wpmath0000009_11

Shannon_wavelet.html

  1. Ψ ( Sha ) ( w ) = ( w - 3 π / 2 π ) + ( w + 3 π / 2 π ) . \Psi^{(\operatorname{Sha})}(w)=\prod\left(\frac{w-3\pi/2}{\pi}\right)+\prod% \left(\frac{w+3\pi/2}{\pi}\right).
  2. ( x ) := { 1 , if | x | 1 / 2 , 0 if otherwise . \prod(x):=\begin{cases}1,&\mbox{if }~{}{|x|\leq 1/2},\\ 0&\mbox{if }~{}\mbox{otherwise}~{}.\\ \end{cases}
  3. ψ ( Sha ) ( t ) = sinc ( t 2 ) cos ( 3 π t 2 ) \psi^{(\operatorname{Sha})}(t)=\operatorname{sinc}\left(\frac{t}{2}\right)% \cdot\cos\left(\frac{3\pi t}{2}\right)
  4. ψ ( Sha ) ( t ) = 2 sinc ( 2 t - 1 ) - sinc ( t ) , \psi^{(\operatorname{Sha})}(t)=2\cdot\operatorname{sinc}(2t-1)-\operatorname{% sinc}(t),
  5. sinc ( t ) := sin π t π t \operatorname{sinc}(t):=\frac{\sin{\pi t}}{\pi t}
  6. C C^{\infty}
  7. ϕ ( S h a ) ( t ) = sin π t π t = sinc ( t ) . \phi^{(Sha)}(t)=\frac{\sin\pi t}{\pi t}=\operatorname{sinc}(t).
  8. ψ ( C S h a ) ( t ) = sinc ( t ) . e - j 2 π t \psi^{(CSha)}(t)=\operatorname{sinc}(t).e^{-j2\pi t}

Shape-memory_polymer.html

  1. R r ( N ) = ε m - ε p ( N ) ε m - ε p ( N - 1 ) R_{r}(N)=\frac{\varepsilon_{m}-\varepsilon_{p}(N)}{\varepsilon_{m}-\varepsilon% _{p}(N-1)}
  2. R f ( N ) = ε p ( N ) ε m R_{f}(N)=\frac{\varepsilon_{p}(N)}{\varepsilon_{m}}
  3. R f ( N ) = 1 - E f E g R_{f}(N)=1-\frac{E_{f}}{E_{g}}
  4. R r ( N ) = 1 - f I R f α ( 1 - E f / E g ) R_{r}(N)=1-\frac{f_{IR}}{f_{\alpha}(1-E_{f}/E_{g})}

Shape_moiré.html

  1. p m = - p b p r p b - p r p_{m}=-\frac{p_{b}\cdot p_{r}}{p_{b}-p_{r}}
  2. v m v r = - p b p b - p r \frac{v_{m}}{v_{r}}=-\frac{p_{b}}{p_{b}-p_{r}}

Shapiro_polynomials.html

  1. P 1 ( x ) \displaystyle P_{1}(x)
  2. P 0 ( z ) = 1 ; Q 0 ( z ) = 1 ; P_{0}(z)=1;~{}~{}Q_{0}(z)=1;
  3. P n + 1 ( z ) = P n ( z ) + z 2 n Q n ( z ) ; P_{n+1}(z)=P_{n}(z)+z^{2^{n}}Q_{n}(z);
  4. Q n + 1 ( z ) = P n ( z ) - z 2 n Q n ( z ) . Q_{n+1}(z)=P_{n}(z)-z^{2^{n}}Q_{n}(z).
  5. P n + 1 ( z ) = P n ( z 2 ) + z P n ( - z 2 ) ; P_{n+1}(z)=P_{n}(z^{2})+zP_{n}(-z^{2});\,
  6. Q n + 1 ( z ) = Q n ( z 2 ) + z Q n ( - z 2 ) ; Q_{n+1}(z)=Q_{n}(z^{2})+zQ_{n}(-z^{2});\,
  7. P n ( z ) P n ( 1 / z ) + Q n ( z ) Q n ( 1 / z ) = 2 n + 1 ; P_{n}(z)P_{n}(1/z)+Q_{n}(z)Q_{n}(1/z)=2^{n+1};\,
  8. P n + k + 1 ( z ) = P k ( z ) P n ( z 2 k + 1 ) + z 2 k Q k ( z ) P n ( - z 2 k + 1 ) ; P_{n+k+1}(z)=P_{k}(z)P_{n}(z^{2k+1})+z^{2k}Q_{k}(z)P_{n}(-z^{2k+1});\,
  9. P n ( 1 ) = 2 ( n + 1 ) / 2 ; P n ( - 1 ) = ( 1 + ( - 1 ) n ) 2 n / 2 - 1 . P_{n}(1)=2^{\lfloor(n+1)/2\rfloor};{~{}}{~{}}P_{n}(-1)=(1+(-1)^{n})2^{\lfloor n% /2\rfloor-1}.\,

Shapley–Shubik_power_index.html

  1. 2 n + 1 2n+1
  2. k k
  3. 2 n 2n
  4. k 2 n + 2 - k \dfrac{k}{2n+2-k}
  5. k k

Shear_strength_(soil).html

  1. τ \tau
  2. τ \tau
  3. τ \tau
  4. μ \mu
  5. τ \tau
  6. τ \tau
  7. τ \tau
  8. τ \tau

Shephard's_problem.html

  1. V k ( π k ( K ) ) V k ( π k ( L ) ) for all 1 k < n V n ( K ) V n ( L ) . V_{k}(\pi_{k}(K))\leq V_{k}(\pi_{k}(L))\mbox{ for all }~{}1\leq k<n\implies V_% {n}(K)\leq V_{n}(L).

Shewhart_individuals_control_chart.html

  1. x ¯ ± 3 M R ¯ d 2 \overline{x}\pm 3\frac{\overline{MR}}{d_{2}}
  2. x i x_{i}
  3. x i - 1 x_{i-1}
  4. M R i = | x i - x i - 1 | {MR}_{i}=\big|x_{i}-x_{i-1}\big|
  5. m m
  6. m - 1 m-1
  7. M R ¯ = i = 2 m M R i m - 1 \overline{MR}=\frac{\sum_{i=2}^{m}{MR_{i}}}{m-1}
  8. σ \sigma
  9. M R ¯ \overline{MR}
  10. d 2 σ = 2 σ / π . d_{2}\sigma=2\sigma/\sqrt{\pi}.
  11. U C L r = 3.267 M R ¯ UCL_{r}=3.267\overline{MR}
  12. x ¯ = i = 1 m x i m \overline{x}=\frac{\sum_{i=1}^{m}{x_{i}}}{m}
  13. U C L = x ¯ + 2.66 M R ¯ UCL=\overline{x}+2.66\overline{MR}
  14. L C L = x ¯ - 2.66 M R ¯ LCL=\overline{x}-2.66\overline{MR}
  15. M R i MR_{i}
  16. x ¯ \overline{x}
  17. x ¯ \overline{x}

Shifted_Gompertz_distribution.html

  1. ( 1 - e - b x ) e - η e - b x \left(1-e^{-bx}\right)e^{-\eta e^{-bx}}
  2. ( - 1 / b ) { E [ ln ( X ) ] - ln ( η ) } (-1/b)\{\mathrm{E}[\ln(X)]-\ln(\eta)\}\,
  3. X = η e - b x X=\eta e^{-bx}\,
  4. E [ ln ( X ) ] = [ 1 + 1 / η ] 0 η e - X [ ln ( X ) ] d X - 1 / η 0 η X e - X [ ln ( X ) ] d X \begin{aligned}\displaystyle\mathrm{E}[\ln(X)]=&\displaystyle[1{+}1/\eta]\!\!% \int_{0}^{\eta}\!\!\!\!e^{-X}[\ln(X)]dX\\ &\displaystyle-1/\eta\!\!\int_{0}^{\eta}\!\!\!\!Xe^{-X}[\ln(X)]dX\end{aligned}
  5. 0 for 0 < η 0.5 0\,\text{ for }0<\eta\leq 0.5
  6. ( - 1 / b ) ln ( z ) , for η > 0.5 (-1/b)\ln(z^{\star})\,\text{, for }\eta>0.5
  7. where z = [ 3 + η - ( η 2 + 2 η + 5 ) 1 / 2 ] / ( 2 η ) \,\text{ where }z^{\star}=[3+\eta-(\eta^{2}+2\eta+5)^{1/2}]/(2\eta)
  8. ( 1 / b 2 ) ( E { [ ln ( X ) ] 2 } - ( E [ ln ( X ) ] ) 2 ) (1/b^{2})(\mathrm{E}\{[\ln(X)]^{2}\}-(\mathrm{E}[\ln(X)])^{2})\,
  9. X = η e - b x X=\eta e^{-bx}\,
  10. E { [ ln ( X ) ] 2 } = [ 1 + 1 / η ] 0 η e - X [ ln ( X ) ] 2 d X - 1 / η 0 η X e - X [ ln ( X ) ] 2 d X \begin{aligned}\displaystyle\mathrm{E}\{[\ln(X)]^{2}\}=&\displaystyle[1{+}1/% \eta]\!\!\int_{0}^{\eta}\!\!\!\!e^{-X}[\ln(X)]^{2}dX\\ &\displaystyle-1/\eta\!\!\int_{0}^{\eta}\!\!\!\!Xe^{-X}[\ln(X)]^{2}dX\end{aligned}
  11. η \eta
  12. f ( x ; b , η ) = b e - b x e - η e - b x [ 1 + η ( 1 - e - b x ) ] for x 0. f(x;b,\eta)=be^{-bx}e^{-\eta e^{-bx}}\left[1+\eta\left(1-e^{-bx}\right)\right]% \,\text{ for }x\geq 0.\,
  13. b > 0 b>0
  14. η > 0 \eta>0
  15. F ( x ; b , η ) = ( 1 - e - b x ) e - η e - b x for x 0. F(x;b,\eta)=\left(1-e^{-bx}\right)e^{-\eta e^{-bx}}\,\text{ for }x\geq 0.\,
  16. η \eta
  17. η \eta
  18. 0 < η 0.5 0<\eta\leq 0.5\,
  19. η > 0.5 \eta>0.5\,
  20. mode = - ln ( z ) b 0 < z < 1 \,\text{mode}=-\frac{\ln(z^{\star})}{b}\,\qquad 0<z^{\star}<1
  21. z z^{\star}\,
  22. η 2 z 2 - η ( 3 + η ) z + η + 1 = 0 , \eta^{2}z^{2}-\eta(3+\eta)z+\eta+1=0\,,
  23. z = [ 3 + η - ( η 2 + 2 η + 5 ) 1 / 2 ] / ( 2 η ) . z^{\star}=[3+\eta-(\eta^{2}+2\eta+5)^{1/2}]/(2\eta).
  24. η \eta
  25. α \alpha
  26. β \beta
  27. α β \alpha\beta
  28. x x
  29. α \alpha

Shimura_variety.html

  1. 𝔤 = 𝔨 𝔭 + 𝔭 - , \mathfrak{g}\otimes\mathbb{C}=\mathfrak{k}\oplus\mathfrak{p}^{+}\oplus% \mathfrak{p}^{-},
  2. z / z ¯ z/\bar{z}
  3. z ¯ / z \bar{z}/z
  4. S h K ( G , X ) = G ( ) \ X × G ( 𝔸 f ) / K Sh_{K}(G,X)=G(\mathbb{Q})\backslash X\times G(\mathbb{A}_{f})/K
  5. ( S h K ( G , X ) ) K (Sh_{K}(G,X))_{K}

Shock_(fluid_dynamics).html

  1. 𝐓 𝟎𝟐 = 𝐓 𝟎𝟏 \mathbf{T_{02}}=\mathbf{T_{01}}
  2. M 2 = ( 2 γ - 1 + M 1 2 2 γ γ - 1 M 1 2 - 1 ) 0.5 M_{2}=(\frac{\frac{2}{\gamma-1}+{M_{1}}^{2}}{\frac{2\gamma}{\gamma-1}{M_{1}}^{% 2}-1})^{0.5}
  3. p 2 p 1 = 1 + γ M 1 2 1 + γ M 2 2 = 2 γ γ + 1 M 1 2 - γ - 1 γ + 1 \frac{p_{2}}{p_{1}}=\frac{1+\gamma M_{1}^{2}}{{1+\gamma M_{2}^{2}}}=\frac{2% \gamma}{\gamma+1}M_{1}^{2}-\frac{\gamma-1}{\gamma+1}
  4. T 2 T 1 = 1 + γ - 1 2 M 1 2 1 + γ - 1 2 M 2 2 = ( 1 + γ - 1 2 M 1 2 ) ( 2 γ γ - 1 M 1 2 - 1 ) ( γ + 1 ) 2 M 1 2 2 ( γ - 1 ) \frac{T_{2}}{T_{1}}=\frac{1+\frac{\gamma-1}{2}M_{1}^{2}}{{1+\frac{\gamma-1}{2}% M_{2}^{2}}}=\frac{(1+\frac{\gamma-1}{2}M_{1}^{2})(\frac{2\gamma}{\gamma-1}M_{1% }^{2}-1)}{\frac{(\gamma+1)^{2}M_{1}^{2}}{2(\gamma-1)}}
  5. a 2 a 1 = ( T 2 T 1 ) 0.5 \frac{a_{2}}{a_{1}}={(\frac{T_{2}}{T_{1}})}^{0.5}
  6. ρ 2 ρ 1 = p 2 p 1 T 1 T 2 \frac{\rho_{2}}{\rho_{1}}=\frac{p_{2}}{p_{1}}\frac{T_{1}}{T_{2}}
  7. p 01 p 1 = ( 1 + γ - 1 2 M 1 2 ) γ γ - 1 \frac{p_{01}}{p_{1}}=(1+\frac{\gamma-1}{2}M_{1}^{2})^{\frac{\gamma}{\gamma-1}}
  8. p 02 p 2 = ( 1 + γ - 1 2 M 2 2 ) γ γ - 1 \frac{p_{02}}{p_{2}}=(1+\frac{\gamma-1}{2}M_{2}^{2})^{\frac{\gamma}{\gamma-1}}

Shocks_and_discontinuities_(magnetohydrodynamics).html

  1. 𝐁 \nabla\cdot\mathbf{B}
  2. ρ 1 v n 1 = ρ 2 v n 2 , \rho_{1}v_{n1}=\rho_{2}v_{n2},
  3. B n 1 = B n 2 , B_{n1}=B_{n2},
  4. ρ 1 v n 1 2 + p 1 + B t 1 2 2 μ 0 = ρ 2 v n 2 2 + p 2 + B t 2 2 2 μ 0 , \rho_{1}v_{n1}^{2}+p_{1}+\frac{B_{t1}^{2}}{2\mu_{0}}=\rho_{2}v_{n2}^{2}+p_{2}+% \frac{B_{t2}^{2}}{2\mu_{0}},
  5. ρ 1 v n 1 𝐯 𝐭𝟏 - 𝐁 𝐭𝟏 B n 1 μ 0 = ρ 2 v n 2 𝐯 𝐭𝟐 - 𝐁 𝐭𝟐 B n 2 μ 0 , \rho_{1}v_{n1}\mathbf{v_{t1}}-\frac{\mathbf{B_{t1}}B_{n1}}{\mu_{0}}=\rho_{2}v_% {n2}\mathbf{v_{t2}}-\frac{\mathbf{B_{t2}}B_{n2}}{\mu_{0}},
  6. ( γ γ - 1 p 1 ρ 1 + v 1 2 2 ) ρ 1 v n 1 + v n 1 B t 1 2 μ 0 - B n 1 ( 𝐁 𝐭𝟏 𝐯 𝐭𝟏 ) μ 0 = ( γ γ - 1 p 2 ρ 2 + v 2 2 2 ) ρ 2 v n 2 + v n 2 B t 2 2 μ 0 - B n 2 ( 𝐁 𝐭𝟐 𝐯 𝐭𝟐 ) μ 0 , \left(\frac{\gamma}{\gamma-1}\frac{p_{1}}{\rho_{1}}+\frac{v_{1}^{2}}{2}\right)% \rho_{1}v_{n1}+\frac{v_{n1}B_{t1}^{2}}{\mu_{0}}-\frac{B_{n1}(\mathbf{B_{t1}}% \cdot\mathbf{v_{t1}})}{\mu_{0}}=\left(\frac{\gamma}{\gamma-1}\frac{p_{2}}{\rho% _{2}}+\frac{v_{2}^{2}}{2}\right)\rho_{2}v_{n2}+\frac{v_{n2}B_{t2}^{2}}{\mu_{0}% }-\frac{B_{n2}(\mathbf{B_{t2}}\cdot\mathbf{v_{t2}})}{\mu_{0}},
  7. ( 𝐯 × 𝐁 ) t 1 = ( 𝐯 × 𝐁 ) t 2 , (\mathbf{v}\times\mathbf{B})_{t1}=(\mathbf{v}\times\mathbf{B})_{t2},
  8. ρ \rho
  9. v n 1 = v n 2 = 0 v_{n1}=v_{n2}=0
  10. V A V_{A}
  11. c s c_{s}
  12. a slow 2 = 1 2 [ ( c s 2 + V A 2 ) - ( c s 2 + V A 2 ) 2 - 4 c s 2 V A 2 cos 2 θ B n ] , a_{\mathrm{slow}}^{2}=\frac{1}{2}\left[\left(c_{s}^{2}+V_{A}^{2}\right)-\sqrt{% \left(c_{s}^{2}+V_{A}^{2}\right)^{2}-4c_{s}^{2}V_{A}^{2}\cos^{2}\theta_{Bn}}\,% \right],
  13. a fast 2 = 1 2 [ ( c s 2 + V A 2 ) + ( c s 2 + V A 2 ) 2 - 4 c s 2 V A 2 cos 2 θ B n ] , a_{\mathrm{fast}}^{2}=\frac{1}{2}\left[\left(c_{s}^{2}+V_{A}^{2}\right)+\sqrt{% \left(c_{s}^{2}+V_{A}^{2}\right)^{2}-4c_{s}^{2}V_{A}^{2}\cos^{2}\theta_{Bn}}\,% \right],
  14. V A V_{A}
  15. θ B n \theta_{Bn}
  16. a slow a_{\mathrm{slow}}
  17. V A n V_{An}
  18. a fast a_{\mathrm{fast}}

Siegel's_lemma.html

  1. a 11 X 1 + + a 1 N X N = 0 a_{11}X_{1}+\cdots+a_{1N}X_{N}=0
  2. \cdots
  3. a M 1 X 1 + + a M N X N = 0 a_{M1}X_{1}+\cdots+a_{MN}X_{N}=0
  4. ( X 1 , X 2 , , X N ) (X_{1},X_{2},\dots,X_{N})
  5. ( N B ) M / ( N - M ) . (NB)^{M/(N-M)}.\,
  6. max | X j | ( D - 1 det ( A A T ) ) 1 / ( N - M ) \max|X_{j}|\leq\left(D^{-1}\sqrt{\det(AA^{T})}\right)^{1/(N-M)}

Siegel_disc.html

  1. f : S S f:S\to S
  2. S S
  3. f f
  4. f n = f ( n ) f f^{n}=f\circ\stackrel{\left(n\right)}{\cdots}\circ f
  5. 𝒪 + ( z 0 ) \mathcal{O}^{+}(z_{0})
  6. z 0 z_{0}
  7. z 0 z_{0}
  8. S S
  9. \mathbb{C}
  10. ^ = { } \mathbb{\hat{C}}=\mathbb{C}\cup\{\infty\}
  11. S S
  12. z 0 z_{0}
  13. f p ( z 0 ) = z 0 f^{p}(z_{0})=z_{0}
  14. p p
  15. p = 1 p=1
  16. z 0 z_{0}
  17. ρ = ( f p ) ( z 0 ) \rho=(f^{p})^{\prime}(z_{0})
  18. | ρ | < 1 |\rho|<1
  19. | ρ | = 0 |\rho|=0
  20. | ρ | > 1 |\rho|>1
  21. ρ = 1 \rho=1
  22. ρ n = 1 \rho^{n}=1
  23. n n\in\mathbb{Z}
  24. ρ n 1 \rho^{n}\neq 1
  25. n n\in\mathbb{Z}
  26. f f
  27. B ( z ) = λ a ( e z / a ( z + 1 - a ) + a - 1 ) B(z)=\lambda a(e^{z/a}(z+1-a)+a-1)
  28. a = 15 - 15 i a=15-15i
  29. λ \lambda
  30. B ( z ) = λ a ( e z / a ( z + 1 - a ) + a - 1 ) B(z)=\lambda a(e^{z/a}(z+1-a)+a-1)
  31. a = - 0.33258 + 0.10324 i a=-0.33258+0.10324i
  32. λ \lambda
  33. f c ( z ) = z * z + c f_{c}(z)=z*z+c
  34. f c ( z ) = z * z + c f_{c}(z)=z*z+c
  35. f : S S f:S\to S
  36. S S
  37. ( f ) \mathcal{F}(f)
  38. ϕ : U 𝔻 \phi:U\to\mathbb{D}
  39. 𝔻 \mathbb{D}
  40. ϕ ( f n ( ϕ - 1 ( z ) ) ) = e 2 π i α z \phi(f^{n}(\phi^{-1}(z)))=e^{2\pi i\alpha}z
  41. α \ \alpha\in\mathbb{R}\backslash\mathbb{Q}
  42. ϕ ( z 0 ) = 0 \phi(z_{0})=0

Sieving_coefficient.html

  1. S = C r C d S=\frac{C_{r}}{C_{d}}

Signal-recognition-particle_GTPase.html

  1. \rightleftharpoons

Simon's_problem.html

  1. O ( n ) O(n)
  2. Ω ( 2 n / 2 ) \Omega(2^{n/2})
  3. Ω ( n ) \Omega(n)
  4. f : { 0 , 1 } n { 0 , 1 } n f:\{0,1\}^{n}\rightarrow\{0,1\}^{n}
  5. s { 0 , 1 } n s\in\{0,1\}^{n}
  6. y , z { 0 , 1 } n y,z\in\{0,1\}^{n}
  7. f ( y ) = f ( z ) f(y)=f(z)
  8. y = z y=z
  9. y z = s y\oplus z=s
  10. s = 0 n s=0^{n}
  11. f f
  12. 2 \mathbb{Z}_{2}
  13. x | x | 0 \sum_{x}\left|x\right\rangle\left|0\right\rangle
  14. x | x | f ( x ) \sum_{x}\left|x\right\rangle\left|f(x)\right\rangle
  15. y x ( - 1 ) x . y | y | f ( x ) \sum_{y}\sum_{x}(-1)^{x.y}\left|y\right\rangle\left|f(x)\right\rangle
  16. s y = 1 s\cdot y=1
  17. s y = 0 s\cdot y=0

Simon_model.html

  1. k k
  2. n n
  3. k i k_{i}
  4. i = 1 , , n i=1,\ldots,n
  5. [ k ] [k]
  6. f ( k ) f(k)
  7. k k
  8. α \alpha
  9. 1 - α 1-\alpha
  10. j j
  11. [ k ] [k]
  12. k f ( k ) kf(k)
  13. P ( k ) k - γ P(k)\propto k^{-\gamma}
  14. γ = 1 + 1 1 - α . \gamma=1+\frac{1}{1-\alpha}.
  15. α = 1 / 2 \alpha=1/2
  16. i i
  17. k i k_{i}
  18. P ( new link to i ) k i P(\mathrm{new\ link\ to\ }i)\propto k_{i}
  19. α \alpha
  20. α \alpha
  21. γ 2 \gamma\approx 2
  22. γ 2.1 \gamma\approx 2.1

Sinclair_Coefficients.html

  1. 10 A ( log 10 ( x / b ) ) 2 10^{A({\log_{10}{(x/b)}})^{2}}

Singularity_function.html

  1. x - a n \langle x-a\rangle^{n}
  2. \langle\rangle
  3. x - a n \langle x-a\rangle^{n}
  4. < 0 <0
  5. d | n + 1 | d x | n + 1 | δ ( x - a ) \frac{d^{|n+1|}}{dx^{|n+1|}}\delta(x-a)\,
  6. d d x δ ( x - a ) \frac{d}{dx}\delta(x-a)\,
  7. δ ( x - a ) \delta(x-a)\,
  8. H ( x - a ) H(x-a)\,
  9. ( x - a ) H ( x - a ) (x-a)H(x-a)\,
  10. ( x - a ) 2 H ( x - a ) (x-a)^{2}H(x-a)
  11. 0 \geq 0
  12. ( x - a ) n H ( x - a ) (x-a)^{n}H(x-a)
  13. H ( x ) H(x)
  14. x - a 1 \langle x-a\rangle^{1}
  15. x - a n \langle x-a\rangle^{n}
  16. x - a n d x = { x - a n + 1 , n 0 x - a n + 1 n + 1 , n 0 \int\langle x-a\rangle^{n}dx=\begin{cases}\langle x-a\rangle^{n+1},&n\leq 0\\ \frac{\langle x-a\rangle^{n+1}}{n+1},&n\geq 0\end{cases}
  17. w = - 3 N x - 0 - 1 + 6 N m - 1 x - 2 m 0 - 9 N x - 4 m - 1 w=-3N\langle x-0\rangle^{-1}\ +\ 6Nm^{-1}\langle x-2m\rangle^{0}\ -\ 9N\langle x% -4m\rangle^{-1}\,
  18. S = w d x S=\int wdx
  19. S = - 3 N x - 0 0 + 6 N m - 1 x - 2 m 1 - 9 N x - 4 m 0 S=-3N\langle x-0\rangle^{0}\ +\ 6Nm^{-1}\langle x-2m\rangle^{1}\ -\ 9N\langle x% -4m\rangle^{0}\,
  20. M = - S d x M=-\int Sdx
  21. M = 3 N x - 0 1 - 3 N m - 1 x - 2 m 2 + 9 N x - 4 m 1 M=3N\langle x-0\rangle^{1}\ -\ 3Nm^{-1}\langle x-2m\rangle^{2}\ +\ 9N\langle x% -4m\rangle^{1}\,
  22. u = 1 E I M d x u^{\prime}=\frac{1}{EI}\int Mdx
  23. u = 1 E I ( 3 2 N x - 0 2 - 1 N m - 1 x - 2 m 3 + 9 2 N x - 4 m 2 + c ) u^{\prime}=\frac{1}{EI}\left(\frac{3}{2}N\langle x-0\rangle^{2}\ -\ 1Nm^{-1}% \langle x-2m\rangle^{3}\ +\ \frac{9}{2}N\langle x-4m\rangle^{2}\ +\ c\right)\,
  24. u = u d x u=\int u^{\prime}dx
  25. u = 1 E I ( 1 2 N x - 0 3 - 1 4 N m - 1 x - 2 m 4 + 3 2 N x - 4 m 3 + c x ) u=\frac{1}{EI}\left(\frac{1}{2}N\langle x-0\rangle^{3}\ -\ \frac{1}{4}Nm^{-1}% \langle x-2m\rangle^{4}\ +\ \frac{3}{2}N\langle x-4m\rangle^{3}\ +\ cx\right)\,

Skew-Hamiltonian_matrix.html

  1. Ω \Omega
  2. A : V V A:\;V\mapsto V
  3. Ω \Omega
  4. x , y Ω ( A ( x ) , y ) x,y\mapsto\Omega(A(x),y)
  5. e 1 , e 2 n e_{1},...e_{2n}
  6. Ω \Omega
  7. i e i e n + i \sum_{i}e_{i}\wedge e_{n+i}
  8. Ω \Omega
  9. A T J = J A A^{T}J=JA
  10. J = [ 0 I n - I n 0 ] J=\begin{bmatrix}0&I_{n}\\ -I_{n}&0\\ \end{bmatrix}
  11. n × n n\times n

Skew_lattice.html

  1. \wedge
  2. \vee
  3. x ( x y ) = x = ( y x ) x x\wedge(x\vee y)=x=(y\vee x)\wedge x
  4. x ( x y ) = x = ( y x ) x x\vee(x\wedge y)=x=(y\wedge x)\vee x
  5. \vee
  6. \wedge
  7. x y = x x\vee y=x
  8. x y = y x\wedge y=y
  9. x y = x x\wedge y=x
  10. x y = y x\vee y=y
  11. ( S ; , ) (S;\wedge,\vee)
  12. \wedge
  13. \vee
  14. \wedge
  15. \vee
  16. x ( y x ) = x = ( x y ) x . x\wedge(y\vee x)=x=(x\wedge y)\vee x.
  17. S S
  18. y x y\leq x
  19. x y = y = y x x\wedge y=y=y\wedge x
  20. x y = x = y x x\vee y=x=y\vee x
  21. S S
  22. y x y\preceq x
  23. y x y = y y\wedge x\wedge y=y
  24. x y x = x x\vee y\vee x=x
  25. \leq
  26. \preceq
  27. \leq
  28. \preceq
  29. D D
  30. x D y xDy
  31. x y x x\preceq y\preceq x
  32. x y x = x x\wedge y\wedge x=x
  33. y x y = y y\wedge x\wedge y=y
  34. x y x = x x\vee y\vee x=x
  35. y x y = y y\vee x\vee y=y
  36. S / D S/D
  37. A > B A>B
  38. a A a\in A
  39. b B b\in B
  40. a > b a>b
  41. a a
  42. b b
  43. D D
  44. D D
  45. 1 1
  46. c c
  47. 0
  48. D D
  49. D D
  50. x y x = x x\wedge y\wedge x=x
  51. y x y = y y\vee x\vee y=y
  52. x y = y x x\vee y=y\wedge x
  53. L L
  54. R R
  55. L × R L\times R
  56. ( x , y ) ( z , w ) = ( z , y ) (x,y)\vee(z,w)=(z,y)
  57. ( x , y ) ( z , w ) = ( x , w ) (x,y)\wedge(z,w)=(x,w)
  58. D D
  59. S S
  60. S S
  61. D D
  62. S / D S/D
  63. S S
  64. S S
  65. x y x = y x x\wedge y\wedge x=y\wedge x
  66. x y x = x y x\vee y\vee x=x\vee y
  67. x y = y x\wedge y=y
  68. x y = x x\vee y=x
  69. D D
  70. S S
  71. S / L S/L
  72. L L
  73. x L y xLy
  74. x y = x x\wedge y=x
  75. y x = y y\wedge x=y
  76. x y = y x\vee y=y
  77. y x = x y\vee x=x
  78. x y = x x\wedge y=x
  79. x y = y x\vee y=y
  80. D D
  81. S S
  82. S / R S/R
  83. R R
  84. L L
  85. R L = D R\vee L=D
  86. R L R\cap L
  87. Δ \Delta
  88. S S / D S\rightarrow S/D
  89. S S / L S\rightarrow S/L
  90. S S / R S\rightarrow S/R
  91. T = S / D T=S/D
  92. k : S S / L × S / R k:S\rightarrow S/L\times S/R
  93. k ( x ) = ( L x , R x ) k(x)=(L_{x},R_{x})
  94. k * : S S / L × T S / R k*:S\sim S/L\times_{T}S/R
  95. S S
  96. x y x z x = x y z x xyxzx=xyzx
  97. \wedge
  98. \vee
  99. D D
  100. R R
  101. L L
  102. x , y S x,y\in S
  103. x y = y x x\wedge y=y\wedge x
  104. x y = y x x\vee y=y\vee x
  105. x y ( x y ) = ( y x ) y x x\vee y\vee(x\wedge y)=(y\wedge x)\vee y\vee x
  106. x y ( x y ) = ( y x ) y x x\wedge y\wedge(x\vee y)=(y\vee x)\wedge y\wedge x
  107. S S
  108. T T
  109. S S
  110. D D
  111. S S
  112. T T
  113. S / D S/D
  114. T S S / D T\subset S\rightarrow S/D
  115. T T
  116. S S
  117. S / L S/L
  118. S / R S/R
  119. T [ R ] = t T R t T[R]=\bigcup_{t\in T}R_{t}
  120. T [ L ] = t T L t T[L]=\bigcup_{t\in T}L_{t}
  121. R t R_{t}
  122. L t Lt
  123. R R
  124. L L
  125. t t
  126. T T
  127. T [ R ] S S / L T[R]\subset S\rightarrow S/L
  128. T [ L ] S S / R T[L]\subset S\rightarrow S/R
  129. x y = x z x\vee y=x\vee z
  130. x y = x z x\wedge y=x\wedge z
  131. y = z y=z
  132. x z = y z x\vee z=y\vee z
  133. x z = y z x\wedge z=y\wedge z
  134. x = y x=y
  135. x ( y z ) x = ( x y x ) ( x z x ) x\wedge(y\vee z)\wedge x=(x\wedge y\wedge x)\vee(x\wedge z\wedge x)
  136. x ( y z ) x = ( x y x ) ( x z x ) . x\vee(y\wedge z)\vee x=(x\vee y\vee x)\wedge(x\vee z\vee x).
  137. x ( y z ) w = ( x y w ) ( x z w ) x\wedge(y\vee z)\wedge w=(x\wedge y\wedge w)\vee(x\wedge z\wedge w)
  138. S S
  139. x ( y z ) w = ( x y w ) ( x z w ) x\vee(y\wedge z)\vee w=(x\vee y\vee w)\wedge(x\vee z\vee w)
  140. x ( y z ) = ( x y ) ( x z ) x\wedge(y\vee z)=(x\wedge y)\vee(x\wedge z)
  141. ( y z ) w = ( y w ) ( z w ) (y\vee z)\wedge w=(y\wedge w)\vee(z\wedge w)
  142. \wedge
  143. \vee
  144. x y x z x = x y z x xyxzx=xyzx
  145. x y z x = x z y x xyzx=xzyx
  146. x y z x = x z y x . ( N ) x\wedge y\wedge z\wedge x=x\wedge z\wedge y\wedge x.(N)
  147. S S
  148. a S a a\wedge S\wedge a
  149. a x a | x S a\wedge x\wedge a|x\in S
  150. x S | x a x\in S|x\leq a
  151. S S
  152. \wedge
  153. \vee
  154. S S
  155. T × D T\times D
  156. T T
  157. D D
  158. ( D 2 ) = ( D 1 ) + ( N ) (D2)=(D1)+(N)
  159. ( D 2 ) (D2)
  160. a > b > c a>b>c
  161. a A a\in A
  162. b B b\in B
  163. c C c\in C
  164. φ \varphi
  165. A A
  166. B B
  167. a a
  168. b b
  169. ψ \psi
  170. B B
  171. C C
  172. b b
  173. c c
  174. χ \chi
  175. A A
  176. C C
  177. a a
  178. c c
  179. S S
  180. ψ φ = χ \psi\circ\varphi=\chi
  181. ψ φ \psi\circ\varphi
  182. C C
  183. A A
  184. A A
  185. C C
  186. ( A b A ) ( C b C ) = ( C a C ) b ( C a C ) = ( A c A ) b ( A c A ) (A\wedge b\wedge A)\cap(C\vee b\vee C)=(C\vee a\vee C)\wedge b\wedge(C\vee a% \vee C)=(A\wedge c\wedge A)\vee b\vee(A\wedge c\wedge A)
  187. x S x\in S
  188. 0 x = 0 = x 0 0\wedge x=0=x\wedge 0
  189. 0 x = x = x 0 0\vee x=x=x\vee 0
  190. ( S ; , , 0 ) (S;\vee,\wedge,0)
  191. a S a a\wedge S\wedge a
  192. a S a\in S
  193. x - x y x x-x\wedge y\wedge x
  194. x S x x\wedge S\wedge x
  195. y x / y = 0 = x / y y y\wedge x/y=0=x/y\wedge y
  196. ( x y x ) x / y = x = x / y ( x y x ) . ( S B ) (x\wedge y\wedge x)\vee x/y=x=x/y\vee(x\wedge y\wedge x).(SB)
  197. ( S ; , , / , 0 ) (S;\vee,\wedge,/,0)
  198. x / y = x x/y=x
  199. y = 0 y=0
  200. 0
  201. A A
  202. E ( A ) E(A)
  203. A A
  204. x , y A x,y\in A
  205. x y = x y x\wedge y=xy
  206. x y = x + y - x y x\vee y=x+y-xy
  207. \wedge
  208. \vee
  209. S E ( A ) S\subseteq E(A)
  210. \wedge
  211. \vee
  212. ( S , , ) (S,\wedge,\vee)
  213. E ( A ) E(A)
  214. E ( A ) E(A)
  215. ( ) ()
  216. \vee
  217. E ( A ) E(A)
  218. E ( A ) E(A)
  219. x y x = x y xyx=xy
  220. E ( A ) E(A)
  221. \vee
  222. \vee
  223. x y = x + y + y x - x y x - y x y x\nabla y=x+y+yx-xyx-yxy
  224. x y x\nabla y
  225. y x yx
  226. ( x y z w = x z y w ) (xyzw=xzyw)
  227. S S
  228. E ( A ) E(A)
  229. \nabla
  230. ( S ; , , / , 0 ) (S;\wedge,\vee,/,0)
  231. x / y = x - x y x x/y=x-xyx
  232. E ( A ) E(A)
  233. 1 1
  234. E ( A ) E(A)
  235. E ( A ) E(A)
  236. S S
  237. D D
  238. A > B A>B
  239. S / D S/D
  240. a A a\in A
  241. b B b\in B
  242. A b A = A\wedge b\wedge A=
  243. u b u : u A u\wedge b\wedge u:u\in A
  244. B \subseteq B
  245. B a B = B\vee a\vee B=
  246. v a v : v B v\vee a\vee v:v\in B
  247. A \subseteq A
  248. b i n A b A binA\wedge b\wedge A
  249. a B a B a\in B\wedge a\wedge B
  250. D D
  251. \geq
  252. φ : B a B A b A \varphi:B\vee a\vee B\rightarrow A\wedge b\wedge A
  253. ϕ ( x ) = y \phi(x)=y
  254. x > y x>y
  255. x B a B x\in B\vee a\vee B
  256. y A b A y\in A\wedge b\wedge A
  257. \geq
  258. A A
  259. B B
  260. \vee
  261. \wedge
  262. D D
  263. a A a\in A
  264. b B b\in B
  265. φ \varphi
  266. B a B B\vee a\vee B
  267. A A
  268. A b A A\wedge b\wedge A
  269. B B
  270. a b = a φ - 1 ( b ) , b a = φ - 1 ( b ) a a\vee b=a\vee\varphi-1(b),b\vee a=\varphi-1(b)\vee a
  271. a b = φ ( a ) b , b a = b φ ( a ) a\wedge b=\varphi(a)\wedge b,b\wedge a=b\wedge\varphi(a)
  272. a , c A a,c\in A
  273. b , d B b,d\in B
  274. a > b a>b
  275. c > d c>d
  276. a , c a,c
  277. B B
  278. A A
  279. b , d b,d
  280. A A
  281. B B
  282. a > b / / c > d a>b//c>d
  283. a > b a>b
  284. S S
  285. S / R × 2 S / L S/R\times_{2}S/L
  286. A = i A i A=\cup_{i}A_{i}
  287. B = j B j B=\cup_{j}B_{j}
  288. A A
  289. B B
  290. A i A_{i}
  291. B j B_{j}
  292. i , j i,j
  293. φ i , j \varphi_{i},j
  294. A i A_{i}
  295. B j B_{j}
  296. A A
  297. B B
  298. x y = y x\wedge y=y
  299. x y = x x\vee y=x
  300. a A a\in A
  301. b B b\in B
  302. a b = a , b a = a , a b = b a\vee b=a,b\vee a=a^{\prime},a\wedge b=b
  303. b a = b b\wedge a=b^{\prime}
  304. φ i , j ( a ) = b \varphi_{i,j}(a^{\prime})=b
  305. φ i , j ( a ) = b \varphi_{i,j}(a)=b^{\prime}
  306. a a^{\prime}
  307. A i A_{i}
  308. a a
  309. b b^{\prime}
  310. B j B_{j}
  311. b b
  312. φ i , j \varphi i,j
  313. | A i | = | B j | = 2 |A_{i}|=|B_{j}|=2
  314. φ i , j \varphi_{i,j}
  315. \geq
  316. A A
  317. B B
  318. S S
  319. D D
  320. A > B A>B
  321. S / D S/D
  322. A B A\cup B
  323. S S
  324. D D
  325. S S
  326. x y x\vee y
  327. x y x\wedge y
  328. x x
  329. y y
  330. \preceq
  331. x y x\vee y
  332. x y x\wedge y
  333. \preceq
  334. M M
  335. S / D S/D
  336. A > B > C A>B>C
  337. S / D S/D
  338. φ : A B \varphi:A\rightarrow B
  339. ψ : B C \psi:B\rightarrow C
  340. ψ φ \psi\varphi
  341. χ : A C \chi:A\rightarrow C
  342. ψ φ χ \psi\varphi\subseteq\chi
  343. χ \chi
  344. A A
  345. C C
  346. ψ φ \psi\varphi\neq

Skewb_Ultimate.html

  1. 4 ! × 3 6 2 \frac{4!\times 3^{6}}{2}
  2. 6 ! × 2 5 × 4 ! × 3 6 4 = 100 , 776 , 960. \frac{6!\times 2^{5}\times 4!\times 3^{6}}{4}=100,776,960.

Skoda–El_Mir_theorem.html

  1. Θ \Theta
  2. X \ E X\backslash E
  3. Θ \Theta

Skorokhod's_embedding_theorem.html

  1. 𝔼 [ τ ] = 𝔼 [ X 2 ] \mathbb{E}[\tau]=\mathbb{E}[X^{2}]
  2. 𝔼 [ τ 2 ] 4 𝔼 [ X 4 ] . \mathbb{E}[\tau^{2}]\leq 4\mathbb{E}[X^{4}].
  3. S n = X 1 + + X n . S_{n}=X_{1}+\cdots+X_{n}.
  4. W τ n W_{\tau_{n}}
  5. 𝔼 [ τ n - τ n - 1 ] = 𝔼 [ X 1 2 ] \mathbb{E}[\tau_{n}-\tau_{n-1}]=\mathbb{E}[X_{1}^{2}]
  6. 𝔼 [ ( τ n - τ n - 1 ) 2 ] 4 𝔼 [ X 1 4 ] . \mathbb{E}[(\tau_{n}-\tau_{n-1})^{2}]\leq 4\mathbb{E}[X_{1}^{4}].

Skype_security.html

  1. [ H ( H ( P ) ) ] [H(H(P))]

SL2(R).html

  1. SL ( 2 , 𝐑 ) = { ( a b c d ) : a , b , c , d 𝐑 and a d - b c = 1 } . \mbox{SL}~{}(2,\mathbf{R})=\left\{\left(\begin{matrix}a&b\\ c&d\end{matrix}\right):a,b,c,d\in\mathbf{R}\mbox{ and }~{}ad-bc=1\right\}.
  2. x a x + b c x + d . x\mapsto\frac{ax+b}{cx+d}.
  3. z a z + b c z + d (where a , b , c , d 𝐑 ) . z\mapsto\frac{az+b}{cz+d}\;\;\;\;\mbox{ (where }~{}a,b,c,d\in\mathbf{R}\mbox{)% }~{}.
  4. [ a b c d ] [ a 2 2 a c c 2 a b a d + b c c d b 2 2 b d d 2 ] . \begin{bmatrix}a&b\\ c&d\end{bmatrix}\mapsto\begin{bmatrix}a^{2}&2ac&c^{2}\\ ab&ad+bc&cd\\ b^{2}&2bd&d^{2}\end{bmatrix}.
  5. λ 2 - tr ( A ) λ + 1 = 0 \lambda^{2}\,-\,\mathrm{tr}(A)\,\lambda\,+\,1\,=\,0
  6. λ = tr ( A ) ± tr ( A ) 2 - 4 2 . \lambda=\frac{\mathrm{tr}(A)\pm\sqrt{\mathrm{tr}(A)^{2}-4}}{2}.
  7. ϵ < 1 \epsilon<1
  8. ϵ = 1 \epsilon=1
  9. ϵ > 1 \epsilon>1
  10. ( 1 λ 1 ) × { ± I } \left(\begin{smallmatrix}1&\lambda\\ &1\end{smallmatrix}\right)\times\{\pm I\}
  11. ( 1 ± 1 1 ) \left(\begin{smallmatrix}1&\pm 1\\ &1\end{smallmatrix}\right)
  12. ( - 1 ± 1 - 1 ) \left(\begin{smallmatrix}-1&\pm 1\\ &-1\end{smallmatrix}\right)
  13. ( λ λ - 1 ) × { ± I } \left(\begin{smallmatrix}\lambda\\ &\lambda^{-1}\end{smallmatrix}\right)\times\{\pm I\}
  14. SL ( 2 , 𝐑 ) ¯ \overline{\mbox{SL}~{}(2,\mathbf{R})}
  15. SL ( 2 , 𝐑 ) ¯ \overline{\mbox{SL}~{}(2,\mathbf{R})}
  16. SL ( 2 , 𝐑 ) ¯ \overline{\mbox{SL}~{}(2,\mathbf{R})}
  17. SL ( 2 , 𝐑 ) ¯ \overline{\mbox{SL}~{}(2,\mathbf{R})}
  18. SL ( 2 , 𝐑 ) ¯ \overline{\mbox{SL}~{}(2,\mathbf{R})}
  19. SL ( 2 , 𝐑 ) ¯ \overline{\mbox{SL}~{}(2,\mathbf{R})}
  20. SL ( 2 , 𝐑 ) ¯ Mp ( 2 , 𝐑 ) SL ( 2 , 𝐑 ) PSL ( 2 , 𝐑 ) . \overline{\mathrm{SL}(2,\mathbf{R})}\to\cdots\to\mathrm{Mp}(2,\mathbf{R})\to% \mathrm{SL}(2,\mathbf{R})\to\mathrm{PSL}(2,\mathbf{R}).

Slater's_rules.html

  1. Z eff = Z - s . Z_{\mathrm{eff}}=Z-s.\,
  2. 4 s : 0.35 × 1 + 0.85 × 14 + 1.00 × 10 = 22.25 Z eff ( 4 s ) = 3.75 3 d : 0.35 × 5 + 1.00 × 18 = 19.75 Z eff ( 3 d ) = 6.25 3 s , 3 p : 0.35 × 7 + 0.85 × 8 + 1.00 × 2 = 11.25 Z eff ( 3 s , 3 p ) = 14.75 2 s , 2 p : 0.35 × 7 + 0.85 × 2 = 4.15 Z eff ( 2 s , 2 p ) = 21.85 1 s : 0.30 × 1 = 0.30 Z eff ( 1 s ) = 25.70 \begin{matrix}4s&:0.35\times 1&+&0.85\times 14&+&1.00\times 10&=&22.25&% \Rightarrow&Z_{\mathrm{eff}}(4s)=3.75\\ 3d&:0.35\times 5&&&+&1.00\times 18&=&19.75&\Rightarrow&Z_{\mathrm{eff}}(3d)=6.% 25\\ 3s,3p&:0.35\times 7&+&0.85\times 8&+&1.00\times 2&=&11.25&\Rightarrow&Z_{% \mathrm{eff}}(3s,3p)=14.75\\ 2s,2p&:0.35\times 7&+&0.85\times 2&&&=&4.15&\Rightarrow&Z_{\mathrm{eff}}(2s,2p% )=21.85\\ 1s&:0.30\times 1&&&&&=&0.30&\Rightarrow&Z_{\mathrm{eff}}(1s)=25.70\end{matrix}
  3. ψ n * s ( r ) = r n * - 1 exp ( - ( Z - s ) r n * ) \psi_{n^{*}s}(r)=r^{n^{*}-1}\exp\left(-\frac{(Z-s)r}{n^{*}}\right)
  4. R n l ( r ) = r l f n l ( r ) exp ( - Z r n ) , R_{nl}(r)=r^{l}f_{nl}(r)\exp\left(-\frac{Zr}{n}\right),
  5. E = - i = 1 N ( Z - s i n i * ) 2 . E=-\sum_{i=1}^{N}\left(\frac{Z-s_{i}}{n^{*}_{i}}\right)^{2}.

SLD_resolution.html

  1. ¬ L 1 ¬ L i ¬ L n \neg L_{1}\cdots\neg L_{i}\cdots\neg L_{n}
  2. ¬ L i \neg L_{i}
  3. L ¬ K 1 ¬ K m L\neg K_{1}\cdots\neg K_{m}
  4. L L\,
  5. L i L_{i}\,
  6. ¬ L i \neg L_{i}\,
  7. θ \theta\,
  8. ( ¬ L 1 ¬ K 1 ¬ K m ¬ L n ) θ (\neg L_{1}\cdots\neg K_{1}\cdots\neg K_{m}\ \cdots\neg L_{n})\theta
  9. L i L_{i}\,
  10. L L\,
  11. θ \theta\,
  12. C 1 , C 2 , , C l C_{1},C_{2},\cdots,C_{l}
  13. C 1 C_{1}\,
  14. C i + 1 C_{i+1}\,
  15. C i C_{i}\,
  16. C l C_{l}\,
  17. C i C_{i}\,
  18. ¬ L 1 ¬ L i ¬ L n \neg L_{1}\cdots\neg L_{i}\cdots\neg L_{n}
  19. L 1 and and L i and and L n L_{1}\and\cdots\and L_{i}\and\cdots\and L_{n}
  20. C i + 1 C_{i+1}\,
  21. C i C_{i}\,
  22. θ \theta\,
  23. q ¬ p q\neg p
  24. p p\,
  25. ¬ q \neg q
  26. ¬ q , ¬ p , 𝑓𝑎𝑙𝑠𝑒 \neg q,\neg p,\mathit{false}
  27. 𝑓𝑎𝑙𝑠𝑒 \mathit{false}\,
  28. n o t ( p ) not(p)\,
  29. p p\,
  30. n o t ( p ) not(p)\,

Slender-body_theory.html

  1. \ell
  2. 2 a 2a
  3. a \ell\gg a
  4. μ \mu
  5. / a \ell/a\rightarrow\infty
  6. s y m b o l X ( s , t ) symbol{X}(s,t)
  7. s s
  8. t t
  9. s y m b o l f ( s ) symbol{f}(s)
  10. s y m b o l f symbol{f}
  11. a a
  12. s y m b o l X ( s , t ) symbol{X}(s,t)
  13. \partialsymbol X / t \partialsymbol{X}/\partial t
  14. s y m b o l u ( s y m b o l x ) symbol{u}(symbol{x})
  15. s y m b o l x symbol{x}
  16. s y m b o l u ( s y m b o l x ) = 0 s y m b o l f ( s ) 8 π μ ( 𝐈 | s y m b o l x - s y m b o l X | + ( s y m b o l x - s y m b o l X ) ( s y m b o l x - s y m b o l X ) | s y m b o l x - s y m b o l X | 3 ) d s symbol{u}(symbol{x})=\int_{0}^{\ell}\frac{symbol{f}(s)}{8\pi\mu}\cdot\left(% \frac{\mathbf{I}}{|symbol{x}-symbol{X}|}+\frac{(symbol{x}-symbol{X})(symbol{x}% -symbol{X})}{|symbol{x}-symbol{X}|^{3}}\right)\,\mathrm{d}s
  17. 𝐈 \mathbf{I}
  18. s y m b o l x symbol{x}
  19. s 0 s_{0}
  20. | s - s 0 | = O ( a ) |s-s_{0}|=O(a)
  21. a a\ll\ell
  22. s y m b o l f ( s ) s y m b o l f ( s 0 ) symbol{f}(s)\approx symbol{f}(s_{0})
  23. s y m b o l X t ln ( / a ) 4 π μ s y m b o l f ( s ) ( 𝐈 + s y m b o l X s y m b o l X ) \frac{\partial symbol{X}}{\partial t}\sim\frac{\ln(\ell/a)}{4\pi\mu}symbol{f}(% s)\cdot\Bigl(\mathbf{I}+symbol{X}^{\prime}symbol{X}^{\prime}\Bigr)
  24. s y m b o l X = s y m b o l X / s symbol{X}^{\prime}=\partial symbol{X}/\partial s
  25. s y m b o l f ( s ) 4 π μ ln ( / a ) s y m b o l X t ( 𝐈 - 1 2 s y m b o l X s y m b o l X ) symbol{f}(s)\sim\frac{4\pi\mu}{\ln(\ell/a)}\frac{\partial symbol{X}}{\partial t% }\cdot\Bigl(\mathbf{I}-\textstyle\frac{1}{2}symbol{X}^{\prime}symbol{X}^{% \prime}\Bigr)
  26. F F
  27. \ell
  28. a a
  29. u u
  30. F 2 π μ u ln ( / a ) F\sim\frac{2\pi\mu\ell u}{\ln(\ell/a)}
  31. F 4 π μ u ln ( / a ) F\sim\frac{4\pi\mu\ell u}{\ln(\ell/a)}
  32. \ell
  33. O ( 1 ) O(1)
  34. / a \ell/a

Slepian's_lemma.html

  1. X = ( X 1 , , X n ) X=(X_{1},\dots,X_{n})
  2. Y = ( Y 1 , , Y n ) Y=(Y_{1},\dots,Y_{n})
  3. n \mathbb{R}^{n}
  4. E [ X ] = E [ Y ] = 0 E[X]=E[Y]=0
  5. E [ X i 2 ] = E [ Y i 2 ] , i = 1 , , n , E [ X i X j ] E [ Y i Y j ] E[X_{i}^{2}]=E[Y_{i}^{2}],i=1,\dots,n,\ E[X_{i}X_{j}]\leq E[Y_{i}Y_{j}]
  6. i j i\neq j
  7. u 1 , , u n u_{1},...,u_{n}
  8. P [ X 1 u 1 , , X n u n ] P [ Y 1 u 1 , , Y n u n ] P[X_{1}\leq u_{1},\dots,X_{n}\leq u_{n}]\leq P[Y_{1}\leq u_{1},\dots,Y_{n}\leq u% _{n}]
  9. ( X t ) t 0 (X_{t})_{t\geq 0}
  10. E [ X 0 X t ] 0 E[X_{0}X_{t}]\geq 0
  11. P [ sup t [ 0 , T + S ] X t c ] P [ sup t [ 0 , T ] X t c ] P [ sup t [ 0 , S ] X t c ] , T , S > 0 P\left[\sup_{t\in[0,T+S]}X_{t}\leq c\right]\geq P\left[\sup_{t\in[0,T]}X_{t}% \leq c\right]P\left[\sup_{t\in[0,S]}X_{t}\leq c\right],\quad T,S>0

Slip_(vehicle_dynamics).html

  1. s l i p = ω r - v v * 100 % , slip=\frac{\omega r-v}{v}*100\%,
  2. ω \omega
  3. r r
  4. v v
  5. ω r = 0 \omega r=0
  6. s l i p slip
  7. v = 0 v=0
  8. ω r \omega r
  9. 0
  10. s l i p = slip=
  11. α = arctan ( v y | v x | ) \alpha=\arctan\left(\frac{v_{y}}{|v_{x}|}\right)

Smale's_problems.html

  1. T T
  2. T : 0 , 11 0 , 11 T:0,11→0,11
  3. r > 1 ? r>1?
  4. N O ( log log N ) N^{O(\log\log N)}

Small-bias_sample_space.html

  1. ϵ \epsilon
  2. ϵ \epsilon
  3. ϵ \epsilon
  4. ϵ \epsilon
  5. X X
  6. { 0 , 1 } n \{0,1\}^{n}
  7. X X
  8. I { 1 , , n } I\subseteq\{1,\dots,n\}
  9. bias I ( X ) = | Pr x X ( i I x i = 0 ) - Pr x X ( i I x i = 1 ) | = | 2 Pr x X ( i I x i = 0 ) - 1 | , \,\text{bias}_{I}(X)=\left|\Pr_{x\sim X}\left(\sum_{i\in I}x_{i}=0\right)-\Pr_% {x\sim X}\left(\sum_{i\in I}x_{i}=1\right)\right|=\left|2\cdot\Pr_{x\sim X}% \left(\sum_{i\in I}x_{i}=0\right)-1\right|\,,
  10. 𝔽 2 \mathbb{F}_{2}
  11. i I x i \sum_{i\in I}x_{i}
  12. 0
  13. x { 0 , 1 } n x\in\{0,1\}^{n}
  14. I I
  15. 1 1
  16. I = I=\emptyset
  17. bias ( X ) = 1 \,\text{bias}_{\emptyset}(X)=1
  18. X X
  19. { 0 , 1 } n \{0,1\}^{n}
  20. ϵ \epsilon
  21. bias I ( X ) ϵ \,\text{bias}_{I}(X)\leq\epsilon
  22. I { 1 , 2 , , n } I\subseteq\{1,2,\ldots,n\}
  23. ϵ \epsilon
  24. X X
  25. X { 0 , 1 } n X\subseteq\{0,1\}^{n}
  26. ϵ \epsilon
  27. s s
  28. ϵ \epsilon
  29. X X
  30. ϵ \epsilon
  31. G : { 0 , 1 } { 0 , 1 } n G:\{0,1\}^{\ell}\to\{0,1\}^{n}
  32. \ell
  33. n n
  34. X G = { G ( y ) | y { 0 , 1 } } X_{G}=\{G(y)\;|\;y\in\{0,1\}^{\ell}\}
  35. ϵ \epsilon
  36. \ell
  37. ϵ \epsilon
  38. X G X_{G}
  39. s = 2 s=2^{\ell}
  40. ϵ \epsilon
  41. ϵ \epsilon
  42. C : { 0 , 1 } n { 0 , 1 } s C:\{0,1\}^{n}\to\{0,1\}^{s}
  43. n n
  44. s s
  45. ϵ \epsilon
  46. C ( x ) C(x)
  47. ( 1 2 - ϵ ) s (\frac{1}{2}-\epsilon)s
  48. ( 1 2 + ϵ ) s (\frac{1}{2}+\epsilon)s
  49. C C
  50. ( n × s ) (n\times s)
  51. A A
  52. 𝔽 2 \mathbb{F}_{2}
  53. C ( x ) = x A C(x)=x\cdot A
  54. X { 0 , 1 } n X\subset\{0,1\}^{n}
  55. ϵ \epsilon
  56. C X C_{X}
  57. X X
  58. ϵ \epsilon
  59. ϵ \epsilon
  60. s s
  61. n n
  62. ϵ \epsilon
  63. s s
  64. s = O ( n / ϵ 2 ) s=O(n/\epsilon^{2})
  65. ϵ \epsilon
  66. ϵ \epsilon
  67. s = Ω ( n / ( ϵ 2 log ( 1 / ϵ ) ) s=\Omega(n/(\epsilon^{2}\log(1/\epsilon))
  68. ϵ \epsilon
  69. ϵ \epsilon
  70. s = n poly ( ϵ ) \displaystyle s=\frac{n}{\,\text{poly}(\epsilon)}
  71. s = O ( n ϵ log ( n / ϵ ) ) 2 \displaystyle s=O\left(\frac{n}{\epsilon\log(n/\epsilon)}\right)^{2}
  72. ϵ \epsilon
  73. ϵ \epsilon
  74. ϵ \epsilon
  75. s = O ( n ϵ 3 log ( 1 / ϵ ) ) \displaystyle s=O\left(\frac{n}{\epsilon^{3}\log(1/\epsilon)}\right)
  76. s = O ( n ϵ 2 log ( 1 / ϵ ) ) 5 / 4 \displaystyle s=O\left(\frac{n}{\epsilon^{2}\log(1/\epsilon)}\right)^{5/4}
  77. ϵ \epsilon
  78. ϵ \epsilon
  79. n n
  80. Y Y
  81. { 0 , 1 } n \{0,1\}^{n}
  82. I { 1 , , n } I\subseteq\{1,\dots,n\}
  83. k k
  84. Y | I Y|_{I}
  85. { 0 , 1 } k \{0,1\}^{k}
  86. I I
  87. z { 0 , 1 } k z\in\{0,1\}^{k}
  88. Y Y
  89. Pr Y ( Y | I = z ) = 2 - k \Pr_{Y}(Y|_{I}=z)=2^{-k}
  90. n k n^{k}
  91. n k / 2 n^{k/2}
  92. n k / 2 n^{k/2}
  93. k k
  94. Y Y
  95. n > k n>k
  96. Y Y
  97. 𝔽 n n \mathbb{F}_{n}^{n}
  98. k k
  99. ( Y 0 , , Y k - 1 ) 𝔽 n k (Y_{0},\dots,Y_{k-1})\sim\mathbb{F}_{n}^{k}
  100. i i
  101. k i < n k\leq i<n
  102. Y i Y_{i}
  103. Y i = Y 0 + Y 1 i + Y 2 i 2 + + Y k - 1 i k - 1 , Y_{i}=Y_{0}+Y_{1}\cdot i+Y_{2}\cdot i^{2}+\dots+Y_{k-1}\cdot i^{k-1}\,,
  104. 𝔽 n \mathbb{F}_{n}
  105. Y Y
  106. k k
  107. 𝔽 n n \mathbb{F}_{n}^{n}
  108. Y Y
  109. Y Y
  110. k k
  111. n k n^{k}
  112. 𝔽 n k \mathbb{F}_{n}^{k}
  113. n n
  114. Y Y
  115. { 0 , 1 } n \{0,1\}^{n}
  116. δ \delta
  117. I { 1 , , n } I\subseteq\{1,\dots,n\}
  118. k k
  119. Y | I Y|_{I}
  120. U k U_{k}
  121. { 0 , 1 } k \{0,1\}^{k}
  122. δ \delta
  123. Y | I - U k | 1 δ \Big\|Y|_{I}-U_{k}\Big\|_{1}\leq\delta
  124. ϵ \epsilon
  125. δ \delta
  126. G 1 : { 0 , 1 } h { 0 , 1 } n G_{1}:\{0,1\}^{h}\to\{0,1\}^{n}
  127. G 2 : { 0 , 1 } { 0 , 1 } h G_{2}:\{0,1\}^{\ell}\to\{0,1\}^{h}
  128. ϵ \epsilon
  129. { 0 , 1 } h \{0,1\}^{h}
  130. G 1 G_{1}
  131. G 2 G_{2}
  132. ϵ \epsilon
  133. G : { 0 , 1 } { 0 , 1 } n G:\{0,1\}^{\ell}\to\{0,1\}^{n}
  134. G ( x ) = G 1 ( G 2 ( x ) ) G(x)=G_{1}(G_{2}(x))
  135. δ \delta
  136. k k
  137. δ = 2 k / 2 ϵ \delta=2^{k/2}\epsilon
  138. G 1 G_{1}
  139. h = k 2 log n h=\tfrac{k}{2}\log n
  140. G 2 G_{2}
  141. = log s = log h + O ( log ( ϵ - 1 ) ) \ell=\log s=\log h+O(\log(\epsilon^{-1}))
  142. G G
  143. G 1 G_{1}
  144. G 2 G_{2}
  145. = log k + log log n + O ( log ( ϵ - 1 ) ) \ell=\log k+\log\log n+O(\log(\epsilon^{-1}))
  146. G G
  147. δ \delta
  148. ϵ = δ 2 - k / 2 \epsilon=\delta 2^{-k/2}
  149. = log log n + O ( k + log ( δ - 1 ) ) \ell=\log\log n+O(k+\log(\delta^{-1}))
  150. 2 log n poly ( 2 k ϵ - 1 ) 2^{\ell}\leq\log n\cdot\,\text{poly}(2^{k}\cdot\epsilon^{-1})

Snub_(geometry).html

  1. { p , q } \begin{Bmatrix}p,q\end{Bmatrix}
  2. t { p , q } t\begin{Bmatrix}p,q\end{Bmatrix}
  3. h t { p , q } = s { p , q } ht\begin{Bmatrix}p,q\end{Bmatrix}=s\begin{Bmatrix}p,q\end{Bmatrix}
  4. { p q } \begin{Bmatrix}p\\ q\end{Bmatrix}
  5. t { p q } t\begin{Bmatrix}p\\ q\end{Bmatrix}
  6. h t { p q } = s { p q } ht\begin{Bmatrix}p\\ q\end{Bmatrix}=s\begin{Bmatrix}p\\ q\end{Bmatrix}
  7. { 4 3 } \begin{Bmatrix}4\\ 3\end{Bmatrix}
  8. s { 4 3 } s\begin{Bmatrix}4\\ 3\end{Bmatrix}
  9. t { 4 3 } t\begin{Bmatrix}4\\ 3\end{Bmatrix}
  10. { 4 3 } \begin{Bmatrix}4\\ 3\end{Bmatrix}
  11. t { 4 3 } t\begin{Bmatrix}4\\ 3\end{Bmatrix}
  12. h t { 4 3 } = s { 4 3 } ht\begin{Bmatrix}4\\ 3\end{Bmatrix}=s\begin{Bmatrix}4\\ 3\end{Bmatrix}
  13. s { 3 , 4 } s\begin{Bmatrix}3,4\end{Bmatrix}
  14. s { 3 3 } s\begin{Bmatrix}3\\ 3\end{Bmatrix}
  15. t { 3 , 4 } t\begin{Bmatrix}3,4\end{Bmatrix}
  16. t { 3 3 } t\begin{Bmatrix}3\\ 3\end{Bmatrix}
  17. s { 2 n } s\begin{Bmatrix}2\\ n\end{Bmatrix}
  18. s { 2 , 2 n } s\begin{Bmatrix}2,2n\end{Bmatrix}
  19. t { 2 n } t\begin{Bmatrix}2\\ n\end{Bmatrix}
  20. t { 2 , 2 n } t\begin{Bmatrix}2,2n\end{Bmatrix}
  21. { 2 , n } \begin{Bmatrix}2,n\end{Bmatrix}
  22. s { 2 2 } s\begin{Bmatrix}2\\ 2\end{Bmatrix}
  23. s { 2 3 } s\begin{Bmatrix}2\\ 3\end{Bmatrix}
  24. s { 2 4 } s\begin{Bmatrix}2\\ 4\end{Bmatrix}
  25. s { 2 5 } s\begin{Bmatrix}2\\ 5\end{Bmatrix}
  26. s { 2 6 } s\begin{Bmatrix}2\\ 6\end{Bmatrix}
  27. s { 2 7 } s\begin{Bmatrix}2\\ 7\end{Bmatrix}
  28. s { 2 8 } s\begin{Bmatrix}2\\ 8\end{Bmatrix}
  29. s { 2 } s\begin{Bmatrix}2\\ \infty\end{Bmatrix}
  30. s { 2 3 } s\begin{Bmatrix}2\\ 3\end{Bmatrix}
  31. s { 3 3 } s\begin{Bmatrix}3\\ 3\end{Bmatrix}
  32. s { 4 3 } s\begin{Bmatrix}4\\ 3\end{Bmatrix}
  33. s { 5 3 } s\begin{Bmatrix}5\\ 3\end{Bmatrix}
  34. s { 6 3 } s\begin{Bmatrix}6\\ 3\end{Bmatrix}
  35. s { 7 3 } s\begin{Bmatrix}7\\ 3\end{Bmatrix}
  36. s { 8 3 } s\begin{Bmatrix}8\\ 3\end{Bmatrix}
  37. s { 3 } s\begin{Bmatrix}\infty\\ 3\end{Bmatrix}
  38. s { 2 4 } s\begin{Bmatrix}2\\ 4\end{Bmatrix}
  39. s { 3 4 } s\begin{Bmatrix}3\\ 4\end{Bmatrix}
  40. s { 4 4 } s\begin{Bmatrix}4\\ 4\end{Bmatrix}
  41. s { 5 4 } s\begin{Bmatrix}5\\ 4\end{Bmatrix}
  42. s { 6 4 } s\begin{Bmatrix}6\\ 4\end{Bmatrix}
  43. s { 7 4 } s\begin{Bmatrix}7\\ 4\end{Bmatrix}
  44. s { 8 4 } s\begin{Bmatrix}8\\ 4\end{Bmatrix}
  45. s { 4 } s\begin{Bmatrix}\infty\\ 4\end{Bmatrix}
  46. s s { 2 2 } ss\begin{Bmatrix}2\\ 2\end{Bmatrix}
  47. s s { 2 3 } ss\begin{Bmatrix}2\\ 3\end{Bmatrix}
  48. s s { 2 4 } ss\begin{Bmatrix}2\\ 4\end{Bmatrix}
  49. s s { 2 5 } ss\begin{Bmatrix}2\\ 5\end{Bmatrix}
  50. { p , q , r } \begin{Bmatrix}p,q,r\end{Bmatrix}
  51. s { p , q , r } s\begin{Bmatrix}p,q,r\end{Bmatrix}
  52. { p q , r } \begin{Bmatrix}p\\ q,r\end{Bmatrix}
  53. s { p q , r } s\begin{Bmatrix}p\\ q,r\end{Bmatrix}
  54. { 3 , 4 , 3 } \begin{Bmatrix}3,4,3\end{Bmatrix}
  55. s { 3 , 4 , 3 } s\begin{Bmatrix}3,4,3\end{Bmatrix}
  56. s { 3 3 3 } s\left\{\begin{array}[]{l}3\\ 3\\ 3\end{array}\right\}
  57. s { 3 3 , 4 } s\begin{Bmatrix}3\\ 3,4\end{Bmatrix}
  58. s { 3 , 4 , 3 , 3 } s\begin{Bmatrix}3,4,3,3\end{Bmatrix}
  59. s { 3 3 , 4 , 3 } s\begin{Bmatrix}3\\ 3,4,3\end{Bmatrix}
  60. s { 3 3 3 3 } s\left\{\begin{array}[]{l}3\\ 3\\ 3\\ 3\end{array}\right\}

Soft-collinear_effective_theory.html

  1. B X s γ B\rightarrow X_{s}\gamma

Sokhotski–Plemelj_theorem.html

  1. 1 2 π i C φ ( ζ ) d ζ ζ - z , \frac{1}{2\pi i}\int_{C}\frac{\varphi(\zeta)d\zeta}{\zeta-z},
  2. 𝒫 \mathcal{P}
  3. ϕ i ( z ) = 1 2 π i 𝒫 C φ ( ζ ) d ζ ζ - z + 1 2 φ ( z ) , \phi_{i}(z)=\frac{1}{2\pi i}\mathcal{P}\int_{C}\frac{\varphi(\zeta)d\zeta}{% \zeta-z}+\frac{1}{2}\varphi(z),\,
  4. ϕ e ( z ) = 1 2 π i 𝒫 C φ ( ζ ) d ζ ζ - z - 1 2 φ ( z ) . \phi_{e}(z)=\frac{1}{2\pi i}\mathcal{P}\int_{C}\frac{\varphi(\zeta)d\zeta}{% \zeta-z}-\frac{1}{2}\varphi(z).\,
  5. 𝒫 \mathcal{P}
  6. lim ε 0 + a b f ( x ) x ± i ε d x = i π lim ε 0 + a b ε π ( x 2 + ε 2 ) f ( x ) d x + lim ε 0 + a b x 2 x 2 + ε 2 f ( x ) x d x . \lim_{\varepsilon\rightarrow 0^{+}}\int_{a}^{b}\frac{f(x)}{x\pm i\varepsilon}% \,dx=\mp i\pi\lim_{\varepsilon\rightarrow 0^{+}}\int_{a}^{b}\frac{\varepsilon}% {\pi(x^{2}+\varepsilon^{2})}f(x)\,dx+\lim_{\varepsilon\rightarrow 0^{+}}\int_{% a}^{b}\frac{x^{2}}{x^{2}+\varepsilon^{2}}\,\frac{f(x)}{x}\,dx.
  7. π \pi
  8. - d E 0 d t f ( E ) exp ( - i E t ) \int_{-\infty}^{\infty}dE\,\int_{0}^{\infty}dt\,f(E)\exp(-iEt)
  9. lim ε 0 + - d E 0 d t f ( E ) exp ( - i E t - ε t ) \lim_{\varepsilon\rightarrow 0^{+}}\int_{-\infty}^{\infty}dE\,\int_{0}^{\infty% }dt\,f(E)\exp(-iEt-\varepsilon t)
  10. = - i lim ε 0 + - f ( E ) E - i ε d E = π f ( 0 ) - i 𝒫 - f ( E ) E d E , =-i\lim_{\varepsilon\rightarrow 0^{+}}\int_{-\infty}^{\infty}\frac{f(E)}{E-i% \varepsilon}\,dE=\pi f(0)-i\mathcal{P}\int_{-\infty}^{\infty}\frac{f(E)}{E}\,dE,

Solar_neutrino_unit.html

  1. Z N A + ν e N - 1 A ( Z + 1 ) + e - {}^{A}_{N}Z+\nu_{e}\longrightarrow^{A}_{N-1}(Z+1)+e^{-}
  2. R = N Φ ( E ) σ ( E ) d E R=N\int\Phi(E)\sigma(E)dE
  3. Φ \Phi
  4. σ \sigma
  5. N N

Solder_form.html

  1. θ x : T x M V o ( x ) E \theta_{x}:T_{x}M\rightarrow V_{o(x)}E
  2. θ : T M V o E , \theta\colon TM\to V_{o}E,
  3. T M E . TM\to E.
  4. g : T M T * M g\colon TM\to T^{*}M

Solid_harmonics.html

  1. R m ( 𝐫 ) R^{m}_{\ell}(\mathbf{r})
  2. I m ( 𝐫 ) I^{m}_{\ell}(\mathbf{r})
  3. R m ( 𝐫 ) 4 π 2 + 1 r Y m ( θ , φ ) R^{m}_{\ell}(\mathbf{r})\equiv\sqrt{\frac{4\pi}{2\ell+1}}\;r^{\ell}Y^{m}_{\ell% }(\theta,\varphi)
  4. I m ( 𝐫 ) 4 π 2 + 1 Y m ( θ , φ ) r + 1 I^{m}_{\ell}(\mathbf{r})\equiv\sqrt{\frac{4\pi}{2\ell+1}}\;\frac{Y^{m}_{\ell}(% \theta,\varphi)}{r^{\ell+1}}
  5. 2 Φ ( 𝐫 ) = ( 1 r 2 r 2 r - l ^ 2 r 2 ) Φ ( 𝐫 ) = 0 , 𝐫 𝟎 , \nabla^{2}\Phi(\mathbf{r})=\left(\frac{1}{r}\frac{\partial^{2}}{\partial r^{2}% }r-\frac{\hat{l}^{2}}{r^{2}}\right)\Phi(\mathbf{r})=0,\qquad\mathbf{r}\neq% \mathbf{0},
  6. 𝐥 ^ = - i ( 𝐫 × ) . \mathbf{\hat{l}}=-i\,(\mathbf{r}\times\mathbf{\nabla}).
  7. l ^ 2 Y m [ l ^ x 2 + l ^ y 2 + l ^ z 2 ] Y m = ( + 1 ) Y m . \hat{l}^{2}Y^{m}_{\ell}\equiv\left[{\hat{l}_{x}}^{2}+\hat{l}^{2}_{y}+\hat{l}^{% 2}_{z}\right]Y^{m}_{\ell}=\ell(\ell+1)Y^{m}_{\ell}.
  8. 1 r 2 r 2 r F ( r ) = ( + 1 ) r 2 F ( r ) F ( r ) = A r + B r - - 1 . \frac{1}{r}\frac{\partial^{2}}{\partial r^{2}}rF(r)=\frac{\ell(\ell+1)}{r^{2}}% F(r)\Longrightarrow F(r)=Ar^{\ell}+Br^{-\ell-1}.
  9. R m ( 𝐫 ) 4 π 2 + 1 r Y m ( θ , φ ) , R^{m}_{\ell}(\mathbf{r})\equiv\sqrt{\frac{4\pi}{2\ell+1}}\;r^{\ell}Y^{m}_{\ell% }(\theta,\varphi),
  10. I m ( 𝐫 ) 4 π 2 + 1 Y m ( θ , φ ) r + 1 . I^{m}_{\ell}(\mathbf{r})\equiv\sqrt{\frac{4\pi}{2\ell+1}}\;\frac{Y^{m}_{\ell}(% \theta,\varphi)}{r^{\ell+1}}.
  11. 0 π sin θ d θ 0 2 π d φ R m ( 𝐫 ) * R m ( 𝐫 ) = 4 π 2 + 1 r 2 \int_{0}^{\pi}\sin\theta\,d\theta\int_{0}^{2\pi}d\varphi\;R^{m}_{\ell}(\mathbf% {r})^{*}\;R^{m}_{\ell}(\mathbf{r})=\frac{4\pi}{2\ell+1}r^{2\ell}
  12. R m ( 𝐫 + 𝐚 ) = λ = 0 ( 2 2 λ ) 1 / 2 μ = - λ λ R λ μ ( 𝐫 ) R - λ m - μ ( 𝐚 ) λ , μ ; - λ , m - μ | m , R^{m}_{\ell}(\mathbf{r}+\mathbf{a})=\sum_{\lambda=0}^{\ell}{\left({{2\ell}% \atop{2\lambda}}\right)}^{1/2}\sum_{\mu=-\lambda}^{\lambda}R^{\mu}_{\lambda}(% \mathbf{r})R^{m-\mu}_{\ell-\lambda}(\mathbf{a})\;\langle\lambda,\mu;\ell-% \lambda,m-\mu|\ell m\rangle,
  13. λ , μ ; - λ , m - μ | m = ( + m λ + μ ) 1 / 2 ( - m λ - μ ) 1 / 2 ( 2 2 λ ) - 1 / 2 . \langle\lambda,\mu;\ell-\lambda,m-\mu|\ell m\rangle={\left({{\ell+m}\atop{% \lambda+\mu}}\right)}^{1/2}{\left({{\ell-m}\atop{\lambda-\mu}}\right)}^{1/2}{% \left({{2\ell}\atop{2\lambda}}\right)}^{-1/2}.
  14. I m ( 𝐫 + 𝐚 ) = λ = 0 ( 2 + 2 λ + 1 2 λ ) 1 / 2 μ = - λ λ R λ μ ( 𝐫 ) I + λ m - μ ( 𝐚 ) λ , μ ; + λ , m - μ | m I^{m}_{\ell}(\mathbf{r}+\mathbf{a})=\sum_{\lambda=0}^{\infty}{\left({{2\ell+2% \lambda+1}\atop{2\lambda}}\right)}^{1/2}\sum_{\mu=-\lambda}^{\lambda}R^{\mu}_{% \lambda}(\mathbf{r})I^{m-\mu}_{\ell+\lambda}(\mathbf{a})\;\langle\lambda,\mu;% \ell+\lambda,m-\mu|\ell m\rangle
  15. | r | | a | |r|\leq|a|\,
  16. λ , μ ; + λ , m - μ | m = ( - 1 ) λ + μ ( + λ - m + μ λ + μ ) 1 / 2 ( + λ + m - μ λ - μ ) 1 / 2 ( 2 + 2 λ + 1 2 λ ) - 1 / 2 . \langle\lambda,\mu;\ell+\lambda,m-\mu|\ell m\rangle=(-1)^{\lambda+\mu}{\left({% {\ell+\lambda-m+\mu}\atop{\lambda+\mu}}\right)}^{1/2}{\left({{\ell+\lambda+m-% \mu}\atop{\lambda-\mu}}\right)}^{1/2}{\left({{2\ell+2\lambda+1}\atop{2\lambda}% }\right)}^{-1/2}.
  17. R m ( r , θ , φ ) = ( - 1 ) ( m + | m | ) / 2 r Θ | m | ( cos θ ) e i m φ , - m , R_{\ell}^{m}(r,\theta,\varphi)=(-1)^{(m+|m|)/2}\;r^{\ell}\;\Theta_{\ell}^{|m|}% (\cos\theta)e^{im\varphi},\qquad-\ell\leq m\leq\ell,
  18. Θ m ( cos θ ) [ ( - m ) ! ( + m ) ! ] 1 / 2 sin m θ d m P ( cos θ ) d cos m θ , m 0 , \Theta_{\ell}^{m}(\cos\theta)\equiv\left[\frac{(\ell-m)!}{(\ell+m)!}\right]^{1% /2}\,\sin^{m}\theta\,\frac{d^{m}P_{\ell}(\cos\theta)}{d\cos^{m}\theta},\qquad m% \geq 0,
  19. P ( cos θ ) P_{\ell}(\cos\theta)
  20. ( C m S m ) 2 r Θ m ( cos m φ sin m φ ) = 1 2 ( ( - 1 ) m 1 - ( - 1 ) m i i ) ( R m R - m ) , m > 0. \begin{pmatrix}C_{\ell}^{m}\\ S_{\ell}^{m}\end{pmatrix}\equiv\sqrt{2}\;r^{\ell}\;\Theta^{m}_{\ell}\begin{% pmatrix}\cos m\varphi\\ \sin m\varphi\end{pmatrix}=\frac{1}{\sqrt{2}}\begin{pmatrix}(-1)^{m}&\quad 1\\ -(-1)^{m}i&\quad i\end{pmatrix}\begin{pmatrix}R_{\ell}^{m}\\ R_{\ell}^{-m}\end{pmatrix},\qquad m>0.
  21. C 0 R 0 . C_{\ell}^{0}\equiv R_{\ell}^{0}.
  22. d m P ( u ) d u m = k = 0 ( - m ) / 2 γ k ( m ) u - 2 k - m \frac{d^{m}P_{\ell}(u)}{du^{m}}=\sum_{k=0}^{\left\lfloor(\ell-m)/2\right% \rfloor}\gamma^{(m)}_{\ell k}\;u^{\ell-2k-m}
  23. γ k ( m ) = ( - 1 ) k 2 - ( k ) ( 2 - 2 k ) ( - 2 k ) ! ( - 2 k - m ) ! . \gamma^{(m)}_{\ell k}=(-1)^{k}2^{-\ell}{\left({{\ell}\atop{k}}\right)}{\left({% {2\ell-2k}\atop{\ell}}\right)}\frac{(\ell-2k)!}{(\ell-2k-m)!}.
  24. Π m ( z ) r - m d m P ( u ) d u m = k = 0 ( - m ) / 2 γ k ( m ) r 2 k z - 2 k - m . \Pi^{m}_{\ell}(z)\equiv r^{\ell-m}\frac{d^{m}P_{\ell}(u)}{du^{m}}=\sum_{k=0}^{% \left\lfloor(\ell-m)/2\right\rfloor}\gamma^{(m)}_{\ell k}\;r^{2k}\;z^{\ell-2k-% m}.
  25. r m sin m θ cos m φ = 1 2 [ ( r sin θ e i φ ) m + ( r sin θ e - i φ ) m ] = 1 2 [ ( x + i y ) m + ( x - i y ) m ] r^{m}\sin^{m}\theta\cos m\varphi=\frac{1}{2}\left[(r\sin\theta e^{i\varphi})^{% m}+(r\sin\theta e^{-i\varphi})^{m}\right]=\frac{1}{2}\left[(x+iy)^{m}+(x-iy)^{% m}\right]
  26. r m sin m θ sin m φ = 1 2 i [ ( r sin θ e i φ ) m - ( r sin θ e - i φ ) m ] = 1 2 i [ ( x + i y ) m - ( x - i y ) m ] . r^{m}\sin^{m}\theta\sin m\varphi=\frac{1}{2i}\left[(r\sin\theta e^{i\varphi})^% {m}-(r\sin\theta e^{-i\varphi})^{m}\right]=\frac{1}{2i}\left[(x+iy)^{m}-(x-iy)% ^{m}\right].
  27. A m ( x , y ) 1 2 [ ( x + i y ) m + ( x - i y ) m ] = p = 0 m ( m p ) x p y m - p cos ( m - p ) π 2 A_{m}(x,y)\equiv\frac{1}{2}\left[(x+iy)^{m}+(x-iy)^{m}\right]=\sum_{p=0}^{m}{% \left({{m}\atop{p}}\right)}x^{p}y^{m-p}\cos(m-p)\frac{\pi}{2}
  28. B m ( x , y ) 1 2 i [ ( x + i y ) m - ( x - i y ) m ] = p = 0 m ( m p ) x p y m - p sin ( m - p ) π 2 . B_{m}(x,y)\equiv\frac{1}{2i}\left[(x+iy)^{m}-(x-iy)^{m}\right]=\sum_{p=0}^{m}{% \left({{m}\atop{p}}\right)}x^{p}y^{m-p}\sin(m-p)\frac{\pi}{2}.
  29. C m ( x , y , z ) = [ ( 2 - δ m 0 ) ( - m ) ! ( + m ) ! ] 1 / 2 Π m ( z ) A m ( x , y ) , m = 0 , 1 , , C^{m}_{\ell}(x,y,z)=\left[\frac{(2-\delta_{m0})(\ell-m)!}{(\ell+m)!}\right]^{1% /2}\Pi^{m}_{\ell}(z)\;A_{m}(x,y),\qquad m=0,1,\ldots,\ell
  30. S m ( x , y , z ) = [ 2 ( - m ) ! ( + m ) ! ] 1 / 2 Π m ( z ) B m ( x , y ) , m = 1 , 2 , , . S^{m}_{\ell}(x,y,z)=\left[\frac{2(\ell-m)!}{(\ell+m)!}\right]^{1/2}\Pi^{m}_{% \ell}(z)\;B_{m}(x,y),\qquad m=1,2,\ldots,\ell.
  31. Π ¯ m ( z ) [ ( 2 - δ m 0 ) ( - m ) ! ( + m ) ! ] 1 / 2 Π m ( z ) . \bar{\Pi}^{m}_{\ell}(z)\equiv\left[\tfrac{(2-\delta_{m0})(\ell-m)!}{(\ell+m)!}% \right]^{1/2}\Pi^{m}_{\ell}(z).
  32. Π ¯ 0 0 = 1 Π ¯ 3 1 = 1 4 6 ( 5 z 2 - r 2 ) Π ¯ 4 4 = 1 8 35 Π ¯ 1 0 = z Π ¯ 3 2 = 1 2 15 z Π ¯ 5 0 = 1 8 z ( 63 z 4 - 70 z 2 r 2 + 15 r 4 ) Π ¯ 1 1 = 1 Π ¯ 3 3 = 1 4 10 Π ¯ 5 1 = 1 8 15 ( 21 z 4 - 14 z 2 r 2 + r 4 ) Π ¯ 2 0 = 1 2 ( 3 z 2 - r 2 ) Π ¯ 4 0 = 1 8 ( 35 z 4 - 30 r 2 z 2 + 3 r 4 ) Π ¯ 5 2 = 1 4 105 ( 3 z 2 - r 2 ) z Π ¯ 2 1 = 3 z Π ¯ 4 1 = 10 4 z ( 7 z 2 - 3 r 2 ) Π ¯ 5 3 = 1 16 70 ( 9 z 2 - r 2 ) Π ¯ 2 2 = 1 2 3 Π ¯ 4 2 = 1 4 5 ( 7 z 2 - r 2 ) Π ¯ 5 4 = 3 8 35 z Π ¯ 3 0 = 1 2 z ( 5 z 2 - 3 r 2 ) Π ¯ 4 3 = 1 4 70 z Π ¯ 5 5 = 3 16 14 \begin{aligned}\displaystyle\bar{\Pi}^{0}_{0}&\displaystyle=1&\displaystyle% \bar{\Pi}^{1}_{3}&\displaystyle=\frac{1}{4}\sqrt{6}(5z^{2}-r^{2})&% \displaystyle\bar{\Pi}^{4}_{4}&\displaystyle=\frac{1}{8}\sqrt{35}\\ \displaystyle\bar{\Pi}^{0}_{1}&\displaystyle=z&\displaystyle\bar{\Pi}^{2}_{3}&% \displaystyle=\frac{1}{2}\sqrt{15}\;z&\displaystyle\bar{\Pi}^{0}_{5}&% \displaystyle=\frac{1}{8}z(63z^{4}-70z^{2}r^{2}+15r^{4})\\ \displaystyle\bar{\Pi}^{1}_{1}&\displaystyle=1&\displaystyle\bar{\Pi}^{3}_{3}&% \displaystyle=\frac{1}{4}\sqrt{10}&\displaystyle\bar{\Pi}^{1}_{5}&% \displaystyle=\frac{1}{8}\sqrt{15}(21z^{4}-14z^{2}r^{2}+r^{4})\\ \displaystyle\bar{\Pi}^{0}_{2}&\displaystyle=\frac{1}{2}(3z^{2}-r^{2})&% \displaystyle\bar{\Pi}^{0}_{4}&\displaystyle=\frac{1}{8}(35z^{4}-30r^{2}z^{2}+% 3r^{4})&\displaystyle\bar{\Pi}^{2}_{5}&\displaystyle=\frac{1}{4}\sqrt{105}(3z^% {2}-r^{2})z\\ \displaystyle\bar{\Pi}^{1}_{2}&\displaystyle=\sqrt{3}z&\displaystyle\bar{\Pi}^% {1}_{4}&\displaystyle=\frac{\sqrt{10}}{4}z(7z^{2}-3r^{2})&\displaystyle\bar{% \Pi}^{3}_{5}&\displaystyle=\frac{1}{16}\sqrt{70}(9z^{2}-r^{2})\\ \displaystyle\bar{\Pi}^{2}_{2}&\displaystyle=\frac{1}{2}\sqrt{3}&\displaystyle% \bar{\Pi}^{2}_{4}&\displaystyle=\frac{1}{4}\sqrt{5}(7z^{2}-r^{2})&% \displaystyle\bar{\Pi}^{4}_{5}&\displaystyle=\frac{3}{8}\sqrt{35}z\\ \displaystyle\bar{\Pi}^{0}_{3}&\displaystyle=\frac{1}{2}z(5z^{2}-3r^{2})&% \displaystyle\bar{\Pi}^{3}_{4}&\displaystyle=\frac{1}{4}\sqrt{70}\;z&% \displaystyle\bar{\Pi}^{5}_{5}&\displaystyle=\frac{3}{16}\sqrt{14}\\ \end{aligned}
  33. A m ( x , y ) A_{m}(x,y)\,
  34. B m ( x , y ) B_{m}(x,y)\,
  35. 1 1\,
  36. 0 0\,
  37. x x\,
  38. y y\,
  39. x 2 - y 2 x^{2}-y^{2}\,
  40. 2 x y 2xy\,
  41. x 3 - 3 x y 2 x^{3}-3xy^{2}\,
  42. 3 x 2 y - y 3 3x^{2}y-y^{3}\,
  43. x 4 - 6 x 2 y 2 + y 4 x^{4}-6x^{2}y^{2}+y^{4}\,
  44. 4 x 3 y - 4 x y 3 4x^{3}y-4xy^{3}\,
  45. x 5 - 10 x 3 y 2 + 5 x y 4 x^{5}-10x^{3}y^{2}+5xy^{4}\,
  46. 5 x 4 y - 10 x 2 y 3 + y 5 5x^{4}y-10x^{2}y^{3}+y^{5}\,

Solid_solution_strengthening.html

  1. τ \tau
  2. Δ τ = G b ϵ 3 2 c \Delta\tau=Gb\epsilon^{\tfrac{3}{2}}\sqrt{c}
  3. ϵ \epsilon
  4. ϵ = | ϵ a - β ϵ G | \epsilon=|\epsilon_{a}-\beta\epsilon_{G}|
  5. ϵ a \epsilon_{a}
  6. β \beta
  7. ϵ G \epsilon_{G}
  8. ϵ a = Δ a a Δ c \epsilon_{a}=\dfrac{\Delta a}{a\Delta c}
  9. ϵ G = Δ G G Δ c \epsilon_{G}=\dfrac{\Delta G}{G\Delta c}

Soliton_(optics).html

  1. φ ( x ) \varphi(x)
  2. φ ( x ) = k 0 n L ( x ) \varphi(x)=k_{0}nL(x)
  3. L ( x ) L(x)
  4. φ ( x ) \varphi(x)
  5. k 0 k_{0}
  6. n ( x ) n(x)
  7. φ ( x ) = k 0 n ( x ) L = k 0 L [ n + n 2 I ( x ) ] \varphi(x)=k_{0}n(x)L=k_{0}L[n+n_{2}I(x)]
  8. I ( x ) I(x)
  9. n 2 n_{2}
  10. n ( I ) = n + n 2 I n(I)=n+n_{2}I
  11. I = | E | 2 2 η I=\frac{|E|^{2}}{2\eta}
  12. η = η 0 / n \eta=\eta_{0}/n
  13. η 0 \eta_{0}
  14. η 0 = μ 0 ϵ 0 377 Ω \eta_{0}=\sqrt{\frac{\mu_{0}}{\epsilon_{0}}}\approx 377\Omega
  15. z z
  16. k 0 n k_{0}n
  17. E ( x , z , t ) = A m a ( x , z ) e i ( k 0 n z - ω t ) E(x,z,t)=A_{m}a(x,z)e^{i(k_{0}nz-\omega t)}
  18. A m A_{m}
  19. a ( x , z ) a(x,z)
  20. 2 E + k 0 2 n 2 ( I ) E = 0 \nabla^{2}E+k_{0}^{2}n^{2}(I)E=0
  21. a ( x , z ) a(x,z)
  22. | 2 a ( x , z ) z 2 | | k 0 a ( x , z ) z | \left|\frac{\partial^{2}a(x,z)}{\partial z^{2}}\right|\ll\left|k_{0}\frac{% \partial a(x,z)}{\partial z}\right|
  23. 2 a x 2 + i 2 k 0 n a z + k 0 2 [ n 2 ( I ) - n 2 ] a = 0 \frac{\partial^{2}a}{\partial x^{2}}+i2k_{0}n\frac{\partial a}{\partial z}+k_{% 0}^{2}[n^{2}(I)-n^{2}]a=0
  24. [ n 2 ( I ) - n 2 ] = [ n ( I ) - n ] [ n ( I ) + n ] = n 2 I ( 2 n + n 2 I ) 2 n n 2 I [n^{2}(I)-n^{2}]=[n(I)-n][n(I)+n]=n_{2}I(2n+n_{2}I)\approx 2nn_{2}I
  25. [ n 2 ( I ) - n 2 ] 2 n n 2 | A m | 2 | a ( x , z ) | 2 2 η 0 / n = n 2 n 2 | A m | 2 | a ( x , z ) | 2 η 0 [n^{2}(I)-n^{2}]\approx 2nn_{2}\frac{|A_{m}|^{2}|a(x,z)|^{2}}{2\eta_{0}/n}=n^{% 2}n_{2}\frac{|A_{m}|^{2}|a(x,z)|^{2}}{\eta_{0}}
  26. 1 2 k 0 n 2 a x 2 + i a z + k 0 n n 2 | A m | 2 2 η 0 | a | 2 a = 0 \frac{1}{2k_{0}n}\frac{\partial^{2}a}{\partial x^{2}}+i\frac{\partial a}{% \partial z}+\frac{k_{0}nn_{2}|A_{m}|^{2}}{2\eta_{0}}|a|^{2}a=0
  27. n 2 > 0 n_{2}>0
  28. n 2 = | n 2 | n_{2}=|n_{2}|
  29. ξ = x X 0 \xi=\frac{x}{X_{0}}
  30. X 0 X_{0}
  31. L d = X 0 2 k 0 n L_{d}=X_{0}^{2}k_{0}n
  32. ζ = z L d \zeta=\frac{z}{L_{d}}
  33. L n l = 2 η 0 k 0 n | n 2 | | A m | 2 L_{nl}=\frac{2\eta_{0}}{k_{0}n|n_{2}|\cdot|A_{m}|^{2}}
  34. N 2 = L d L n l N^{2}=\frac{L_{d}}{L_{nl}}
  35. 1 2 2 a ξ 2 + i a ζ + N 2 | a | 2 a = 0 \frac{1}{2}\frac{\partial^{2}a}{\partial\xi^{2}}+i\frac{\partial a}{\partial% \zeta}+N^{2}|a|^{2}a=0
  36. N 1 N\ll 1
  37. L d L n l L_{d}\ll L_{nl}
  38. N 1 N\gg 1
  39. N 1 N\approx 1
  40. N = 1 N=1
  41. a ( ξ , ζ ) = sech ( ξ ) e i ζ / 2 a(\xi,\zeta)=\operatorname{sech}(\xi)e^{i\zeta/2}
  42. N = 2 N=2
  43. a ( ξ , ζ ) = 4 [ cosh ( 3 ξ ) + 3 e 4 i ζ cosh ( ξ ) ] e i ζ / 2 cosh ( 4 ξ ) + 4 cosh ( 2 ξ ) + 3 cos ( 4 ζ ) a(\xi,\zeta)=\frac{4[\cosh(3\xi)+3e^{4i\zeta}\cosh(\xi)]e^{i\zeta/2}}{\cosh(4% \xi)+4\cosh(2\xi)+3\cos(4\zeta)}
  44. ζ = π / 2 \zeta=\pi/2
  45. a ( ξ , ζ = 0 ) = N sech ( ξ ) a(\xi,\zeta=0)=N\operatorname{sech}(\xi)
  46. N = 1 N=1
  47. 1 = N = L d L n l = X 0 2 k 0 2 n 2 | n 2 | | A m | 2 2 η 0 1=N=\frac{L_{d}}{L_{nl}}=\frac{X_{0}^{2}k_{0}^{2}n^{2}|n_{2}||A_{m}|^{2}}{2% \eta_{0}}
  48. I m a x = | A m | 2 2 η 0 / n = 1 X 0 2 k 0 2 n | n 2 | I_{max}=\frac{|A_{m}|^{2}}{2\eta_{0}/n}=\frac{1}{X_{0}^{2}k_{0}^{2}n|n_{2}|}
  49. I max I_{\max}
  50. X 0 X_{0}
  51. Δ τ D L Δ λ \Delta\tau\approx DL\,\Delta\lambda
  52. Δ λ \Delta\lambda
  53. D > 0 D>0
  54. φ ( t ) = ω 0 t - k z = ω 0 t - k 0 z [ n + n 2 I ( t ) ] \varphi(t)=\omega_{0}t-kz=\omega_{0}t-k_{0}z[n+n_{2}I(t)]
  55. ω ( t ) = φ ( t ) t = ω 0 - k 0 z n 2 I ( t ) t \omega(t)=\frac{\partial\varphi(t)}{\partial t}=\omega_{0}-k_{0}zn_{2}\frac{% \partial I(t)}{\partial t}
  56. β 0 \beta_{0}
  57. E ( 𝐫 , t ) = A m a ( t , z ) f ( x , y ) e i ( β 0 z - ω 0 t ) E(\mathbf{r},t)=A_{m}a(t,z)f(x,y)e^{i(\beta_{0}z-\omega_{0}t)}
  58. A m A_{m}
  59. a ( t , z ) a(t,z)
  60. f ( x , y ) f(x,y)
  61. A m A_{m}
  62. E ~ ( 𝐫 , ω - ω 0 ) = - E ( 𝐫 , t ) e - i ( ω - ω 0 ) t d t \tilde{E}(\mathbf{r},\omega-\omega_{0})=\int\limits_{-\infty}^{\infty}\!\!E(% \mathbf{r},t)e^{-i(\omega-\omega_{0})t}dt
  63. t E - i ( ω - ω 0 ) E ~ \frac{\partial}{\partial t}E\Longleftrightarrow-i(\omega-\omega_{0})\tilde{E}
  64. E ~ ( 𝐫 , ω - ω 0 ) = A m a ~ ( ω - ω 0 , z ) f ( x , y ) e i β 0 z \tilde{E}(\mathbf{r},\omega-\omega_{0})=A_{m}\tilde{a}(\omega-\omega_{0},z)f(x% ,y)e^{i\beta_{0}z}
  65. 2 E ~ + n 2 ( ω ) k 0 2 E ~ = 0 \nabla^{2}\tilde{E}+n^{2}(\omega)k_{0}^{2}\tilde{E}=0
  66. n ( ω ) k 0 = β ( ω ) = β 0 linear non dispersive + β l ( ω ) linear dispersive + β n l non linear = β 0 + Δ β ( ω ) n(\omega)k_{0}=\beta(\omega)=\underbrace{\beta_{0}}_{\mbox{linear non % dispersive}~{}}+\underbrace{\beta_{l}(\omega)}_{\mbox{linear dispersive}~{}}+% \underbrace{\beta_{nl}}_{\mbox{non linear}~{}}=\beta_{0}+\Delta\beta(\omega)
  67. Δ β \Delta\beta
  68. | β 0 | | Δ β ( ω ) | |\beta_{0}|\gg|\Delta\beta(\omega)|
  69. ω 0 \omega_{0}
  70. β ( ω ) β 0 + ( ω - ω 0 ) β 1 + ( ω - ω 0 ) 2 2 β 2 + β n l \beta(\omega)\approx\beta_{0}+(\omega-\omega_{0})\beta_{1}+\frac{(\omega-% \omega_{0})^{2}}{2}\beta_{2}+\beta_{nl}
  71. β u = d u β ( ω ) d ω u | ω = ω 0 \beta_{u}=\left.\frac{d^{u}\beta(\omega)}{d\omega^{u}}\right|_{\omega=\omega_{% 0}}
  72. | 2 a ~ z 2 | | β 0 a ~ z | \left|\frac{\partial^{2}\tilde{a}}{\partial z^{2}}\right|\ll\left|\beta_{0}% \frac{\partial\tilde{a}}{\partial z}\right|
  73. 2 i β 0 a ~ z + [ β 2 ( ω ) - β 0 2 ] a ~ = 0 2i\beta_{0}\frac{\partial\tilde{a}}{\partial z}+[\beta^{2}(\omega)-\beta_{0}^{% 2}]\tilde{a}=0
  74. f ( x , y ) f(x,y)
  75. β 2 ( ω ) - β 0 2 = [ β ( ω ) - β 0 ] [ β ( ω ) + β 0 ] = [ β 0 + Δ β ( ω ) - β 0 ] [ 2 β 0 + Δ β ( ω ) ] 2 β 0 Δ β ( ω ) \beta^{2}(\omega)-\beta_{0}^{2}=[\beta(\omega)-\beta_{0}][\beta(\omega)+\beta_% {0}]=[\beta_{0}+\Delta\beta(\omega)-\beta_{0}][2\beta_{0}+\Delta\beta(\omega)]% \approx 2\beta_{0}\Delta\beta(\omega)
  76. i a ~ z + Δ β ( ω ) a ~ = 0 i\frac{\partial\tilde{a}}{\partial z}+\Delta\beta(\omega)\tilde{a}=0
  77. Δ β ( ω ) i β 1 t - β 2 2 2 t 2 + β n l \Delta\beta(\omega)\Longleftrightarrow i\beta_{1}\frac{\partial}{\partial t}-% \frac{\beta_{2}}{2}\frac{\partial^{2}}{\partial t^{2}}+\beta_{nl}
  78. β n l = k 0 n 2 I = k 0 n 2 | E | 2 2 η 0 / n = k 0 n 2 n | A m | 2 2 η 0 | a | 2 \beta_{nl}=k_{0}n_{2}I=k_{0}n_{2}\frac{|E|^{2}}{2\eta_{0}/n}=k_{0}n_{2}n\frac{% |A_{m}|^{2}}{2\eta_{0}}|a|^{2}
  79. L n l = 2 η 0 k 0 n n 2 | A m | 2 L_{nl}=\frac{2\eta_{0}}{k_{0}nn_{2}|A_{m}|^{2}}
  80. i a z + i β 1 a t - β 2 2 2 a t 2 + 1 L n l | a | 2 a = 0 i\frac{\partial a}{\partial z}+i\beta_{1}\frac{\partial a}{\partial t}-\frac{% \beta_{2}}{2}\frac{\partial^{2}a}{\partial t^{2}}+\frac{1}{L_{nl}}|a|^{2}a=0
  81. v g = 1 / β 1 v_{g}=1/\beta_{1}
  82. T = t - β 1 z T=t-\beta_{1}z
  83. i a z - β 2 2 2 a T 2 + 1 L n l | a | 2 a = 0 i\frac{\partial a}{\partial z}-\frac{\beta_{2}}{2}\frac{\partial^{2}a}{% \partial T^{2}}+\frac{1}{L_{nl}}|a|^{2}a=0
  84. β 2 < 0 \beta_{2}<0
  85. D = - 2 π c λ 2 β 2 > 0 D=\frac{-2\pi c}{\lambda^{2}}\beta_{2}>0
  86. β 2 = - | β 2 | \beta_{2}=-|\beta_{2}|
  87. L d = T 0 2 | β 2 | ; τ = T T 0 ; ζ = z L d ; N 2 = L d L n l L_{d}=\frac{T_{0}^{2}}{|\beta_{2}|};\qquad\tau=\frac{T}{T_{0}};\qquad\zeta=% \frac{z}{L_{d}};\qquad N^{2}=\frac{L_{d}}{L_{nl}}
  88. 1 2 2 a τ 2 + i a ζ + N 2 | a | 2 a = 0 \frac{1}{2}\frac{\partial^{2}a}{\partial\tau^{2}}+i\frac{\partial a}{\partial% \zeta}+N^{2}|a|^{2}a=0
  89. a ( τ , ζ ) = sech ( τ ) e i ζ / 2 a(\tau,\zeta)=\operatorname{sech}(\tau)e^{i\zeta/2}
  90. N = 1 N=1
  91. | A m | 2 = 2 η 0 | β 2 | T 0 2 n 2 k 0 n |A_{m}|^{2}=\frac{2\eta_{0}|\beta_{2}|}{T_{0}^{2}n_{2}k_{0}n}
  92. I m a x = | A m | 2 2 η 0 / n = | β 2 | T 0 2 n 2 k 0 I_{max}=\frac{|A_{m}|^{2}}{2\eta_{0}/n}=\frac{|\beta_{2}|}{T_{0}^{2}n_{2}k_{0}}
  93. A 𝑒𝑓𝑓 A_{\mathit{eff}}
  94. P = I A 𝑒𝑓𝑓 P=IA_{\mathit{eff}}
  95. P = | β 2 | A 𝑒𝑓𝑓 T 0 2 n 2 k 0 P=\frac{|\beta_{2}|A_{\mathit{eff}}}{T_{0}^{2}n_{2}k_{0}}
  96. 0.5 < N < 1.5 0.5<N<1.5
  97. n ( I ) = n + n 2 I n(I)=n+n_{2}I
  98. P ( z ) = P 0 e - α z P(z)=P_{0}e^{-\alpha z}
  99. P 0 P_{0}
  100. T 0 T_{0}
  101. P = | β 2 | A 𝑒𝑓𝑓 T 0 2 n 2 k 0 P=\frac{|\beta_{2}|A_{\mathit{eff}}}{T_{0}^{2}n_{2}k_{0}}
  102. T 0 T_{0}
  103. T 0 T_{0}
  104. T ( z ) = T 0 e α 2 z T(z)=T_{0}e^{\frac{\alpha}{2}z}
  105. α z 1 \alpha z\ll 1
  106. n 2 > 0 n_{2}>0
  107. β 2 < 0 \beta_{2}<0
  108. D > 0 D>0
  109. n 2 < 0 n_{2}<0
  110. β 2 > 0 \beta_{2}>0
  111. - 1 2 2 a τ 2 + i a ζ + N 2 | a | 2 a = 0. \frac{-1}{2}\frac{\partial^{2}a}{\partial\tau^{2}}+i\frac{\partial a}{\partial% \zeta}+N^{2}|a|^{2}a=0.
  112. a ( τ , ζ ) = tanh ( τ ) e i ζ . a(\tau,\zeta)=\tanh(\tau)e^{i\zeta}.
  113. | a ( τ , ζ ) | 2 |a(\tau,\zeta)|^{2}
  114. N > 1 N>1
  115. a ( τ , ζ = 0 ) = N tanh ( τ ) . a(\tau,\zeta=0)=N\tanh(\tau).

Soliton_model_in_neuroscience.html

  1. 2 Δ ρ t 2 = x [ ( c 0 2 + p Δ ρ + q Δ ρ 2 ) Δ ρ x ] - h 4 Δ ρ x 4 , \frac{\partial^{2}\Delta\rho}{\partial t^{2}}=\frac{\partial}{\partial x}\left% [\left(c_{0}^{2}+p\Delta\rho+q\Delta\rho^{2}\right)\frac{\partial\Delta\rho}{% \partial x}\right]-h\frac{\partial^{4}\Delta\rho}{\partial x^{4}},
  2. t t
  3. x x
  4. Δ ρ Δρ
  5. c < s u b > 0 c<sub>0

Solving_quadratic_equations_with_continued_fractions.html

  1. a x 2 + b x + c = 0 , ax^{2}+bx+c=0,\,\!
  2. x 2 = 2 x^{2}=2\,
  3. x 2 - 1 = 1. x^{2}-1=1.\,
  4. ( x + 1 ) ( x - 1 ) = 1 (x+1)(x-1)=1\,
  5. ( x - 1 ) = 1 1 + x (x-1)=\frac{1}{1+x}\,
  6. x = 1 + 1 1 + x . x=1+\frac{1}{1+x}.\,
  7. x = 1 + 1 1 + ( 1 + 1 1 + x ) = 1 + 1 2 + 1 1 + x . x=1+\cfrac{1}{1+\left(1+\cfrac{1}{1+x}\right)}=1+\cfrac{1}{2+\cfrac{1}{1+x}}.\,
  8. x = 1 + 1 2 + 1 2 + 1 2 + 1 2 + 1 2 + = 2 . x=1+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{2+\ddots}}}}}=% \sqrt{2}.\,
  9. ω = 2 - 1. \omega=\sqrt{2}-1.\,
  10. ω 2 = 3 - 2 2 , ω 3 = 5 2 - 7 , ω 4 = 17 - 12 2 , ω 5 = 29 2 - 41 , ω 6 = 99 - 70 2 , ω 7 = 169 2 - 239 , \begin{aligned}\displaystyle\omega^{2}&\displaystyle=3-2\sqrt{2},&% \displaystyle\omega^{3}&\displaystyle=5\sqrt{2}-7,&\displaystyle\omega^{4}&% \displaystyle=17-12\sqrt{2},\\ \displaystyle\omega^{5}&\displaystyle=29\sqrt{2}-41,&\displaystyle\omega^{6}&% \displaystyle=99-70\sqrt{2},&\displaystyle\omega^{7}&\displaystyle=169\sqrt{2}% -239,\end{aligned}
  11. ω - 1 = 2 + 1. \omega^{-1}=\sqrt{2}+1.\,
  12. x 2 + b x + c = 0 x^{2}+bx+c=0\,
  13. x 2 + b x = - c x + b = - c x x = - b - c x \begin{aligned}\displaystyle x^{2}+bx&\displaystyle=-c\\ \displaystyle x+b&\displaystyle=\frac{-c}{x}\\ \displaystyle x&\displaystyle=-b-\frac{c}{x}\end{aligned}
  14. x = - b - c - b - c - b - c - b - c - b - x=-b-\cfrac{c}{-b-\cfrac{c}{-b-\cfrac{c}{-b-\cfrac{c}{-b-\ddots\,}}}}
  15. x 2 + b x + c = 0 x^{2}+bx+c=0\,
  16. x = - b - c - b - c - b - c - b - c - b - x=-b-\cfrac{c}{-b-\cfrac{c}{-b-\cfrac{c}{-b-\cfrac{c}{-b-\ddots\,}}}}
  17. \infty
  18. x 2 = c ( c > 0 ) x^{2}=c\qquad(c>0)\,
  19. p 2 < c . p^{2}<c.\,
  20. x 2 - p 2 = c - p 2 ( x + p ) ( x - p ) = c - p 2 x - p = c - p 2 p + x x = p + c - p 2 p + x = p + c - p 2 p + ( p + c - p 2 p + x ) = p + c - p 2 2 p + c - p 2 2 p + c - p 2 2 p + \begin{aligned}\displaystyle x^{2}-p^{2}&\displaystyle=c-p^{2}\\ \displaystyle(x+p)(x-p)&\displaystyle=c-p^{2}\\ \displaystyle x-p&\displaystyle=\frac{c-p^{2}}{p+x}\\ \displaystyle x&\displaystyle=p+\frac{c-p^{2}}{p+x}\\ &\displaystyle=p+\cfrac{c-p^{2}}{p+\left(p+\cfrac{c-p^{2}}{p+x}\right)}&% \displaystyle=p+\cfrac{c-p^{2}}{2p+\cfrac{c-p^{2}}{2p+\cfrac{c-p^{2}}{2p+% \ddots\,}}}\end{aligned}
  21. x 2 + b x + c = 0 ( b 0 ) x^{2}+bx+c=0\qquad(b\neq 0)\,
  22. x = - b - c - b - c - b - c - b - c - b - x=-b-\cfrac{c}{-b-\cfrac{c}{-b-\cfrac{c}{-b-\cfrac{c}{-b-\ddots\,}}}}

Sonic_logging.html

  1. Δ t {\Delta}t
  2. V m a t V_{mat}
  3. V l V_{l}
  4. Δ t {{\Delta}t}
  5. t f a r - t n e a r {t_{far}}-{t_{near}}
  6. t f a r {t_{far}}
  7. t n e a r {t_{near}}
  8. 1 V = ϕ V f + 1 - ϕ V m a t \frac{1}{V}=\frac{\phi}{V_{f}}+\frac{1-{\phi}}{V_{mat}}
  9. V V
  10. V f V_{f}
  11. V m a t V_{mat}
  12. ϕ {\phi}

Sophomore's_dream.html

  1. 0 1 x - x d x = n = 1 n - n ( = 1.29128599706266354040728259059560054149861936827 ) 0 1 x x d x = n = 1 ( - 1 ) n + 1 n - n = - n = 1 ( - n ) - n ( = 0.78343051071213440705926438652697546940768199014 ) \begin{aligned}\displaystyle\int_{0}^{1}x^{-x}\,\mathrm{d}x&\displaystyle=\sum% _{n=1}^{\infty}n^{-n}&&\displaystyle(\scriptstyle{=1.2912859970626635404072825% 9059560054149861936827\dots)}\\ \displaystyle\int_{0}^{1}x^{x}\,\mathrm{d}x&\displaystyle=\sum_{n=1}^{\infty}(% -1)^{n+1}n^{-n}=-\sum_{n=1}^{\infty}(-n)^{-n}&&\displaystyle(\scriptstyle{=0.7% 8343051071213440705926438652697546940768199014\dots})\end{aligned}
  2. x x = exp ( x log x ) = n = 0 x n ( log x ) n n ! . x^{x}=\exp(x\log x)=\sum_{n=0}^{\infty}\frac{x^{n}(\log x)^{n}}{n!}.
  3. 0 1 x x d x = 0 1 n = 0 x n ( log x ) n n ! d x . \int_{0}^{1}x^{x}\,\mathrm{d}x=\int_{0}^{1}\sum_{n=0}^{\infty}\frac{x^{n}(\log x% )^{n}}{n!}\,\mathrm{d}x.
  4. 0 1 x x d x = n = 0 0 1 x n ( log x ) n n ! d x . \int_{0}^{1}x^{x}\,\mathrm{d}x=\sum_{n=0}^{\infty}\int_{0}^{1}\frac{x^{n}(\log x% )^{n}}{n!}\,\mathrm{d}x.
  5. x = exp ( - u n + 1 ) \scriptstyle x=\exp\,\left(-\frac{u}{n+1}\right)
  6. 0 < u < \scriptstyle 0<u<\infty
  7. 0 1 x n ( log x ) n d x = ( - 1 ) n ( n + 1 ) - ( n + 1 ) 0 u n e - u d u . \int_{0}^{1}x^{n}(\log\,x)^{n}\,\mathrm{d}x=(-1)^{n}(n+1)^{-(n+1)}\int_{0}^{% \infty}u^{n}e^{-u}\,\mathrm{d}u.
  8. 0 u n e - u d u = n ! \int_{0}^{\infty}u^{n}e^{-u}\,\mathrm{d}u=n!
  9. 0 1 x n ( log x ) n n ! d x = ( - 1 ) n ( n + 1 ) - ( n + 1 ) . \int_{0}^{1}\frac{x^{n}(\log x)^{n}}{n!}\,\mathrm{d}x=(-1)^{n}(n+1)^{-(n+1)}.
  10. 0 1 x n ( log x ) n d x \int_{0}^{1}x^{n}(\log\,x)^{n}\,\mathrm{d}x
  11. + C +C
  12. x m ( ln x ) n d x \scriptstyle\int x^{m}(\ln x)^{n}\,\mathrm{d}x
  13. x m ( ln x ) n d x \displaystyle\int x^{m}(\ln x)^{n}\,\mathrm{d}x
  14. n n
  15. n - 1 n-1
  16. x m ( ln x ) n d x = x m + 1 m + 1 i = 0 n ( - 1 ) i ( n ) i ( m + 1 ) i ( ln x ) n - i \int x^{m}(\ln x)^{n}\,\mathrm{d}x=\frac{x^{m+1}}{m+1}\cdot\sum_{i=0}^{n}(-1)^% {i}\frac{(n)_{i}}{(m+1)^{i}}(\ln x)^{n-i}
  17. x n ( ln x ) n d x = x n + 1 n + 1 i = 0 n ( - 1 ) i ( n ) i ( n + 1 ) i ( ln x ) n - i . \int x^{n}(\ln x)^{n}\,\mathrm{d}x=\frac{x^{n+1}}{n+1}\cdot\sum_{i=0}^{n}(-1)^% {i}\frac{(n)_{i}}{(n+1)^{i}}(\ln x)^{n-i}.
  18. 0 1 x n ( ln x ) n n ! d x = 1 n ! 1 n + 1 n + 1 ( - 1 ) n ( n ) n ( n + 1 ) n = ( - 1 ) n ( n + 1 ) - ( n + 1 ) . \int_{0}^{1}\frac{x^{n}(\ln x)^{n}}{n!}\,\mathrm{d}x=\frac{1}{n!}\frac{1^{n+1}% }{n+1}(-1)^{n}\frac{(n)_{n}}{(n+1)^{n}}=(-1)^{n}(n+1)^{-(n+1)}.
  19. Γ ( n + 1 ) = n ! \Gamma(n+1)=n!\,
  20. lim x 0 + x m ( ln x ) n = 0 \scriptstyle\lim_{x\to 0^{+}}x^{m}(\ln x)^{n}\,=\,0

Spacetime_algebra.html

  1. γ 0 \gamma_{0}
  2. { γ 1 , γ 2 , γ 3 } \{\gamma_{1},\gamma_{2},\gamma_{3}\}
  3. γ μ γ ν + γ ν γ μ = 2 η μ ν \gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu}=2\eta_{\mu\nu}
  4. η μ ν \eta_{\mu\nu}\,
  5. γ 0 2 = + 1 \gamma_{0}^{2}={+1}
  6. γ 1 2 = γ 2 2 = γ 3 2 = - 1 \gamma_{1}^{2}=\gamma_{2}^{2}=\gamma_{3}^{2}={-1}
  7. γ μ γ ν = - γ ν γ μ \displaystyle\gamma_{\mu}\gamma_{\nu}=-\gamma_{\nu}\gamma_{\mu}
  8. γ k \gamma_{k}
  9. { 1 } \{1\}
  10. { γ 0 , γ 1 , γ 2 , γ 3 } \{\gamma_{0},\gamma_{1},\gamma_{2},\gamma_{3}\}
  11. { γ 0 γ 1 , γ 0 γ 2 , γ 0 γ 3 , γ 1 γ 2 , γ 2 γ 3 , γ 3 γ 1 } \{\gamma_{0}\gamma_{1},\,\gamma_{0}\gamma_{2},\,\gamma_{0}\gamma_{3},\,\gamma_% {1}\gamma_{2},\,\gamma_{2}\gamma_{3},\,\gamma_{3}\gamma_{1}\}
  12. { i γ 0 , i γ 1 , i γ 2 , i γ 3 } \{i\gamma_{0},i\gamma_{1},i\gamma_{2},i\gamma_{3}\}
  13. { i } \{i\}
  14. i = γ 0 γ 1 γ 2 γ 3 i=\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}
  15. { γ μ } \{\gamma_{\mu}\}
  16. { γ μ = 1 < m t p l > γ μ } \{\gamma^{\mu}=\frac{1}{<}mtpl>{{\gamma_{\mu}}}\}
  17. μ \mu
  18. γ μ γ ν = δ μ ν \gamma_{\mu}\cdot\gamma^{\nu}={\delta_{\mu}}^{\nu}
  19. γ 0 = γ 0 \gamma^{0}=\gamma_{0}
  20. γ k = - γ k \gamma^{k}=-\gamma_{k}
  21. a = a μ γ μ = a μ γ μ a=a^{\mu}\gamma_{\mu}=a_{\mu}\gamma^{\mu}
  22. μ \mu
  23. a γ ν = a ν a γ ν = a ν \begin{aligned}\displaystyle a\cdot\gamma^{\nu}&\displaystyle=a^{\nu}\\ \displaystyle a\cdot\gamma_{\nu}&\displaystyle=a_{\nu}\end{aligned}
  24. a F ( x ) = lim τ 0 F ( x + a τ ) - F ( x ) τ a\cdot\nabla F(x)=\lim_{\tau\rightarrow 0}\frac{F(x+a\tau)-F(x)}{\tau}
  25. = γ μ x μ = γ μ μ . \nabla=\gamma^{\mu}\frac{\partial}{\partial x^{\mu}}=\gamma^{\mu}\partial_{\mu}.
  26. x = c t γ 0 + x k γ k x=ct\gamma_{0}+x^{k}\gamma_{k}
  27. 0 = 1 c t , k = x k \partial_{0}=\frac{1}{c}\frac{\partial}{\partial t},\quad\partial_{k}=\frac{% \partial}{\partial{x^{k}}}
  28. x γ 0 = x 0 + 𝐱 x\gamma_{0}=x^{0}+\mathbf{x}
  29. p γ 0 = E + 𝐩 p\gamma_{0}=E+\mathbf{p}
  30. v γ 0 = γ ( 1 + 𝐯 ) v\gamma_{0}=\gamma(1+\mathbf{v})
  31. γ 0 = t - \nabla\gamma_{0}=\partial_{t}-\nabla
  32. γ 0 \gamma_{0}
  33. x = x μ γ μ x=x^{\mu}\gamma_{\mu}
  34. x γ 0 = x 0 + x k γ k γ 0 γ 0 x = x 0 - x k γ k γ 0 \begin{aligned}\displaystyle x\gamma_{0}&\displaystyle=x^{0}+x^{k}\gamma_{k}% \gamma_{0}\\ \displaystyle\gamma_{0}x&\displaystyle=x^{0}-x^{k}\gamma_{k}\gamma_{0}\end{aligned}
  35. γ k γ 0 \gamma_{k}\gamma_{0}
  36. σ k = γ k γ 0 \sigma_{k}=\gamma_{k}\gamma_{0}
  37. 𝐱 = x k σ k \mathbf{x}=x^{k}\sigma_{k}
  38. γ 0 \gamma_{0}
  39. x γ 0 x\gamma_{0}
  40. γ 0 x \gamma_{0}x
  41. x γ 0 \displaystyle x\gamma_{0}
  42. 1 2 ( 1 ± γ 0 γ i ) \tfrac{1}{2}(1\pm\gamma_{0}\gamma_{i})
  43. ( 1 + γ 0 γ i ) ( 1 - γ 0 γ i ) = 0 (1+\gamma_{0}\gamma_{i})(1-\gamma_{0}\gamma_{i})=0\,\!
  44. i t Ψ = H S Ψ - e 2 m c σ ^ 𝐁 Ψ i\hbar\,\partial_{t}\Psi=H_{S}\Psi-\frac{e\hbar}{2mc}\,\hat{\sigma}\cdot% \mathbf{B}\Psi
  45. σ ^ i \hat{\sigma}_{i}
  46. σ ^ \hat{\sigma}
  47. H S H_{S}
  48. t ψ i σ 3 = H S ψ - e 2 m c 𝐁 ψ σ 3 \partial_{t}\psi\,i\sigma_{3}\,\hbar=H_{S}\psi-\frac{e\hbar}{2mc}\,\mathbf{B}% \psi\sigma_{3}
  49. i = σ 1 σ 2 σ 3 i=\sigma_{1}\sigma_{2}\sigma_{3}
  50. ψ \psi
  51. σ 3 \sigma_{3}
  52. ψ \psi
  53. H S H_{S}
  54. ψ = e 1 2 ( μ + β i + ϕ ) \psi=e^{\frac{1}{2}(\mu+\beta i+\phi)}
  55. ψ = R ( ρ e i β ) 1 2 \psi=R(\rho e^{i\beta})^{\frac{1}{2}}
  56. ψ = ψ ( x ) \psi=\psi(x)
  57. R = R ( x ) R=R(x)
  58. ρ = ρ ( x ) \rho=\rho(x)
  59. β = β ( x ) \beta=\beta(x)
  60. γ μ \gamma_{\mu}
  61. e μ e_{\mu}
  62. e μ = R γ μ R ~ e_{\mu}=R\gamma_{\mu}\tilde{R}
  63. ψ \psi
  64. ψ = e i Φ λ / \psi=e^{i\Phi_{\lambda}/\hbar}
  65. Φ λ \Phi_{\lambda}
  66. λ \lambda
  67. γ ^ μ ( 𝐣 μ - e 𝐀 μ ) | ψ = m | ψ \hat{\gamma}^{\mu}(\mathbf{j}\partial_{\mu}-e\mathbf{A}_{\mu})|\psi\rangle=m|\psi\rangle
  68. γ ^ \hat{\gamma}
  69. ψ i σ 3 - 𝐀 ψ = m ψ γ 0 \nabla\psi\,i\sigma_{3}-\mathbf{A}\psi=m\psi\gamma_{0}
  70. ψ \psi
  71. σ 3 \sigma_{3}
  72. = γ μ μ \nabla=\gamma^{\mu}\partial_{\mu}
  73. d d τ R = 1 2 ( Ω - ω ) R \frac{d}{d\tau}R=\frac{1}{2}(\Omega-\omega)R
  74. D τ = τ + 1 2 ω D_{\tau}=\partial_{\tau}+\frac{1}{2}\omega

Span_(category_theory).html

  1. Λ = ( - 1 0 + 1 ) , \Lambda=(-1\leftarrow 0\rightarrow+1),
  2. Y X Z Y\leftarrow X\rightarrow Z
  3. X × Y π X X X\times Y\overset{\pi_{X}}{\to}X
  4. X × Y π Y Y X\times Y\overset{\pi_{Y}}{\to}Y
  5. A = A = A ; A=A=A;
  6. ϕ : A B \phi\colon A\to B
  7. X Y Z , X\leftarrow Y\rightarrow Z,
  8. Λ op = ( - 1 0 + 1 ) , \Lambda\text{op}=(-1\rightarrow 0\leftarrow+1),
  9. Y X Z Y\rightarrow X\leftarrow Z

Specification_(regression).html

  1. y = f ( s , x ) y=f(s,x)
  2. y y
  3. s s
  4. x x
  5. ln y = ln y 0 + ρ s + β 1 x + β 2 x 2 + ϵ \ln y=\ln y_{0}+\rho s+\beta_{1}x+\beta_{2}x^{2}+\epsilon
  6. ϵ \epsilon
  7. ρ \rho
  8. β \beta

Specificity_constant.html

  1. k c a t / K M k_{cat}/K_{M}
  2. E + S k f k r E S k c a t E + P E+S\,\overset{k_{f}}{\underset{}{}}{k_{r}}\rightleftharpoons\,ES\,\overset{k_{% cat}}{\longrightarrow}\,E+P
  3. k f k_{f}
  4. k r k_{r}
  5. k cat k_{\mathrm{cat}}
  6. K M = k r + k c a t k f K_{M}=\frac{k_{r}+k_{cat}}{k_{f}}
  7. k c a t k_{cat}
  8. k c a t / K M k_{cat}/K_{M}

Spectral_centroid.html

  1. C e n t r o i d = n = 0 N - 1 f ( n ) x ( n ) n = 0 N - 1 x ( n ) Centroid=\frac{\sum_{n=0}^{N-1}f\left(n\right)x\left(n\right)}{\sum_{n=0}^{N-1% }x\left(n\right)}

Spectral_element_method.html

  1. L 2 ( Ω ) L^{2}(\Omega)
  2. u H s + 1 ( Ω ) u\in H^{s+1}(\Omega)
  3. u - u N H 1 ( Ω ) C s N - s u H s + 1 ( Ω ) \|u-u_{N}\|_{H^{1}(\Omega)}\leqq C_{s}N^{-s}\|u\|_{H^{s+1}(\Omega)}
  4. u - u N H 1 ( Ω ) C exp ( - γ N ) \|u-u_{N}\|_{H^{1}(\Omega)}\leqq C\exp(-\gamma N)
  5. γ \gamma
  6. u u
  7. a ( , ) a(\cdot,\cdot)
  8. F F

Spectral_risk_measure.html

  1. X X
  2. M ϕ : M_{\phi}:\mathcal{L}\to\mathbb{R}
  3. ϕ \phi
  4. [ 0 , 1 ] [0,1]
  5. 0 1 ϕ ( p ) d p = 1 \int_{0}^{1}\phi(p)dp=1
  6. M ϕ ( X ) = - 0 1 ϕ ( p ) F X - 1 ( p ) d p M_{\phi}(X)=-\int_{0}^{1}\phi(p)F_{X}^{-1}(p)dp
  7. F X F_{X}
  8. S S
  9. X 1 : S , X S : S X_{1:S},...X_{S:S}
  10. ϕ S \phi\in\mathbb{R}^{S}
  11. M ϕ : S M_{\phi}:\mathbb{R}^{S}\rightarrow\mathbb{R}
  12. M ϕ ( X ) = - δ s = 1 S ϕ s X s : S M_{\phi}(X)=-\delta\sum_{s=1}^{S}\phi_{s}X_{s:S}
  13. ϕ S \phi\in\mathbb{R}^{S}
  14. ϕ s 0 \phi_{s}\geq 0
  15. s = 1 , , S s=1,\dots,S
  16. s = 1 S ϕ s = 1 \sum_{s=1}^{S}\phi_{s}=1
  17. ϕ s \phi_{s}
  18. ϕ s 1 ϕ s 2 \phi_{s_{1}}\geq\phi_{s_{2}}
  19. s 1 < s 2 {s_{1}}<{s_{2}}
  20. s 1 , s 2 { 1 , , S } {s_{1}},{s_{2}}\in\{1,\dots,S\}
  21. ρ : \rho:\mathcal{L}\to\mathbb{R}
  22. λ > 0 \lambda>0
  23. ρ ( λ X ) = λ ρ ( X ) \rho(\lambda X)=\lambda\rho(X)
  24. α \alpha\in\mathbb{R}
  25. ρ ( X + a ) = ρ ( X ) - a \rho(X+a)=\rho(X)-a
  26. X Y X\geq Y
  27. ρ ( X ) ρ ( Y ) \rho(X)\leq\rho(Y)
  28. ρ ( X + Y ) ρ ( X ) + ρ ( Y ) \rho(X+Y)\leq\rho(X)+\rho(Y)
  29. F X F_{X}
  30. F Y F_{Y}
  31. F X = F Y F_{X}=F_{Y}
  32. ρ ( X ) = ρ ( Y ) \rho(X)=\rho(Y)
  33. ρ ( X + Y ) = ρ ( X ) + ρ ( Y ) \rho(X+Y)=\rho(X)+\rho(Y)
  34. ω 1 , ω 2 Ω : ( X ( ω 2 ) - X ( ω 1 ) ) ( Y ( ω 2 ) - Y ( ω 1 ) ) 0 \omega_{1},\omega_{2}\in\Omega:\;(X(\omega_{2})-X(\omega_{1}))(Y(\omega_{2})-Y% (\omega_{1}))\geq 0

Spectral_set.html

  1. X X\subseteq\mathbb{C}
  2. T T
  3. T T
  4. X X
  5. T T
  6. X X
  7. r ( x ) r(x)
  8. X X
  9. r ( T ) r X = sup { | r ( x ) | : x X } \left\|r(T)\right\|\leq\left\|r\right\|_{X}=\sup\left\{\left|r(x)\right|:x\in X\right\}

Spectroscopic_parallax.html

  1. M - m = - 5 l o g ( d / 10 ) M-m=-5log(d/10)

Speeded_up_robust_features.html

  1. S ( x , y ) = i = 0 x j = 0 y I ( i , j ) S(x,y)=\sum_{i=0}^{x}\sum_{j=0}^{y}I(i,j)
  2. H ( p , σ ) = ( L x x ( p , σ ) L x y ( p , σ ) L x y ( p , σ ) L y y ( p , σ ) ) H(p,\sigma)=\begin{pmatrix}L_{xx}(p,\sigma)&L_{xy}(p,\sigma)\\ L_{xy}(p,\sigma)&L_{yy}(p,\sigma)\end{pmatrix}
  3. L x x ( p , σ ) L_{xx}(p,\sigma)
  4. σ approx = Current filter size * ( Base Filter Scale Base Filter Size ) \sigma\text{approx}=\,\text{Current filter size}*\left(\frac{\,\text{Base % Filter Scale}}{\,\text{Base Filter Size}}\right)
  5. 6 s 6s
  6. s s

Spherical_design.html

  1. N ( d , t ) < C d t d N(d,t)<C_{d}t^{d}

Spherical_mean.html

  1. 1 ω n - 1 ( r ) B ( x , r ) u ( y ) d S ( y ) \frac{1}{\omega_{n-1}(r)}\int\limits_{\partial B(x,r)}\!u(y)\,\mathrm{d}S(y)
  2. 1 ω n - 1 y = 1 u ( x + r y ) d S ( y ) \frac{1}{\omega_{n-1}}\int\limits_{\|y\|=1}\!u(x+ry)\,\mathrm{d}S(y)
  3. B ( x , r ) - u ( y ) d S ( y ) . \int\limits_{\partial B(x,r)}\!\!\!\!\!\!\!\!\!\!\!-\,u(y)\,\mathrm{d}S(y).
  4. u u
  5. r B ( x , r ) - u ( y ) d S ( y ) r\to\int\limits_{\partial B(x,r)}\!\!\!\!\!\!\!\!\!\!\!-\,u(y)\,\mathrm{d}S(y)
  6. r 0 r\to 0
  7. u ( x ) . u(x).
  8. u t t = c 2 Δ u u_{tt}=c^{2}\Delta u
  9. t > 0 t>0
  10. t = 0. t=0.
  11. U U
  12. n \mathbb{R}^{n}
  13. u u
  14. U U
  15. u u
  16. x x
  17. U U
  18. r > 0 r>0
  19. B ( x , r ) B(x,r)
  20. U U
  21. u ( x ) = B ( x , r ) - u ( y ) d S ( y ) . u(x)=\int\limits_{\partial B(x,r)}\!\!\!\!\!\!\!\!\!\!\!-\,u(y)\,\mathrm{d}S(y).

Spin_representation.html

  1. V V
  2. Q Q
  3. Q Q
  4. O ( V , Q ) O(V,Q)
  5. S O ( V , Q ) SO(V,Q)
  6. V V
  7. S O ( V , Q ) SO(V,Q)
  8. S p i n ( V , Q ) Spin(V,Q)
  9. S p i n ( V , Q ) S O ( V , Q ) Spin(V,Q)→SO(V,Q)
  10. 1 1
  11. O ( V , Q ) , S O ( V , Q ) O(V,Q),SO(V,Q)
  12. S p i n ( V , Q ) Spin(V,Q)
  13. ( V , Q ) (V,Q)
  14. 𝐬𝐨 ( V , Q ) \mathbf{so}(V,Q)
  15. V V
  16. V V
  17. Q Q
  18. S O ( V , Q ) SO(V,Q)
  19. S p i n ( V , Q ) Spin(V,Q)
  20. 𝐬𝐨 ( V , Q ) \mathbf{so}(V,Q)
  21. n n
  22. V V
  23. Q ( z 1 , z n ) = z 1 2 + z 2 2 + + z n 2 . Q(z_{1},\ldots z_{n})=z_{1}^{2}+z_{2}^{2}+\cdots+z_{n}^{2}.
  24. O ( n , 𝐂 ) , S O ( n , 𝐂 ) , S p i n ( n , 𝐂 ) O(n,\mathbf{C}),SO(n,\mathbf{C}),Spin(n,\mathbf{C})
  25. 𝐬𝐨 ( n , 𝐂 ) \mathbf{so}(n,\mathbf{C})
  26. ( p , q ) (p,q)
  27. n = p + q n=p+q
  28. V V
  29. p q p−q
  30. Q ( x 1 , x n ) = x 1 2 + x 2 2 + + x p 2 - ( x p + 1 2 + + x p + q 2 ) . Q(x_{1},\ldots x_{n})=x_{1}^{2}+x_{2}^{2}+\cdots+x_{p}^{2}-(x_{p+1}^{2}+\cdots% +x_{p+q}^{2}).
  31. O ( p , q ) , S O ( p , q ) , S p i n ( p , q ) O(p,q),SO(p,q),Spin(p,q)
  32. 𝐬𝐨 ( p , q ) \mathbf{so}(p,q)
  33. S p i n ( n , 𝐂 ) Spin(n,\mathbf{C})
  34. S p i n ( p , q ) Spin(p,q)
  35. S O ( n , 𝐂 ) SO(n,\mathbf{C})
  36. S O ( p , q ) SO(p,q)
  37. S S
  38. ρ ρ
  39. S p i n ( n , 𝐂 ) Spin(n,\mathbf{C})
  40. S p i n ( p , q ) Spin(p,q)
  41. G L ( S ) GL(S)
  42. 1 −1
  43. ρ ρ
  44. S S
  45. 𝐬𝐨 ( n , C ) \mathbf{so}(n,C)
  46. 𝐬𝐨 ( p , q ) \mathbf{so}(p,q)
  47. 𝐠𝐥 ( S ) \mathbf{gl}(S)
  48. S S
  49. S S
  50. S p i n ( p , q ) Spin(p,q)
  51. S p i n ( p , q ) Spin(p,q)
  52. 𝐬𝐨 ( p , q ) \mathbf{so}(p,q)
  53. 𝐬𝐨 ( n , 𝐂 ) \mathbf{so}(n,\mathbf{C})
  54. S p i n ( n , 𝐂 ) Spin(n,\mathbf{C})
  55. 𝐬𝐨 ( n , 𝐂 ) \mathbf{so}(n,\mathbf{C})
  56. 𝐬𝐨 ( p , q ) \mathbf{so}(p,q)
  57. S p i n ( p , q ) Spin(p,q)
  58. Q Q
  59. 𝔰 𝔬 ( V , Q ) = 𝔰 𝔬 ( n , ) . \mathfrak{so}(V,Q)=\mathfrak{so}(n,\mathbb{C}).
  60. V V
  61. Q Q
  62. . , . \langle.,.\rangle
  63. 𝐬𝐨 ( n , 𝐂 ) \mathbf{so}(n,\mathbf{C})
  64. V V
  65. n = 2 m n=2m
  66. n = 2 m + 1 n=2m+1
  67. W W
  68. m m
  69. n = 2 m n=2m
  70. n = 2 m + 1 n=2m+1
  71. U U
  72. . , . \langle.,.\rangle
  73. W W
  74. W W
  75. Q Q
  76. W W
  77. W W
  78. α i , a j = δ i j . \langle\alpha_{i},a_{j}\rangle=\delta_{ij}.
  79. A A
  80. m × m m×m
  81. A A
  82. W W
  83. A w , w * = w , A T w * \langle Aw,w^{*}\rangle=\langle w,A^{\mathrm{T}}w^{*}\rangle
  84. w w
  85. W W
  86. V V
  87. A A
  88. W W
  89. U U
  90. n n
  91. ρ A u , v = - u , ρ A v \langle\rho_{A}u,v\rangle=-\langle u,\rho_{A}v\rangle
  92. u , v u,v
  93. V V
  94. 𝐬𝐨 ( n , 𝐂 ) E n d ( V ) \mathbf{so}(n,\mathbf{C})⊂End(V)
  95. 𝐡 \mathbf{h}
  96. 𝐬𝐨 ( n , 𝐂 ) \mathbf{so}(n,\mathbf{C})
  97. 𝐬𝐨 ( n , 𝐂 ) \mathbf{so}(n,\mathbf{C})
  98. m m
  99. n × n n×n
  100. m m
  101. k k
  102. A A
  103. 𝐬𝐨 ( n , 𝐂 ) \mathbf{so}(n,\mathbf{C})
  104. 2 V \wedge^{2}V
  105. x y φ x y , φ x y ( v ) = 2 ( y , v x - x , v y ) , x y 2 V , x , y , v V , φ x y 𝔰 𝔬 ( n , ) , x\wedge y\mapsto\varphi_{x\wedge y},\quad\varphi_{x\wedge y}(v)=2(\langle y,v% \rangle x-\langle x,v\rangle y),\quad x\wedge y\in\wedge^{2}V,\quad x,y,v\in V% ,\quad\varphi_{x\wedge y}\in\mathfrak{so}(n,\mathbb{C}),
  106. 𝐡 \mathbf{h}
  107. 𝐡 \mathbf{h}
  108. a i a j , i j , a_{i}\wedge a_{j},\;i\neq j,
  109. ε i + ε j \varepsilon_{i}+\varepsilon_{j}
  110. a i α j a_{i}\wedge\alpha_{j}
  111. 𝐡 \mathbf{h}
  112. i = j ) i=j)
  113. ε i - ε j \varepsilon_{i}-\varepsilon_{j}
  114. α i α j , i j , \alpha_{i}\wedge\alpha_{j},\;i\neq j,
  115. - ε i - ε j , -\varepsilon_{i}-\varepsilon_{j},
  116. n n
  117. u u
  118. U U
  119. a i u , a_{i}\wedge u,
  120. ε i \varepsilon_{i}
  121. α i u , \alpha_{i}\wedge u,
  122. - ε i . -\varepsilon_{i}.
  123. ( ± 1 , ± 1 , 0 , 0 , , 0 ) (\pm 1,\pm 1,0,0,\dots,0)
  124. ( ± 1 , 0 , 0 , , 0 ) (\pm 1,0,0,\dots,0)
  125. n = 2 m + 1 n=2m+1
  126. W W
  127. V V
  128. E n d ( S ) End(S)
  129. S S′
  130. S S
  131. S S′
  132. 𝐬𝐨 ( n , 𝐂 ) \mathbf{so}(n,\mathbf{C})
  133. S p i n ( n ) S O ( n ) Spin(n)→SO(n)
  134. v w 1 4 [ v , w ] . v\wedge w\mapsto\tfrac{1}{4}[v,w].
  135. S S
  136. S S′
  137. 𝐬𝐨 ( n , 𝐂 ) \mathbf{so}(n,\mathbf{C})
  138. 𝐡 \mathbf{h}
  139. S S
  140. ( α i a i ) ψ = 1 4 ( 2 1 2 ) 2 ( ι ( α i ) ( a i ψ ) - a i ( ι ( α i ) ψ ) ) = 1 2 ψ - a i ( ι ( α i ) ψ ) . (\alpha_{i}\wedge a_{i})\cdot\psi=\tfrac{1}{4}(2^{\tfrac{1}{2}})^{2}(\iota(% \alpha_{i})(a_{i}\wedge\psi)-a_{i}\wedge(\iota(\alpha_{i})\psi))=\tfrac{1}{2}% \psi-a_{i}\wedge(\iota(\alpha_{i})\psi).
  141. S S
  142. a i 1 a i 2 a i k a_{i_{1}}\wedge a_{i_{2}}\wedge\cdots\wedge a_{i_{k}}
  143. 0 k m 0≤k≤m
  144. 𝐡 \mathbf{h}
  145. j j
  146. 1 / 2 1/2
  147. S S
  148. ( ± 1 2 , ± 1 2 , ± 1 2 ) \bigl(\pm\tfrac{1}{2},\pm\tfrac{1}{2},\ldots\pm\tfrac{1}{2}\bigr)
  149. S S
  150. n n
  151. S S
  152. S + = even W S_{+}=\wedge^{\mathrm{even}}W
  153. S - = odd W S_{-}=\wedge^{\mathrm{odd}}W
  154. ( 1 2 , 1 2 , 1 2 , 1 2 ) \bigl(\tfrac{1}{2},\tfrac{1}{2},\ldots\tfrac{1}{2},\tfrac{1}{2}\bigr)
  155. ( 1 2 , 1 2 , 1 2 , - 1 2 ) \bigl(\tfrac{1}{2},\tfrac{1}{2},\ldots\tfrac{1}{2},-\tfrac{1}{2}\bigr)
  156. u ψ = { ψ if ψ even W - ψ if ψ odd W u\cdot\psi=\left\{\begin{matrix}\psi&\hbox{if }\psi\in\wedge^{\mathrm{even}}W% \\ -\psi&\hbox{if }\psi\in\wedge^{\mathrm{odd}}W\end{matrix}\right.
  157. ( 1 2 , 1 2 , 1 2 ) . \bigl(\tfrac{1}{2},\tfrac{1}{2},\ldots\tfrac{1}{2}\bigr).
  158. β ( φ , ψ ) = B ( φ ) ( ψ ) . \beta(\varphi,\psi)=B(\varphi)(\psi).
  159. β ( ξ φ , ψ ) + β ( φ , ξ ψ ) = 0 \beta(\xi\cdot\varphi,\psi)+\beta(\varphi,\xi\cdot\psi)=0
  160. β ( A φ , ψ ) = β ( φ , τ ( A ) ψ ) ( 1 ) \quad\beta(A\cdot\varphi,\psi)=\beta(\varphi,\tau(A)\cdot\psi)\qquad(1)
  161. β ( A φ , ψ ) = ε ε k β ( A ψ , φ ) \beta(A\cdot\varphi,\psi)=\varepsilon\varepsilon_{k}\beta(A\cdot\psi,\varphi)
  162. S S j = 0 m 2 j V * S\otimes S\cong\bigoplus_{j=0}^{m}\wedge^{2j}V^{*}
  163. j = 0 m ( - 1 ) j dim 2 j \C 2 m + 1 = ( - 1 ) 1 2 m ( m + 1 ) 2 m = ( - 1 ) 1 2 m ( m + 1 ) ( dim S 2 S - dim 2 S ) \sum_{j=0}^{m}(-1)^{j}\dim\wedge^{2j}\C^{2m+1}=(-1)^{\frac{1}{2}m(m+1)}2^{m}=(% -1)^{\frac{1}{2}m(m+1)}(\dim\mathrm{S}^{2}S-\dim\wedge^{2}S)
  164. β ( ϕ , ψ ) = ( - 1 ) 1 2 m ( m + 1 ) β ( ψ , ϕ ) , \beta(\phi,\psi)=(-1)^{\frac{1}{2}m(m+1)}\beta(\psi,\phi),
  165. β ( v ϕ , ψ ) = ( - 1 ) m ( - 1 ) 1 2 m ( m + 1 ) β ( v ψ , ϕ ) = ( - 1 ) m β ( ϕ , v ψ ) \beta(v\cdot\phi,\psi)=(-1)^{m}(-1)^{\frac{1}{2}m(m+1)}\beta(v\cdot\psi,\phi)=% (-1)^{m}\beta(\phi,v\cdot\psi)
  166. 𝔰 𝔬 ( S + ) 𝔰 𝔬 ( S - ) \mathfrak{so}(S_{+})\oplus\mathfrak{so}(S_{-})
  167. 𝔰 𝔬 ( S ) \mathfrak{so}(S)
  168. 𝔤 𝔩 ( S ± ) \mathfrak{gl}(S_{\pm})
  169. 𝔰 𝔭 ( S ) \mathfrak{sp}(S)
  170. 𝔰 𝔭 ( S + ) 𝔰 𝔭 ( S - ) \mathfrak{sp}(S_{+})\oplus\mathfrak{sp}(S_{-})
  171. 𝔰 𝔭 ( S ) \mathfrak{sp}(S)
  172. 𝔤 𝔩 ( S ± ) \mathfrak{gl}(S_{\pm})
  173. 𝔰 𝔬 ( S ) \mathfrak{so}(S)
  174. 𝔰 𝔬 ( 2 , ) 𝔤 𝔩 ( 1 , ) ( = ) \mathfrak{so}(2,\mathbb{C})\cong\mathfrak{gl}(1,\mathbb{C})\qquad(=\mathbb{C})
  175. 𝔰 𝔬 ( 3 , ) 𝔰 𝔭 ( 2 , ) ( = 𝔰 𝔩 ( 2 , ) ) \mathfrak{so}(3,\mathbb{C})\cong\mathfrak{sp}(2,\mathbb{C})\qquad(=\mathfrak{% sl}(2,\mathbb{C}))
  176. 𝔰 𝔬 ( 4 , ) 𝔰 𝔭 ( 2 , ) 𝔰 𝔭 ( 2 , ) \mathfrak{so}(4,\mathbb{C})\cong\mathfrak{sp}(2,\mathbb{C})\oplus\mathfrak{sp}% (2,\mathbb{C})
  177. 𝔰 𝔬 ( 5 , ) 𝔰 𝔭 ( 4 , ) \mathfrak{so}(5,\mathbb{C})\cong\mathfrak{sp}(4,\mathbb{C})
  178. 𝔰 𝔬 ( 6 , ) 𝔰 𝔩 ( 4 , ) . \mathfrak{so}(6,\mathbb{C})\cong\mathfrak{sl}(4,\mathbb{C}).
  179. N N
  180. n > 3 n>3
  181. n = 3 n=3
  182. 𝐬𝐩 ( 1 ) \mathbf{sp}(1)
  183. n > 2 n>2
  184. 𝐬𝐥 ( 2 N , 𝐂 ) \mathbf{sl}(2N,\mathbf{C})
  185. 𝐬𝐥 ( 2 N , 𝐑 ) \mathbf{sl}(2N,\mathbf{R})
  186. 𝐬𝐮 ( K , L ) \mathbf{su}(K,L)
  187. K + L = 2 N K+L=2N
  188. 𝐬𝐥 ( N , 𝐇 ) \mathbf{sl}(N,\mathbf{H})
  189. p q = 0 pq=0
  190. 𝐬𝐨 ( 2 N ) + 𝐬𝐨 ( 2 N ) \mathbf{so}(2N)+\mathbf{so}(2N)
  191. N N
  192. n > 4 n>4
  193. p q = 0 pq=0
  194. n = 4 n=4
  195. N = 1 N=1
  196. 𝐬𝐩 ( N ) + 𝐬𝐩 ( N ) \mathbf{sp}(N)+\mathbf{sp}(N)
  197. 𝔰 𝔬 ( 2 ) 𝔲 ( 1 ) \mathfrak{so}(2)\cong\mathfrak{u}(1)
  198. 𝔰 𝔬 ( 1 , 1 ) \mathfrak{so}(1,1)\cong\mathbb{R}
  199. 𝔰 𝔬 ( 3 ) 𝔰 𝔭 ( 1 ) \mathfrak{so}(3)\cong\mathfrak{sp}(1)
  200. 𝔰 𝔬 ( 2 , 1 ) 𝔰 𝔩 ( 2 , ) \mathfrak{so}(2,1)\cong\mathfrak{sl}(2,\mathbb{R})
  201. 𝔰 𝔬 ( 4 ) 𝔰 𝔭 ( 1 ) 𝔰 𝔭 ( 1 ) \mathfrak{so}(4)\cong\mathfrak{sp}(1)\oplus\mathfrak{sp}(1)
  202. 𝔰 𝔬 ( 3 , 1 ) 𝔰 𝔩 ( 2 , ) \mathfrak{so}(3,1)\cong\mathfrak{sl}(2,\mathbb{C})
  203. 𝔰 𝔬 ( 2 , 2 ) 𝔰 𝔩 ( 2 , ) 𝔰 𝔩 ( 2 , ) \mathfrak{so}(2,2)\cong\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{sl}(2,% \mathbb{R})
  204. 𝔰 𝔬 ( 5 ) 𝔰 𝔭 ( 2 ) \mathfrak{so}(5)\cong\mathfrak{sp}(2)
  205. 𝔰 𝔬 ( 4 , 1 ) 𝔰 𝔭 ( 1 , 1 ) \mathfrak{so}(4,1)\cong\mathfrak{sp}(1,1)
  206. 𝔰 𝔬 ( 3 , 2 ) 𝔰 𝔭 ( 4 , ) \mathfrak{so}(3,2)\cong\mathfrak{sp}(4,\mathbb{R})
  207. 𝔰 𝔬 ( 6 ) 𝔰 𝔲 ( 4 ) \mathfrak{so}(6)\cong\mathfrak{su}(4)
  208. 𝔰 𝔬 ( 5 , 1 ) 𝔰 𝔩 ( 2 , ) \mathfrak{so}(5,1)\cong\mathfrak{sl}(2,\mathbb{H})
  209. 𝔰 𝔬 ( 4 , 2 ) 𝔰 𝔲 ( 2 , 2 ) \mathfrak{so}(4,2)\cong\mathfrak{su}(2,2)
  210. 𝔰 𝔬 ( 3 , 3 ) 𝔰 𝔩 ( 4 , ) \mathfrak{so}(3,3)\cong\mathfrak{sl}(4,\mathbb{R})
  211. 𝔰 𝔬 * ( 3 , ) 𝔰 𝔲 ( 3 , 1 ) \mathfrak{so}^{*}(3,\mathbb{H})\cong\mathfrak{su}(3,1)
  212. 𝔰 𝔬 * ( 4 , ) 𝔰 𝔬 ( 6 , 2 ) . \mathfrak{so}^{*}(4,\mathbb{H})\cong\mathfrak{so}(6,2).

Spinors_in_three_dimensions.html

  1. x X = ( x 3 x 1 - i x 2 x 1 + i x 2 - x 3 ) . {x}\rightarrow X=\left(\begin{matrix}x_{3}&x_{1}-ix_{2}\\ x_{1}+ix_{2}&-x_{3}\end{matrix}\right).
  2. 1 2 ( X Y + Y X ) = ( x y ) I \frac{1}{2}(XY+YX)=({x}\cdot{y})I
  3. 1 2 ( X Y - Y X ) = i Z \frac{1}{2}(XY-YX)=iZ
  4. ξ = [ ξ 1 ξ 2 ] , \xi=\left[\begin{matrix}\xi_{1}\\ \xi_{2}\end{matrix}\right],
  5. x x = x 1 2 + x 2 2 + x 3 2 = 0. {x}\cdot{x}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=0.
  6. X = 2 [ ξ 1 ξ 2 ] [ - ξ 2 ξ 1 ] . X=2\left[\begin{matrix}\xi_{1}\\ \xi_{2}\end{matrix}\right]\left[\begin{matrix}-\xi_{2}&\xi_{1}\end{matrix}% \right].
  7. ξ 1 2 - ξ 2 2 = x 1 i ( ξ 1 2 + ξ 2 2 ) = x 2 - 2 ξ 1 ξ 2 = x 3 } \left.\begin{matrix}\xi_{1}^{2}-\xi_{2}^{2}&=x_{1}\\ i(\xi_{1}^{2}+\xi_{2}^{2})&=x_{2}\\ -2\xi_{1}\xi_{2}&=x_{3}\end{matrix}\right\}
  8. ξ 1 = ± x 1 - i x 2 2 , ξ 2 = ± - x 1 - i x 2 2 . \xi_{1}=\pm\sqrt{\frac{x_{1}-ix_{2}}{2}},\quad\xi_{2}=\pm\sqrt{\frac{-x_{1}-ix% _{2}}{2}}.
  9. C = ( 0 1 - 1 0 ) , C=\left(\begin{matrix}0&1\\ -1&0\end{matrix}\right),
  10. μ | ξ = μ ¯ 1 ξ 1 + μ ¯ 2 ξ 2 \langle\mu|\xi\rangle=\bar{\mu}_{1}\xi_{1}+\bar{\mu}_{2}\xi_{2}
  11. | 𝐱 | 2 = x 1 2 - x 2 2 + x 3 2 \left|\mathbf{x}\right|^{2}=x_{1}^{2}-x^{2}_{2}+x_{3}^{2}
  12. ( x 3 x 1 - x 2 x 1 + x 2 - x 3 ) \left(\begin{matrix}x_{3}&x_{1}-x_{2}\\ x_{1}+x_{2}&-x_{3}\end{matrix}\right)
  13. μ | ξ = μ ¯ 1 ξ 2 - μ ¯ 2 ξ 1 \langle\mu|\xi\rangle=\bar{\mu}_{1}\xi_{2}-\bar{\mu}_{2}\xi_{1}
  14. μ | ξ 2 = length ( Q ( μ ¯ , ξ ) ) 2 \langle\mu|\xi\rangle^{2}=\hbox{length}\left(Q(\bar{\mu},\xi)\right)^{2}
  15. ξ * = K ξ ¯ . \xi^{*}=K\bar{\xi}.
  16. X ¯ = K X K \bar{X}=KXK\,
  17. X ¯ = K X K \bar{X}=KXK\,
  18. μ | ξ = i t μ * C ξ \langle\mu|\xi\rangle=i\,^{t}\mu^{*}C\xi
  19. S u = ( 0.8 , - 0.6 , 0.0 ) σ = [ 0.0 0.8 + 0.6 i 0.8 - 0.6 i 0.0 ] S_{u}=(0.8,-0.6,0.0)\cdot\vec{\sigma}=\begin{bmatrix}0.0&0.8+0.6i\\ 0.8-0.6i&0.0\end{bmatrix}
  20. S u ( 1 ± S u ) = ± 1 ( 1 ± S u ) S_{u}(1\pm S_{u})=\pm 1(1\pm S_{u})
  21. [ 1.0 + ( 0.0 ) 0.0 + ( 0.8 - 0.6 i ) ] , [ 1.0 - ( 0.0 ) 0.0 - ( 0.8 - 0.6 i ) ] \begin{bmatrix}1.0+(0.0)\\ 0.0+(0.8-0.6i)\end{bmatrix},\begin{bmatrix}1.0-(0.0)\\ 0.0-(0.8-0.6i)\end{bmatrix}
  22. 1 2 [ 1 + c a - i b a + i b 1 - c ] \frac{1}{2}\begin{bmatrix}1+c&a-ib\\ a+ib&1-c\end{bmatrix}
  23. σ = - S , - ( S - 1 ) , , + ( S - 1 ) , + S \sigma=-S\cdot\hbar,-(S-1)\cdot\hbar,...,+(S-1)\cdot\hbar,+S\cdot\hbar
  24. 𝕁 \vec{\mathbb{J}}

Spin–lattice_relaxation.html

  1. M z ( t ) = M z , eq - [ M z , eq - M z ( 0 ) ] e - t / T 1 M_{z}(t)=M_{z,\mathrm{eq}}-\left[M_{z,\mathrm{eq}}-M_{z}(0)\right]e^{-t/T_{1}}
  2. M z ( 0 ) = 0 M_{z}(0)=0
  3. M z ( t ) = M z , eq ( 1 - e - t / T 1 ) M_{z}(t)=M_{z,\mathrm{eq}}\left(1-e^{-t/T_{1}}\right)

Spin–spin_relaxation.html

  1. T 1 T_{1}
  2. T 2 T_{2}
  3. T T
  4. T T
  5. M x y ( t ) = M x y ( 0 ) e - t / T 2 M_{xy}(t)=M_{xy}(0)e^{-t/T_{2}}\,
  6. T T
  7. T T
  8. T T
  9. T T
  10. T T
  11. ω 1 \omega_{1}
  12. ( 1 / τ c ) (1/\tau_{c})
  13. ω 0 \omega_{0}
  14. τ c \tau_{c}

Split_graph.html

  1. i = 1 m d i = m ( m - 1 ) + i = m + 1 n d i . \sum_{i=1}^{m}d_{i}=m(m-1)+\sum_{i=m+1}^{n}d_{i}.

Spray_nozzle.html

  1. Q f = Q w a t e r 1 S g {Q_{f}}={Q_{water}}\sqrt{\frac{1}{Sg}}

Springer_correspondence.html

  1. ( u , ϕ ) E u , ϕ u U ( G ) , ϕ A ( u ) ^ , E u , ϕ W ^ (u,\phi)\mapsto E_{u,\phi}\quad u\in U(G),\phi\in\widehat{A(u)},E_{u,\phi}\in% \widehat{W}

Square_root_of_3.html

  1. 1 + 1 1 + 1 2 + 1 1 + 1 2 + 1 1 + 1+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1+\ddots}}}}}
  2. 3 . \sqrt{3}.
  3. 97 56 \tfrac{97}{56}
  4. [ 2 ; - 4 , - 4 , - 4 , ] = 2 - 1 4 - 1 4 - 1 4 - [2;-4,-4,-4,...]=2-\cfrac{1}{4-\cfrac{1}{4-\cfrac{1}{4-\ddots}}}
  5. 3 \sqrt{3}
  6. m n \frac{m}{n}
  7. m m
  8. n n
  9. m ( 3 - q ) n ( 3 - q ) \frac{m(\sqrt{3}-q)}{n(\sqrt{3}-q)}
  10. q q
  11. 3 \sqrt{3}
  12. m 3 - m q n 3 - n q \frac{m\sqrt{3}-mq}{n\sqrt{3}-nq}
  13. m m
  14. 3 n \sqrt{3}n
  15. n 3 2 - m q n 3 - n q \frac{n\sqrt{3}^{2}-mq}{n\sqrt{3}-nq}
  16. 3 \sqrt{3}
  17. m n \frac{m}{n}
  18. n 3 2 - m q n m n - n q \frac{n\sqrt{3}^{2}-mq}{n\frac{m}{n}-nq}
  19. 3 \sqrt{3}
  20. m n \frac{m}{n}
  21. n n
  22. m m
  23. 3 n - m q m - n q \frac{3n-mq}{m-nq}
  24. 3 \sqrt{3}
  25. m / n m/n
  26. 3 n - m q m - n q \frac{3n-mq}{m-nq}
  27. m / n m/n
  28. 3 = m n \sqrt{3}=\frac{m}{n}
  29. m n \frac{m}{n}
  30. n n
  31. 3 n 2 = m 2 . 3n^{2}=m^{2}.
  32. m m
  33. m m
  34. 3 k 3k
  35. 3 n 2 = ( 3 k ) 2 3n^{2}=(3k)^{2}
  36. 3 n 2 = 9 k 2 3n^{2}=9k^{2}
  37. n 2 = 3 k 2 n^{2}=3k^{2}
  38. n n
  39. n n
  40. m m
  41. m / n m/n
  42. 3 \sqrt{3}
  43. 3 \sqrt{3}
  44. 3 \sqrt{3}
  45. i i
  46. 3 \sqrt{3}
  47. 3 \sqrt{3}
  48. - 3 = ± 3 i \sqrt{-3}=\pm\sqrt{3}i
  49. 3 \sqrt{3}
  50. 3 \sqrt{3}
  51. sin ( π 3 ) \sin\left(\frac{\pi}{3}\right)

Square_root_of_5.html

  1. 2 + 1 4 + 1 4 + 1 4 + 1 4 + 2+\cfrac{1}{4+\cfrac{1}{4+\cfrac{1}{4+\cfrac{1}{4+\ddots}}}}
  2. 5 . \sqrt{5}.\,
  3. m n \frac{m}{n}
  4. 5 n - 2 m m - 2 n \frac{5n-2m}{m-2n}
  5. 5 n 2 = m 2 5n^{2}=m^{2}
  6. m n = 5 \tfrac{m}{n}=\sqrt{5}
  7. m - 2 n < n m-2n<n
  8. m < 3 n m<3n
  9. m n < 3 \tfrac{m}{n}<3
  10. 5 < 3 \sqrt{5}<3
  11. 2 < 5 < 5 2 2<\sqrt{5}<\tfrac{5}{2}
  12. m n = 5 \tfrac{m}{n}=\sqrt{5}
  13. \color b l u e 2 1 , 7 3 , \color b l u e 9 4 , 20 9 , 29 13 , \color b l u e 38 17 , 123 55 , \color b l u e 161 72 , 360 161 , 521 233 , \color b l u e 682 305 , 2207 987 , \color b l u e 2889 1292 , {\color{blue}{\frac{2}{1}}},\frac{7}{3},{\color{blue}{\frac{9}{4}}},\frac{20}{% 9},\frac{29}{13},{\color{blue}{\frac{38}{17}}},\frac{123}{55},{\color{blue}{% \frac{161}{72}}},\frac{360}{161},\frac{521}{233},{\color{blue}{\frac{682}{305}% }},\frac{2207}{987},{\color{blue}{\frac{2889}{1292}}},\dots
  14. 5 \sqrt{5}
  15. 2 1 = 2.0 , 9 4 = 2.25 , 161 72 = 2.23611 , 51841 23184 = 2.2360679779 \frac{2}{1}=2.0,\quad\frac{9}{4}=2.25,\quad\frac{161}{72}=2.23611\dots,\quad% \frac{51841}{23184}=2.2360679779\ldots
  16. φ \varphi
  17. Φ = 1 / φ = φ - 1 \Phi=1/\varphi=\varphi-1
  18. 5 = φ + Φ = 2 φ - 1 = 2 Φ + 1 \sqrt{5}=\varphi+\Phi=2\varphi-1=2\Phi+1
  19. φ = 5 + 1 2 \varphi=\frac{\sqrt{5}+1}{2}
  20. Φ = 5 - 1 2 . \Phi=\frac{\sqrt{5}-1}{2}.
  21. F ( n ) = φ n - ( 1 - φ ) n 5 . F\left(n\right)={{\varphi^{n}-(1-\varphi)^{n}}\over{\sqrt{5}}}.
  22. φ \varphi
  23. Φ \Phi
  24. 5 φ = Φ 5 = 5 - 5 2 = 1.3819660112501051518 = [ 1 ; 2 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , ] \frac{\sqrt{5}}{\varphi}=\Phi\cdot\sqrt{5}=\frac{5-\sqrt{5}}{2}=1.381966011250% 1051518\dots=[1;2,1,1,1,1,1,1,1,\dots]
  25. φ 5 = 1 Φ 5 = 5 + 5 10 = 0.72360679774997896964 = [ 0 ; 1 , 2 , 1 , 1 , 1 , 1 , 1 , 1 , ] . \frac{\varphi}{\sqrt{5}}=\frac{1}{\Phi\cdot\sqrt{5}}=\frac{5+\sqrt{5}}{10}=0.7% 2360679774997896964\dots=[0;1,2,1,1,1,1,1,1,\dots].
  26. 1 , 3 2 , 4 3 , 7 5 , 11 8 , 18 13 , 29 21 , 47 34 , 76 55 , 123 89 , [ 1 ; 2 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , ] {1,\frac{3}{2},\frac{4}{3},\frac{7}{5},\frac{11}{8},\frac{18}{13},\frac{29}{21% },\frac{47}{34},\frac{76}{55},\frac{123}{89}},\dots\dots[1;2,1,1,1,1,1,1,1,\dots]
  27. 1 , 2 3 , 3 4 , 5 7 , 8 11 , 13 18 , 21 29 , 34 47 , 55 76 , 89 123 , [ 0 ; 1 , 2 , 1 , 1 , 1 , 1 , 1 , 1 , ] . {1,\frac{2}{3},\frac{3}{4},\frac{5}{7},\frac{8}{11},\frac{13}{18},\frac{21}{29% },\frac{34}{47},\frac{55}{76},\frac{89}{123}},\dots\dots[0;1,2,1,1,1,1,1,1,% \dots].
  28. sin π 10 = sin 18 = 1 4 ( 5 - 1 ) = 1 5 + 1 , \sin\frac{\pi}{10}=\sin 18^{\circ}=\tfrac{1}{4}(\sqrt{5}-1)=\frac{1}{\sqrt{5}+% 1},\,
  29. sin π 5 = sin 36 = 1 4 2 ( 5 - 5 ) , \sin\frac{\pi}{5}=\sin 36^{\circ}=\tfrac{1}{4}\sqrt{2(5-\sqrt{5})},\,
  30. sin 3 π 10 = sin 54 = 1 4 ( 5 + 1 ) = 1 5 - 1 , \sin\frac{3\pi}{10}=\sin 54^{\circ}=\tfrac{1}{4}(\sqrt{5}+1)=\frac{1}{\sqrt{5}% -1},\,
  31. sin 2 π 5 = sin 72 = 1 4 2 ( 5 + 5 ) . \sin\frac{2\pi}{5}=\sin 72^{\circ}=\tfrac{1}{4}\sqrt{2(5+\sqrt{5})}\,.
  32. | x - m n | < 1 5 n 2 \left|x-\frac{m}{n}\right|<\frac{1}{\sqrt{5}\,n^{2}}
  33. | α - p i q i | < 1 5 q i 2 , | α - p i + 1 q i + 1 | < 1 5 q i + 1 2 , | α - p i + 2 q i + 2 | < 1 5 q i + 2 2 . \left|\alpha-{p_{i}\over q_{i}}\right|<{1\over\sqrt{5}q_{i}^{2}},\qquad\left|% \alpha-{p_{i+1}\over q_{i+1}}\right|<{1\over\sqrt{5}q_{i+1}^{2}},\qquad\left|% \alpha-{p_{i+2}\over q_{i+2}}\right|<{1\over\sqrt{5}q_{i+2}^{2}}.
  34. [ - 5 ] \scriptstyle\mathbb{Z}\left[\,\sqrt{-5}\,\right]
  35. a + b - 5 \scriptstyle a\,+\,b\sqrt{-5}
  36. - 5 \scriptstyle\sqrt{-5}
  37. i 5 \scriptstyle i\sqrt{5}
  38. 6 = 2 3 = ( 1 - - 5 ) ( 1 + - 5 ) . 6=2\cdot 3=(1-\sqrt{-5})(1+\sqrt{-5}).\,
  39. [ 5 ] \scriptstyle\mathbb{Q}\left[\,\sqrt{5}\,\right]
  40. 5 = e 2 π i / 5 - e 4 π i / 5 - e 6 π i / 5 + e 8 π i / 5 . \sqrt{5}=e^{2\pi i/5}-e^{4\pi i/5}-e^{6\pi i/5}+e^{8\pi i/5}.\,
  41. 1 1 + e - 2 π 1 + e - 4 π 1 + e - 6 π 1 + = ( 5 + 5 2 - 5 + 1 2 ) e 2 π / 5 = e 2 π / 5 ( φ 5 - φ ) . \cfrac{1}{1+\cfrac{e^{-2\pi}}{1+\cfrac{e^{-4\pi}}{1+\cfrac{e^{-6\pi}}{1+\ddots% }}}}=\left(\sqrt{\frac{5+\sqrt{5}}{2}}-\frac{\sqrt{5}+1}{2}\right)e^{2\pi/5}=e% ^{2\pi/5}\left(\sqrt{\varphi\sqrt{5}}-\varphi\right).
  42. 1 1 + e - 2 π 5 1 + e - 4 π 5 1 + e - 6 π 5 1 + = ( 5 1 + [ 5 3 / 4 ( φ - 1 ) 5 / 2 - 1 ] 1 / 5 - φ ) e 2 π / 5 . \cfrac{1}{1+\cfrac{e^{-2\pi\sqrt{5}}}{1+\cfrac{e^{-4\pi\sqrt{5}}}{1+\cfrac{e^{% -6\pi\sqrt{5}}}{1+\ddots}}}}=\left({\sqrt{5}\over 1+\left[5^{3/4}(\varphi-1)^{% 5/2}-1\right]^{1/5}}-\varphi\right)e^{2\pi/\sqrt{5}}.
  43. 4 0 x e - x 5 cosh x d x = 1 1 + 1 2 1 + 1 2 1 + 2 2 1 + 2 2 1 + 3 2 1 + 3 2 1 + . 4\int_{0}^{\infty}\frac{xe^{-x\sqrt{5}}}{\cosh x}\,dx=\cfrac{1}{1+\cfrac{1^{2}% }{1+\cfrac{1^{2}}{1+\cfrac{2^{2}}{1+\cfrac{2^{2}}{1+\cfrac{3^{2}}{1+\cfrac{3^{% 2}}{1+\ddots}}}}}}}.

Squared_deviations.html

  1. E ( X 2 ) . \operatorname{E}(X^{2}).
  2. X X
  3. μ \mu
  4. σ 2 \sigma^{2}
  5. σ 2 = E ( X 2 ) - μ 2 \sigma^{2}=\operatorname{E}(X^{2})-\mu^{2}
  6. E ( X 2 ) = σ 2 + μ 2 . \operatorname{E}(X^{2})=\sigma^{2}+\mu^{2}.
  7. E ( ( X 2 ) ) = n σ 2 + n μ 2 \operatorname{E}\left(\sum\left(X^{2}\right)\right)=n\sigma^{2}+n\mu^{2}
  8. E ( ( X ) 2 ) = n σ 2 + n 2 μ 2 \operatorname{E}\left(\left(\sum X\right)^{2}\right)=n\sigma^{2}+n^{2}\mu^{2}
  9. Y ^ \hat{Y}
  10. Y Y
  11. S S E = 1 2 i = 1 n ( Y i ^ - Y i ) 2 SSE=\frac{1}{2}\sum_{i=1}^{n}(\hat{Y_{i}}-Y_{i})^{2}
  12. S = x 2 - ( x ) 2 n S=\sum x^{2}-\frac{\left(\sum x\right)^{2}}{n}
  13. E ( S ) = n σ 2 + n μ 2 - n σ 2 + n 2 μ 2 n \operatorname{E}(S)=n\sigma^{2}+n\mu^{2}-\frac{n\sigma^{2}+n^{2}\mu^{2}}{n}
  14. E ( S ) = ( n - 1 ) σ 2 . \operatorname{E}(S)=(n-1)\sigma^{2}.
  15. E ( μ i ) = μ + T i \operatorname{E}(\mu_{i})=\mu+T_{i}
  16. σ 2 \sigma^{2}
  17. T i T_{i}
  18. I = x 2 I=\sum x^{2}
  19. E ( I ) = n σ 2 + n μ 2 \operatorname{E}(I)=n\sigma^{2}+n\mu^{2}
  20. T = i = 1 k ( ( x ) 2 / n i ) T=\sum_{i=1}^{k}\left(\left(\sum x\right)^{2}/n_{i}\right)
  21. E ( T ) = k σ 2 + i = 1 k n i ( μ + T i ) 2 \operatorname{E}(T)=k\sigma^{2}+\sum_{i=1}^{k}n_{i}(\mu+T_{i})^{2}
  22. E ( T ) = k σ 2 + n μ 2 + 2 μ i = 1 k ( n i T i ) + i = 1 k n i ( T i ) 2 \operatorname{E}(T)=k\sigma^{2}+n\mu^{2}+2\mu\sum_{i=1}^{k}(n_{i}T_{i})+\sum_{% i=1}^{k}n_{i}(T_{i})^{2}
  23. T i T_{i}
  24. E ( T ) = k σ 2 + n μ 2 . \operatorname{E}(T)=k\sigma^{2}+n\mu^{2}.
  25. C = ( x ) 2 / n C=\left(\sum x\right)^{2}/n
  26. E ( C ) = σ 2 + n μ 2 \operatorname{E}(C)=\sigma^{2}+n\mu^{2}
  27. μ \mu
  28. σ 2 \sigma^{2}
  29. E ( I - C ) = ( n - 1 ) σ 2 \operatorname{E}(I-C)=(n-1)\sigma^{2}
  30. E ( T - C ) = ( k - 1 ) σ 2 \operatorname{E}(T-C)=(k-1)\sigma^{2}
  31. E ( I - T ) = ( n - k ) σ 2 \operatorname{E}(I-T)=(n-k)\sigma^{2}
  32. I = 1 2 1 + 2 2 1 + 3 2 1 + 4 2 1 + 6 2 1 = 66 I=\frac{1^{2}}{1}+\frac{2^{2}}{1}+\frac{3^{2}}{1}+\frac{4^{2}}{1}+\frac{6^{2}}% {1}=66
  33. T = ( 1 + 2 + 3 ) 2 3 + ( 4 + 6 ) 2 2 = 12 + 50 = 62 T=\frac{(1+2+3)^{2}}{3}+\frac{(4+6)^{2}}{2}=12+50=62
  34. C = ( 1 + 2 + 3 + 4 + 6 ) 2 5 = 256 / 5 = 51.2 C=\frac{(1+2+3+4+6)^{2}}{5}=256/5=51.2
  35. σ 2 \sigma^{2}
  36. 7 2 + 2 2 + 1 2 + 7 2 + 6 2 + 11 2 + 6 2 + 10 2 + 7 2 + 3 2 + 5 2 + 3 2 + 4 2 + 11 2 + 4 2 7^{2}+2^{2}+1^{2}+7^{2}+6^{2}+11^{2}+6^{2}+10^{2}+7^{2}+3^{2}+5^{2}+3^{2}+4^{2% }+11^{2}+4^{2}
  37. ( 7 + 2 + 1 ) 2 3 + ( 7 + 6 ) 2 2 + ( 11 + 6 ) 2 2 + ( 10 + 7 + 3 ) 2 3 + ( 5 + 3 + 4 ) 2 3 + ( 11 + 4 ) 2 2 \frac{(7+2+1)^{2}}{3}+\frac{(7+6)^{2}}{2}+\frac{(11+6)^{2}}{2}+\frac{(10+7+3)^% {2}}{3}+\frac{(5+3+4)^{2}}{3}+\frac{(11+4)^{2}}{2}
  38. ( 7 + 2 + 1 + 7 + 6 ) 2 5 + ( 11 + 6 + 10 + 7 + 3 ) 2 5 + ( 5 + 3 + 4 + 11 + 4 ) 2 5 \frac{(7+2+1+7+6)^{2}}{5}+\frac{(11+6+10+7+3)^{2}}{5}+\frac{(5+3+4+11+4)^{2}}{5}
  39. ( 7 + 2 + 1 + 11 + 6 + 5 + 3 + 4 ) 2 8 + ( 7 + 6 + 10 + 7 + 3 + 11 + 4 ) 2 7 \frac{(7+2+1+11+6+5+3+4)^{2}}{8}+\frac{(7+6+10+7+3+11+4)^{2}}{7}
  40. ( 7 + 2 + 1 + 11 + 6 + 5 + 3 + 4 + 7 + 6 + 10 + 7 + 3 + 11 + 4 ) 2 15 \frac{(7+2+1+11+6+5+3+4+7+6+10+7+3+11+4)^{2}}{15}
  41. σ 2 \sigma^{2}

Stable_normal_bundle.html

  1. i , i : X 𝐑 m i,i^{\prime}\colon X\hookrightarrow\mathbf{R}^{m}
  2. i : X 𝐑 m , i\colon X\hookrightarrow\mathbf{R}^{m},
  3. j : X 𝐑 n , j\colon X\hookrightarrow\mathbf{R}^{n},
  4. m m
  5. n n
  6. 𝐑 N \mathbf{R}^{N}
  7. N - m N-m
  8. i : X 𝐑 m 𝐑 m × { ( 0 , , 0 ) } 𝐑 m × 𝐑 N - m 𝐑 N . i\colon X\hookrightarrow\mathbf{R}^{m}\cong\mathbf{R}^{m}\times\left\{(0,\dots% ,0)\right\}\subset\mathbf{R}^{m}\times\mathbf{R}^{N-m}\cong\mathbf{R}^{N}.
  9. N = max ( m , n ) N=\max(m,n)
  10. N N
  11. n n
  12. N N
  13. τ M : M B O ( n ) . \tau_{M}\colon M\to BO(n).
  14. B O ( n ) B O BO(n)\to BO
  15. M R n + k M\subset R^{n+k}
  16. k k
  17. ν M : M B O ( k ) \nu_{M}\colon M\to BO(k)
  18. τ M \tau_{M}
  19. τ M ν M : M B O ( n + k ) \tau_{M}\oplus\nu_{M}\colon M\to BO(n+k)
  20. ν M : M B O ( k ) B O \nu_{M}\colon M\to BO(k)\to BO
  21. ν M \nu_{M}
  22. X B G X\to BG
  23. B G BG
  24. π * ( B G ) = π * - 1 S . \pi_{*}(BG)=\pi_{*-1}^{S}.
  25. B O B G BO\to BG
  26. B O B G B ( G / O ) . BO\to BG\to B(G/O).
  27. X B G X\to BG
  28. X B O X\to BO
  29. X B ( G / O ) X\to B(G/O)
  30. X B ( G / O ) X\to B(G/O)
  31. X B ( G / O ) X\to B(G/O)
  32. f : M N f\colon M\to N

Standard_map.html

  1. 2 π 2\pi
  2. p n + 1 = p n + K sin ( θ n ) p_{n+1}=p_{n}+K\sin(\theta_{n})
  3. θ n + 1 = θ n + p n + 1 \theta_{n+1}=\theta_{n}+p_{n+1}
  4. p n p_{n}
  5. θ n \theta_{n}
  6. 2 π 2\pi
  7. θ n \theta_{n}
  8. p n p_{n}
  9. K = 0 K=0
  10. K > 0 K>0
  11. θ n + 1 = θ n + Ω - K sin ( θ n ) \theta_{n+1}=\theta_{n}+\Omega-K\sin(\theta_{n})
  12. θ n + 1 = θ n + p n + K sin ( θ n ) \theta_{n+1}=\theta_{n}+p_{n}+K\sin(\theta_{n})
  13. p n + 1 = θ n + 1 - θ n p_{n+1}=\theta_{n+1}-\theta_{n}

Standard_probability_space.html

  1. ( Ω 1 , 1 , P 1 ) \textstyle(\Omega_{1},\mathcal{F}_{1},P_{1})
  2. ( Ω 2 , 2 , P 2 ) \textstyle(\Omega_{2},\mathcal{F}_{2},P_{2})
  3. f : Ω 1 Ω 2 \textstyle f:\Omega_{1}\to\Omega_{2}
  4. f \textstyle f
  5. f - 1 \textstyle f^{-1}
  6. ( Ω 1 , 1 , P 1 ) \textstyle(\Omega_{1},\mathcal{F}_{1},P_{1})
  7. ( Ω 2 , 2 , P 2 ) \textstyle(\Omega_{2},\mathcal{F}_{2},P_{2})
  8. mod 0 \textstyle\operatorname{mod}\,0
  9. A 1 Ω 1 \textstyle A_{1}\subset\Omega_{1}
  10. A 2 Ω 2 \textstyle A_{2}\subset\Omega_{2}
  11. Ω 1 A 1 \textstyle\Omega_{1}\setminus A_{1}
  12. Ω 2 A 2 \textstyle\Omega_{2}\setminus A_{2}
  13. mod 0 \textstyle\operatorname{mod}\,0
  14. f : \textstyle f:\mathbb{R}\to\mathbb{R}
  15. \textstyle\mathbb{R}^{\mathbb{R}}
  16. \textstyle\mathbb{R}
  17. \textstyle\mathbb{R}
  18. γ = N ( 0 , 1 ) \textstyle\gamma=N(0,1)
  19. ( , γ ) \textstyle(\mathbb{R},\gamma)^{\mathbb{R}}
  20. ( , γ ) \textstyle(\mathbb{R},\gamma)
  21. γ \textstyle\gamma^{\mathbb{R}}
  22. \textstyle\mathbb{R}^{\mathbb{R}}
  23. γ \textstyle\gamma^{\mathbb{R}}
  24. f ( , γ ) \textstyle f\in\textstyle(\mathbb{R},\gamma)^{\mathbb{R}}
  25. Z ( 0 , 1 ) \textstyle Z\subset(0,1)
  26. Z \textstyle Z
  27. m \textstyle m
  28. Z \textstyle Z
  29. m ( Z A ) = mes ( A ) \textstyle m(Z\cap A)=\operatorname{mes}(A)
  30. A ( 0 , 1 ) \textstyle A\subset(0,1)
  31. mes \textstyle\operatorname{mes}
  32. ( Z , m ) \textstyle(Z,m)
  33. mod 0 \textstyle\operatorname{mod}\,0
  34. ( ( 0 , 1 ) , mes ) \textstyle((0,1),\operatorname{mes})
  35. ( Z , m ) \textstyle(Z,m)
  36. ( ( 0 , 1 ) , mes ) \textstyle((0,1),\operatorname{mes})
  37. X \textstyle X
  38. X ( ω ) = ω \textstyle X(\omega)=\omega
  39. ( 0 , 1 ) \textstyle(0,1)
  40. X = x \textstyle X=x
  41. x \textstyle x
  42. ( ( 0 , 1 ) , mes ) \textstyle((0,1),\operatorname{mes})
  43. ( Z , m ) \textstyle(Z,m)
  44. x Z \textstyle x\notin Z
  45. Z ( 0 , 1 ) \textstyle Z\subset(0,1)
  46. ( A Z ) ( B Z ) , \textstyle(A\cap Z)\cup(B\setminus Z),
  47. A \textstyle A
  48. B \textstyle B
  49. ; \textstyle\mathcal{F};
  50. Z . \textstyle Z.
  51. m ( ( A Z ) ( B Z ) ) = p mes ( A ) + ( 1 - p ) mes ( B ) \displaystyle m\big((A\cap Z)\cup(B\setminus Z)\big)=p\,\operatorname{mes}(A)+% (1-p)\operatorname{mes}(B)
  52. m \textstyle m
  53. ( ( 0 , 1 ) , ) \textstyle\big((0,1),\mathcal{F}\big)
  54. p [ 0 , 1 ] \textstyle p\in[0,1]
  55. p = 0.5. \textstyle p=0.5.
  56. f ( x ) = { 0.5 x for x Z , 0.5 + 0.5 x for x ( 0 , 1 ) Z \displaystyle f(x)=\begin{cases}0.5x&\,\text{for }x\in Z,\\ 0.5+0.5x&\,\text{for }x\in(0,1)\setminus Z\end{cases}
  57. ( ( 0 , 1 ) , , m ) \textstyle\big((0,1),\mathcal{F},m\big)
  58. Z 1 = { 0.5 x : x Z } { 0.5 + 0.5 x : x ( 0 , 1 ) Z } , \displaystyle Z_{1}=\{0.5x:x\in Z\}\cup\{0.5+0.5x:x\in(0,1)\setminus Z\}\,,
  59. ( Ω , , P ) \textstyle(\Omega,\mathcal{F},P)
  60. f \textstyle f
  61. ( Ω , , P ) \textstyle(\Omega,\mathcal{F},P)
  62. ( X , Σ ) . \textstyle(X,\Sigma).
  63. ( X , Σ ) \textstyle(X,\Sigma)
  64. f \textstyle f
  65. ( X , Σ ) \textstyle(X,\Sigma)
  66. f \textstyle f
  67. ( Ω , , P ) . \textstyle(\Omega,\mathcal{F},P).
  68. f : Ω , \textstyle f:\Omega\to\mathbb{R},
  69. f : Ω n , \textstyle f:\Omega\to\mathbb{R}^{n},
  70. f : Ω , \textstyle f:\Omega\to\mathbb{R}^{\infty},
  71. ( A 1 , A 2 , ) \textstyle(A_{1},A_{2},\dots)
  72. f : Ω { 0 , 1 } . \textstyle f:\Omega\to\{0,1\}^{\infty}.
  73. f \textstyle f
  74. f \textstyle f
  75. ( Ω , , P ) \textstyle(\Omega,\mathcal{F},P)
  76. f : Ω \textstyle f:\Omega\to\mathbb{R}
  77. μ \textstyle\mu
  78. , \textstyle\mathbb{R},
  79. μ ( B ) = P ( f - 1 ( B ) ) \displaystyle\mu(B)=P\big(f^{-1}(B)\big)
  80. B . \textstyle B\subset\mathbb{R}.
  81. f ( Ω ) \textstyle f(\Omega)
  82. μ * ( f ( Ω ) ) = 1 , \displaystyle\mu^{*}\big(f(\Omega)\big)=1,
  83. f ( Ω ) \textstyle f(\Omega)
  84. μ . \textstyle\mu.
  85. f : Ω \textstyle f:\Omega\to\mathbb{R}
  86. \textstyle\mathcal{F}
  87. f - 1 ( B ) , \textstyle f^{-1}(B),
  88. B \textstyle B\subset\mathbb{R}
  89. f \textstyle f
  90. A \textstyle A\in\mathcal{F}
  91. B \textstyle B\subset\mathbb{R}
  92. P ( A Δ f - 1 ( B ) ) = 0. \textstyle P(A\Delta f^{-1}(B))=0.
  93. Δ \textstyle\Delta
  94. f : Ω \textstyle f:\Omega\to\mathbb{R}
  95. f ( Ω ) \textstyle f(\Omega)
  96. μ ; \textstyle\mu;
  97. ( Ω , , P ) (\Omega,\mathcal{F},P)\,
  98. n \mathbb{R}^{n}\,
  99. \mathbb{R}\,
  100. f : Ω n f:\Omega\to\mathbb{R}^{n}\,
  101. X 1 , , X n : Ω , X_{1},\dots,X_{n}:\Omega\to\mathbb{R},\,
  102. f f\,
  103. \mathcal{F}\,
  104. X 1 , , X n . X_{1},\dots,X_{n}.\,
  105. \mathbb{R}^{\infty}\,
  106. f : Ω f:\Omega\to\mathbb{R}^{\infty}\,
  107. X 1 , X 2 , : Ω , X_{1},X_{2},\dots:\Omega\to\mathbb{R},\,
  108. f f\,
  109. \mathcal{F}\,
  110. X 1 , X 2 , . X_{1},X_{2},\dots.\,
  111. X n X_{n}\,
  112. f : Ω { 0 , 1 } f:\Omega\to\{0,1\}^{\infty}\,
  113. A 1 , A 2 , . A_{1},A_{2},\ldots\in\mathcal{F}.\,
  114. f f\,
  115. \mathcal{F}\,
  116. A 1 , A 2 , . A_{1},A_{2},\dots.\,
  117. A 1 , A 2 , A_{1},A_{2},\ldots\,
  118. f f\,
  119. ( Ω , , P ) (\Omega,\mathcal{F},P)\,
  120. f ( Ω ) f(\Omega)\,
  121. μ , \mu,\,
  122. , \mathbb{R},\,
  123. n , \mathbb{R}^{n},\,
  124. \mathbb{R}^{\infty}\,
  125. { 0 , 1 } \{0,1\}^{\infty}\,
  126. ( Ω , , P ) \textstyle(\Omega,\mathcal{F},P)
  127. ( X , Σ ) \textstyle(X,\Sigma)
  128. ( X , Σ ) . \textstyle(X,\Sigma).
  129. ( X , Σ ) = { 0 , 1 } \textstyle(X,\Sigma)=\{0,1\}^{\infty}
  130. ( Ω , , P ) \textstyle(\Omega,\mathcal{F},P)
  131. ( X , Σ ) \textstyle(X,\Sigma)
  132. ( X , Σ ) . \textstyle(X,\Sigma).
  133. ( X , Σ ) = { 0 , 1 } \textstyle(X,\Sigma)=\{0,1\}^{\infty}
  134. ( Ω , , P ) \textstyle(\Omega,\mathcal{F},P)
  135. Ω \textstyle\Omega
  136. ( Ω , , P ) \textstyle(\Omega,\mathcal{F},P)
  137. Ω \textstyle\Omega
  138. ( Ω , , P ) \textstyle(\Omega,\mathcal{F},P)
  139. ( Ω , , P ) \textstyle(\Omega,\mathcal{F},P)
  140. ( Ω , , P ) \textstyle(\Omega,\mathcal{F},P)
  141. \mathbb{R}\,
  142. \textstyle\mathcal{F}
  143. \mathbb{R}\,
  144. ( Ω , , P ) \textstyle(\Omega,\mathcal{F},P)
  145. τ \textstyle\tau
  146. Ω \textstyle\Omega
  147. ( Ω , τ ) \textstyle(\Omega,\tau)
  148. \textstyle\mathcal{F}
  149. τ \textstyle\tau
  150. ε > 0 \textstyle\varepsilon>0
  151. K \textstyle K
  152. ( Ω , τ ) \textstyle(\Omega,\tau)
  153. P ( K ) 1 - ε . \textstyle P(K)\geq 1-\varepsilon.
  154. n \textstyle\mathbb{R}^{n}
  155. C [ 0 , ) \textstyle C[0,\infty)
  156. [ 0 , ) , \textstyle[0,\infty)\to\mathbb{R},
  157. \textstyle\mathbb{R}^{\infty}
  158. Y \textstyle Y
  159. ( Ω , , P ) \textstyle(\Omega,\mathcal{F},P)
  160. P y \textstyle P_{y}
  161. ω Ω \textstyle\omega\in\Omega
  162. Y ( ω ) = y \textstyle Y(\omega)=y
  163. ( Ω , , P ) \textstyle(\Omega,\mathcal{F},P)
  164. ( Ω 1 , 1 , P 1 ) \textstyle(\Omega_{1},\mathcal{F}_{1},P_{1})
  165. ( Ω 2 , 2 , P 2 ) \textstyle(\Omega_{2},\mathcal{F}_{2},P_{2})
  166. f : Ω 1 Ω 2 \textstyle f:\Omega_{1}\to\Omega_{2}
  167. f ( Ω 1 ) \textstyle f(\Omega_{1})
  168. Ω 2 \textstyle\Omega_{2}
  169. P 2 ( f ( Ω 1 ) ) \textstyle P_{2}(f(\Omega_{1}))
  170. f ( Ω 1 ) \textstyle f(\Omega_{1})
  171. ( Ω 1 , 1 , P 1 ) \textstyle(\Omega_{1},\mathcal{F}_{1},P_{1})
  172. ( Ω 2 , 2 , P 2 ) \textstyle(\Omega_{2},\mathcal{F}_{2},P_{2})
  173. P 2 ( f ( Ω 1 ) ) = 1 \textstyle P_{2}(f(\Omega_{1}))=1
  174. f \textstyle f
  175. A 1 \textstyle A\in\mathcal{F}_{1}
  176. f ( A ) 2 \textstyle f(A)\in\mathcal{F}_{2}
  177. P 2 ( f ( A ) ) = P 1 ( A ) \textstyle P_{2}(f(A))=P_{1}(A)
  178. f - 1 \textstyle f^{-1}
  179. f : Ω 1 Ω 2 \textstyle f:\Omega_{1}\to\Omega_{2}
  180. F \textstyle F
  181. F ( B ) = f - 1 ( B ) \textstyle F(B)=f^{-1}(B)
  182. B 2 \textstyle B\in\mathcal{F}_{2}
  183. F \textstyle F
  184. f \textstyle f

Standard_RAID_levels.html

  1. G F ( m ) GF(m)
  2. m = 2 k m=2^{k}
  3. G F ( m ) F 2 [ x ] / ( p ( x ) ) GF(m)\cong F_{2}[x]/(p(x))
  4. p ( x ) p(x)
  5. k k
  6. d k - 1 d k - 2 d 0 d_{k-1}d_{k-2}...d_{0}
  7. d i d_{i}
  8. d k - 1 x k - 1 + d k - 2 x k - 2 + + d 1 x + d 0 d_{k-1}x^{k-1}+d_{k-2}x^{k-2}+...+d_{1}x+d_{0}
  9. D 0 , , D n - 1 G F ( m ) D_{0},...,D_{n-1}\in GF(m)
  10. g g
  11. \oplus
  12. 𝐏 \mathbf{P}
  13. 𝐐 \mathbf{Q}
  14. n n
  15. 𝐏 = i D i = 𝐃 0 𝐃 1 𝐃 2 𝐃 n - 1 \mathbf{P}=\bigoplus_{i}{D_{i}}=\mathbf{D}_{0}\;\oplus\;\mathbf{D}_{1}\;\oplus% \;\mathbf{D}_{2}\;\oplus\;...\;\oplus\;\mathbf{D}_{n-1}
  16. 𝐐 = i g i D i = g 0 𝐃 0 g 1 𝐃 1 g 2 𝐃 2 g n - 1 𝐃 n - 1 \mathbf{Q}=\bigoplus_{i}{g^{i}D_{i}}=g^{0}\mathbf{D}_{0}\;\oplus\;g^{1}\mathbf% {D}_{1}\;\oplus\;g^{2}\mathbf{D}_{2}\;\oplus\;...\;\oplus\;g^{n-1}\mathbf{D}_{% n-1}
  17. \oplus
  18. g i g^{i}
  19. g i g^{i}
  20. i i
  21. i < n i<n
  22. D i D_{i}
  23. D j D_{j}
  24. i j i\neq j
  25. D D
  26. A A
  27. B B
  28. D i D j = A D_{i}\oplus D_{j}=A
  29. g i D i g j D j = B g^{i}D_{i}\oplus g^{j}D_{j}=B
  30. A = : i and j D = 𝐏 𝐃 0 𝐃 1 𝐃 i - 1 𝐃 i + 1 𝐃 j - 1 𝐃 j + 1 𝐃 n - 1 A=\bigoplus_{\ell:\;\ell\not=i\;\mathrm{and}\;\ell\not=j}{D_{\ell}}=\mathbf{P}% \;\oplus\;\mathbf{D}_{0}\;\oplus\;\mathbf{D}_{1}\;\oplus\;\dots\;\oplus\;% \mathbf{D}_{i-1}\;\oplus\;\mathbf{D}_{i+1}\;\oplus\;\dots\;\oplus\;\mathbf{D}_% {j-1}\;\oplus\;\mathbf{D}_{j+1}\;\oplus\;\dots\;\oplus\;\mathbf{D}_{n-1}
  31. B = : i and j g D = 𝐐 g 0 𝐃 0 g 1 𝐃 1 g i - 1 𝐃 i - 1 g i + 1 𝐃 i + 1 g j - 1 𝐃 j - 1 g j + 1 𝐃 j + 1 g n - 1 𝐃 n - 1 B=\bigoplus_{\ell:\;\ell\not=i\;\mathrm{and}\;\ell\not=j}{g^{\ell}D_{\ell}}=% \mathbf{Q}\;\oplus\;g^{0}\mathbf{D}_{0}\;\oplus\;g^{1}\mathbf{D}_{1}\;\oplus\;% \dots\;\oplus\;g^{i-1}\mathbf{D}_{i-1}\;\oplus\;g^{i+1}\mathbf{D}_{i+1}\;% \oplus\;\dots\;\oplus\;g^{j-1}\mathbf{D}_{j-1}\;\oplus\;g^{j+1}\mathbf{D}_{j+1% }\;\oplus\;\dots\;\oplus\;g^{n-1}\mathbf{D}_{n-1}
  32. B B
  33. g n - i g^{n-i}
  34. ( g n - i + j 1 ) D j = g n - i B A (g^{n-i+j}\oplus 1)D_{j}=g^{n-i}B\oplus A
  35. D j D_{j}
  36. D i D_{i}
  37. n n
  38. 1 1 / n = 1 1 / 3 = 2 / 3 67 % 1−1/n=1−1/3=2/3≈67\%
  39. n n
  40. r r
  41. 1 - ( 1 - r ) n - n r ( 1 - r ) n - 1 \displaystyle 1-(1-r)^{n}-nr(1-r)^{n-1}
  42. 1 1
  43. 1 ( 1 r ) < s u p > n 1−(1−r)<sup>n

State-Action-Reward-State-Action.html

  1. Q ( s t , a t ) Q ( s t , a t ) + α [ r t + 1 + γ Q ( s t + 1 , a t + 1 ) - Q ( s t , a t ) ] Q(s_{t},a_{t})\leftarrow Q(s_{t},a_{t})+\alpha[r_{t+1}+\gamma Q(s_{t+1},a_{t+1% })-Q(s_{t},a_{t})]
  2. Q Q
  3. Q ( s 0 , a 0 ) Q(s_{0},a_{0})
  4. r r
  5. Q Q

Static_light_scattering.html

  1. N ( θ ) = I R ( θ ) - I S ( θ ) I R ( 90 ) - I S ( 90 ) \ N(\theta)=\frac{I_{R}(\theta)-I_{S}(\theta)}{I_{R}(90)-I_{S}(90)}
  2. K c Δ R ( θ , c ) = 1 M w ( 1 + q 2 R g 2 3 + O ( q 4 ) ) ( 1 + 2 M w A 2 c + O ( c 2 ) ) \frac{Kc}{\Delta R(\theta,c)}=\frac{1}{M_{w}}\left(1+\frac{q^{2}R_{g}^{2}}{3}+% O(q^{4})\right)\left(1+2M_{w}A_{2}c+O(c^{2})\right)
  3. K = 4 π 2 n 0 2 ( d n / d c ) 2 / N A λ 4 \ K=4\pi^{2}n_{0}^{2}(dn/dc)^{2}/N_{A}\lambda^{4}
  4. Δ R ( θ , c ) = R A ( θ ) - R 0 ( θ ) \ \Delta R(\theta,c)=R_{A}(\theta)-R_{0}(\theta)
  5. R ( θ ) = I A ( θ ) n 0 2 I T ( θ ) n T 2 R T N ( θ ) \ R(\theta)=\frac{I_{A}(\theta)n_{0}^{2}}{I_{T}(\theta)n_{T}^{2}}\frac{R_{T}}{% N(\theta)}
  6. q = 4 π n 0 sin ( θ / 2 ) / λ \ q=4\pi n_{0}\sin(\theta/2)/\lambda
  7. K c / Δ R ( θ 0 , c 0 ) = 1 / M w \ Kc/\Delta R(\theta\rightarrow 0,c\rightarrow 0)=1/M_{w}
  8. l n ( Δ R ( θ ) ) = 1 - ( R g 2 / 3 ) q 2 \ ln(\Delta R(\theta))=1-(R_{g}^{2}/3)q^{2}
  9. μ 2 / Γ ¯ 2 < 0.3 \scriptstyle\mu_{2}/\bar{\Gamma}^{2}<0.3
  10. K c Δ R ( θ , c ) = 1 M w ( 1 + q 2 R g 2 3 + O ( q 4 ) ) ( 1 + 2 M w A 2 c + O ( c 2 ) ) \frac{Kc}{\Delta R(\theta,c)}=\frac{1}{M_{w}}\left(1+\frac{q^{2}R_{g}^{2}}{3}+% O(q^{4})\right)\left(1+2M_{w}A_{2}c+O(c^{2})\right)
  11. q R g < 1 qR_{g}<1

Statistical_benchmarking.html

  1. k k
  2. Y ( k ) Y(k)
  3. Y ( k ) Y(k)
  4. k k
  5. k k
  6. Y Y
  7. Y ( k ) Y(k)
  8. k k
  9. k k
  10. W ( k ) Y ( k ) W(k)\cdot Y(k)
  11. k k
  12. W ( k ) W(k)
  13. k k
  14. k k
  15. Y Y
  16. W ( k ) Y ( k ) W(k)\cdot Y(k)
  17. k k
  18. Y Y
  19. W ( k ) W(k)
  20. Y ( k ) Y(k)
  21. C C
  22. W ( C ) W(C)
  23. W ( k ) W(k)
  24. k k
  25. C C
  26. C C
  27. T ( C ) T(C)
  28. C C
  29. C C
  30. F ( C ) = T ( C ) / W ( C ) F(C)=T(C)/W(C)
  31. W ( k ) W(k)
  32. F ( C ) F(C)
  33. C C
  34. W W
  35. F ( C ) W ( k ) F(C)\cdot W(k)
  36. T T
  37. Y Y
  38. F ( C ) F ( k ) Y ( k ) F(C)\cdot F(k)\cdot Y(k)
  39. W ( C ) = 0 W(C)=0

Statistical_shape_analysis.html

  1. Φ \Phi
  2. y = Φ ( x ) y=\Phi(x)

Steel_design.html

  1. R a R n Ω R_{a}\leq{R_{n}\over\Omega}
  2. R u ϕ * R n R_{u}\leq\phi*R_{n}
  3. Ω = 1.5 ϕ \Omega=\frac{1.5}{\phi}

Stefan's_formula.html

  1. Δ H * \scriptstyle\Delta H^{*}
  2. σ = γ 0 ( Δ H * N A 1 / 3 V m 2 / 3 ) , \sigma=\gamma_{0}\left(\frac{\Delta H^{*}}{N_{A}^{1/3}V_{m}^{2/3}}\right),
  3. γ 0 \gamma_{0}

Stefan_number.html

  1. Ste = c p Δ T L , \mathrm{Ste}=\frac{c_{p}\Delta T}{L},

Steffensen's_method.html

  1. f f
  2. x x_{\star}
  3. f ( x ) = 0 f(x_{\star})=0
  4. x x_{\star}
  5. f f
  6. - 1 < f ( x ) < 0 -1<f^{\prime}(x_{\star})<0
  7. f f
  8. x 0 x_{0}
  9. x x_{\star}
  10. x 0 x_{0}
  11. x 0 , x 1 , x 2 , , x n , x_{0},\ x_{1},\ x_{2},\dots,\ x_{n},\dots
  12. x x_{\star}
  13. x n x_{n}
  14. x n + 1 x_{n+1}
  15. x n + 1 = x n - f ( x n ) g ( x n ) x_{n+1}=x_{n}-\frac{f(x_{n})}{g(x_{n})}
  16. g ( x n ) g(x_{n})
  17. f f
  18. g ( x n ) = f ( x n + f ( x n ) ) - f ( x n ) f ( x n ) . g(x_{n})=\frac{f(x_{n}+f(x_{n}))-f(x_{n})}{f(x_{n})}.
  19. g g
  20. f f
  21. ( x , y ) = ( x n , f ( x n ) ) (x,y)=(x_{n},\ f(x_{n}))
  22. ( x , y ) = ( x n + h , f ( x n + h ) ) (x,y)=(x_{n}+h,\ f(x_{n}+h))
  23. h = f ( x n ) h=f(x_{n})
  24. f f
  25. h h
  26. f f
  27. - 1 < f ( x ) < 0 -1<f^{\prime}(x_{\star})<0
  28. f f
  29. h h
  30. g g
  31. f f
  32. f f
  33. f ( x n ) f(x_{n})
  34. f ( x n + h ) f(x_{n}+h)
  35. f f
  36. x 0 x_{0}
  37. x 0 x_{0}
  38. x x_{\star}
  39. x 0 , x 1 , x 2 , x 3 , x_{0},x_{1},x_{2},x_{3},\dots
  40. x n = p - p n x_{n}=p\ -\ p_{n}
  41. p n , p n + 1 , p n + 2 p_{n},\ p_{n+1},\ p_{n+2}
  42. p n p_{n}
  43. p p
  44. p n + 1 - p p n - p p n + 2 - p p n + 1 - p \frac{p_{n+1}-p}{p_{n}-p}\approx\frac{p_{n+2}-p}{p_{n+1}-p}
  45. ( p n + 1 - p ) 2 ( p n + 2 - p ) ( p n - p ) (p_{n+1}-p)^{2}\approx(p_{n+2}-p)(p_{n}-p)
  46. p n + 1 2 - 2 p n + 1 p + p 2 p n + 2 p n - ( p n + p n + 2 ) p + p 2 p_{n+1}^{2}-2p_{n+1}p+p^{2}\approx p_{n+2}p_{n}-(p_{n}+p_{n+2})p+p^{2}
  47. ( p n + 2 - 2 p n + 1 + p n ) p p n + 2 p n - p n + 1 2 (p_{n+2}-2p_{n+1}+p_{n})p\approx p_{n+2}p_{n}-p_{n+1}^{2}
  48. p p
  49. p p n + 2 p n - p n + 1 2 p n + 2 - 2 p n + 1 + p n p\approx\frac{p_{n+2}p_{n}-p_{n+1}^{2}}{p_{n+2}-2p_{n+1}+p_{n}}
  50. = p n 2 + p n p n + 2 + 2 p n p n + 1 - 2 p n p n + 1 - p n 2 - p n + 1 2 p n + 2 - 2 p n + 1 + p n =\frac{p_{n}^{2}+p_{n}p_{n+2}+2p_{n}p_{n+1}-2p_{n}p_{n+1}-p_{n}^{2}-p_{n+1}^{2% }}{p_{n+2}-2p_{n+1}+p_{n}}
  51. = ( p n 2 + p n p n + 2 - 2 p n p n + 1 ) - ( p n 2 - 2 p n p n + 1 + p n + 1 2 ) p n + 2 - 2 p n + 1 + p n =\frac{(p_{n}^{2}+p_{n}p_{n+2}-2p_{n}p_{n+1})-(p_{n}^{2}-2p_{n}p_{n+1}+p_{n+1}% ^{2})}{p_{n+2}-2p_{n+1}+p_{n}}
  52. = p n - ( p n + 1 - p n ) 2 p n + 2 - 2 p n + 1 + p n , =p_{n}-\frac{(p_{n+1}-p_{n})^{2}}{p_{n+2}-2p_{n+1}+p_{n}},
  53. p p n + 3 = p n - ( p n + 1 - p n ) 2 p n + 2 - 2 p n + 1 + p n . p\approx p_{n+3}=p_{n}-\frac{(p_{n+1}-p_{n})^{2}}{p_{n+2}-2p_{n+1}+p_{n}}.
  54. x = x x=x_{\star}
  55. F F
  56. x = F ( x ) x_{\star}=F(x_{\star})
  57. x x_{\star}
  58. x x_{\star}
  59. F : X X F:X\to X
  60. X X
  61. { L ( u , v ) : u , v X } \{L(u,v):u,v\in X\}
  62. u u
  63. v v
  64. F ( u ) - F ( v ) = L ( u , v ) ( u - v ) . F(u)-F(v)=L(u,v)\ (u-v).
  65. f f
  66. g g
  67. L L
  68. L L
  69. u u
  70. v v
  71. L ( F ( x ) , x ) L(F(x),x)
  72. F ( x ) F^{\prime}(x)
  73. x n + 1 = x n + [ I - L ( F ( x n ) , x n ) ] - 1 ( F ( x n ) - x n ) , x_{n+1}=x_{n}+[I-L(F(x_{n}),x_{n})]^{-1}(F(x_{n})-x_{n}),
  74. n = 1 , 2 , 3 , n=1,\ 2,\ 3,\ ...
  75. I I
  76. L L
  77. L ( u , v ) - L ( x , y ) K ( u - x + v - y ) \|L(u,v)-L(x,y)\|\leq K\big(\|u-x\|+\|v-y\|\big)
  78. K K
  79. F F
  80. x 0 x_{0}
  81. x x_{\star}
  82. x = F ( x ) x_{\star}=F(x_{\star})

Steiner's_problem.html

  1. f ( x ) = x 1 / x . f(x)=x^{1/x}.\,
  2. x = e x=e
  3. g ( x ) = ln f ( x ) = ln x x . g(x)=\ln f(x)=\frac{\ln x}{x}.
  4. g g
  5. g ( x ) = 1 - ln x x 2 . g^{\prime}(x)=\frac{1-\ln x}{x^{2}}.
  6. g ( x ) g^{\prime}(x)
  7. 0 < x < e 0<x<e
  8. x > e x>e
  9. g ( x ) g(x)
  10. f ( x ) f(x)
  11. 0 < x < e 0<x<e
  12. x > e . x>e.
  13. x = e x=e
  14. f ( x ) . f(x).

Steinhaus_theorem.html

  1. A - A = { a - b a , b A } A-A=\{a-b\mid a,b\in A\}\,
  2. A A - 1 = { a b - 1 a , b A } AA^{-1}=\{ab^{-1}\mid a,b\in A\}\,
  3. 0 < μ ( A ) < , 0<\mu(A)<\infty,\,
  4. K A U , μ ( K ) + ϵ > μ ( A ) > μ ( U ) - ϵ . K\subset A\subset U,\quad\mu(K)+\epsilon>\mu(A)>\mu(U)-\epsilon.
  5. 2 μ ( K ) > μ ( U ) . 2\mu(K)>\mu(U).\,
  6. k K k\in K
  7. V k V_{k}
  8. k + V k \sub U k+V_{k}\sub U
  9. { k + V k : k K } \{k+V_{k}:k\in K\}
  10. { k 1 + V k 1 , , k n + V k n } \{k_{1}+V_{k_{1}},\dots,k_{n}+V_{k_{n}}\}
  11. V := V k 1 V k n V:=V_{k_{1}}\cap\dots\cap V_{k_{n}}
  12. ( K + v ) K = . (K+v)\cap K=\varnothing.\,
  13. 2 μ ( K ) = μ ( K + v ) + μ ( K ) < μ ( U ) 2\mu(K)=\mu(K+v)+\mu(K)<\mu(U)\,
  14. k 1 , k 2 K A k_{1},k_{2}\in K\subset A\,
  15. v + k 1 = k 2 , v+k_{1}=k_{2},\,

Stellar_rotation.html

  1. v e sin i v_{e}\cdot\sin i
  2. v e sin i v_{e}\cdot\sin i
  3. Ω e t - 1 2 \Omega_{e}\propto t^{-\frac{1}{2}}
  4. Ω e \Omega_{e}

Step_recovery_diode.html

  1. Q S I A τ Q_{S}\cong I_{A}\cdot\tau
  2. t S Q S I R t_{S}\cong\frac{Q_{S}}{I_{R}}

Stern–Volmer_relationship.html

  1. A * + Q A + Q \mathrm{A}^{*}+\mathrm{Q}\rightarrow\mathrm{A}+\mathrm{Q}
  2. A * + Q A + Q * \mathrm{A}^{*}+\mathrm{Q}\rightarrow\mathrm{A}+\mathrm{Q}^{*}
  3. I f 0 I f = 1 + k q τ 0 [ Q ] \frac{I_{f}^{0}}{I_{f}}=1+k_{q}\tau_{0}\cdot[\mathrm{Q}]
  4. I f 0 I_{f}^{0}
  5. I f I_{f}
  6. k q k_{q}
  7. τ 0 \tau_{0}
  8. [ Q ] [\mathrm{Q}]
  9. k q = 8 R T / 3 η k_{q}={8RT}/{3\eta}
  10. R R
  11. T T
  12. η \eta

Stick_number.html

  1. stick ( T ( p , q ) ) = 2 q , if 2 p < q 2 p . \,\text{stick}(T(p,q))=2q\,\text{, if }2\leq p<q\leq 2p.\,
  2. stick ( K 1 # K 2 ) stick ( K 1 ) + stick ( K 2 ) - 3 \,\text{stick}(K_{1}\#K_{2})\leq\,\text{stick}(K_{1})+\,\text{stick}(K_{2})-3\,
  3. 1 2 ( 7 + 8 cr ( K ) + 1 ) stick ( K ) 3 2 ( c ( K ) + 1 ) . \frac{1}{2}(7+\sqrt{8\,\,\text{cr}(K)+1})\leq\,\text{stick}(K)\leq\frac{3}{2}(% c(K)+1).

Stieltjes_transformation.html

  1. S ρ ( z ) = I ρ ( t ) d t z - t , t I , z \ S_{\rho}(z)=\int_{I}\frac{\rho(t)\,dt}{z-t},\qquad t\in I\subset\mathbb{R},\;z% \in\mathbb{C}\backslash\mathbb{R}
  2. ρ ( x ) = lim ε 0 + S ρ ( x - i ε ) - S ρ ( x + i ε ) 2 i π . \rho(x)=\underset{\varepsilon\rightarrow 0^{+}}{\,\text{lim}}\frac{S_{\rho}(x-% i\varepsilon)-S_{\rho}(x+i\varepsilon)}{2i\pi}.
  3. m n = I t n ρ ( t ) d t , m_{n}=\int_{I}t^{n}\,\rho(t)\,dt,
  4. S ρ ( z ) = k = 0 n m k z k + 1 + o ( 1 z n + 1 ) . S_{\rho}(z)=\sum_{k=0}^{n}\frac{m_{k}}{z^{k+1}}+o\left(\frac{1}{z^{n+1}}\right).
  5. S ρ ( z ) = n = 0 m n z n + 1 . S_{\rho}(z)=\sum_{n=0}^{\infty}\frac{m_{n}}{z^{n+1}}.
  6. ( f , g ) I f ( t ) g ( t ) ρ ( t ) d t (f,g)\mapsto\int_{I}f(t)g(t)\rho(t)\,dt
  7. Q n ( x ) = I P n ( t ) - P n ( x ) t - x ρ ( t ) d t . Q_{n}(x)=\int_{I}\frac{P_{n}(t)-P_{n}(x)}{t-x}\rho(t)\,dt.
  8. F n ( z ) = Q n ( z ) P n ( z ) F_{n}(z)=\frac{Q_{n}(z)}{P_{n}(z)}
  9. S ρ ( z ) - Q n ( z ) P n ( z ) = O ( 1 z 2 n ) . S_{\rho}(z)-\frac{Q_{n}(z)}{P_{n}(z)}=O\left(\frac{1}{z^{2n}}\right).

Stimpmeter.html

  1. 2 × S × S S + S \frac{2\times S\uparrow\times\ S\downarrow}{S\uparrow+\ S\downarrow}

Stirling_numbers_and_exponential_generating_functions_in_symbolic_combinatorics.html

  1. [ z n ] [z^{n}]
  2. \mathfrak{C}
  3. 𝔓 \mathfrak{P}
  4. 𝒫 = 𝔓 ( ( 𝒵 ) ) , \mathcal{P}=\mathfrak{P}(\mathfrak{C}(\mathcal{Z})),
  5. = 𝔓 ( 𝔓 1 ( 𝒵 ) ) , \mathcal{B}=\mathfrak{P}(\mathfrak{P}_{\geq 1}(\mathcal{Z})),
  6. 𝒵 \mathcal{Z}
  7. 𝒫 \mathcal{P}\,
  8. 𝒫 = 𝔓 ( 𝒰 ( 𝒵 ) ) , \mathcal{P}=\mathfrak{P}(\mathcal{U}\mathfrak{C}(\mathcal{Z})),\,
  9. 𝒰 \mathcal{U}
  10. G ( z , u ) = exp ( u log 1 1 - z ) = ( 1 1 - z ) u = n = 0 k = 0 n | [ n k ] | u k z n n ! . G(z,u)=\exp\left(u\log\frac{1}{1-z}\right)=\left(\frac{1}{1-z}\right)^{u}=\sum% _{n=0}^{\infty}\sum_{k=0}^{n}\left|\left[\begin{matrix}n\\ k\end{matrix}\right]\right|u^{k}\,\frac{z^{n}}{n!}.
  11. [ n k ] = ( - 1 ) n - k | [ n k ] | . \left[\begin{matrix}n\\ k\end{matrix}\right]=(-1)^{n-k}\left|\left[\begin{matrix}n\\ k\end{matrix}\right]\right|.
  12. H ( z , u ) H(z,u)
  13. H ( z , u ) = G ( - z , - u ) = ( 1 1 + z ) - u = ( 1 + z ) u = n = 0 k = 0 n [ n k ] u k z n n ! . H(z,u)=G(-z,-u)=\left(\frac{1}{1+z}\right)^{-u}=(1+z)^{u}=\sum_{n=0}^{\infty}% \sum_{k=0}^{n}\left[\begin{matrix}n\\ k\end{matrix}\right]u^{k}\,\frac{z^{n}}{n!}.
  14. ( 1 + z ) u = n = 0 ( u n ) z n = n = 0 z n n ! k = 0 n [ n k ] u k = k = 0 u k n = k z n n ! [ n k ] = e u log ( 1 + z ) . (1+z)^{u}=\sum_{n=0}^{\infty}{u\choose n}z^{n}=\sum_{n=0}^{\infty}\frac{z^{n}}% {n!}\sum_{k=0}^{n}\left[\begin{matrix}n\\ k\end{matrix}\right]u^{k}=\sum_{k=0}^{\infty}u^{k}\sum_{n=k}^{\infty}\frac{z^{% n}}{n!}\left[\begin{matrix}n\\ k\end{matrix}\right]=e^{u\log(1+z)}.
  15. k = 0 n ( - 1 ) k [ n k ] = ( - 1 ) n n ! . \sum_{k=0}^{n}(-1)^{k}\left[\begin{matrix}n\\ k\end{matrix}\right]=(-1)^{n}n!.
  16. H ( z , - 1 ) = 1 1 + z and hence n ! [ z n ] H ( z , - 1 ) = ( - 1 ) n n ! . H(z,-1)=\frac{1}{1+z}\quad\mbox{and hence}~{}\quad n![z^{n}]H(z,-1)=(-1)^{n}n!.
  17. n = k [ n k ] z n n ! = ( log ( 1 + z ) ) k k ! \sum_{n=k}^{\infty}\left[\begin{matrix}n\\ k\end{matrix}\right]\frac{z^{n}}{n!}=\frac{\left(\log(1+z)\right)^{k}}{k!}
  18. | z | < 1 |z|<1
  19. z = 0 z=0
  20. log ( 1 + z ) \log(1+z)
  21. z = - 1. z=-1.
  22. [ u k ] H ( z , u ) = [ u k ] exp ( u log ( 1 + z ) ) = ( log ( 1 + z ) ) k k ! . [u^{k}]H(z,u)=[u^{k}]\exp\left(u\log(1+z)\right)=\frac{\left(\log(1+z)\right)^% {k}}{k!}.
  23. B n = k = 1 n { n k } and B 0 = 1 , B_{n}=\sum_{k=1}^{n}\left\{\begin{matrix}n\\ k\end{matrix}\right\}\mbox{ and }~{}B_{0}=1,
  24. \mathcal{B}\,
  25. = 𝔓 ( 𝔓 1 ( 𝒵 ) ) . \mathcal{B}=\mathfrak{P}(\mathfrak{P}_{\geq 1}(\mathcal{Z})).
  26. 𝒫 \mathcal{P}\,
  27. 𝒫 = 𝔓 ( ( 𝒵 ) ) . \mathcal{P}=\mathfrak{P}(\mathfrak{C}(\mathcal{Z})).
  28. B ( z ) = exp ( exp z - 1 ) . B(z)=\exp\left(\exp z-1\right).
  29. d d z B ( z ) = exp ( exp z - 1 ) exp z = B ( z ) exp z , \frac{d}{dz}B(z)=\exp\left(\exp z-1\right)\exp z=B(z)\exp z,
  30. B n + 1 = k = 0 n ( n k ) B k , B_{n+1}=\sum_{k=0}^{n}{n\choose k}B_{k},
  31. 𝒰 \mathcal{U}\,
  32. = 𝔓 ( 𝒰 𝔓 1 ( 𝒵 ) ) . \mathcal{B}=\mathfrak{P}(\mathcal{U}\;\mathfrak{P}_{\geq 1}(\mathcal{Z})).
  33. B ( z , u ) = exp ( u ( exp z - 1 ) ) . B(z,u)=\exp\left(u\left(\exp z-1\right)\right).
  34. { n k } = n ! [ u k ] [ z n ] B ( z , u ) = n ! [ z n ] ( exp z - 1 ) k k ! \left\{\begin{matrix}n\\ k\end{matrix}\right\}=n![u^{k}][z^{n}]B(z,u)=n![z^{n}]\frac{(\exp z-1)^{k}}{k!}
  35. n ! [ z n ] 1 k ! j = 0 k ( k j ) exp ( j z ) ( - 1 ) k - j n![z^{n}]\frac{1}{k!}\sum_{j=0}^{k}{k\choose j}\exp(jz)(-1)^{k-j}
  36. n ! k ! j = 0 k ( k j ) ( - 1 ) k - j j n n ! = 1 k ! j = 0 k ( k j ) ( - 1 ) k - j j n . \frac{n!}{k!}\sum_{j=0}^{k}{k\choose j}(-1)^{k-j}\frac{j^{n}}{n!}=\frac{1}{k!}% \sum_{j=0}^{k}{k\choose j}(-1)^{k-j}j^{n}.

STO-nG_basis_sets.html

  1. n n
  2. n n
  3. ϕ i \mathbf{\phi}_{i}
  4. ψ S T O - 3 G ( s ) = c 1 ϕ 1 + c 2 ϕ 2 + c 3 ϕ 3 \mathbf{\psi}_{STO-3G}(s)=c_{1}\phi_{1}+c_{2}\phi_{2}+c_{3}\phi_{3}
  5. ϕ 1 = ( 2 α 1 π ) 3 / 4 e - α 1 r 2 \mathbf{\phi}_{1}=\left(\frac{2\alpha_{1}}{\pi}\right)^{3/4}e^{-\alpha_{1}r^{2}}
  6. ϕ 2 = ( 2 α 2 π ) 3 / 4 e - α 2 r 2 \mathbf{\phi}_{2}=\left(\frac{2\alpha_{2}}{\pi}\right)^{3/4}e^{-\alpha_{2}r^{2}}
  7. ϕ 3 = ( 2 α 3 π ) 3 / 4 e - α 3 r 2 \mathbf{\phi}_{3}=\left(\frac{2\alpha_{3}}{\pi}\right)^{3/4}e^{-\alpha_{3}r^{2}}

Stochastic_approximation.html

  1. min x Θ f ( x ) = min x Θ 𝔼 [ F ( x , ξ ) ] \min_{x\in\Theta}\;f(x)=\min_{x\in\Theta}\mathbb{E}[F(x,\xi)]
  2. x Θ x\in\Theta
  3. f ( x ) f(x)
  4. ξ \xi
  5. d d
  6. x x
  7. Θ d \Theta\subset\mathbb{R}^{d}
  8. f ( x ) f(x)
  9. f ( x ) f(x)
  10. f ( x ) f(x)
  11. F ( x , ξ ) F(x,\xi)
  12. ξ \xi
  13. ξ \xi
  14. F ( x , ξ ) F(x,\xi)
  15. f ( x ) f(x)
  16. F ( x , ξ ) F(x,\xi)
  17. F ( x , ξ ) F(x,\xi)
  18. 𝔼 [ F ( x , ξ ) ] \mathbb{E}[F(x,\xi)]
  19. M ( x ) M(x)
  20. α \alpha
  21. M ( x ) = α M(x)=\alpha
  22. x = θ x=\theta
  23. M ( x ) M(x)
  24. N ( x ) N(x)
  25. 𝔼 [ N ( x ) ] = M ( x ) \mathbb{E}[N(x)]=M(x)
  26. x n + 1 - x n = a n ( α - N ( x n ) ) x_{n+1}-x_{n}=a_{n}(\alpha-N(x_{n}))
  27. a 1 , a 2 , a_{1},a_{2},\dots
  28. x n x_{n}
  29. L 2 L^{2}
  30. θ \theta
  31. N ( x ) N(x)
  32. M ( x ) M(x)
  33. M ( θ ) M^{\prime}(\theta)
  34. a n a_{n}
  35. n = 0 a n = and n = 0 a n 2 < \qquad\sum^{\infty}_{n=0}a_{n}=\infty\quad\mbox{ and }~{}\quad\sum^{\infty}_{n% =0}a^{2}_{n}<\infty\quad
  36. a n = a / n a_{n}=a/n
  37. a > 0 a>0
  38. N ( x ) N(x)
  39. f ( x ) f(x)
  40. f ( x ) f(x)
  41. Θ \Theta
  42. 𝔼 [ f ( x n ) - f * ] = O ( 1 / n ) \mathbb{E}[f(x_{n})-f^{*}]=O(1/n)
  43. f * f^{*}
  44. f ( x ) f(x)
  45. x Θ x\in\Theta
  46. O ( 1 / n ) O(1/\sqrt{n})
  47. O ( 1 / n ) O(1/n)
  48. x n + 1 - x n = b n ( α - N ( x n ) ) , x ¯ n = 1 n i = 0 n - 1 x i x_{n+1}-x_{n}=b_{n}(\alpha-N(x_{n})),\qquad\bar{x}_{n}=\frac{1}{n}\sum^{n-1}_{% i=0}x_{i}
  49. x ¯ n \bar{x}_{n}
  50. θ \theta
  51. { b n } \{b_{n}\}
  52. b n 0 , b n - b n + 1 b n = o ( b n ) b_{n}\rightarrow 0,\qquad\frac{b_{n}-b_{n+1}}{b_{n}}=o(b_{n})
  53. b n = n - α b_{n}=n^{-\alpha}
  54. 0 < α < 1 0<\alpha<1
  55. α = 1 \alpha=1
  56. O ( 1 / n ) O(1/n)
  57. O ( 1 / n ) O(1/\sqrt{n})
  58. M ( x ) M(x)
  59. θ \theta
  60. M ( x ) M(x)
  61. N ( x ) N(x)
  62. 𝔼 [ N ( x ) ] = M ( x ) \mathbb{E}[N(x)]=M(x)
  63. x x
  64. x n + 1 = x n + a n ( N ( x n + c n ) - N ( x n - c n ) c n ) x_{n+1}=x_{n}+a_{n}\bigg(\frac{N(x_{n}+c_{n})-N(x_{n}-c_{n})}{c_{n}}\bigg)
  65. M ( x ) M(x)
  66. { c n } \{c_{n}\}
  67. { a n } \{a_{n}\}
  68. M ( x ) M(x)
  69. x n x_{n}
  70. θ \theta
  71. f ( x ) f(x)
  72. f ( ) f(\cdot)
  73. C 0 d C_{0}\subset\mathbb{R}^{d}
  74. { a n } \{a_{n}\}
  75. { c n } \{c_{n}\}
  76. 1. c n 0 , a n 0 as n \mbox{1. }~{}\quad c_{n}\rightarrow 0,\quad a_{n}\rightarrow 0\quad\mbox{ as }% ~{}\quad n\rightarrow\infty
  77. 2. n = 0 a n = , n = 0 a n 2 c n 2 < \mbox{2. }~{}\quad\sum^{\infty}_{n=0}a_{n}=\infty,\qquad\sum^{\infty}_{n=0}% \frac{a^{2}_{n}}{c^{2}_{n}}<\infty
  78. a n = 1 / n a_{n}=1/n
  79. c n = n - 1 / 3 c_{n}=n^{-1/3}
  80. d + 1 d+1
  81. d d
  82. d d
  83. d d
  84. a n a_{n}

Stochastic_frontier_analysis.html

  1. y i = f ( x i ; β ) T E i y_{i}=f(x_{i};\beta)\cdot TE_{i}
  2. β \beta
  3. y i = f ( x i ; β ) T E i exp { v i } y_{i}=f(x_{i};\beta)\cdot TE_{i}\cdot\exp\left\{{v_{i}}\right\}
  4. T E i = exp { - u i } TE_{i}=\exp\left\{{-u_{i}}\right\}
  5. y i = f ( x i ; β ) exp { - u i } exp { v i } y_{i}=f(x_{i};\beta)\cdot\exp\left\{{-u_{i}}\right\}\cdot\exp\left\{{v_{i}}\right\}
  6. ln y i = β 0 + n β n ln x n i + v i - u i \ln y_{i}=\beta_{0}+\sum\limits_{n}{\beta_{n}\ln x_{ni}+v_{i}-u_{i}}

Stochastic_game.html

  1. I I
  2. M M
  3. ( M , 𝒜 ) (M,{\mathcal{A}})
  4. i I i\in I
  5. S i S^{i}
  6. ( S i , 𝒮 i ) (S^{i},{\mathcal{S}}^{i})
  7. P P
  8. M × S M\times S
  9. S = × i I S i S=\times_{i\in I}S^{i}
  10. M M
  11. P ( A m , s ) P(A\mid m,s)
  12. A A
  13. m m
  14. s s
  15. g g
  16. M × S M\times S
  17. R I R^{I}
  18. i i
  19. g g
  20. g i g^{i}
  21. i i
  22. m m
  23. s s
  24. m 1 m_{1}
  25. t t
  26. m t m_{t}
  27. s t i S i s^{i}_{t}\in S^{i}
  28. s t = ( s t i ) i s_{t}=(s^{i}_{t})_{i}
  29. m t + 1 m_{t+1}
  30. P ( m t , s t ) P(\cdot\mid m_{t},s_{t})
  31. m 1 , s 1 , , m t , s t , m_{1},s_{1},\ldots,m_{t},s_{t},\ldots
  32. g 1 , g 2 , g_{1},g_{2},\ldots
  33. g t = g ( m t , s t ) g_{t}=g(m_{t},s_{t})
  34. Γ λ \Gamma_{\lambda}
  35. λ \lambda
  36. 0 < λ 1 0<\lambda\leq 1
  37. i i
  38. λ t = 1 ( 1 - λ ) t - 1 g t i \lambda\sum_{t=1}^{\infty}(1-\lambda)^{t-1}g^{i}_{t}
  39. n n
  40. i i
  41. g ¯ n i := 1 n t = 1 n g t i \bar{g}^{i}_{n}:=\frac{1}{n}\sum_{t=1}^{n}g^{i}_{t}
  42. v n ( m 1 ) v_{n}(m_{1})
  43. v λ ( m 1 ) v_{\lambda}(m_{1})
  44. Γ n \Gamma_{n}
  45. Γ λ \Gamma_{\lambda}
  46. v n ( m 1 ) v_{n}(m_{1})
  47. n n
  48. v λ ( m 1 ) v_{\lambda}(m_{1})
  49. λ \lambda
  50. 0
  51. Γ \Gamma_{\infty}
  52. i i
  53. Γ \Gamma_{\infty}
  54. Γ \Gamma_{\infty}
  55. v v_{\infty}
  56. Γ \Gamma_{\infty}
  57. ε > 0 \varepsilon>0
  58. N N
  59. σ ε \sigma_{\varepsilon}
  60. τ ε \tau_{\varepsilon}
  61. σ \sigma
  62. τ \tau
  63. n N n\geq N
  64. g ¯ n i \bar{g}^{i}_{n}
  65. σ ε \sigma_{\varepsilon}
  66. τ \tau
  67. v - ε v_{\infty}-\varepsilon
  68. g ¯ n i \bar{g}^{i}_{n}
  69. σ \sigma
  70. τ ε \tau_{\varepsilon}
  71. v + ε v_{\infty}+\varepsilon
  72. Γ \Gamma_{\infty}
  73. v v_{\infty}
  74. ε > 0 \varepsilon>0
  75. N N
  76. σ \sigma
  77. i i
  78. τ \tau
  79. σ j = τ j \sigma^{j}=\tau^{j}
  80. j i j\neq i
  81. n N n\geq N
  82. g ¯ n i \bar{g}^{i}_{n}
  83. σ \sigma
  84. v i - ε v^{i}_{\infty}-\varepsilon
  85. g ¯ n i \bar{g}^{i}_{n}
  86. τ \tau
  87. v i + ε v^{i}_{\infty}+\varepsilon
  88. Γ \Gamma_{\infty}
  89. v v_{\infty}
  90. ε > 0 \varepsilon>0
  91. σ \sigma
  92. i i
  93. σ \sigma
  94. v i - ε v^{i}_{\infty}-\varepsilon
  95. τ \tau
  96. v i + ε v^{i}_{\infty}+\varepsilon

Stochastic_oscillator.html

  1. % K = 100 * ( ( P r i c e - L 5 ) / ( H 5 - L 5 ) ) \%K=100*((Price-L5)/(H5-L5))
  2. % D = 100 * ( ( K 1 + K 2 + K 3 ) / 3 ) \%D=100*((K1+K2+K3)/3)
  3. P r i c e Price
  4. L O W N ( P r i c e ) LOW_{N}(Price)
  5. H I G H N ( P r i c e ) HIGH_{N}(Price)
  6. % D \%D
  7. S M A 3 ( % K ) SMA_{3}(\%K)
  8. % D - S l o w \%D-Slow
  9. S M A 3 ( % D ) SMA_{3}(\%D)

Stochastic_partial_differential_equation.html

  1. t u = Δ u + ξ , \partial_{t}u=\Delta u+\xi\;,
  2. ξ \xi
  3. Δ \Delta

Stochastic_processes_and_boundary_value_problems.html

  1. { - Δ u ( x ) = 0 , x D ; lim y x u ( y ) = g ( x ) , x D . \begin{cases}-\Delta u(x)=0,&x\in D;\\ \displaystyle{\lim_{y\to x}u(y)}=g(x),&x\in\partial D.\end{cases}
  2. L = i = 1 n b i ( x ) x i + i , j = 1 n a i j ( x ) 2 x i x j , L=\sum_{i=1}^{n}b_{i}(x)\frac{\partial}{\partial x_{i}}+\sum_{i,j=1}^{n}a_{ij}% (x)\frac{\partial^{2}}{\partial x_{i}\,\partial x_{j}},
  3. { - L u ( x ) = f ( x ) , x D ; lim y x u ( y ) = g ( x ) , x D . (P1) \begin{cases}-Lu(x)=f(x),&x\in D;\\ \displaystyle{\lim_{y\to x}u(y)}=g(x),&x\in\partial D.\end{cases}\quad\mbox{(P% 1)}~{}
  4. d X t = b ( X t ) d t + σ ( X t ) d B t , \mathrm{d}X_{t}=b(X_{t})\,\mathrm{d}t+\sigma(X_{t})\,\mathrm{d}B_{t},
  5. 1 2 σ ( x ) σ ( x ) = a ( x ) for all x 𝐑 n . \frac{1}{2}\sigma(x)\sigma(x)^{\top}=a(x)\mbox{ for all }~{}x\in\mathbf{R}^{n}.
  6. u ( x ) = 𝐄 x [ g ( X τ D ) χ { τ D < + } ] + 𝐄 X [ 0 τ D f ( X t ) d t ] u(x)=\mathbf{E}^{x}\left[g\big(X_{\tau_{D}}\big)\cdot\chi_{\{\tau_{D}<+\infty% \}}\right]+\mathbf{E}^{X}\left[\int_{0}^{\tau_{D}}f(X_{t})\,\mathrm{d}t\right]
  7. 𝐄 x [ 0 τ D | f ( X t ) | d t ] < + . \mathbf{E}^{x}\left[\int_{0}^{\tau_{D}}\big|f(X_{t})\big|\,\mathrm{d}t\right]<% +\infty.
  8. 𝐏 x [ τ D < + ] = 1 for all x D , \mathbf{P}^{x}\big[\tau_{D}<+\infty\big]=1\mbox{ for all }~{}x\in D,
  9. u ( x ) = 𝐄 x [ g ( X τ D ) ] + 𝐄 x [ 0 τ D f ( X t ) d t ] u(x)=\mathbf{E}^{x}\left[g\big(X_{\tau_{D}}\big)\right]+\mathbf{E}^{x}\left[% \int_{0}^{\tau_{D}}f(X_{t})\,\mathrm{d}t\right]
  10. 𝒜 \mathcal{A}
  11. { - 𝒜 u ( x ) = f ( x ) , x D ; lim t τ D u ( X t ) = g ( X τ D ) , 𝐏 x -a.s., for all x D . (P2) \begin{cases}-\mathcal{A}u(x)=f(x),&x\in D;\\ \displaystyle{\lim_{t\uparrow\tau_{D}}u(X_{t})}=g\big(X_{\tau_{D}}\big),&% \mathbf{P}^{x}\mbox{-a.s., for all }~{}x\in D.\end{cases}\quad\mbox{(P2)}~{}
  12. | v ( x ) | C ( 1 + 𝐄 x [ 0 τ D | g ( X s ) | d s ] ) , |v(x)|\leq C\left(1+\mathbf{E}^{x}\left[\int_{0}^{\tau_{D}}\big|g(X_{s})\big|% \,\mathrm{d}s\right]\right),

Stokesian_dynamics.html

  1. m d u d t = F H + F B + F P . m\frac{du}{dt}=F^{H}+F^{B}+F^{P}.
  2. F H F^{H}
  3. F B F^{B}
  4. F P F^{P}
  5. O ( N 3 ) O(N^{3})
  6. O ( N 1.25 log N ) . O(N^{1.25}\,\log N).

Storage_effect.html

  1. r ( t ) = g ( e ( t ) , c ( t ) ) r(t)=g(e(t),c(t))\,
  2. E ( t ) = g ( e ( t ) , c * ) E(t)=g(e(t),c^{*})\,
  3. C ( t ) = - g ( e * , c ( t ) ) C(t)=-g(e^{*},c(t))\,
  4. r ( t ) = E ( t ) - C ( t ) + γ E ( t ) C ( t ) r(t)=E(t)-C(t)+\gamma E(t)C(t)\,
  5. γ = 2 r E C \gamma=\frac{\partial^{2}r}{\partial E\,\partial C}
  6. r ¯ E ¯ - C ¯ + γ Cov ( E ( t ) , C ( t ) ) \bar{r}\approx\bar{E}-\bar{C}+\gamma\,\text{Cov}(E(t),C(t))
  7. r i ¯ = Δ E - Δ C + Δ I \bar{r_{i}}=\Delta E-\Delta C+\Delta I\,
  8. Δ E = E i ¯ - i r q i r E r ¯ \Delta E=\bar{E_{i}}-\sum_{i\neq r}q_{ir}\bar{E_{r}}\,
  9. Δ C = C i ¯ - i r q i r C r ¯ \Delta C=\bar{C_{i}}-\sum_{i\neq r}q_{ir}\bar{C_{r}}\,
  10. Δ I = γ i Cov ( E i , C i ) - i r q i r γ r Cov ( E r , C r ) \Delta I=\gamma_{i}\,\text{Cov}(E_{i},C_{i})-\sum_{i\neq r}q_{ir}\gamma_{r}\,% \text{Cov}(E_{r},C_{r})\,
  11. q i r = C i C r q_{ir}=\frac{\partial C_{i}}{\partial C_{r}}

Strain_scanning.html

  1. ϵ = Δ d d 0 = Δ θ θ 0 = Δ E E 0 = Δ t t 0 \epsilon=\frac{\Delta d}{d_{0}}=\frac{\Delta\theta}{\theta_{0}}=\frac{\Delta E% }{E_{0}}=\frac{\Delta t}{t_{0}}\,

Strategic_complements.html

  1. Π ( x i , x j ) \,\Pi(x_{i},x_{j})\,
  2. x i \,x_{i}\,
  3. x j \,x_{j}\,
  4. Π \,\Pi\,
  5. x i \,x_{i}\,
  6. x i \,x_{i}\,
  7. Π x j \frac{\partial\Pi}{\partial x_{j}}
  8. 2 Π x i x j \frac{\partial^{2}\Pi}{\partial x_{i}\partial x_{j}}
  9. i j i\neq j
  10. Π \,\Pi\,
  11. 2 Π x i x j \frac{\partial^{2}\Pi}{\partial x_{i}\partial x_{j}}
  12. Π \,\Pi\,
  13. ( x 1 , x 2 ) (x_{1},x_{2})
  14. U x ( x 1 , x 2 ; y 2 ) = p 1 x 1 + ( 1 - x 2 - y 2 ) x 2 - ( x 1 + x 2 ) 2 / 2 - F U_{x}(x_{1},x_{2};y_{2})=p_{1}x_{1}+(1-x_{2}-y_{2})x_{2}-(x_{1}+x_{2})^{2}/2-F
  15. y 2 y_{2}
  16. U y ( y 2 ; x 1 , x 2 ) = ( 1 - x 2 - y 2 ) y 2 - y 2 2 / 2 - F U_{y}(y_{2};x_{1},x_{2})=(1-x_{2}-y_{2})y_{2}-y_{2}^{2}/2-F
  17. ( x 1 * , x 2 * , y 2 * ) (x_{1}^{*},x_{2}^{*},y_{2}^{*})
  18. U x x 1 = 0 , U x x 2 = 0 , U y y 2 = 0. \dfrac{\partial U_{x}}{\partial x_{1}}=0,\dfrac{\partial U_{x}}{\partial x_{2}% }=0,\dfrac{\partial U_{y}}{\partial y_{2}}=0.
  19. p 1 p_{1}
  20. [ d x 1 * d p 1 d x 2 * d p 1 d y 2 * d p 1 ] = [ 2 U x x 1 x 1 2 U x x 1 x 2 2 U x x 1 y 2 2 U x x 1 x 2 2 U x x 2 x 2 2 U x y 2 x 2 2 U y x 1 y 2 2 U y x 2 y 2 2 U y y 2 y 2 ] - 1 [ - 2 U x p 1 x 1 - 2 U x p 1 x 2 - 2 U y p 1 y 2 ] \begin{bmatrix}\dfrac{dx_{1}^{*}}{dp_{1}}\\ \dfrac{dx_{2}^{*}}{dp_{1}}\\ \dfrac{dy_{2}^{*}}{dp_{1}}\end{bmatrix}=\begin{bmatrix}\dfrac{\partial^{2}U_{x% }}{\partial x_{1}\partial x_{1}}&\dfrac{\partial^{2}U_{x}}{\partial x_{1}% \partial x_{2}}&\dfrac{\partial^{2}U_{x}}{\partial x_{1}\partial y_{2}}\\ \dfrac{\partial^{2}U_{x}}{\partial x_{1}\partial x_{2}}&\dfrac{\partial^{2}U_{% x}}{\partial x_{2}\partial x_{2}}&\dfrac{\partial^{2}U_{x}}{\partial y_{2}% \partial x_{2}}\\ \dfrac{\partial^{2}U_{y}}{\partial x_{1}\partial y_{2}}&\dfrac{\partial^{2}U_{% y}}{\partial x_{2}\partial y_{2}}&\dfrac{\partial^{2}U_{y}}{\partial y_{2}% \partial y_{2}}\end{bmatrix}^{-1}\begin{bmatrix}-\dfrac{\partial^{2}U_{x}}{% \partial p_{1}\partial x_{1}}\\ -\dfrac{\partial^{2}U_{x}}{\partial p_{1}\partial x_{2}}\\ -\dfrac{\partial^{2}U_{y}}{\partial p_{1}\partial y_{2}}\end{bmatrix}
  21. 1 / 4 p 1 2 / 3 1/4\leq p_{1}\leq 2/3
  22. [ d x 1 * d p 1 d x 2 * d p 1 d y 2 * d p 1 ] = [ - 1 - 1 0 - 1 - 3 - 1 0 - 1 - 3 ] - 1 [ - 1 0 0 ] = 1 5 [ 8 - 3 1 ] \begin{bmatrix}\dfrac{dx_{1}^{*}}{dp_{1}}\\ \dfrac{dx_{2}^{*}}{dp_{1}}\\ \dfrac{dy_{2}^{*}}{dp_{1}}\end{bmatrix}=\begin{bmatrix}-1&-1&0\\ -1&-3&-1\\ 0&-1&-3\end{bmatrix}^{-1}\begin{bmatrix}-1\\ 0\\ 0\end{bmatrix}=\frac{1}{5}\begin{bmatrix}8\\ -3\\ 1\end{bmatrix}
  23. x 1 * = max { 0 , 8 p 1 - 2 5 } , x 2 * = max { 0 , 2 - 3 p 1 5 } , y 2 * = p 1 + 1 5 . x_{1}^{*}=\max\left\{0,\frac{8p_{1}-2}{5}\right\},x_{2}^{*}=\max\left\{0,\frac% {2-3p_{1}}{5}\right\},y_{2}^{*}=\frac{p_{1}+1}{5}.