 13 + H 1 N 14 + γ then N 14 + H 1 O 15 + γ then O 15 N 15 + e + + ν then N 15 + H 1 C 12 + H 4 e + γ \left\{\begin{aligned}&&\displaystyle{}^{12}\!C+{}^{1}\!H&\displaystyle% \rightarrow{}^{13}\!N+\gamma\\ \displaystyle\,\text{then}&&\displaystyle{}^{13}\!N&\displaystyle\rightarrow{}% ^{13}\!C+e^{+}+\nu\\ \displaystyle\,\text{then}&&\displaystyle{}^{13}\!C+{}^{1}\!H&\displaystyle% \rightarrow{}^{14}\!N+\gamma\\ \displaystyle\,\text{then}&&\displaystyle{}^{14}\!N+{}^{1}\!H&\displaystyle% \rightarrow{}^{15}\!O+\gamma\\ \displaystyle\,\text{then}&&\displaystyle{}^{15}\!O&\displaystyle\rightarrow{}% ^{15}\!N+e^{+}+\nu\\ \displaystyle\,\text{then}&&\displaystyle{}^{15}\!N+{}^{1}\!H&\displaystyle% \rightarrow{}^{12}\!C+{}^{4}\!He+\gamma\\ \end{aligned}\right.

Solid_Klein_bottle.html

  1. D 2 × I \scriptstyle D^{2}\times I
  2. M o ¨ × I \scriptstyle M\ddot{o}\times I
  3. I = [ 0 , 1 ] \scriptstyle I=[0,1]
  4. M o ¨ × [ 1 2 - ε , 1 2 + ε ] \scriptstyle M\ddot{o}\times[\frac{1}{2}-\varepsilon,\frac{1}{2}+\varepsilon]

Solving_the_geodesic_equations.html

  1. M M
  2. x a x^{a}
  3. d 2 x a d s 2 + Γ b c a d x b d s d x c d s = 0 \frac{d^{2}x^{a}}{ds^{2}}+\Gamma^{a}_{bc}\frac{dx^{b}}{ds}\frac{dx^{c}}{ds}=0
  4. M M
  5. Γ b c a \Gamma^{a}_{bc}
  6. Γ b c a = 1 2 g a d ( g c d , b + g b d , c - g b c , d ) \Gamma^{a}_{bc}=\frac{1}{2}g^{ad}\left(g_{cd,b}+g_{bd,c}-g_{bc,d}\right)
  7. g a b , c = g a b x c g_{ab,c}=\frac{\partial{g_{ab}}}{\partial{x^{c}}}
  8. n n
  9. n n
  10. n n
  11. L = g μ ν d x μ d s d x ν d s L=g_{\mu\nu}\frac{dx^{\mu}}{ds}\frac{dx^{\nu}}{ds}
  12. d s 2 ds^{2}
  13. - 1 = g μ ν x μ ˙ x ν ˙ -1=g_{\mu\nu}\dot{x^{\mu}}\dot{x^{\nu}}
  14. s s

Solvmanifold.html

  1. 𝔤 \mathfrak{g}
  2. ( X ) : 𝔤 𝔤 , X 𝔤 (X):\mathfrak{g}\to\mathfrak{g},X\in\mathfrak{g}
  3. 𝔤 \mathfrak{g}

Sommerfeld_identity.html

  1. e i k R R = 0 I 0 ( λ r ) e - μ | z | < m t p l > λ d λ μ \frac{{e^{ikR}}}{R}=\int\limits_{0}^{\infty}I_{0}(\lambda r)e^{-\mu\left|z% \right|}\frac{<}{m}tpl>{{\lambda d\lambda}}{{\mu}}
  2. μ = λ 2 - k 2 \mu=\sqrt{\lambda^{2}-k^{2}}
  3. z ± z\rightarrow\pm\infty
  4. R 2 = r 2 + z 2 R^{2}=r^{2}+z^{2}
  5. R R
  6. r r
  7. ( r , ϕ , z ) (r,\phi,z)
  8. I 0 I_{0}
  9. I n ( ρ ) = J n ( i ρ ) I_{n}(\rho)=J_{n}(i\rho)
  10. e i k 0 r r = i 0 d k ρ < m t p l > k ρ k z J 0 ( k ρ ρ ) e i k z | z | \frac{{e^{ik_{0}r}}}{r}=i\int\limits_{0}^{\infty}{dk_{\rho}\frac{<}{m}tpl>{{k_% {\rho}}}{{k_{z}}}J_{0}(k_{\rho}\rho)e^{ik_{z}\left|z\right|}}
  11. k z = ( k 0 2 - k ρ 2 ) 1 / 2 k_{z}=(k_{0}^{2}-k_{\rho}^{2})^{1/2}
  12. r r
  13. ρ \rho
  14. ( ρ , ϕ , z ) (\rho,\phi,z)
  15. ρ \rho
  16. z z
  17. k ρ k_{\rho}

Sorptivity.html

  1. I = S t I=S\sqrt{t}
  2. I = S t + A 1 t I=S\sqrt{t}\ +A_{1}t
  3. i = 0.5 S / t + A 1 i=0.5S/\sqrt{t}\ +A_{1}
  4. S ( θ 0 , θ i ) = ( θ 0 - θ i ) L f t 1 / 2 S(\theta_{0},\theta_{i})=\frac{(\theta_{0}-\theta_{i})L_{f}}{t^{1/2}}

Source_field.html

  1. J J
  2. S s o u r c e = J Φ S_{source}=J\Phi
  3. Φ \Phi
  4. Φ \Phi
  5. Φ \Phi

Source_transformation.html

  1. V = I * Z V=I*Z
  2. V = I Z , I = V Z V=I\cdot Z,\qquad I=\cfrac{V}{Z}

Spatial_analysis.html

  1. I I
  2. C C
  3. G G
  4. I I
  5. C C
  6. G G
  7. I I
  8. C C

Spatial_cutoff_frequency.html

  1. f o = 1 λ × ( f / # ) cycles / millimeter , f_{o}={1\over{\lambda\times\mathrm{(f/\#)}}}\ \ \mathrm{cycles/millimeter}\ ,
  2. λ \lambda

Species_evenness.html

  1. J = H H max J^{\prime}={H^{\prime}\over H_{\max}^{\prime}}
  2. H H^{\prime}
  3. H max H_{\max}^{\prime}
  4. H H^{\prime}
  5. H max = - i = 1 S 1 S ln 1 S = ln S . H^{\prime}_{\max}=-\sum_{i=1}^{S}{1\over S}\ln{1\over S}=\ln S.
  6. H min > 0 H_{\min}^{\prime}>0

Specific_kinetic_energy.html

  1. e k = 1 2 v 2 \begin{matrix}e_{k}=\frac{1}{2}\end{matrix}v^{2}
  2. e k e_{k}
  3. v v

Spectral_asymmetry.html

  1. ω n \omega_{n}
  2. B = lim t 0 1 2 n sgn ( ω n ) exp - t | ω n | B=\lim_{t\to 0}\frac{1}{2}\sum_{n}\operatorname{sgn}(\omega_{n})\exp-t|\omega_% {n}|
  3. sgn ( x ) \operatorname{sgn}(x)
  4. ω n = n + θ \omega_{n}=n+\theta
  5. B = θ B=\theta
  6. E = lim t 0 1 2 n | ω n | exp - t | ω n | E=\lim_{t\to 0}\frac{1}{2}\sum_{n}|\omega_{n}|\exp-t|\omega_{n}|

Spectral_phase_interferometry_for_direct_electric-field_reconstruction.html

  1. S ( ω ) = | E ( ω ) + E ( ω - Ω ) e i ω τ | 2 = I ( ω ) + I ( ω - Ω ) + 2 I ( ω ) I ( ω - Ω ) cos [ ϕ ( ω ) - ϕ ( ω - Ω ) - ω τ ] \begin{aligned}\displaystyle S(\omega)&\displaystyle=|E(\omega)+E(\omega-% \Omega)e^{i\omega\tau}|^{2}\\ &\displaystyle=I(\omega)+I(\omega-\Omega)+2\sqrt{I(\omega)I(\omega-\Omega)}% \cos[\phi(\omega)-\phi(\omega-\Omega)-\omega\tau]\end{aligned}
  2. E ( ω ) E(\omega)
  3. Ω \Omega
  4. τ \tau
  5. I ( ω ) = | E ( ω ) | 2 I(\omega)=|E(\omega)|^{2}
  6. ϕ ( ω ) \phi(\omega)
  7. δ ω 2 π / τ \delta\omega\sim 2\pi/\tau
  8. S ~ ( t ~ ) \displaystyle\widetilde{S}(\widetilde{t})
  9. E ~ d c ( t ~ ) = 𝔉 [ I ( ω ) + I ( ω - Ω ) ] \widetilde{E}^{dc}(\widetilde{t})=\mathfrak{F}[I(\omega)+I(\omega-\Omega)]
  10. t ~ \widetilde{t}
  11. E ~ ± a c ( t ~ τ ) = 𝔉 { I ( ω ) I ( ω - Ω ) e ± i [ ϕ ( ω ) - ϕ ( ω - Ω ) ] e ± i ω τ } \widetilde{E}^{\pm ac}(\widetilde{t}\mp\tau)=\mathfrak{F}\{\sqrt{I(\omega)I(% \omega-\Omega)}e^{\pm i[\phi(\omega)-\phi(\omega-\Omega)]}e^{\pm i\omega\tau}\}
  12. D ( ω , Ω ) \displaystyle D(\omega,\Omega)
  13. θ ( ω ) \displaystyle\theta(\omega)
  14. ϕ ( ω N - Ω / 2 ) - n = 0 N ω n + 1 - ω n 2 Ω [ θ ( ω n ) + θ ( ω n + 1 ) ] \phi(\omega_{N}-\Omega/2)\approxeq-\sum_{n=0}^{N}\frac{\omega_{n+1}-\omega_{n}% }{2\Omega}[\theta(\omega_{n})+\theta(\omega_{n+1})]
  15. ϕ ( ω 0 + N | Ω | ) \displaystyle\phi(\omega_{0}+N|\Omega|)
  16. N N
  17. { ω N } = { ω 0 + N | Ω | } \{\omega_{N}\}=\{\omega_{0}+N|\Omega|\}
  18. D ( ω ) D(\omega)

Spectral_theory_of_compact_operators.html

  1. 𝐂 n = i = 1 k Y i . \mathbf{C}^{n}=\bigoplus_{i=1}^{k}Y_{i}.
  2. 1 - ε d ( x , Y ) 1 1-\varepsilon\leq d(x,Y)\leq 1
  3. Y 1 Y n Y m Y_{1}\supset\cdots\supset Y_{n}\cdots\supset Y_{m}\cdots
  4. ( C - I ) y n - ( C - I ) y m - y m Y n + 1 , (C-I)y_{n}-(C-I)y_{m}-y_{m}\in Y_{n+1},
  5. ( C - I ) y n + y n - ( C - I ) y m Y m - 1 , (C-I)y_{n}+y_{n}-(C-I)y_{m}\in Y_{m-1},
  6. ( C - λ n ) y n + λ n y n - ( C - λ m ) y m Y m - 1 , (C-\lambda_{n})y_{n}+\lambda_{n}y_{n}-(C-\lambda_{m})y_{m}\in Y_{m-1},
  7. n { | λ | > 1 n } = n S n . \bigcup_{n}\left\{|\lambda|>\tfrac{1}{n}\right\}=\bigcup_{n}S_{n}.
  8. E ( λ ) = 1 2 π i γ ( ξ - C ) - 1 d ξ E(\lambda)={1\over 2\pi i}\int_{\gamma}(\xi-C)^{-1}d\xi
  9. E ( λ ) ( λ - C ) ν = ( λ - C ) ν E ( λ ) = 0. E(\lambda)(\lambda-C)^{\nu}=(\lambda-C)^{\nu}E(\lambda)=0.

Spectrum_of_a_theory.html

  1. I ( T , ξ ) < ω 1 ( | ξ | ) \textstyle I(T,\aleph_{\xi})<\beth_{\omega_{1}}(|\xi|)
  2. d + 1 ( | α + ω | ) \beth_{d+1}(|\alpha+\omega|)
  3. d - 1 ( | α + ω | 2 0 ) \beth_{d-1}(|\alpha+\omega|^{2^{\aleph_{0}}})
  4. d - 1 ( | α + ω | 0 + 2 ) \beth_{d-1}(|\alpha+\omega|^{\aleph_{0}}+\beth_{2})
  5. d - 1 ( | α + ω | + 2 ) \beth_{d-1}(|\alpha+\omega|+\beth_{2})
  6. d - 1 ( | α + ω | 0 ) \beth_{d-1}(|\alpha+\omega|^{\aleph_{0}})
  7. d - 1 ( | α + ω | + 1 ) \beth_{d-1}(|\alpha+\omega|+\beth_{1})
  8. d - 1 ( | α + ω | ) \beth_{d-1}(|\alpha+\omega|)
  9. d - 2 ( | α + ω | | α + 1 | ) \beth_{d-2}(|\alpha+\omega|^{|\alpha+1|})
  10. 2 \beth_{2}
  11. | ( α + 1 ) n / G | - | α n / G | |(\alpha+1)^{n}/G|-|\alpha^{n}/G|
  12. 1 1
  13. 0
  14. n ( T ) \beth_{n}(T)
  15. 0 ( T ) = T \beth_{0}(T)=T
  16. n + 1 ( T ) = 2 n ( T ) \beth_{n+1}(T)=2^{\beth_{n}(T)}
  17. I ( n ( T ) , λ ) = min ( n ( I ( T , λ ) ) , 2 λ ) I(\beth_{n}(T),\lambda)=\min(\beth_{n}(I(T,\lambda)),2^{\lambda})

Sphere_theorem.html

  1. ( 1 , 4 ] (1,4]
  2. ( 1 , 4 ] (1,4]
  3. [ 1 , 4 ] [1,4]

Sphere_theorem_(3-manifolds).html

  1. M M
  2. π 2 ( M ) \pi_{2}(M)
  3. π 2 ( M ) \pi_{2}(M)
  4. S 2 M S^{2}\to M
  5. M M
  6. N N
  7. π 1 ( M ) \pi_{1}(M)
  8. π 2 ( M ) \pi_{2}(M)
  9. f : S 2 M f\colon S^{2}\to M
  10. [ f ] N [f]\notin N
  11. U U
  12. Σ ( f ) \Sigma(f)
  13. g : S 2 M g\colon S^{2}\to M
  14. [ g ] N [g]\notin N
  15. g ( S 2 ) f ( S 2 ) U g(S^{2})\subset f(S^{2})\cup U
  16. g : S 2 g ( S 2 ) g\colon S^{2}\to g(S^{2})
  17. g ( S 2 ) g(S^{2})
  18. M M

Spherical_law_of_cosines.html

  1. 𝐮 , 𝐯 \mathbf{u},\mathbf{v}
  2. 𝐰 \mathbf{w}
  3. a a
  4. 𝐮 \mathbf{u}
  5. 𝐯 ) , b \mathbf{v}),b
  6. 𝐮 \mathbf{u}
  7. 𝐰 \mathbf{w}
  8. c c
  9. 𝐯 \mathbf{v}
  10. 𝐰 \mathbf{w}
  11. c c
  12. C C
  13. cos ( c ) = cos ( a ) cos ( b ) + sin ( a ) sin ( b ) cos ( C ) . \cos(c)=\cos(a)\cos(b)+\sin(a)\sin(b)\cos(C).\,
  14. a , b a,b
  15. c c
  16. C = π / 2 C={π}/{2}
  17. c o s C = 0 cosC=0
  18. cos ( c ) = cos ( a ) cos ( b ) . \cos(c)=\cos(a)\cos(b).\,
  19. cos ( A ) = - cos ( B ) cos ( C ) + sin ( B ) sin ( C ) cos ( a ) \cos(A)=-\cos(B)\cos(C)+\sin(B)\sin(C)\cos(a)\,
  20. A A
  21. B B
  22. a a
  23. b b
  24. c c
  25. c c
  26. 𝐮 , 𝐯 \mathbf{u},\mathbf{v}
  27. 𝐰 \mathbf{w}
  28. cos ( a ) = 𝐮 𝐯 \cos(a)=\mathbf{u}\cdot\mathbf{v}
  29. cos ( b ) = 𝐮 𝐰 \cos(b)=\mathbf{u}\cdot\mathbf{w}
  30. cos ( c ) = 𝐯 𝐰 \cos(c)=\mathbf{v}\cdot\mathbf{w}
  31. C C
  32. 𝐮 \mathbf{u}
  33. a a
  34. b b
  35. 𝐮 \mathbf{u}
  36. 𝐮 - 𝐯 \mathbf{u}-\mathbf{v}
  37. 𝐯 \mathbf{v}
  38. 𝐮 \mathbf{u}
  39. 𝐭 a = 𝐯 - 𝐮 ( 𝐮 𝐯 ) 𝐯 - 𝐮 ( 𝐮 𝐯 ) = 𝐯 - 𝐮 cos ( a ) sin ( a ) \mathbf{t}_{a}=\frac{\mathbf{v}-\mathbf{u}(\mathbf{u}\cdot\mathbf{v})}{\left\|% \mathbf{v}-\mathbf{u}(\mathbf{u}\cdot\mathbf{v})\right\|}=\frac{\mathbf{v}-% \mathbf{u}\cos(a)}{\sin(a)}
  40. 𝐭 b = 𝐰 - 𝐮 cos ( b ) sin ( b ) . \mathbf{t}_{b}=\frac{\mathbf{w}-\mathbf{u}\cos(b)}{\sin(b)}.
  41. C C
  42. cos ( C ) = 𝐭 a 𝐭 b = cos ( c ) - cos ( a ) cos ( b ) sin ( a ) sin ( b ) \cos(C)=\mathbf{t}_{a}\cdot\mathbf{t}_{b}=\frac{\cos(c)-\cos(a)\cos(b)}{\sin(a% )\sin(b)}
  43. 𝐮 \mathbf{u}
  44. 𝐎 \mathbf{O}
  45. 𝐯 \mathbf{v}
  46. 𝐰 \mathbf{w}
  47. 𝐲 \mathbf{y}
  48. 𝐳 \mathbf{z}
  49. 𝐮 , 𝐲 \mathbf{u},\mathbf{y}
  50. 𝐳 \mathbf{z}
  51. 𝐎 , 𝐲 \mathbf{O},\mathbf{y}
  52. 𝐳 \mathbf{z}
  53. t a n a tana
  54. t a n b tanb
  55. C C
  56. s e c a seca
  57. s e c b secb
  58. c c
  59. tan 2 a + tan 2 b - 2 tan a tan b cos C \displaystyle\tan^{2}a+\tan^{2}b-2\tan a\tan b\cos C
  60. - tan a tan b cos C = 1 - sec a sec b cos c -\tan a\tan b\cos C=1-\sec a\sec b\cos c
  61. c o s a c o s b cosacosb
  62. a , b a,b
  63. c c
  64. c 2 a 2 + b 2 - 2 a b cos ( C ) . c^{2}\approx a^{2}+b^{2}-2ab\cos(C).\,\!
  65. cos ( a ) = 1 - a 2 2 + O ( a 4 ) , sin ( a ) = a + O ( a 3 ) \cos(a)=1-\frac{a^{2}}{2}+O(a^{4}),\,\sin(a)=a+O(a^{3})
  66. 1 - c 2 2 + O ( c 4 ) = 1 - a 2 2 - b 2 2 + a 2 b 2 4 + O ( a 4 ) + O ( b 4 ) + cos ( C ) ( a b + O ( a 3 b ) + O ( a b 3 ) + O ( a 3 b 3 ) ) 1-\frac{c^{2}}{2}+O(c^{4})=1-\frac{a^{2}}{2}-\frac{b^{2}}{2}+\frac{a^{2}b^{2}}% {4}+O(a^{4})+O(b^{4})+\cos(C)(ab+O(a^{3}b)+O(ab^{3})+O(a^{3}b^{3}))
  67. c 2 = a 2 + b 2 - 2 a b cos ( C ) + O ( c 4 ) + O ( a 4 ) + O ( b 4 ) + O ( a 2 b 2 ) + O ( a 3 b ) + O ( a b 3 ) + O ( a 3 b 3 ) . c^{2}=a^{2}+b^{2}-2ab\cos(C)+O(c^{4})+O(a^{4})+O(b^{4})+O(a^{2}b^{2})+O(a^{3}b% )+O(ab^{3})+O(a^{3}b^{3}).
  68. a a
  69. b b
  70. 1 1
  71. O ( c 4 ) + O ( a 3 b ) + O ( a b 3 ) . O(c^{4})+O(a^{3}b)+O(ab^{3}).\,\!

Spherical_multipole_moments.html

  1. ρ ( 𝐫 ) \rho(\mathbf{r^{\prime}})
  2. 𝐫 \mathbf{r^{\prime}}
  3. 𝐫 \mathbf{r}
  4. 𝐫 \mathbf{r^{\prime}}
  5. ( r , θ , ϕ ) (r^{\prime},\theta^{\prime},\phi^{\prime})
  6. r r^{\prime}
  7. θ \theta^{\prime}
  8. ϕ \phi^{\prime}
  9. 𝐫 \mathbf{r^{\prime}}
  10. Φ ( 𝐫 ) = q 4 π ε 1 R = q 4 π ε 1 r 2 + r 2 - 2 r r cos γ . \Phi(\mathbf{r})=\frac{q}{4\pi\varepsilon}\frac{1}{R}=\frac{q}{4\pi\varepsilon% }\frac{1}{\sqrt{r^{2}+r^{\prime 2}-2r^{\prime}r\cos\gamma}}.
  11. R = def | 𝐫 - 𝐫 | R\ \stackrel{\mathrm{def}}{=}\ \left|\mathbf{r}-\mathbf{r^{\prime}}\right|
  12. γ \gamma
  13. 𝐫 \mathbf{r}
  14. 𝐫 \mathbf{r^{\prime}}
  15. r r
  16. r r^{\prime}
  17. ( r / r ) < 1 (r^{\prime}/r)<1
  18. Φ ( 𝐫 ) = q 4 π ε r l = 0 ( r r ) l P l ( cos γ ) \Phi(\mathbf{r})=\frac{q}{4\pi\varepsilon r}\sum_{l=0}^{\infty}\left(\frac{r^{% \prime}}{r}\right)^{l}P_{l}(\cos\gamma)
  19. cos γ \cos\gamma
  20. cos γ = cos θ cos θ + sin θ sin θ cos ( ϕ - ϕ ) \cos\gamma=\cos\theta\cos\theta^{\prime}+\sin\theta\sin\theta^{\prime}\cos(% \phi-\phi^{\prime})
  21. 𝐳 ^ \mathbf{\hat{z}}
  22. 𝐫 ^ \mathbf{\hat{r}}
  23. 𝐫 ^ \mathbf{\hat{r}^{\prime}}
  24. cos γ \cos\gamma
  25. P l ( cos γ ) = 4 π 2 l + 1 m = - l l Y l m ( θ , ϕ ) Y l m * ( θ , ϕ ) P_{l}(\cos\gamma)=\frac{4\pi}{2l+1}\sum_{m=-l}^{l}Y_{lm}(\theta,\phi)Y_{lm}^{*% }(\theta^{\prime},\phi^{\prime})
  26. Y l m Y_{lm}
  27. Φ ( 𝐫 ) = q 4 π ε r l = 0 ( r r ) l ( 4 π 2 l + 1 ) m = - l l Y l m ( θ , ϕ ) Y l m * ( θ , ϕ ) \Phi(\mathbf{r})=\frac{q}{4\pi\varepsilon r}\sum_{l=0}^{\infty}\left(\frac{r^{% \prime}}{r}\right)^{l}\left(\frac{4\pi}{2l+1}\right)\sum_{m=-l}^{l}Y_{lm}(% \theta,\phi)Y_{lm}^{*}(\theta^{\prime},\phi^{\prime})
  28. Φ ( 𝐫 ) = 1 4 π ε l = 0 m = - l l ( Q l m r l + 1 ) 4 π 2 l + 1 Y l m ( θ , ϕ ) \Phi(\mathbf{r})=\frac{1}{4\pi\varepsilon}\sum_{l=0}^{\infty}\sum_{m=-l}^{l}% \left(\frac{Q_{lm}}{r^{l+1}}\right)\sqrt{\frac{4\pi}{2l+1}}Y_{lm}(\theta,\phi)
  29. Q l m = def q ( r ) l 4 π 2 l + 1 Y l m * ( θ , ϕ ) Q_{lm}\ \stackrel{\mathrm{def}}{=}\ q\left(r^{\prime}\right)^{l}\sqrt{\frac{4% \pi}{2l+1}}Y_{lm}^{*}(\theta^{\prime},\phi^{\prime})
  30. r r
  31. r r^{\prime}
  32. Φ ( 𝐫 ) = q 4 π ε r l = 0 ( r r ) l ( 4 π 2 l + 1 ) m = - l l Y l m ( θ , ϕ ) Y l m * ( θ , ϕ ) \Phi(\mathbf{r})=\frac{q}{4\pi\varepsilon r^{\prime}}\sum_{l=0}^{\infty}\left(% \frac{r}{r^{\prime}}\right)^{l}\left(\frac{4\pi}{2l+1}\right)\sum_{m=-l}^{l}Y_% {lm}(\theta,\phi)Y_{lm}^{*}(\theta^{\prime},\phi^{\prime})
  33. Φ ( 𝐫 ) = 1 4 π ε l = 0 m = - l l I l m r l 4 π 2 l + 1 Y l m ( θ , ϕ ) \Phi(\mathbf{r})=\frac{1}{4\pi\varepsilon}\sum_{l=0}^{\infty}\sum_{m=-l}^{l}I_% {lm}r^{l}\sqrt{\frac{4\pi}{2l+1}}Y_{lm}(\theta,\phi)
  34. I l m = def q ( r ) l + 1 4 π 2 l + 1 Y l m * ( θ , ϕ ) I_{lm}\ \stackrel{\mathrm{def}}{=}\ \frac{q}{\left(r^{\prime}\right)^{l+1}}% \sqrt{\frac{4\pi}{2l+1}}Y_{lm}^{*}(\theta^{\prime},\phi^{\prime})
  35. r < r_{<}
  36. r > r_{>}
  37. r r
  38. r r^{\prime}
  39. Φ ( 𝐫 ) = q 4 π ε l = 0 r < l r > l + 1 ( 4 π 2 l + 1 ) m = - l l Y l m ( θ , ϕ ) Y l m * ( θ , ϕ ) \Phi(\mathbf{r})=\frac{q}{4\pi\varepsilon}\sum_{l=0}^{\infty}\frac{r_{<}^{l}}{% r_{>}^{l+1}}\left(\frac{4\pi}{2l+1}\right)\sum_{m=-l}^{l}Y_{lm}(\theta,\phi)Y_% {lm}^{*}(\theta^{\prime},\phi^{\prime})
  40. q q
  41. ρ ( 𝐫 ) d 𝐫 \rho(\mathbf{r}^{\prime})d\mathbf{r}^{\prime}
  42. Φ ( 𝐫 ) = 1 4 π ε l = 0 m = - l l ( Q l m r l + 1 ) 4 π 2 l + 1 Y l m ( θ , ϕ ) \Phi(\mathbf{r})=\frac{1}{4\pi\varepsilon}\sum_{l=0}^{\infty}\sum_{m=-l}^{l}% \left(\frac{Q_{lm}}{r^{l+1}}\right)\sqrt{\frac{4\pi}{2l+1}}Y_{lm}(\theta,\phi)
  43. Q l m = def d 𝐫 ρ ( 𝐫 ) ( r ) l 4 π 2 l + 1 Y l m * ( θ , ϕ ) Q_{lm}\ \stackrel{\mathrm{def}}{=}\ \int d\mathbf{r}^{\prime}\rho(\mathbf{r}^{% \prime})\left(r^{\prime}\right)^{l}\sqrt{\frac{4\pi}{2l+1}}Y_{lm}^{*}(\theta^{% \prime},\phi^{\prime})
  44. Φ ( 𝐫 ) = 1 4 π ε l = 0 m = - l l I l m r l 4 π 2 l + 1 Y l m ( θ , ϕ ) \Phi(\mathbf{r})=\frac{1}{4\pi\varepsilon}\sum_{l=0}^{\infty}\sum_{m=-l}^{l}I_% {lm}r^{l}\sqrt{\frac{4\pi}{2l+1}}Y_{lm}(\theta,\phi)
  45. I l m = def d 𝐫 ρ ( 𝐫 ) ( r ) l + 1 4 π 2 l + 1 Y l m * ( θ , ϕ ) I_{lm}\ \stackrel{\mathrm{def}}{=}\ \int d\mathbf{r}^{\prime}\frac{\rho(% \mathbf{r}^{\prime})}{\left(r^{\prime}\right)^{l+1}}\sqrt{\frac{4\pi}{2l+1}}Y_% {lm}^{*}(\theta^{\prime},\phi^{\prime})
  46. ρ 1 ( 𝐫 ) \rho_{1}(\mathbf{r}^{\prime})
  47. ρ 2 ( 𝐫 ) \rho_{2}(\mathbf{r}^{\prime})
  48. U = def d 𝐫 ρ 2 ( 𝐫 ) Φ 1 ( 𝐫 ) U\ \stackrel{\mathrm{def}}{=}\ \int d\mathbf{r}\rho_{2}(\mathbf{r})\Phi_{1}(% \mathbf{r})
  49. Φ 1 ( 𝐫 ) \Phi_{1}(\mathbf{r})
  50. Φ ( 𝐫 ) = 1 4 π ε l = 0 m = - l l Q 1 l m ( 1 r l + 1 ) 4 π 2 l + 1 Y l m ( θ , ϕ ) \Phi(\mathbf{r})=\frac{1}{4\pi\varepsilon}\sum_{l=0}^{\infty}\sum_{m=-l}^{l}Q_% {1lm}\left(\frac{1}{r^{l+1}}\right)\sqrt{\frac{4\pi}{2l+1}}Y_{lm}(\theta,\phi)
  51. Q 1 l m Q_{1lm}
  52. l m lm
  53. U = 1 4 π ε l = 0 m = - l l Q 1 l m d 𝐫 ρ 2 ( 𝐫 ) ( 1 r l + 1 ) 4 π 2 l + 1 Y l m ( θ , ϕ ) U=\frac{1}{4\pi\varepsilon}\sum_{l=0}^{\infty}\sum_{m=-l}^{l}Q_{1lm}\int d% \mathbf{r}\ \rho_{2}(\mathbf{r})\left(\frac{1}{r^{l+1}}\right)\sqrt{\frac{4\pi% }{2l+1}}Y_{lm}(\theta,\phi)
  54. I 2 l m I_{2lm}
  55. U = 1 4 π ε l = 0 m = - l l Q 1 l m I 2 l m * U=\frac{1}{4\pi\varepsilon}\sum_{l=0}^{\infty}\sum_{m=-l}^{l}Q_{1lm}I_{2lm}^{*}
  56. ϕ \phi^{\prime}
  57. ϕ \phi^{\prime}
  58. Q l m Q_{lm}
  59. I l m I_{lm}
  60. m = 0 m=0
  61. P l ( cos θ ) = def 4 π 2 l + 1 Y l 0 ( θ , ϕ ) P_{l}(\cos\theta)\ \stackrel{\mathrm{def}}{=}\ \sqrt{\frac{4\pi}{2l+1}}Y_{l0}(% \theta,\phi)
  62. Φ ( 𝐫 ) = 1 4 π ε l = 0 ( Q l r l + 1 ) P l ( cos θ ) \Phi(\mathbf{r})=\frac{1}{4\pi\varepsilon}\sum_{l=0}^{\infty}\left(\frac{Q_{l}% }{r^{l+1}}\right)P_{l}(\cos\theta)
  63. Q l = def d 𝐫 ρ ( 𝐫 ) ( r ) l P l ( cos θ ) Q_{l}\ \stackrel{\mathrm{def}}{=}\ \int d\mathbf{r}^{\prime}\rho(\mathbf{r}^{% \prime})\left(r^{\prime}\right)^{l}P_{l}(\cos\theta^{\prime})
  64. z z
  65. Φ ( 𝐫 ) = 1 4 π ε l = 0 I l r l P l ( cos θ ) \Phi(\mathbf{r})=\frac{1}{4\pi\varepsilon}\sum_{l=0}^{\infty}I_{l}r^{l}P_{l}(% \cos\theta)
  66. I l = def d 𝐫 ρ ( 𝐫 ) ( r ) l + 1 P l ( cos θ ) I_{l}\ \stackrel{\mathrm{def}}{=}\ \int d\mathbf{r}^{\prime}\frac{\rho(\mathbf% {r}^{\prime})}{\left(r^{\prime}\right)^{l+1}}P_{l}(\cos\theta^{\prime})
  67. z z

Spin_density_wave.html

  1. N ( E F ) N(E_{F})
  2. N ( E F ) Δ 2 N(E_{F})\Delta^{2}
  3. Δ \Delta
  4. q q
  5. N ( E F ) N(E_{F})
  6. Γ \Gamma
  7. q q
  8. 2 π / q 2\pi/q

Spin_magnetic_moment.html

  1. μ S = g q 2 m S = γ S \vec{\mu}\text{S}\ =\ g\ \frac{q}{2m}\ \vec{S}=\gamma\vec{S}
  2. γ g q 2 m \gamma\equiv g\frac{q}{2m}
  3. μ S - 2 - e 2 m e 2 = μ B \mu\text{S}\approx-2\frac{-e}{2m\text{e}}\frac{\hbar}{2}=\mu\text{B}
  4. μ S = - g μ B σ 2 \vec{\mu}\text{S}=-\frac{g\mu\text{B}\vec{\sigma}}{2}
  5. s ( s + 1 ) \sqrt{s}{(}{s}{+1)}
  6. | μ S | = g μ B s ( s + 1 ) |\vec{\mu}\text{S}|=g\mu\text{B}\sqrt{s(s+1)}

Spinor_field.html

  1. π 𝐒 : 𝐒 M \pi_{\mathbf{S}}:{\mathbf{S}}\to M\,
  2. π 𝐏 : 𝐏 M \pi_{\mathbf{P}}:{\mathbf{P}}\to M\,
  3. F S O ( M ) M \mathrm{F}_{SO}(M)\to M
  4. ρ : Spin ( n ) SO ( n ) . \rho:{\mathrm{Spin}}(n)\to{\mathrm{SO}}(n)\,.
  5. π 𝐒 : 𝐒 M \pi_{\mathbf{S}}:{\mathbf{S}}\to M\,
  6. 𝐒 = 𝐏 × κ Δ n {\mathbf{S}}={\mathbf{P}}\times_{\kappa}\Delta_{n}\,
  7. κ : Spin ( n ) U ( Δ n ) , \kappa:{\mathrm{Spin}}(n)\to{\mathrm{U}}(\Delta_{n}),\,
  8. ψ : M 𝐒 \psi:M\to{\mathbf{S}}\,
  9. π 𝐒 ψ : M M \pi_{\mathbf{S}}\circ\psi:M\to M\,

Split-biquaternion.html

  1. q = w + x i + y j + z k q=w+xi+yj+zk\!
  2. e i e j = { - 1 i = j , - e j e i i j e_{i}e_{j}=\Bigg\{\begin{matrix}-1&i=j,\\ -e_{j}e_{i}&i\not=j\end{matrix}
  3. C 0 , 3 ( ) = 𝔻 C\ell_{0,3}(\mathbb{R})=\mathbb{H}\otimes\mathbb{D}
  4. C 0 , 3 ( ) = . C\ell_{0,3}(\mathbb{R})=\mathbb{H}\oplus\mathbb{H}.
  5. \mathbb{H}\oplus\mathbb{H}
  6. ( a b ) (a\oplus b)
  7. ( c d ) (c\oplus d)
  8. a c b d ac\oplus bd
  9. . \mathbb{H}\oplus\mathbb{H}.
  10. p q = u w + v z + ( u z + v w ) ω . pq=uw+vz+(uz+vw)\omega.\!
  11. \mathbb{H}\oplus\mathbb{H}
  12. p ( u + v ) ( u - v ) , q ( w + z ) ( w - z ) . p\mapsto(u+v)\oplus(u-v),\quad q\mapsto(w+z)\oplus(w-z).
  13. \mathbb{H}\oplus\mathbb{H}
  14. \mathbb{H}\oplus\mathbb{H}
  15. ( u + v ) ( w + z ) ( u - v ) ( w - z ) . (u+v)(w+z)\oplus(u-v)(w-z).
  16. . \mathbb{H}\oplus\mathbb{H}.
  17. C 2 ( ) = . C\ell_{2}(\mathbb{C})=\mathbb{H}\otimes\mathbb{C}.
  18. 𝔻 \mathbb{D}\otimes\mathbb{H}
  19. \mathbb{H}\oplus\mathbb{H}

Split-octonion.html

  1. ( a + b ) ( c + d ) = ( a c + λ d b ¯ ) + ( a ¯ d + c b ) (a+\ell b)(c+\ell d)=(ac+\lambda d\bar{b})+\ell(\bar{a}d+cb)
  2. λ = 2 . \lambda=\ell^{2}.
  3. x = x 0 + x 1 i + x 2 j + x 3 k + x 4 + x 5 i + x 6 j + x 7 k , x=x_{0}+x_{1}\,i+x_{2}\,j+x_{3}\,k+x_{4}\,\ell+x_{5}\,\ell i+x_{6}\,\ell j+x_{% 7}\,\ell k,
  4. 1 1\,
  5. i i\,
  6. j j\,
  7. k k\,
  8. \ell\,
  9. i \ell i\,
  10. j \ell j\,
  11. k \ell k\,
  12. i i\,
  13. - 1 -1\,
  14. k k\,
  15. - j -j\,
  16. - i -\ell i\,
  17. \ell\,
  18. - k -\ell k\,
  19. j \ell j\,
  20. j j\,
  21. - k -k\,
  22. - 1 -1\,
  23. i i\,
  24. - j -\ell j\,
  25. k \ell k\,
  26. \ell\,
  27. - i -\ell i\,
  28. k k\,
  29. j j\,
  30. - i -i\,
  31. - 1 -1\,
  32. - k -\ell k\,
  33. - j -\ell j\,
  34. i \ell i\,
  35. \ell\,
  36. \ell\,
  37. i \ell i\,
  38. j \ell j\,
  39. k \ell k\,
  40. 1 1\,
  41. i i\,
  42. j j\,
  43. k k\,
  44. i \ell i\,
  45. - -\ell\,
  46. - k -\ell k\,
  47. j \ell j\,
  48. - i -i\,
  49. 1 1\,
  50. k k\,
  51. - j -j\,
  52. j \ell j\,
  53. k \ell k\,
  54. - -\ell\,
  55. - i -\ell i\,
  56. - j -j\,
  57. - k -k\,
  58. 1 1\,
  59. i i\,
  60. k \ell k\,
  61. - j -\ell j\,
  62. i \ell i\,
  63. - -\ell\,
  64. - k -k\,
  65. j j\,
  66. - i -i\,
  67. 1 1\,
  68. e i e j = - δ i j e 0 + ε i j k e k , e_{i}e_{j}=-\delta_{ij}e_{0}+\varepsilon_{ijk}e_{k},\,
  69. ε i j k \varepsilon_{ijk}
  70. e i e 0 = e 0 e i = e i ; e 0 e 0 = e 0 , e_{i}e_{0}=e_{0}e_{i}=e_{i};\,\,\,\,e_{0}e_{0}=e_{0},\,
  71. x ¯ = x 0 - x 1 i - x 2 j - x 3 k - x 4 - x 5 i - x 6 j - x 7 k \bar{x}=x_{0}-x_{1}\,i-x_{2}\,j-x_{3}\,k-x_{4}\,\ell-x_{5}\,\ell i-x_{6}\,\ell j% -x_{7}\,\ell k
  72. N ( x ) = x ¯ x = ( x 0 2 + x 1 2 + x 2 2 + x 3 2 ) - ( x 4 2 + x 5 2 + x 6 2 + x 7 2 ) N(x)=\bar{x}x=(x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2})-(x_{4}^{2}+x_{5}^{2}+x% _{6}^{2}+x_{7}^{2})
  73. x - 1 = x ¯ N ( x ) . x^{-1}=\frac{\bar{x}}{N(x)}.
  74. N ( x y ) = N ( x ) N ( y ) . N(xy)=N(x)N(y).\,
  75. [ a 𝐯 𝐰 b ] \begin{bmatrix}a&\mathbf{v}\\ \mathbf{w}&b\end{bmatrix}
  76. [ a 𝐯 𝐰 b ] [ a 𝐯 𝐰 b ] = [ a a + 𝐯 𝐰 a 𝐯 + b 𝐯 + 𝐰 × 𝐰 a 𝐰 + b 𝐰 - 𝐯 × 𝐯 b b + 𝐯 𝐰 ] \begin{bmatrix}a&\mathbf{v}\\ \mathbf{w}&b\end{bmatrix}\begin{bmatrix}a^{\prime}&\mathbf{v}^{\prime}\\ \mathbf{w}^{\prime}&b^{\prime}\end{bmatrix}=\begin{bmatrix}aa^{\prime}+\mathbf% {v}\cdot\mathbf{w}^{\prime}&a\mathbf{v}^{\prime}+b^{\prime}\mathbf{v}+\mathbf{% w}\times\mathbf{w}^{\prime}\\ a^{\prime}\mathbf{w}+b\mathbf{w}^{\prime}-\mathbf{v}\times\mathbf{v}^{\prime}&% bb^{\prime}+\mathbf{v}^{\prime}\cdot\mathbf{w}\end{bmatrix}
  77. det [ a 𝐯 𝐰 b ] = a b - 𝐯 𝐰 \det\begin{bmatrix}a&\mathbf{v}\\ \mathbf{w}&b\end{bmatrix}=ab-\mathbf{v}\cdot\mathbf{w}
  78. det ( A B ) = det ( A ) det ( B ) . \det(AB)=\det(A)\det(B).\,
  79. x = ( a + 𝐚 ) + ( b + 𝐛 ) x=(a+\mathbf{a})+\ell(b+\mathbf{b})
  80. a a
  81. x ϕ ( x ) = [ a + b 𝐚 + 𝐛 - 𝐚 + 𝐛 a - b ] . x\mapsto\phi(x)=\begin{bmatrix}a+b&\mathbf{a}+\mathbf{b}\\ -\mathbf{a}+\mathbf{b}&a-b\end{bmatrix}.
  82. N ( x ) = det ( ϕ ( x ) ) N(x)=\det(\phi(x))

Splitting_circle_method.html

  1. p ( x ) = x n + p n - 1 x n - 1 + + p 0 p(x)=x^{n}+p_{n-1}x^{n-1}+\cdots+p_{0}
  2. p ( x ) = x n + p n - 1 x n - 1 + + p 0 = ( x - z 1 ) ( x - z n ) p(x)=x^{n}+p_{n-1}x^{n-1}+\cdots+p_{0}=(x-z_{1})\cdots(x-z_{n})
  3. t m = z 1 m + + z n m t_{m}=z_{1}^{m}+\cdots+z_{n}^{m}
  4. m = 0 , 1 , , n m=0,1,\dots,n
  5. a n - 1 + 2 a n - 2 u + + ( n - 1 ) a 1 u n - 2 + n a 0 u n - 1 a_{n-1}+2\,a_{n-2}\,u+\cdots+(n-1)\,a_{1}\,u^{n-2}+n\,a_{0}\,u^{n-1}
  6. = - ( 1 + a n - 1 u + + a 1 u n - 1 + a 0 u n ) ( t 1 + t 2 u + t 3 u 2 + + t n u n - 1 + ) . =-(1+a_{n-1}\,u+\cdots+a_{1}\,u^{n-1}+a_{0}\,u^{n})\cdot(t_{1}+t_{2}\,u+t_{3}% \,u^{2}+\dots+t_{n}\,u^{n-1}+\cdots).
  7. G G\subset\mathbb{C}
  8. 1 2 π i C p ( z ) p ( z ) z m d z = z G : p ( z ) = 0 p ( z ) z m p ( z ) = z G : p ( z ) = 0 z m . \frac{1}{2\pi\,i}\oint_{C}\frac{p^{\prime}(z)}{p(z)}z^{m}\,dz=\sum_{z\in G:\,p% (z)=0}\frac{p^{\prime}(z)z^{m}}{p^{\prime}(z)}=\sum_{z\in G:\,p(z)=0}z^{m}.
  9. f ( x ) := z G : p ( z ) = 0 ( x - z ) f(x):=\prod_{z\in G:\,p(z)=0}(x-z)
  10. p [ X ] p\in\mathbb{C}[X]
  11. f 0 f_{0}
  12. f - f 0 2 2 k - N n k 100 / 98 \|f-f_{0}\|\leq 2^{2k-N}\,nk\,100/98
  13. f 0 f_{0}
  14. g 0 g_{0}
  15. p - f 0 g 0 = ( f - f 0 ) g 0 + ( g - g 0 ) f 0 + ( f - f 0 ) ( g - g 0 ) p-f_{0}g_{0}=(f-f_{0})g_{0}+(g-g_{0})f_{0}+(f-f_{0})(g-g_{0})
  16. p - f 0 g 0 = f 0 Δ g + g 0 Δ f p-f_{0}g_{0}=f_{0}\Delta g+g_{0}\Delta f
  17. f 1 = f 0 + Δ f f_{1}=f_{0}+\Delta f
  18. g 1 = g 0 + Δ g g_{1}=g_{0}+\Delta g
  19. p 0 = p , p j + 1 ( x ) = ( - 1 ) deg p p ( x ) p ( - x ) , p_{0}=p,\qquad p_{j+1}(x)=(-1)^{\deg p}p(\sqrt{x})\,p(-\sqrt{x}),
  20. p j ( x ) p_{j}(x)
  21. 2 j 2^{j}
  22. p j ( x ) = e ( x ) + x o ( x ) p_{j}(x)=e(x)+x\,o(x)
  23. p j + 1 ( x ) = ( - 1 ) deg p ( e ( x ) 2 - x o ( x ) 2 ) p_{j+1}(x)=(-1)^{\deg p}(e(x)^{2}-x\,o(x)^{2})
  24. 2 j 2^{j}
  25. p j ( x ) f j ( x ) g j ( x ) p_{j}(x)\approx f_{j}(x)\,g_{j}(x)
  26. p j - 1 ( x ) g j ( x 2 ) f j - 1 ( x ) g j - 1 ( - x ) \frac{p_{j-1}(x)}{g_{j}(x^{2})}\approx\frac{f_{j-1}(x)}{g_{j-1}(-x)}
  27. R j > r j > 0 R_{j}>r_{j}>0
  28. R j / r j R_{j}/r_{j}
  29. R j / r j > e 0.3 1.35 R_{j}/r_{j}>e^{0{.}3}\approx 1.35
  30. e 0.3 / n 1 + 0.3 n \textstyle e^{0{.}3/n}\approx 1+\frac{0{.}3}{n}
  31. 3 + log 2 ( n ) \textstyle 3+\log_{2}(n)
  32. 4 + log 2 ( n ) + log 2 ( 2 + log 2 ( n ) ) \textstyle 4+\log_{2}(n)+\log_{2}(2+\log_{2}(n))
  33. 2 13.8 n 6.9 > ( 64 n 3 ) 2 \textstyle 2^{13{.}8}\cdot n^{6{.}9}>(64\cdot n^{3})^{2}
  34. 0 = j k | p j | u j - | p k | u k , \,0=\sum_{j\neq k}|p_{j}|u^{j}-|p_{k}|u^{k},
  35. u k < v k u_{k}<v_{k}
  36. D ( 0 , u k ) D(0,u_{k})
  37. A ( 0 , u k , v k ) A(0,u_{k},v_{k})

Spurline.html

  1. λ g \lambda_{g}

Square-free_polynomial.html

  1. f k [ X ] f\in k[X]
  2. b 2 f b^{2}\nmid f
  3. b k [ X ] b\in k[X]
  4. f = a 1 a 2 2 a 3 3 a n n f=a_{1}a_{2}^{2}a_{3}^{3}\cdots a_{n}^{n}\,
  5. a i a_{i}
  6. f f
  7. T n T_{n}
  8. n n
  9. 2 T n 2T_{n}
  10. f = a 1 a 2 2 a 3 3 a k k f=a_{1}a_{2}^{2}a_{3}^{3}\cdots a_{k}^{k}
  11. a 0 = a 2 1 a 3 2 a k k - 1 , a_{0}=a_{2}^{1}a_{3}^{2}\cdots a_{k}^{k-1},
  12. f / a 0 = a 1 a 2 a 3 a k f/a_{0}=a_{1}a_{2}a_{3}\cdots a_{k}
  13. f / a 0 = i = 1 k i a i a 1 a i - 1 a i + 1 a k . f^{\prime}/a_{0}=\sum_{i=1}^{k}ia_{i}^{\prime}a_{1}\cdots a_{i-1}a_{i+1}\cdots a% _{k}.
  14. b 1 = f / a 0 b_{1}=f/a_{0}
  15. c 1 = f / a 0 c_{1}=f^{\prime}/a_{0}
  16. d 1 = c 1 - b 1 d_{1}=c_{1}-b_{1}^{\prime}
  17. gcd ( b 1 , d 1 ) = a 1 , \gcd(b_{1},d_{1})=a_{1},
  18. b 2 = b 1 / a 1 = a 2 a 3 a n , b_{2}=b_{1}/a_{1}=a_{2}a_{3}\cdots a_{n},
  19. c 2 = d 1 / a 1 = i = 2 k ( i - 1 ) a i a 2 a i - 1 a i + 1 a k . c_{2}=d_{1}/a_{1}=\sum_{i=2}^{k}(i-1)a_{i}^{\prime}a_{2}\cdots a_{i-1}a_{i+1}% \cdots a_{k}.
  20. b k + 1 = 1 b_{k+1}=1
  21. a i . a_{i}.
  22. c i c_{i}
  23. d i d_{i}
  24. b i . b_{i}.
  25. f f
  26. b i , b_{i},
  27. b i b_{i}
  28. f . f.
  29. f f
  30. f f^{\prime}
  31. f f
  32. f f^{\prime}

Square-free_word.html

  1. w 1 w_{1}
  2. { w i i } \{w_{i}\mid i\in\mathbb{N}\}
  3. w i + 1 w_{i+1}
  4. w i w_{i}
  5. w i w_{i}
  6. w 1 w p w_{1}\cdots w_{p}
  7. w i w_{i}
  8. w 1 w_{1}

Squared_triangular_number.html

  1. n n
  2. n n
  3. 1 3 + 2 3 + 3 3 + + n 3 = ( 1 + 2 + 3 + + n ) 2 . 1^{3}+2^{3}+3^{3}+\cdots+n^{3}=\left(1+2+3+\cdots+n\right)^{2}.
  4. k = 1 n k 3 = ( k = 1 n k ) 2 . \sum_{k=1}^{n}k^{3}=\left(\sum_{k=1}^{n}k\right)^{2}.
  5. 0 , 1 , 9 , 36 , 100 , 225 , 0,1,9,36,100,225,
  6. 441 , 784 , 1296 , 2025 , 3025 , 4356 , 6084 , 8281 , 441,784,1296,2025,3025,4356,6084,8281,
  7. n × n n×n
  8. 4 × 4 4×4
  9. X , Y , Z , W X,Y,Z,W
  10. 1 1
  11. n n
  12. W W
  13. Y Y
  14. X X
  15. W W
  16. Z Z
  17. k = 1 n k 3 \displaystyle\sum_{k=1}^{n}k^{3}
  18. 1 1
  19. 1 + 2 + 3 + 4 + + n 1+2+3+4+...+n

Squid_giant_axon.html

  1. λ = r m / r i \lambda=\sqrt{r_{m}/r_{i}}
  2. E = E o e - x / λ E=E_{o}e^{-x/\lambda}

Sridhara.html

  1. a x 2 + b x + c = 0 ax^{2}+bx+c=0
  2. 4 a 2 x 2 + 4 a b x + 4 a c = 0 4a^{2}x^{2}+4abx+4ac=0
  3. 4 a 2 x 2 + 4 a b x = - 4 a c 4a^{2}x^{2}+4abx=-4ac
  4. b 2 b^{2}
  5. 4 a 2 x 2 + 4 a b x + b 2 = - 4 a c + b 2 4a^{2}x^{2}+4abx+b^{2}=-4ac+b^{2}
  6. ( m + n ) 2 = m 2 + 2 m n + n 2 (m+n)^{2}=m^{2}+2mn+n^{2}
  7. ( 2 a x + b ) 2 = b 2 - 4 a c = D (2ax+b)^{2}=b^{2}-4ac={D}
  8. 2 a x + b = ± D 2ax+b=\pm\sqrt{D}
  9. 2 a x = - b ± D 2ax=-b\pm\sqrt{D}
  10. x = - b ± b 2 - 4 a c 2 a . x=\frac{-b\pm\sqrt{b^{2}-4ac\ }}{2a}.
  11. N x 2 ± 1 = y 2 Nx^{2}\pm\ 1=y^{2}
  12. 1 - N x 2 = y 2 1-Nx^{2}=y^{2}
  13. N x 2 ± C = y 2 Nx^{2}\pm\ C=y^{2}
  14. C - N x 2 = y 2 C-Nx^{2}=y^{2}
  15. a x 2 + b x + c = 0 ax^{2}+bx+c=0
  16. 4 a 2 x 2 + 4 a b x + 4 a c = 0 4a^{2}x^{2}+4abx+4ac=0
  17. 4 a c 4ac
  18. 4 a 2 x 2 + 4 a b x = - 4 a c 4a^{2}x^{2}+4abx=-4ac
  19. b 2 b^{2}
  20. 4 a 2 x 2 + 4 a b x + b 2 = - 4 a c + b 2 4a^{2}x^{2}+4abx+b^{2}=-4ac+b^{2}
  21. ( a + b ) 2 = a 2 + 2 a b + b 2 (a+b)^{2}=a^{2}+2ab+b^{2}
  22. ( 2 a x + b ) 2 = b 2 - 4 a c (2ax+b)^{2}=b^{2}-4ac
  23. 2 a x + b = ± D 2ax+b=\pm\sqrt{D}
  24. 2 a x = - b ± D 2ax=-b\pm\sqrt{D}
  25. 2 a 2a
  26. x = - b ± b 2 - 4 a c 2 a . x=\frac{-b\pm\sqrt{b^{2}-4ac\ }}{2a}.

Stabilizer_code.html

  1. Π \Pi
  2. Π = { I , X , Y , Z } \Pi=\left\{I,X,Y,Z\right\}
  3. I [ 1 0 0 1 ] , X [ 0 1 1 0 ] , Y [ 0 - i i 0 ] , Z [ 1 0 0 - 1 ] . I\equiv\begin{bmatrix}1&0\\ 0&1\end{bmatrix},\ X\equiv\begin{bmatrix}0&1\\ 1&0\end{bmatrix},\ Y\equiv\begin{bmatrix}0&-i\\ i&0\end{bmatrix},\ Z\equiv\begin{bmatrix}1&0\\ 0&-1\end{bmatrix}.
  4. Π \Pi
  5. ± 1 \pm 1
  6. Π n \Pi^{n}
  7. n n
  8. Π n = { e i ϕ A 1 A n : j { 1 , , n } A j Π , ϕ { 0 , π / 2 , π , 3 π / 2 } } . \Pi^{n}=\left\{\begin{array}[c]{c}e^{i\phi}A_{1}\otimes\cdots\otimes A_{n}:% \forall j\in\left\{1,\ldots,n\right\}A_{j}\in\Pi,\ \ \phi\in\left\{0,\pi/2,\pi% ,3\pi/2\right\}\end{array}\right\}.
  9. Π n \Pi^{n}
  10. n n
  11. A 1 A n A 1 A n . A_{1}\cdots A_{n}\equiv A_{1}\otimes\cdots\otimes A_{n}.
  12. n n
  13. Π n \Pi^{n}
  14. n n
  15. [ n , k ] \left[n,k\right]
  16. k k
  17. n n
  18. k / n k/n
  19. 𝒮 \mathcal{S}
  20. n n
  21. Π n \Pi^{n}
  22. 𝒮 Π n \mathcal{S}\subset\Pi^{n}
  23. 𝒮 \mathcal{S}
  24. - I n -I^{\otimes n}
  25. + 1 +1
  26. 2 k 2^{k}
  27. k k
  28. 𝒮 \mathcal{S}
  29. n - k n-k
  30. { g 1 , , g n - k | i { 1 , , n - k } , g i 𝒮 } . \left\{g_{1},\ldots,g_{n-k}\ |\ \forall i\in\left\{1,\ldots,n-k\right\},\ g_{i% }\in\mathcal{S}\right\}.
  31. g 1 , , g n - k g_{1},\ldots,g_{n-k}
  32. Π n \Pi^{n}
  33. \mathcal{E}
  34. Π n \Pi^{n}
  35. Π n . \mathcal{E}\subset\Pi^{n}.
  36. E E\in\mathcal{E}
  37. g g
  38. 𝒮 \mathcal{S}
  39. E E
  40. g g
  41. 𝒮 \mathcal{S}
  42. E E
  43. g g
  44. 𝒮 \mathcal{S}
  45. 𝐫 \mathbf{r}
  46. E E
  47. 𝐫 \mathbf{r}
  48. n - k n-k
  49. E E
  50. g 𝒮 g\in\mathcal{S}
  51. E E
  52. g g
  53. 𝒮 \mathcal{S}
  54. 𝒮 \mathcal{S}
  55. 𝒮 \mathcal{S}
  56. 𝒮 \mathcal{S}
  57. E 1 , E 2 E_{1},E_{2}
  58. \mathcal{E}
  59. E 1 E 2 𝒵 ( 𝒮 ) E_{1}^{\dagger}E_{2}\notin\mathcal{Z}\left(\mathcal{S}\right)
  60. E 1 E 2 𝒮 E_{1}^{\dagger}E_{2}\in\mathcal{S}
  61. 𝒵 ( 𝒮 ) \mathcal{Z}\left(\mathcal{S}\right)
  62. 𝒮 \mathcal{S}
  63. Π \Pi
  64. ( 2 ) 2 \left(\mathbb{Z}_{2}\right)^{2}
  65. [ A ] \left[A\right]
  66. A A
  67. [ A ] = { β A | β , | β | = 1 } . \left[A\right]=\left\{\beta A\ |\ \beta\in\mathbb{C},\ \left|\beta\right|=1% \right\}.
  68. [ Π ] \left[\Pi\right]
  69. [ Π ] = { [ A ] | A Π } \left[\Pi\right]=\left\{\left[A\right]\ |\ A\in\Pi\right\}
  70. N : ( 2 ) 2 Π N:\left(\mathbb{Z}_{2}\right)^{2}\rightarrow\Pi
  71. 00 I , 01 X , 11 Y , 10 Z 00\to I,\,\,01\to X,\,\,11\to Y,\,\,10\to Z
  72. u , v ( 2 ) 2 u,v\in\left(\mathbb{Z}_{2}\right)^{2}
  73. u = ( z | x ) u=\left(z|x\right)
  74. v = ( z | x ) v=\left(z^{\prime}|x^{\prime}\right)
  75. z z
  76. x x
  77. z z^{\prime}
  78. x 2 x^{\prime}\in\mathbb{Z}_{2}
  79. u = ( 0 | 1 ) u=\left(0|1\right)
  80. N ( u ) = X N\left(u\right)=X
  81. N N
  82. [ N ] : ( 2 ) 2 [ Π ] \left[N\right]:\left(\mathbb{Z}_{2}\right)^{2}\rightarrow\left[\Pi\right]
  83. ( 2 ) 2 \left(\mathbb{Z}_{2}\right)^{2}
  84. [ N ( u + v ) ] = [ N ( u ) ] [ N ( v ) ] . \left[N\left(u+v\right)\right]=\left[N\left(u\right)\right]\left[N\left(v% \right)\right].
  85. \odot
  86. u , v ( 2 ) 2 u,v\in\left(\mathbb{Z}_{2}\right)^{2}
  87. u v z x - x z . u\odot v\equiv zx^{\prime}-xz^{\prime}.
  88. \odot
  89. Π \Pi
  90. N ( u ) N ( v ) = ( - 1 ) ( u v ) N ( v ) N ( u ) . N\left(u\right)N\left(v\right)=\left(-1\right)^{\left(u\odot v\right)}N\left(v% \right)N\left(u\right).
  91. N N
  92. N N
  93. 𝐀 = A 1 A n \mathbf{A}=A_{1}\otimes\cdots\otimes A_{n}
  94. Π n \Pi^{n}
  95. n n
  96. [ Π n ] = { [ 𝐀 ] | 𝐀 Π n } \left[\Pi^{n}\right]=\left\{\left[\mathbf{A}\right]\ |\ \mathbf{A}\in\Pi^{n}\right\}
  97. [ 𝐀 ] = { β 𝐀 | β , | β | = 1 } . \left[\mathbf{A}\right]=\left\{\beta\mathbf{A}\ |\ \beta\in\mathbb{C},\ \left|% \beta\right|=1\right\}.
  98. \ast
  99. [ 𝐀 ] [ 𝐁 ] [ A 1 ] [ B 1 ] [ A n ] [ B n ] = [ A 1 B 1 ] [ A n B n ] = [ 𝐀𝐁 ] . \left[\mathbf{A}\right]\ast\left[\mathbf{B}\right]\equiv\left[A_{1}\right]\ast% \left[B_{1}\right]\otimes\cdots\otimes\left[A_{n}\right]\ast\left[B_{n}\right]% =\left[A_{1}B_{1}\right]\otimes\cdots\otimes\left[A_{n}B_{n}\right]=\left[% \mathbf{AB}\right].
  100. [ Π n ] \left[\Pi^{n}\right]
  101. \ast
  102. 2 n 2n
  103. ( 2 ) 2 n = { ( 𝐳 , 𝐱 ) : 𝐳 , 𝐱 ( 2 ) n } . \left(\mathbb{Z}_{2}\right)^{2n}=\left\{\left(\mathbf{z,x}\right):\mathbf{z},% \mathbf{x}\in\left(\mathbb{Z}_{2}\right)^{n}\right\}.
  104. ( ( 2 ) 2 n , + ) (\left(\mathbb{Z}_{2}\right)^{2n},+)
  105. + +
  106. 𝐮 = ( 𝐳 | 𝐱 ) , 𝐯 = ( 𝐳 | 𝐱 ) \mathbf{u}=\left(\mathbf{z}|\mathbf{x}\right),\mathbf{v}=\left(\mathbf{z}^{% \prime}|\mathbf{x}^{\prime}\right)
  107. 𝐮 , 𝐯 ( 2 ) 2 n \mathbf{u,v}\in\left(\mathbb{Z}_{2}\right)^{2n}
  108. 𝐳 \mathbf{z}
  109. 𝐱 \mathbf{x}
  110. ( z 1 , , z n ) \left(z_{1},\ldots,z_{n}\right)
  111. ( x 1 , , x n ) \left(x_{1},\ldots,x_{n}\right)
  112. 𝐳 \mathbf{z}^{\prime}
  113. 𝐱 \mathbf{x}^{\prime}
  114. \odot
  115. 𝐮 \mathbf{u}
  116. 𝐯 \mathbf{v}
  117. 𝐮 𝐯 i = 1 n z i x i - x i z i , \mathbf{u}\odot\mathbf{v\equiv}\sum_{i=1}^{n}z_{i}x_{i}^{\prime}-x_{i}z_{i}^{% \prime},
  118. 𝐮 𝐯 i = 1 n u i v i , \mathbf{u}\odot\mathbf{v\equiv}\sum_{i=1}^{n}u_{i}\odot v_{i},
  119. u i = ( z i | x i ) u_{i}=\left(z_{i}|x_{i}\right)
  120. v i = ( z i | x i ) v_{i}=\left(z_{i}^{\prime}|x_{i}^{\prime}\right)
  121. 𝐍 : ( 2 ) 2 n Π n \mathbf{N}:\left(\mathbb{Z}_{2}\right)^{2n}\rightarrow\Pi^{n}
  122. 𝐍 ( 𝐮 ) N ( u 1 ) N ( u n ) . \mathbf{N}\left(\mathbf{u}\right)\equiv N\left(u_{1}\right)\otimes\cdots% \otimes N\left(u_{n}\right).
  123. 𝐗 ( 𝐱 ) X x 1 X x n , 𝐙 ( 𝐳 ) Z z 1 Z z n , \mathbf{X}\left(\mathbf{x}\right)\equiv X^{x_{1}}\otimes\cdots\otimes X^{x_{n}% },\,\,\,\,\,\,\,\mathbf{Z}\left(\mathbf{z}\right)\equiv Z^{z_{1}}\otimes\cdots% \otimes Z^{z_{n}},
  124. 𝐍 ( 𝐮 ) \mathbf{N}\left(\mathbf{u}\right)
  125. 𝐙 ( 𝐳 ) 𝐗 ( 𝐱 ) \mathbf{Z}\left(\mathbf{z}\right)\mathbf{X}\left(\mathbf{x}\right)
  126. [ 𝐍 ( 𝐮 ) ] = [ 𝐙 ( 𝐳 ) 𝐗 ( 𝐱 ) ] . \left[\mathbf{N}\left(\mathbf{u}\right)\right]=\left[\mathbf{Z}\left(\mathbf{z% }\right)\mathbf{X}\left(\mathbf{x}\right)\right].
  127. [ 𝐍 ] : ( 2 ) 2 n [ Π n ] \left[\mathbf{N}\right]:\left(\mathbb{Z}_{2}\right)^{2n}\rightarrow\left[\Pi^{% n}\right]
  128. [ 𝐍 ( 𝐮 + 𝐯 ) ] = [ 𝐍 ( 𝐮 ) ] [ 𝐍 ( 𝐯 ) ] , \left[\mathbf{N}\left(\mathbf{u+v}\right)\right]=\left[\mathbf{N}\left(\mathbf% {u}\right)\right]\left[\mathbf{N}\left(\mathbf{v}\right)\right],
  129. 𝐮 , 𝐯 ( 2 ) 2 n \mathbf{u,v}\in\left(\mathbb{Z}_{2}\right)^{2n}
  130. 𝐍 ( 𝐮 ) \mathbf{N}\left(\mathbf{u}\right)
  131. 𝐍 ( 𝐯 ) \mathbf{N}\left(\mathbf{v}\right)
  132. 𝐍 ( 𝐮 ) 𝐍 ( 𝐯 ) = ( - 1 ) ( 𝐮 𝐯 ) 𝐍 ( 𝐯 ) 𝐍 ( 𝐮 ) . \mathbf{N\left(\mathbf{u}\right)N}\left(\mathbf{v}\right)=\left(-1\right)^{% \left(\mathbf{u}\odot\mathbf{v}\right)}\mathbf{N}\left(\mathbf{v}\right)% \mathbf{N}\left(\mathbf{u}\right).
  133. [ 5 , 1 ] \left[5,1\right]
  134. k = 1 k=1
  135. n = 5 n=5
  136. n - k = 4 n-k=4
  137. g 1 = X Z Z X I g 2 = I X Z Z X g 3 = X I X Z Z g 4 = Z X I X Z \begin{array}[c]{ccccccc}g_{1}&=&X&Z&Z&X&I\\ g_{2}&=&I&X&Z&Z&X\\ g_{3}&=&X&I&X&Z&Z\\ g_{4}&=&Z&X&I&X&Z\end{array}
  138. { X i , Y i , Z i } \left\{X_{i},Y_{i},Z_{i}\right\}
  139. A i A_{i}
  140. i i

Stable_model_semantics.html

  1. p p
  2. r p , q r\leftarrow p,\ q
  3. s p , not q . s\leftarrow p,\ \hbox{not }q.
  4. p p
  5. p p
  6. q q
  7. r r
  8. r r
  9. q q
  10. s s
  11. p p
  12. not q \hbox{not }q
  13. q q
  14. p p
  15. q q
  16. r r
  17. s s
  18. , \land,
  19. $\hbox{not}$
  20. ¬ , \neg,
  21. F G F\leftarrow G
  22. G F G\rightarrow F
  23. p ¬ q s . p\land\neg q\rightarrow s.
  24. p p
  25. q q
  26. r r
  27. s s
  28. not p \hbox{not}\ p
  29. p p
  30. s p , not q s\leftarrow p,\ \hbox{not }q
  31. s s
  32. p p
  33. ¬ q \neg q
  34. p p
  35. q q
  36. r r
  37. s s
  38. { p , s } \{p,s\}
  39. \emptyset
  40. e v e n ( 0 ) even(0)
  41. e v e n ( s ( X ) ) not e v e n ( X ) even(s(X))\leftarrow\hbox{not }even(X)
  42. X X
  43. 0 , s ( 0 ) , s ( s ( 0 ) ) , . 0,\ s(0),\ s(s(0)),\dots.
  44. e v e n ( 0 ) even(0)
  45. e v e n ( s ( 0 ) ) not e v e n ( 0 ) even(s(0))\leftarrow\hbox{not }even(0)
  46. e v e n ( s ( s ( 0 ) ) ) not e v e n ( s ( 0 ) ) even(s(s(0)))\leftarrow\hbox{not }even(s(0))
  47. \dots
  48. P P
  49. A B 1 , , B m , not C 1 , , not C n A\leftarrow B_{1},\dots,B_{m},\hbox{not }C_{1},\dots,\hbox{not }C_{n}
  50. A , B 1 , , B m , C 1 , , C n A,B_{1},\dots,B_{m},C_{1},\dots,C_{n}
  51. P P
  52. n = 0 n=0
  53. P P
  54. I I
  55. P P
  56. I I
  57. P P
  58. C i C_{i}
  59. B 1 , , B m , not C 1 , , not C n B_{1},\dots,B_{m},\hbox{not }C_{1},\dots,\hbox{not }C_{n}
  60. I I
  61. not C 1 , , not C n \hbox{not }C_{1},\dots,\hbox{not }C_{n}
  62. I I
  63. P P
  64. I I
  65. P P
  66. I I
  67. P P
  68. P P
  69. { p , s } \{p,s\}
  70. p p
  71. r p , q r\leftarrow p,\ q
  72. s p , not q . s\leftarrow p,\ \hbox{not }q.
  73. { p , s } \{p,s\}
  74. p p
  75. r p , q r\leftarrow p,\ q
  76. s p . s\leftarrow p.
  77. q { p , s } q\not\in\{p,s\}
  78. not q . \hbox{not }q.
  79. { p , s } \{p,s\}
  80. { p , s } \{p,s\}
  81. p , q , r , s p,\ q,\ r,\ s
  82. { p , q , r } \{p,q,r\}
  83. p p
  84. r p , q . r\leftarrow p,\ q.
  85. { p } \{p\}
  86. { p , q , r } \{p,q,r\}
  87. p not q p\leftarrow\hbox{not }q
  88. q not p q\leftarrow\hbox{not }p
  89. { p } \{p\}
  90. { q } \{q\}
  91. p not p p\leftarrow\hbox{not }p
  92. A B 1 , , B m , not C 1 , , not C n A\leftarrow B_{1},\dots,B_{m},\hbox{not }C_{1},\dots,\hbox{not }C_{n}
  93. A , B 1 , , B m , C 1 , , C n A,B_{1},\dots,B_{m},C_{1},\dots,C_{n}
  94. A A
  95. P P
  96. A A
  97. P P
  98. P P
  99. P P
  100. I I
  101. J J
  102. I I
  103. J J
  104. p p p\leftarrow p
  105. p p p\leftrightarrow p
  106. \emptyset
  107. p p p\leftarrow p
  108. { p } \{p\}
  109. P P
  110. P P
  111. p not q p\leftarrow\hbox{not }q
  112. q not p q\leftarrow\hbox{not }p
  113. r p r\leftarrow p
  114. r q r\leftarrow q
  115. { p , r } \{p,r\}
  116. { q , r } \{q,r\}
  117. r r
  118. p p
  119. p p
  120. \sim
  121. Cross not Train \hbox{Cross}\leftarrow\hbox{not Train}
  122. Cross Train . \hbox{Cross}\leftarrow\,\sim\hbox{Train}.
  123. A A
  124. B i B_{i}
  125. C i C_{i}
  126. A B 1 , , B m , not C 1 , , not C n A\leftarrow B_{1},\dots,B_{m},\hbox{not }C_{1},\dots,\hbox{not }C_{n}
  127. A \sim A
  128. A A
  129. A , A \ A,\sim A
  130. p not q p\leftarrow\hbox{not }q
  131. q not p q\leftarrow\hbox{not }p
  132. r r
  133. { p , r } \{p,r\}
  134. { q , r , r } \ \{q,r,\sim r\}
  135. r \ r
  136. r \ \sim r
  137. p p
  138. p ( X 1 , , X n ) not p ( X 1 , , X n ) \sim p(X_{1},\dots,X_{n})\leftarrow\hbox{not }p(X_{1},\dots,X_{n})
  139. p p
  140. X 1 , , X n X_{1},\dots,X_{n}
  141. p ( a , b ) p(a,b)
  142. p ( c , d ) p(c,d)
  143. p ( X , Y ) not p ( X , Y ) \sim p(X,Y)\leftarrow\hbox{not }p(X,Y)
  144. p ( a , b ) , p ( c , d ) p(a,b),\ p(c,d)
  145. p ( a , a ) , p ( a , c ) , \sim p(a,a),\ \sim p(a,c),\ \dots
  146. p , a , b , c , d p,\ a,\ b,\ c,\ d
  147. A B 1 , , B m , not C 1 , , not C n A\leftarrow B_{1},\dots,B_{m},\hbox{not }C_{1},\dots,\hbox{not }C_{n}
  148. A , B 1 , , B m , C 1 , , C n A,B_{1},\dots,B_{m},C_{1},\dots,C_{n}
  149. B 1 , , B m , not C 1 , , not C n . \leftarrow B_{1},\dots,B_{m},\hbox{not }C_{1},\dots,\hbox{not }C_{n}.
  150. , \land,
  151. $\hbox{not}$
  152. ¬ , \neg,
  153. F G F\leftarrow G
  154. G F G\rightarrow F
  155. ¬ ( B 1 B m ¬ C 1 ¬ C n ) . \neg(B_{1}\land\cdots\land B_{m}\land\neg C_{1}\land\cdots\land\neg C_{n}).
  156. P P
  157. P P
  158. P P
  159. I I
  160. P P
  161. P P
  162. I I
  163. I I
  164. P P
  165. P P
  166. P P
  167. C C
  168. P { C } P\cup\{C\}
  169. P P
  170. C C
  171. A 1 ; ; A k B 1 , , B m , not C 1 , , not C n A_{1};\dots;A_{k}\leftarrow B_{1},\dots,B_{m},\hbox{not }C_{1},\dots,\hbox{not% }C_{n}
  172. \lor
  173. k = 1 k=1
  174. k = 0 k=0
  175. n = 0 n=0
  176. I I
  177. P P
  178. I I
  179. P P
  180. I I
  181. { p , r } \{p,r\}
  182. p ; q p;q
  183. r not q r\leftarrow\hbox{not}\ q
  184. p ; q p;q
  185. r . r.
  186. { q } \{q\}
  187. P P
  188. P P
  189. P P
  190. Σ 2 P \Sigma_{2}^{\rm P}
  191. F F
  192. I I
  193. F F
  194. I I
  195. \bot
  196. P P
  197. I I
  198. P P
  199. I I
  200. I I
  201. P P
  202. I I
  203. P P
  204. I I
  205. { p , p q r , p ¬ q s } \{p,\ p\land q\rightarrow r,\ p\land\neg q\rightarrow s\}
  206. { p , s } \{p,s\}
  207. { p , , p ¬ s } . \{p,\ \bot\rightarrow\bot,\ p\land\neg\bot\rightarrow s\}.
  208. { p , s } \{p,s\}
  209. { p , s } \{p,s\}
  210. { p , s } \{p,s\}
  211. P P
  212. P P
  213. A A
  214. P P
  215. A A
  216. P P
  217. p ¬ p p\lor\neg p
  218. $\empty$
  219. { p } \{p\}
  220. Σ 2 P \Sigma_{2}^{\rm P}
  221. ¬ \neg

Stack_effect.html

  1. Δ P = C a h ( 1 T o - 1 T i ) \Delta P=\;C\,a\;h\;\bigg(\frac{1}{T_{o}}-\frac{1}{T_{i}}\bigg)
  2. Q = C A 2 g h T i - T o T i Q=C\;A\;\sqrt{2\;g\;h\;\frac{T_{i}-T_{o}}{T_{i}}}

Standard_asteroid_physical_characteristics.html

  1. d = ( a b c ) 1 3 d=(abc)^{\frac{1}{3}}\,\!
  2. a = d ( μ 2 ν ) 1 3 a=d\,(\mu^{2}\nu)^{\frac{1}{3}}\,\!
  3. b = d ( ν μ ) 1 3 b=d\,\left(\frac{\nu}{\mu}\right)^{\frac{1}{3}}\,\!
  4. c = d ( ν 2 μ ) 1 3 c=\frac{d}{(\nu^{2}\mu)^{\frac{1}{3}}}\,\!
  5. M = π a b c ρ 6 M=\frac{\pi abc\rho}{6}\,\!
  6. g spherical = G M r 2 g_{\rm spherical}=\frac{GM}{r^{2}}\,\!
  7. × 10 11 \times 10^{−}11
  8. g centrifugal = - ( 2 π T ) 2 r sin θ g_{\rm centrifugal}=-\left(\frac{2\pi}{T}\right)^{2}r\sin\theta
  9. g effective = g gravitational + g centrifugal . g_{\rm effective}=g_{\rm gravitational}+g_{\rm centrifugal}\ .
  10. v e = 2 g r v_{e}=\sqrt{2gr}
  11. R in = ( 1 - A ) L 0 π r 2 4 π a 2 , R_{\mathrm{in}}=\frac{(1-A)L_{0}\pi r^{2}}{4\pi a^{2}},
  12. A A\,\!
  13. a a\,\!
  14. L 0 L_{0}\,\!
  15. r r
  16. 1 - A 1-A
  17. R out = 4 π r 2 ϵ σ T 4 , R_{\mathrm{out}}=4\pi r^{2}\epsilon\sigma T^{4}\frac{}{},
  18. σ \sigma\,\!
  19. T T
  20. ϵ \epsilon\,\!
  21. R in = R out R_{\mathrm{in}}=R_{\mathrm{out}}
  22. T = ( ( 1 - A ) L 0 ϵ σ 16 π a 2 ) 1 / 4 T=\left(\frac{(1-A)L_{0}}{\epsilon\sigma 16\pi a^{2}}\right)^{1/4}\,\!
  23. ϵ \epsilon
  24. T s s = 2 T 1.41 T , T_{ss}=\sqrt{2}\,T\approx 1.41\,T,
  25. T T
  26. T s s max = 2 1 - e T , T_{ss}^{\rm max}=\sqrt{\frac{2}{1-e}}\ T,
  27. e e\,\!
  28. T = constant × 1 d , T={\rm constant}\times\frac{1}{\sqrt{d}},
  29. d d\,\!

Standard_ruler.html

  1. θ r D \theta\approx\frac{r}{D}

Standard_time_(manufacturing).html

  1. Standard Time = ( Observed Time ) ( Rating Factor ) ( 1 + PFD Allowance ) \,\text{Standard Time}=(\,\text{Observed Time})(\,\text{Rating Factor})(1+\,% \text{PFD Allowance})

Stanton_number.html

  1. St = h G c p = h ρ u c p \mathrm{St}=\frac{h}{Gc_{p}}=\frac{h}{\rho uc_{p}}
  2. St = Nu Re Pr \mathrm{St}=\frac{\mathrm{Nu}}{\mathrm{Re}\,\mathrm{Pr}}

Star-free_language.html

  1. { a , b } \{a,\,b\}
  2. ( c a a c ) c (\emptyset^{c}aa\emptyset^{c})^{c}
  3. X c X^{c}
  4. X X
  5. { a , b } * \{a,\,b\}^{*}

Starred_transform.html

  1. x ( t ) x(t)
  2. X * ( s ) = [ x ( t ) δ T ( t ) ] = [ x * ( t ) ] , \begin{aligned}\displaystyle X^{*}(s)=\mathcal{L}[x(t)\cdot\delta_{T}(t)]=% \mathcal{L}[x^{*}(t)],\end{aligned}
  3. x ( t ) x(t)
  4. x * ( t ) = def x ( t ) δ T ( t ) \displaystyle x^{*}(t)\ \stackrel{\mathrm{def}}{=}\ x(t)\cdot\delta_{T}(t)
  5. X * ( s ) = 1 2 π j c - j c + j X ( p ) 1 1 - e - T ( s - p ) d p . X^{*}(s)=\frac{1}{2\pi j}\int_{c-j\infty}^{c+j\infty}{X(p)\cdot\frac{1}{1-e^{-% T(s-p)}}\cdot dp}.
  6. X * ( s ) = λ = poles of X ( s ) Res p = λ [ X ( p ) 1 1 - e - T ( s - p ) ] . X^{*}(s)=\sum_{\lambda=\,\text{poles of }X(s)}\operatorname{Res}\limits_{p=% \lambda}\bigg[X(p)\frac{1}{1-e^{-T(s-p)}}\bigg].
  7. X * ( s ) = 1 T k = - X ( s - j 2 π T k ) + x ( 0 ) 2 . X^{*}(s)=\frac{1}{T}\sum_{k=-\infty}^{\infty}X\left(s-j\tfrac{2\pi}{T}k\right)% +\frac{x(0)}{2}.
  8. . X * ( s ) = X ( z ) | z = e s T \bigg.X^{*}(s)=X(z)\bigg|_{\displaystyle z=e^{sT}}
  9. X * ( s ) X^{*}(s)
  10. s s
  11. j 2 π T . j\tfrac{2\pi}{T}.
  12. X * ( s + j 2 π T k ) = X * ( s ) X^{*}(s+j\tfrac{2\pi}{T}k)=X^{*}(s)
  13. s = s 1 s=s_{1}
  14. s = s 1 + j 2 π T k s=s_{1}+j\tfrac{2\pi}{T}k
  15. k = 0 , ± 1 , ± 2 , \scriptstyle k=0,\pm 1,\pm 2,\ldots

State_variable.html

  1. A A\in
  2. B \quad B\in
  3. C \quad C\in
  4. D \quad D\in
  5. x [ n ] x[n]\,
  6. x [ n + 1 ] = A x [ n ] + B u [ n ] x[n+1]=Ax[n]+Bu[n]\,\!
  7. y [ n ] = C x [ n ] + D u [ n ] y[n]=Cx[n]+Du[n]\,\!
  8. x ( t ) x(t)\,
  9. d x ( t ) d t = A x ( t ) + B u ( t ) \frac{dx(t)}{dt}\ =Ax(t)+Bu(t)\,\!
  10. d x ( t ) d t \frac{dx(t)}{dt}\,\!
  11. y ( t ) = C x ( t ) + D u ( t ) y(t)=Cx(t)+Du(t)\,\!

State_vector_(geographical).html

  1. N \mathbb{R}^{N}
  2. 𝐱 \,\textbf{x}
  3. 2 N \mathbb{R}^{2N}
  4. 𝐱 ( t ) = ( x 1 ( t ) x 2 ( t ) x 3 ( t ) v 1 ( t ) v 2 ( t ) v 3 ( t ) ) T \mathbf{x}(t)=(x_{1}(t)\;\;x_{2}(t)\;\;x_{3}(t)\;\;v_{1}(t)\;\;v_{2}(t)\;\;v_{% 3}(t))^{T}
  5. 𝐱 ( t ) = ( 𝐫 ( t ) 𝐯 ( t ) ) \mathbf{x}(t)={\left({{\mathbf{r}(t)}\atop{\mathbf{v}(t)}}\right)}
  6. 𝐫 = ( x 1 x 2 x 3 ) T \mathbf{r}=(x_{1}\;x_{2}\;x_{3})^{T}
  7. 𝐯 = 𝐫 ˙ = ( v 1 v 2 v 3 ) T \mathbf{v}=\dot{\mathbf{r}}=(v_{1}\;v_{2}\;v_{3})^{T}

Static_pressure.html

  1. P + 1 2 ρ v 2 = P 0 , P+\tfrac{1}{2}\rho v^{2}=P_{0},
  2. P P\;
  3. 1 2 ρ v 2 \tfrac{1}{2}\rho v^{2}
  4. q q\;
  5. ρ \rho\,
  6. v v\,
  7. P 0 P_{0}\;
  8. P P
  9. q q
  10. P 0 P_{0}

Stationary_state.html

  1. H ^ | Ψ = E Ψ | Ψ \hat{H}|\Psi\rangle=E_{\Psi}|\Psi\rangle
  2. | Ψ |\Psi\rangle
  3. H ^ \hat{H}
  4. E Ψ E_{\Psi}
  5. | Ψ |\Psi\rangle
  6. H ^ \hat{H}
  7. | Ψ |\Psi\rangle
  8. H ^ \hat{H}
  9. E Ψ E_{\Psi}
  10. | Ψ |\Psi\rangle
  11. i t | Ψ = E Ψ | Ψ i\hbar\frac{\partial}{\partial t}|\Psi\rangle=E_{\Psi}|\Psi\rangle
  12. H ^ \hat{H}
  13. | Ψ |\Psi\rangle
  14. | Ψ ( t ) = e - i E Ψ t / | Ψ ( 0 ) |\Psi(t)\rangle=e^{-iE_{\Psi}t/\hbar}|\Psi(0)\rangle
  15. \hbar
  16. ψ N ( ψ 0 + ψ 1 ) / 2 \psi_{N}\equiv(\psi_{0}+\psi_{1})/\sqrt{2}
  17. | Ψ ( t ) = e - i E Ψ t / | Ψ ( 0 ) |\Psi(t)\rangle=e^{-iE_{\Psi}t/\hbar}|\Psi(0)\rangle
  18. | Ψ ( t ) |\Psi(t)\rangle
  19. Ψ ( x , t ) \Psi(x,t)
  20. | Ψ ( x , t ) | 2 = | e - i E Ψ t / Ψ ( x , 0 ) | 2 = | e - i E Ψ t / | 2 | Ψ ( x , 0 ) | 2 = | Ψ ( x , 0 ) | 2 |\Psi(x,t)|^{2}=\left|e^{-iE_{\Psi}t/\hbar}\Psi(x,0)\right|^{2}=\left|e^{-iE_{% \Psi}t/\hbar}\right|^{2}\left|\Psi(x,0)\right|^{2}=\left|\Psi(x,0)\right|^{2}

Statistical_machine_translation.html

  1. p ( e | f ) p(e|f)
  2. e e
  3. f f
  4. p ( e | f ) p(e|f)
  5. p ( e | f ) p ( f | e ) p ( e ) p(e|f)\propto p(f|e)p(e)
  6. p ( f | e ) p(f|e)
  7. p ( e ) p(e)
  8. e ~ \tilde{e}
  9. e ~ = a r g max e e * p ( e | f ) = a r g max e e * p ( f | e ) p ( e ) \tilde{e}=arg\max_{e\in e^{*}}p(e|f)=arg\max_{e\in e^{*}}p(f|e)p(e)
  10. e * e^{*}

STED_microscopy.html

  1. D = λ 2 N A \mathrm{D}=\frac{\lambda}{\mathrm{2NA}}
  2. D = λ 2 n sin α 1 + I I sat \mathrm{D}=\frac{\lambda}{2n\sin{\alpha}\sqrt{1+\frac{I}{I\text{sat}}}}

Stepwise_regression.html

  1. 2 log p \sqrt{2\log p}

Stern_prime.html

  1. p + 2 b 2 p+2b^{2}

Steve_Jackson_(mathematician).html

  1. δ 𝟓 𝟏 = ω ω ω + 1 \mathbf{\delta^{1}_{5}}=\aleph_{\omega^{\omega^{\omega}}+1}

Stieltjes_moment_problem.html

  1. m n = 0 x n d μ ( x ) m_{n}=\int_{0}^{\infty}x^{n}\,d\mu(x)\,
  2. Δ n = [ m 0 m 1 m 2 m n m 1 m 2 m 3 m n + 1 m 2 m 3 m 4 m n + 2 m n m n + 1 m n + 2 m 2 n ] \Delta_{n}=\left[\begin{matrix}m_{0}&m_{1}&m_{2}&\cdots&m_{n}\\ m_{1}&m_{2}&m_{3}&\cdots&m_{n+1}\\ m_{2}&m_{3}&m_{4}&\cdots&m_{n+2}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ m_{n}&m_{n+1}&m_{n+2}&\cdots&m_{2n}\end{matrix}\right]
  3. Δ n ( 1 ) = [ m 1 m 2 m 3 m n + 1 m 2 m 3 m 4 m n + 2 m 3 m 4 m 5 m n + 3 m n + 1 m n + 2 m n + 3 m 2 n + 1 ] . \Delta_{n}^{(1)}=\left[\begin{matrix}m_{1}&m_{2}&m_{3}&\cdots&m_{n+1}\\ m_{2}&m_{3}&m_{4}&\cdots&m_{n+2}\\ m_{3}&m_{4}&m_{5}&\cdots&m_{n+3}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ m_{n+1}&m_{n+2}&m_{n+3}&\cdots&m_{2n+1}\end{matrix}\right].
  4. [ 0 , ) [0,\infty)
  5. det ( Δ n ) > 0 and det ( Δ n ( 1 ) ) > 0. \det(\Delta_{n})>0\ \mathrm{and}\ \det\left(\Delta_{n}^{(1)}\right)>0.
  6. [ 0 , ) [0,\infty)
  7. n m n\leq m
  8. det ( Δ n ) > 0 and det ( Δ n ( 1 ) ) > 0 \det(\Delta_{n})>0\ \mathrm{and}\ \det\left(\Delta_{n}^{(1)}\right)>0
  9. n n
  10. det ( Δ n ) = 0 and det ( Δ n ( 1 ) ) = 0. \det(\Delta_{n})=0\ \mathrm{and}\ \det\left(\Delta_{n}^{(1)}\right)=0.
  11. n 1 m n - 1 / ( 2 n ) = . \sum_{n\geq 1}m_{n}^{-1/(2n)}=\infty~{}.

Stirling_polynomials.html

  1. ( t 1 - e - t ) x + 1 = k = 0 S k ( x ) t k k ! . \left({t\over{1-e^{-t}}}\right)^{x+1}=\sum_{k=0}^{\infty}S_{k}(x){t^{k}\over k% !}.
  2. S k ( x ) S_{k}(x)\,
  3. 1 1\,
  4. 1 2 ( x + 1 ) {\scriptstyle\frac{1}{2}}(x+1)\,
  5. 1 12 ( 3 x 2 + 5 x + 2 ) {\scriptstyle\frac{1}{12}}(3x^{2}+5x+2)\,
  6. 1 8 ( x 3 + 2 x 2 + x ) {\scriptstyle\frac{1}{8}}(x^{3}+2x^{2}+x)\,
  7. 1 240 ( 15 x 4 + 30 x 3 + 5 x 2 - 18 x - 8 ) {\scriptstyle\frac{1}{240}}(15x^{4}+30x^{3}+5x^{2}-18x-8)\,
  8. 1 96 ( 3 x 5 + 5 x 4 - 5 x 3 - 13 x 2 - 6 x ) {\scriptstyle\frac{1}{96}}(3x^{5}+5x^{4}-5x^{3}-13x^{2}-6x)\,
  9. 1 4032 ( 63 x 6 + 63 x 5 - 315 x 4 - 539 x 3 - 84 x 2 + 236 x + 96 ) {\scriptstyle\frac{1}{4032}}(63x^{6}+63x^{5}-315x^{4}-539x^{3}-84x^{2}+236x+96)\,
  10. 1 1152 ( 9 x 7 - 84 x 5 - 98 x 4 + 91 x 3 + 194 x 2 + 80 x ) {\scriptstyle\frac{1}{1152}}(9x^{7}-84x^{5}-98x^{4}+91x^{3}+194x^{2}+80x)\,
  11. 1 34560 ( 135 x 8 - 180 x 7 - 1890 x 6 - 840 x 5 + 6055 x 4 + 8140 x 3 + 884 x 2 - 3088 x - 1152 ) {\scriptstyle\frac{1}{34560}}(135x^{8}-180x^{7}-1890x^{6}-840x^{5}+6055x^{4}+8% 140x^{3}+884x^{2}-3088x-1152)\,
  12. 1 7680 ( 15 x 9 - 45 x 8 - 270 x 7 + 182 x 6 + 1687 x 5 + 1395 x 4 - 1576 x 3 - 2684 x 2 - 1008 x ) {\scriptstyle\frac{1}{7680}}(15x^{9}-45x^{8}-270x^{7}+182x^{6}+1687x^{5}+1395x% ^{4}-1576x^{3}-2684x^{2}-1008x)\,
  13. S k ( - m ) = ( - 1 ) k ( k + m - 1 k ) S k + m - 1 , m - 1 S_{k}(-m)={(-1)^{k}\over{k+m-1\choose k}}S_{k+m-1,m-1}
  14. S m , n S_{m,n}
  15. S n , m = ( - 1 ) n - m ( n m ) S n - m ( - m - 1 ) S_{n,m}=(-1)^{n-m}{n\choose m}S_{n-m}(-m-1)
  16. S k ( - 1 ) = δ k , 0 ; S_{k}(-1)=\delta_{k,0};
  17. S k ( 0 ) = ( - 1 ) k B k S_{k}(0)=(-1)^{k}B_{k}
  18. B k B_{k}
  19. S k ( 1 ) = ( - 1 ) k + 1 ( ( k - 1 ) B k + k B k - 1 ) S_{k}(1)=(-1)^{k+1}((k-1)B_{k}+kB_{k-1})
  20. S k ( 2 ) = ( - 1 ) k 2 ( ( k - 1 ) ( k - 2 ) B k + 3 k ( k - 2 ) B k - 1 + 2 k ( k - 1 ) B k - 2 ) S_{k}(2)={(-1)^{k}\over 2}((k-1)(k-2)B_{k}+3k(k-2)B_{k-1}+2k(k-1)B_{k-2})
  21. S k ( k ) = k ! S_{k}(k)=k!
  22. S k ( m ) = ( - 1 ) k ( m k ) s m + 1 , m + 1 - k S_{k}(m)={(-1)^{k}\over{m\choose k}}s_{m+1,m+1-k}
  23. s m , n s_{m,n}
  24. s n , m = ( - 1 ) n - m ( n - 1 n - m ) S n - m ( n - 1 ) s_{n,m}=(-1)^{n-m}{n-1\choose n-m}S_{n-m}(n-1)
  25. S k ( x - 1 ) S_{k}(x-1)
  26. S k ( x + y - 1 ) = i = 0 k ( k i ) S i ( x - 1 ) S k - i ( y - 1 ) S_{k}(x+y-1)=\sum_{i=0}^{k}{k\choose i}S_{i}(x-1)S_{k-i}(y-1)
  27. S k ( x ) = ( x - k ) S k ( x - 1 ) x + k S k - 1 ( x + 1 ) S_{k}(x)=(x-k){S_{k}(x-1)\over x}+kS_{k-1}(x+1)
  28. S k ( x ) \displaystyle S_{k}(x)
  29. L n ( α ) L_{n}^{(\alpha)}
  30. ( k + m k ) S k ( x - m ) = i = 0 k ( - 1 ) k - i ( k + m i ) S k - i + m , m S i ( x ) , {k+m\choose k}S_{k}(x-m)=\sum_{i=0}^{k}(-1)^{k-i}{k+m\choose i}S_{k-i+m,m}% \cdot S_{i}(x),
  31. S k , n S_{k,n}
  32. ( k - m k ) S k ( x + m ) = i = 0 k ( k - m i ) s m , m - k + i S i ( x ) , {k-m\choose k}S_{k}(x+m)=\sum_{i=0}^{k}{k-m\choose i}s_{m,m-k+i}\cdot S_{i}(x),
  33. s k , n s_{k,n}
  34. S k ( x ) = - j = 0 k - 1 ( k j ) S j ( x ) B k - j k - j . S_{k}^{\prime}(x)=-\sum_{j=0}^{k-1}{k\choose j}S_{j}(x)\frac{B_{k-j}}{k-j}.
  35. ( t e t - 1 ) a e z t = k = 0 B k ( a ) ( z ) t k k ! . \left({t\over{e^{t}-1}}\right)^{a}e^{zt}=\sum_{k=0}^{\infty}B^{(a)}_{k}(z){t^{% k}\over k!}.
  36. S k ( x ) = B k ( x + 1 ) ( x + 1 ) S_{k}(x)=B_{k}^{(x+1)}(x+1)

Stochastic_dominance.html

  1. P [ A x ] P [ B x ] P[A\geq x]\geq P[B\geq x]
  2. P [ A x ] > P [ B x ] P[A\geq x]>P[B\geq x]
  3. F A ( x ) F B ( x ) F_{A}(x)\leq F_{B}(x)
  4. y y
  5. x B = 𝑑 ( x A + y ) x_{B}\overset{d}{=}(x_{A}+y)
  6. y 0 y\leq 0
  7. = 𝑑 \overset{d}{=}
  8. State (die result) 1 2 3 4 5 6 \,\text{State (die result)}\quad 1\quad 2\quad 3\quad 4\quad 5\quad 6
  9. Gamble A wins $ 1 1 2 2 2 2 \,\text{Gamble A wins}\ \$\quad 1\quad 1\quad 2\quad 2\quad 2\quad 2
  10. Gamble B wins $ 1 1 1 2 2 2 \,\text{Gamble B wins}\ \$\quad 1\quad 1\quad 1\quad 2\quad 2\quad 2
  11. Gamble C wins $ 3 3 3 1 1 1 \,\text{Gamble C wins}\ \$\quad 3\quad 3\quad 3\quad 1\quad 1\quad 1
  12. F A F_{A}
  13. F B F_{B}
  14. F A F_{A}
  15. x x
  16. F B F_{B}
  17. x x
  18. x x
  19. x x
  20. - x [ F B ( t ) - F A ( t ) ] d t 0 \int_{-\infty}^{x}[F_{B}(t)-F_{A}(t)]dt\geq 0
  21. x x
  22. x x
  23. A A
  24. B B
  25. E [ u ( A ) ] E [ u ( B ) ] E[u(A)]\geq E[u(B)]
  26. u ( x ) u(x)
  27. y y
  28. z z
  29. x B = 𝑑 ( x A + y + z ) x_{B}\overset{d}{=}(x_{A}+y+z)
  30. y y
  31. E ( z | x A + y ) = 0 E(z|x_{A}+y)=0
  32. x A + y x_{A}+y
  33. y y
  34. z z
  35. y y
  36. E A ( x ) E B ( x ) E_{A}(x)\geq E_{B}(x)
  37. min A ( x ) min B ( x ) \min_{A}(x)\geq\min_{B}(x)
  38. F B F_{B}
  39. F A F_{A}
  40. F A F_{A}
  41. F B F_{B}
  42. A A
  43. B B
  44. A A
  45. B B
  46. - x - z [ F B ( t ) - F A ( t ) ] d t d z 0 \int_{-\infty}^{x}\int_{-\infty}^{z}[F_{B}(t)-F_{A}(t)]\,dt\,dz\geq 0
  47. x x
  48. E A ( x ) E B ( x ) , E_{A}(x)\geq E_{B}(x),\,
  49. A A
  50. B B
  51. E A U ( x ) E B U ( x ) E_{A}U(x)\geq E_{B}U(x)
  52. U U
  53. E A ( log ( x ) ) E B ( log ( x ) ) E_{A}(\log(x))\geq E_{B}(\log(x))
  54. A A
  55. B B
  56. min A ( x ) min B ( x ) \min_{A}(x)\geq\min_{B}(x)
  57. F B F_{B}
  58. F A F_{A}
  59. f ( X ) f(X)
  60. X X
  61. X 0 X_{0}
  62. X X
  63. B B
  64. f ( X ) + E [ u ( X ) - u ( B ) ] f(X)+E[u(X)-u(B)]
  65. X X
  66. X 0 X_{0}
  67. u ( x ) u(x)
  68. u ( x ) u(x)
  69. u ( x ) u(x)

Stokes_number.html

  1. Stk = t 0 u 0 l 0 \mathrm{Stk}=\frac{t_{0}\,u_{0}}{l_{0}}
  2. t 0 t_{0}
  3. u 0 u_{0}
  4. l 0 l_{0}
  5. t 0 = ρ d d d 2 18 μ g t_{0}=\frac{\rho_{d}d_{d}^{2}}{18\mu_{g}}
  6. ρ d \rho_{d}
  7. d d d_{d}
  8. μ g \mu_{g}
  9. Stk 1 \mathrm{Stk}\gg 1
  10. Stk 1 \mathrm{Stk}\ll 1
  11. Stk 0.1 \mathrm{Stk}\ll 0.1
  12. c / c 0 = 1 + ( u 0 / u - 1 ) ( 1 - 1 1 + Stk ( 2 + 0.617 u / u 0 ) ) c/c_{0}=1+(u_{0}/u-1)\left(1-\frac{1}{1+\mathrm{Stk}(2+0.617u/u_{0})}\right)
  13. c c
  14. u u
  15. Stk = u 0 V s d g \mathrm{Stk}=\frac{u_{0}V_{s}}{dg}
  16. V s V_{s}
  17. d d
  18. g g

Stopping_power_(particle_radiation).html

  1. S ( E ) = - d E / d x S(E)=-dE/dx
  2. Δ x = 0 E 0 1 S ( E ) d E \Delta x=\int_{0}^{E_{0}}\frac{1}{S(E)}\,dE
  3. F e ( E ) F_{e}(E)
  4. E ( r ) E(r)
  5. F n ( E ) F_{n}(E)
  6. F ( E ) = F e ( E ) + F n ( E ) F(E)=F_{e}(E)+F_{n}(E)
  7. V ( r ) = 1 4 π ε 0 Z 1 Z 2 e 2 r φ ( r / a ) V(r)={1\over 4\pi\varepsilon_{0}}{Z_{1}Z_{2}e^{2}\over r}\varphi(r/a)
  8. Z 1 Z_{1}
  9. Z 2 Z_{2}
  10. a = a u = 0.8854 a 0 Z 1 0.23 + Z 2 0.23 a=a_{u}={0.8854a_{0}\over Z_{1}^{0.23}+Z_{2}^{0.23}}
  11. φ ( x ) = 0.1818 e - 3.2 x + 0.5099 e - 0.9423 x + 0.2802 e - 0.4029 x + 0.02817 e - 0.2016 x \varphi(x)=0.1818e^{-3.2x}+0.5099e^{-0.9423x}+0.2802e^{-0.4029x}+0.02817e^{-0.% 2016x}

Strehl_ratio.html

  1. S S
  2. S S
  3. δ ( x , y ) \delta(x,y)
  4. S = | e i ϕ | 2 = | e i 2 π δ / λ | 2 S=|\langle e^{i\phi}\rangle|^{2}=|\langle e^{i2\pi\delta/\lambda}\rangle|^{2}
  5. ϕ = 2 π δ / λ \phi=2\pi\delta/\lambda
  6. ϕ \phi
  7. S e - σ 2 S\approx{e^{-\sigma^{2}}}
  8. σ 2 = ( ϕ - ϕ ¯ ) 2 \sigma^{2}=\langle(\phi-\bar{\phi})^{2}\rangle
  9. λ \lambda
  10. λ - 2 \lambda^{-2}

Strictly_simple_group.html

  1. G G
  2. G G
  3. { e } \{e\}
  4. G G

String_diagram.html

  1. f : A B f:A\to B
  2. α : f g : A B \alpha:f\Rightarrow g:A\to B
  3. ( F , G , η , ε ) (F,G,\eta,\varepsilon)
  4. 𝒞 \mathcal{C}
  5. 𝒟 \mathcal{D}
  6. F : 𝒞 𝒟 F:\mathcal{C}\leftarrow\mathcal{D}
  7. G : 𝒞 𝒟 G:\mathcal{C}\rightarrow\mathcal{D}
  8. η : I G F \eta:I\rightarrow GF
  9. ε : F G I \varepsilon:FG\rightarrow I
  10. ( ε F ) F ( η ) \displaystyle(\varepsilon F)\circ F(\eta)

Stripline.html

  1. v p = c 0 ϵ r , eff . v_{\mathrm{p}}=\frac{c_{0}}{\sqrt{\epsilon_{\mathrm{r,eff}}}}.

Strong_monad.html

  1. t A , B = T ( γ B , A ) t B , A γ T A , B : T A B T ( A B ) t^{\prime}_{A,B}=T(\gamma_{B,A})\circ t_{B,A}\circ\gamma_{TA,B}:TA\otimes B\to T% (A\otimes B)
  2. A A
  3. B B
  4. ( T , η , μ , t ) (T,\eta,\mu,t)
  5. ( T , η , μ , m ) (T,\eta,\mu,m)
  6. m A , B = μ A B T t A , B t T A , B : T A T B T ( A B ) m_{A,B}=\mu_{A\otimes B}\circ Tt^{\prime}_{A,B}\circ t_{TA,B}:TA\otimes TB\to T% (A\otimes B)
  7. ( T , η , μ , m ) (T,\eta,\mu,m)
  8. ( T , η , μ , t ) (T,\eta,\mu,t)
  9. t A , B = m A , B ( η A 1 T B ) : A T B T ( A B ) t_{A,B}=m_{A,B}\circ(\eta_{A}\otimes 1_{TB}):A\otimes TB\to T(A\otimes B)

Strong_prime.html

  1. p p
  2. p p
  3. p p
  4. p p
  5. p - 1 p-1
  6. p = a 1 q 1 + 1 p=a_{1}q_{1}+1
  7. a 1 a_{1}
  8. q 1 q_{1}
  9. q 1 - 1 q_{1}-1
  10. q 1 = a 2 q 2 + 1 q_{1}=a_{2}q_{2}+1
  11. a 2 a_{2}
  12. q 2 q_{2}
  13. p + 1 p+1
  14. p = a 3 q 3 - 1 p=a_{3}q_{3}-1
  15. a 3 a_{3}
  16. q 3 q_{3}
  17. p n p_{n}
  18. p n > p n - 1 + p n + 1 2 p_{n}>{{p_{n-1}+p_{n+1}}\over 2}
  19. n n
  20. n = p q n=pq
  21. log c p \log^{c}p

Structure_(mathematical_logic).html

  1. 𝒜 = ( A , σ , I ) \mathcal{A}=(A,\sigma,I)
  2. dom ( 𝒜 ) \operatorname{dom}(\mathcal{A})
  3. | 𝒜 | |\mathcal{A}|
  4. 𝒜 \mathcal{A}
  5. 𝒜 \mathcal{A}
  6. n = ar ( s ) n=\operatorname{ar}(s)
  7. 𝒜 \mathcal{A}
  8. f 𝒜 = I ( f ) f^{\mathcal{A}}=I(f)
  9. R 𝒜 = I ( R ) A ar ( R ) R^{\mathcal{A}}=I(R)\subseteq A^{\operatorname{ar(R)}}
  10. 𝒜 \mathcal{A}
  11. f : 𝒜 2 𝒜 f:\mathcal{A}^{2}\rightarrow\mathcal{A}
  12. f 𝒜 : | 𝒜 | 2 | 𝒜 | f^{\mathcal{A}}:|\mathcal{A}|^{2}\rightarrow|\mathcal{A}|
  13. 𝒬 = ( Q , σ f , I 𝒬 ) \mathcal{Q}=(Q,\sigma_{f},I_{\mathcal{Q}})
  14. = ( R , σ f , I ) \mathcal{R}=(R,\sigma_{f},I_{\mathcal{R}})
  15. 𝒞 = ( C , σ f , I 𝒞 ) \mathcal{C}=(C,\sigma_{f},I_{\mathcal{C}})
  16. I 𝒬 ( + ) : Q × Q Q I_{\mathcal{Q}}(+)\colon Q\times Q\to Q
  17. I 𝒬 ( × ) : Q × Q Q I_{\mathcal{Q}}(\times)\colon Q\times Q\to Q
  18. I 𝒬 ( - ) : Q Q I_{\mathcal{Q}}(-)\colon Q\to Q
  19. I 𝒬 ( 0 ) Q I_{\mathcal{Q}}(0)\in Q
  20. I 𝒬 ( 1 ) Q I_{\mathcal{Q}}(1)\in Q
  21. I I_{\mathcal{R}}
  22. I 𝒞 I_{\mathcal{C}}
  23. 𝒜 \mathcal{A}
  24. \mathcal{B}
  25. 𝒜 \mathcal{A}
  26. \mathcal{B}
  27. σ ( 𝒜 ) = σ ( ) \sigma(\mathcal{A})=\sigma(\mathcal{B})
  28. 𝒜 \mathcal{A}
  29. \mathcal{B}
  30. | 𝒜 | | | |\mathcal{A}|\subseteq|\mathcal{B}|
  31. | | |\mathcal{B}|
  32. 𝒜 \mathcal{A}\subseteq\mathcal{B}
  33. B | 𝒜 | B\subseteq|\mathcal{A}|
  34. 𝒜 \mathcal{A}
  35. 𝒜 \mathcal{A}
  36. 𝒜 \mathcal{A}
  37. b 1 , b 2 , , b n B b_{1},b_{2},\dots,b_{n}\in B
  38. b 1 b 2 b n b_{1}b_{2}\dots b_{n}
  39. f ( b 1 , b 2 , , b n ) B f(b_{1},b_{2},\dots,b_{n})\in B
  40. B | 𝒜 | B\subseteq|\mathcal{A}|
  41. | 𝒜 | |\mathcal{A}|
  42. B \langle B\rangle
  43. B 𝒜 \langle B\rangle_{\mathcal{A}}
  44. \langle\rangle
  45. | 𝒜 | |\mathcal{A}|
  46. 𝒜 = ( A , σ , I ) \mathcal{A}=(A,\sigma,I)
  47. B A B\subseteq A
  48. ( B , σ , I ) (B,\sigma,I^{\prime})
  49. 𝒜 \mathcal{A}
  50. I I^{\prime}
  51. 𝒜 \mathcal{A}
  52. ( a , b ) E (a,b)\!\in\,\text{E}
  53. 𝒜 \mathcal{A}
  54. \mathcal{B}
  55. 𝒜 \mathcal{A}
  56. \mathcal{B}
  57. h : | 𝒜 | | | h:|\mathcal{A}|\rightarrow|\mathcal{B}|
  58. a 1 , a 2 , , a n | 𝒜 | a_{1},a_{2},\dots,a_{n}\in|\mathcal{A}|
  59. h ( f ( a 1 , a 2 , , a n ) ) = f ( h ( a 1 ) , h ( a 2 ) , , h ( a n ) ) h(f(a_{1},a_{2},\dots,a_{n}))=f(h(a_{1}),h(a_{2}),\dots,h(a_{n}))
  60. a 1 , a 2 , , a n | 𝒜 | a_{1},a_{2},\dots,a_{n}\in|\mathcal{A}|
  61. ( a 1 , a 2 , , a n ) R ( h ( a 1 ) , h ( a 2 ) , , h ( a n ) ) R (a_{1},a_{2},\dots,a_{n})\in R\implies(h(a_{1}),h(a_{2}),\dots,h(a_{n}))\in R
  62. 𝒜 \mathcal{A}
  63. \mathcal{B}
  64. h : 𝒜 h:\mathcal{A}\rightarrow\mathcal{B}
  65. h : 𝒜 h:\mathcal{A}\rightarrow\mathcal{B}
  66. b 1 , b 2 , , b n | | b_{1},b_{2},\dots,b_{n}\in|\mathcal{B}|
  67. ( b 1 , b 2 , , b n ) R (b_{1},b_{2},\dots,b_{n})\in R
  68. a 1 , a 2 , , a n | 𝒜 | a_{1},a_{2},\dots,a_{n}\in|\mathcal{A}|
  69. ( a 1 , a 2 , , a n ) R (a_{1},a_{2},\dots,a_{n})\in R
  70. b 1 = h ( a 1 ) , b 2 = h ( a 2 ) , , b n = h ( a n ) . b_{1}=h(a_{1}),\,b_{2}=h(a_{2}),\,\dots,\,b_{n}=h(a_{n}).
  71. h : 𝒜 h:\mathcal{A}\rightarrow\mathcal{B}
  72. a 1 , a 2 , , a n a_{1},a_{2},\dots,a_{n}
  73. ( a 1 , a 2 , , a n ) R ( h ( a 1 ) , h ( a 2 ) , , h ( a n ) ) R (a_{1},a_{2},\dots,a_{n})\in R\iff(h(a_{1}),h(a_{2}),\dots,h(a_{n}))\in R
  74. 𝒜 \mathcal{A}
  75. \mathcal{B}
  76. h : 𝒜 h:\mathcal{A}\rightarrow\mathcal{B}
  77. \mathcal{M}
  78. ϕ \mathcal{M}\vDash\phi
  79. ϕ \,\phi
  80. \mathcal{M}
  81. \mathcal{M}
  82. \mathcal{M}
  83. \mathcal{M}
  84. \mathcal{M}
  85. \emptyset
  86. R = { ( a 1 , , a n ) M n : ϕ ( a 1 , , a n ) } . R=\{(a_{1},\ldots,a_{n})\in M^{n}:\mathcal{M}\vDash\phi(a_{1},\ldots,a_{n})\}.
  87. ( a 1 , , a n ) R ϕ ( a 1 , , a n ) (a_{1},\ldots,a_{n})\in R\Leftrightarrow\mathcal{M}\vDash\phi(a_{1},\ldots,a_{% n})
  88. \mathcal{M}
  89. x ( x = m ϕ ( x ) ) . \mathcal{M}\vDash\forall x(x=m\leftrightarrow\phi(x)).
  90. | | |\mathcal{M}|
  91. \mathcal{M}
  92. \mathcal{M}
  93. R = { ( a 1 , , a n ) M n : ϕ ( a 1 , , a n ) } R=\{(a_{1},\ldots,a_{n})\in M^{n}:\mathcal{M}\vDash\phi(a_{1},\ldots,a_{n})\}
  94. \mathcal{M}
  95. \mathcal{M}
  96. \mathcal{M}
  97. \mathcal{M}
  98. ϕ \mathcal{M}\vDash\phi
  99. \mathcal{M}
  100. 𝒱 \mathcal{V}
  101. | 𝒱 | V = V |\mathcal{V}|_{V}=V
  102. | 𝒱 | S = F |\mathcal{V}|_{S}=F
  103. 0 V 𝒱 = 0 | 𝒱 | V 0_{V}^{\mathcal{V}}=0\in|\mathcal{V}|_{V}
  104. 0 S 𝒱 = 0 | 𝒱 | S 0_{S}^{\mathcal{V}}=0\in|\mathcal{V}|_{S}
  105. × 𝒱 : | 𝒱 | S × | 𝒱 | V | 𝒱 | V \times^{\mathcal{V}}:|\mathcal{V}|_{S}\times|\mathcal{V}|_{V}\rightarrow|% \mathcal{V}|_{V}
  106. | 𝒜 | |\mathcal{A}|
  107. 𝒜 \mathcal{A}

Structure_factor.html

  1. ϕ ( 𝐫 ) \phi(\mathbf{r})
  2. V V
  3. ϕ ( 𝐪 ) = V ϕ ( 𝐫 ) exp ( - i 𝐪𝐫 ) d 𝐫 \textstyle\phi(\mathbf{q})=\int_{V}\phi(\mathbf{r})\exp(-i\mathbf{q}\mathbf{r}% )\,\mathrm{d}\mathbf{r}
  4. ϕ \phi
  5. 𝐪 \mathbf{q}
  6. 𝐫 \mathbf{r}
  7. 𝐪 \mathbf{q}
  8. ϕ ( 𝐪 ) \textstyle\phi(\mathbf{q})
  9. I ( 𝐪 ) \textstyle I(\mathbf{q})
  10. I ( 𝐪 ) | ϕ ( 𝐪 ) | 2 \textstyle I(\mathbf{q})\sim\left|\phi(\mathbf{q})\right|^{2}
  11. N N
  12. ϕ \phi
  13. f ( 𝐫 ) f(\mathbf{r})
  14. 𝐑 j , j = 1 , , N \textstyle\mathbf{R}_{j},j=1,\,\ldots,\,N
  15. \ast
  16. f f
  17. ϕ ( 𝐪 ) = f ( 𝐪 ) j = 1 N exp ( - i 𝐪𝐑 j ) \textstyle\phi(\mathbf{q})=f(\mathbf{q})\sum_{j=1}^{N}\exp(-i\mathbf{q}\mathbf% {R}_{j})
  18. I ( 𝐪 ) \textstyle\langle I(\mathbf{q})\rangle
  19. \langle\cdot\rangle
  20. S ( 𝐪 ) = 1 N | j = 1 N e - i 𝐪𝐑 j | 2 S(\mathbf{q})=\frac{1}{N}\left|\sum_{j=1}^{N}\mathrm{e}^{-i\mathbf{q}\mathbf{R% }_{j}}\right|^{2}
  21. N N
  22. a a
  23. R j = a ( j - ( N - 1 ) / 2 ) \textstyle R_{j}=a(j-(N-1)/2)
  24. N N
  25. S ( q ) = 1 N | 1 - e - i N q a 1 - e - i q a | 2 = 1 N [ sin ( N q a / 2 ) sin ( q a / 2 ) ] 2 S(q)=\frac{1}{N}\left|\frac{1-\mathrm{e}^{-iNqa}}{1-\mathrm{e}^{-iqa}}\right|^% {2}=\frac{1}{N}\left[\frac{\sin(Nqa/2)}{\sin(qa/2)}\right]^{2}
  26. N N
  27. S ( q ) S(q)
  28. 2 π / a 2\pi/a
  29. S ( q = 2 k π / a ) = N S(q=2k\pi/a)=N
  30. S ( q 0 ) S(q\to 0)
  31. S ( q = ( 2 k + 1 ) π / a ) = 1 / N S(q=(2k+1)\pi/a)=1/N
  32. 1 / N 1/N
  33. N N
  34. 𝐤 i \mathbf{k}_{i}
  35. 𝐤 o \mathbf{k}_{o}
  36. 𝐪 \mathbf{q}
  37. 𝐤 i \mathbf{k}_{i}
  38. a a
  39. 𝐤 o \mathbf{k}_{o}
  40. | 𝐤 o | = | 𝐤 i | |\mathbf{k}_{o}|=|\mathbf{k}_{i}|
  41. 𝐪 = 𝐤 o - 𝐤 i \mathbf{q}=\mathbf{k}_{o}-\mathbf{k}_{i}
  42. exp ( i 𝐪𝐫 ) \exp(i\mathbf{q}\mathbf{r})
  43. q = 2 π / a q=2\pi/a
  44. 𝐤 o \mathbf{k}_{o}
  45. 𝐪 \mathbf{q}
  46. a x ^ , a y ^ , a z ^ a\hat{x},a\hat{y},a\hat{z}
  47. 𝐫 0 = 0 \mathbf{r}_{0}=\vec{0}
  48. 𝐫 1 = ( a / 2 ) ( x ^ + y ^ + z ^ ) \mathbf{r}_{1}=(a/2)(\hat{x}+\hat{y}+\hat{z})
  49. 2 π / a 2\pi/a
  50. f f
  51. 𝐊 = h x ^ * + k y ^ * + l z ^ * = ( 2 π / a ) ( h x ^ + k y ^ + l z ^ ) \mathbf{K}=h\hat{x}^{*}+k\hat{y}^{*}+l\hat{z}^{*}=(2\pi/a)(h\hat{x}+k\hat{y}+l% \hat{z})
  52. ( h k l ) (hkl)
  53. F 𝐊 = f [ e - i 𝐊 0 + e - i 𝐊 ( a / 2 ) ( x ^ + y ^ + z ^ ) ] = f [ 1 + e - i 𝐊 ( a / 2 ) ( x ^ + y ^ + z ^ ) ] = f [ 1 + e - i π ( h + k + l ) ] = f [ 1 + ( - 1 ) h + k + l ] \begin{matrix}F_{\mathbf{K}}&=&f\left[e^{-i\mathbf{K}\cdot\vec{0}}+e^{-i% \mathbf{K}\cdot(a/2)(\hat{x}+\hat{y}+\hat{z})}\right]\\ &=&f\left[1+e^{-i\mathbf{K}\cdot(a/2)(\hat{x}+\hat{y}+\hat{z})}\right]\\ &=&f\left[1+e^{-i\pi(h+k+l)}\right]\\ &=&f\left[1+(-1)^{h+k+l}\right]\\ \end{matrix}
  54. ( h k l ) (hkl)
  55. F h k l = { 2 f , h + k + l even 0 , h + k + l odd F_{hkl}=\begin{cases}2f,&h+k+l\ \ \mbox{even}\\ 0,&h+k+l\ \ \mbox{odd}\end{cases}
  56. 𝐫 0 = 0 \mathbf{r}_{0}=\vec{0}
  57. 𝐫 1 = ( a / 2 ) ( x ^ + y ^ ) \mathbf{r}_{1}=(a/2)(\hat{x}+\hat{y})
  58. 𝐫 2 = ( a / 2 ) ( y ^ + z ^ ) \mathbf{r}_{2}=(a/2)(\hat{y}+\hat{z})
  59. 𝐫 3 = ( a / 2 ) ( x ^ + z ^ ) \mathbf{r}_{3}=(a/2)(\hat{x}+\hat{z})
  60. F 𝐊 = f [ e - i 𝐊 0 + e - i 𝐊 ( a / 2 ) ( x ^ + y ^ ) + e - i 𝐊 ( a / 2 ) ( y ^ + z ^ ) + e - i 𝐊 ( a / 2 ) ( x ^ + z ^ ) ] = f [ 1 + ( - 1 ) h + k + ( - 1 ) k + l + ( - 1 ) h + l ] \begin{matrix}F_{\mathbf{K}}&=&f\left[e^{-i\mathbf{K}\cdot\vec{0}}+e^{-i% \mathbf{K}\cdot(a/2)(\hat{x}+\hat{y})}+e^{-i\mathbf{K}\cdot(a/2)(\hat{y}+\hat{% z})}+e^{-i\mathbf{K}\cdot(a/2)(\hat{x}+\hat{z})}\right]\\ &=&f\left[1+(-1)^{h+k}+(-1)^{k+l}+(-1)^{h+l}\right]\\ \end{matrix}
  61. F h k l = { 4 f , h , k , l all even or all odd 0 , h , k , l mixed parity F_{hkl}=\begin{cases}4f,&h,k,l\ \ \mbox{all even or all odd}\\ 0,&h,k,l\ \ \mbox{mixed parity}\end{cases}
  62. 𝐫 0 = 0 \mathbf{r}_{0}=\vec{0}
  63. 𝐫 1 = ( a / 4 ) ( x ^ + y ^ + z ^ ) \mathbf{r}_{1}=(a/4)(\hat{x}+\hat{y}+\hat{z})
  64. 𝐫 2 = ( a / 4 ) ( 2 x ^ + 2 y ^ ) \mathbf{r}_{2}=(a/4)(2\hat{x}+2\hat{y})
  65. 𝐫 3 = ( a / 4 ) ( 3 x ^ + 3 y ^ + z ^ ) \mathbf{r}_{3}=(a/4)(3\hat{x}+3\hat{y}+\hat{z})
  66. 𝐫 4 = ( a / 4 ) ( 2 x ^ + 2 z ^ ) \mathbf{r}_{4}=(a/4)(2\hat{x}+2\hat{z})
  67. 𝐫 5 = ( a / 4 ) ( 2 y ^ + 2 z ^ ) \mathbf{r}_{5}=(a/4)(2\hat{y}+2\hat{z})
  68. 𝐫 6 = ( a / 4 ) ( 3 x ^ + y ^ + 3 z ^ ) \mathbf{r}_{6}=(a/4)(3\hat{x}+\hat{y}+3\hat{z})
  69. 𝐫 7 = ( a / 4 ) ( x ^ + 3 y ^ + 3 z ^ ) \mathbf{r}_{7}=(a/4)(\hat{x}+3\hat{y}+3\hat{z})
  70. F 𝐊 = f [ e - i 𝐊 0 + e - i 𝐊 ( a / 2 ) ( x ^ + y ^ ) + e - i 𝐊 ( a / 2 ) ( y ^ + z ^ ) + e - i 𝐊 ( a / 2 ) ( x ^ + z ^ ) + e - i 𝐊 ( a / 4 ) ( x ^ + y ^ + z ^ ) + e - i 𝐊 ( a / 4 ) ( 3 x ^ + y ^ + 3 z ^ ) + e - i 𝐊 ( a / 4 ) ( 3 x ^ + 3 y ^ + z ^ ) + e - i 𝐊 ( a / 4 ) ( x ^ + 3 y ^ + 3 z ^ ) ] = f [ 1 + ( - 1 ) h + k + ( - 1 ) k + l + ( - 1 ) h + l + ( - i ) h + k + l + ( - i ) 3 h + k + 3 l + ( - i ) 3 h + 3 k + l + ( - i ) h + 3 k + 3 l ] = f [ 1 + ( - 1 ) h + k + ( - 1 ) k + l + ( - 1 ) h + l ] [ 1 + ( - i ) h + k + l ] \begin{matrix}F_{\mathbf{K}}&=&f\left[\begin{matrix}e^{-i\mathbf{K}\cdot\vec{0% }}+e^{-i\mathbf{K}\cdot(a/2)(\hat{x}+\hat{y})}+e^{-i\mathbf{K}\cdot(a/2)(\hat{% y}+\hat{z})}+e^{-i\mathbf{K}\cdot(a/2)(\hat{x}+\hat{z})}+\\ e^{-i\mathbf{K}\cdot(a/4)(\hat{x}+\hat{y}+\hat{z})}+e^{-i\mathbf{K}\cdot(a/4)(% 3\hat{x}+\hat{y}+3\hat{z})}+e^{-i\mathbf{K}\cdot(a/4)(3\hat{x}+3\hat{y}+\hat{z% })}+e^{-i\mathbf{K}\cdot(a/4)(\hat{x}+3\hat{y}+3\hat{z})}\end{matrix}\right]\\ &=&f\left[\begin{matrix}1+(-1)^{h+k}+(-1)^{k+l}+(-1)^{h+l}+\\ (-i)^{h+k+l}+(-i)^{3h+k+3l}+(-i)^{3h+3k+l}+(-i)^{h+3k+3l}\end{matrix}\right]\\ &=&f\left[1+(-1)^{h+k}+(-1)^{k+l}+(-1)^{h+l}\right]\cdot\left[1+(-i)^{h+k+l}% \right]\\ \end{matrix}
  71. q = | 𝐪 | q=\left|\mathbf{q}\right|
  72. j = k j=k
  73. S ( q ) S(q)
  74. g ( r ) g(r)
  75. S ( q ) = 1 S(q)=1
  76. 𝐑 j \mathbf{R}_{j}
  77. 𝐑 k \mathbf{R}_{k}
  78. exp [ - i 𝐪 ( 𝐑 j - 𝐑 k ) ] = exp ( - i 𝐪𝐑 j ) exp ( i 𝐪𝐑 k ) = 0 \langle\exp[-i\mathbf{q}(\mathbf{R}_{j}-\mathbf{R}_{k})]\rangle=\langle\exp(-i% \mathbf{q}\mathbf{R}_{j})\rangle\langle\exp(i\mathbf{q}\mathbf{R}_{k})\rangle=0
  79. q q
  80. S ( q ) - 1 S(q)-1
  81. g ( r ) g(r)
  82. q q
  83. q q
  84. q q
  85. χ T \chi_{T}
  86. lim q 0 S ( q ) = ρ k B T χ T = k B T ( ρ p ) \lim_{q\rightarrow 0}S(q)=\rho k\text{B}T\chi_{T}=k\text{B}T\left(\frac{% \partial\rho}{\partial p}\right)
  87. R R
  88. r 2 R r\geq 2R
  89. V ( r ) = { for r < 2 R 0 for r 2 R V(r)=\left\{\begin{array}[]{l l}\infty&\,\text{for}\,\,r<2R\\ 0&\,\text{for}\,\,r\geq 2R\\ \end{array}\right.
  90. Φ \Phi
  91. V V
  92. N c N_{c}
  93. N p N_{p}
  94. N c N p = N N_{c}N_{p}=N
  95. N p N_{p}
  96. α , β \alpha,\beta
  97. j , k j,k
  98. α = β \alpha=\beta
  99. α β \alpha\neq\beta
  100. S 1 ( q ) S_{1}(q)