wpmath0000013_12

Septic_equation.html

  1. a x 7 + b x 6 + c x 5 + d x 4 + e x 3 + f x 2 + g x + h = 0 , ax^{7}+bx^{6}+cx^{5}+dx^{4}+ex^{3}+fx^{2}+gx+h=0,\,
  2. y ( x ) = a x 7 + b x 6 + c x 5 + d x 4 + e x 3 + f x 2 + g x + h y(x)=ax^{7}+bx^{6}+cx^{5}+dx^{4}+ex^{3}+fx^{2}+gx+h\,
  3. x 7 + 7 a x 5 + 14 a 2 x 3 + 7 a 3 x + b = 0 x^{7}+7ax^{5}+14a^{2}x^{3}+7a^{3}x+b=0\,
  4. y 2 + b y - a 7 = 0 y^{2}+by-a^{7}=0\,
  5. x = u + v x=u+v
  6. u v + a = 0 uv+a=0
  7. u 7 + v 7 + b = 0 u^{7}+v^{7}+b=0
  8. x k = ω k y 1 7 + ω k 6 y 2 7 x_{k}=\omega_{k}\sqrt[7]{y_{1}}+\omega_{k}^{6}\sqrt[7]{y_{2}}
  9. x 7 - 2 x 6 + ( a + 1 ) x 5 + ( a - 1 ) x 4 - a x 3 - ( a + 5 ) x 2 - 6 x - 4 = 0 x^{7}-2x^{6}+(a+1)x^{5}+(a-1)x^{4}-ax^{3}-(a+5)x^{2}-6x-4=0\,
  10. d = - 4 4 ( 4 a 3 + 99 a 2 - 34 a + 467 ) 3 d=-4^{4}(4a^{3}+99a^{2}-34a+467)^{3}\,

Sequential_decoding.html

  1. p p
  2. P i P_{i}
  3. X X
  4. 𝐫 {\mathbf{r}}
  5. i i
  6. Pr ( P i | X , 𝐫 ) Pr ( 𝐫 | P i , X ) Pr ( P i | X ) \Pr(P_{i}|X,{\mathbf{r}})\propto\Pr({\mathbf{r}}|P_{i},X)\Pr(P_{i}|X)
  7. N N
  8. b b
  9. R R
  10. d i d_{i}
  11. P i P_{i}
  12. n i n_{i}
  13. P i P_{i}
  14. Pr ( 𝐫 | P i , X ) \Pr({\mathbf{r}}|P_{i},X)
  15. p d i ( 1 - p ) n i b - d i 2 - ( N - n i ) b p^{d_{i}}(1-p)^{n_{i}b-d_{i}}2^{-(N-n_{i})b}
  16. n i b n_{i}b
  17. Pr ( P i | X ) \Pr(P_{i}|X)
  18. n i n_{i}
  19. 2 R b 2^{Rb}
  20. Pr ( P i | X , 𝐫 ) \displaystyle\Pr(P_{i}|X,{\mathbf{r}})
  21. d i log 2 p + ( n i b - d i ) log 2 ( 1 - p ) + n i b - n i R b = d i ( log 2 p + 1 - R ) + ( n i b - d i ) ( log 2 ( 1 - p ) + 1 - R ) \begin{aligned}&\displaystyle d_{i}\log_{2}p+(n_{i}b-d_{i})\log_{2}(1-p)+n_{i}% b-n_{i}Rb\\ \displaystyle=&\displaystyle d_{i}(\log_{2}p+1-R)+(n_{i}b-d_{i})(\log_{2}(1-p)% +1-R)\end{aligned}
  22. log 2 p + 1 - R \log_{2}p+1-R
  23. log 2 ( 1 - p ) + 1 - R \log_{2}(1-p)+1-R
  24. R 0 R_{0}
  25. R 0 = 1 - log 2 ( 1 + 2 p ( 1 - p ) ) R_{0}=1-\log_{2}(1+2\sqrt{p(1-p)})
  26. N N
  27. N N

Set_function.html

  1. d ( A ) = lim n | A { 1 , , n } | n , d(A)=\lim_{n\to\infty}\frac{|A\cap\{1,\dots,n\}|}{n},

Setepenre_(princess).html

  1. α \alpha
  2. γ \gamma
  3. α \alpha

Sethi_model.html

  1. d X t = ( r U t 1 - X t - δ X t ) d t + σ ( X t ) d z t , X 0 = x dX_{t}=\left(rU_{t}\sqrt{1-X_{t}}-\delta X_{t}\right)\,dt+\sigma(X_{t})\,dz_{t% },\qquad X_{0}=x
  2. X t X_{t}
  3. t t
  4. U t U_{t}
  5. t t
  6. r r
  7. δ \delta
  8. σ ( X t ) \sigma(X_{t})
  9. z t z_{t}
  10. d z t dz_{t}
  11. r r
  12. δ \delta
  13. r r
  14. δ \delta
  15. σ ( X t ) d z t \sigma(X_{t})dz_{t}
  16. x x
  17. V ( x ) = max U t 0 E [ 0 e - ρ t ( π X t - U t 2 ) d t ] , V(x)=\max_{U_{t}\geq 0}\;E\left[\int_{0}^{\infty}e^{-\rho t}(\pi X_{t}-U_{t}^{% 2})\,dt\right],
  18. π \pi
  19. x = 1 x=1
  20. ρ > 0 \rho>0
  21. V ( x ) V(x)
  22. V ( x ) = λ ¯ x + λ ¯ 2 r 2 4 ρ , V(x)=\bar{\lambda}x+\frac{\bar{\lambda}^{2}r^{2}}{4\rho},
  23. λ ¯ = ( ρ + δ ) 2 + r 2 π - ( ρ + δ ) r 2 / 2 . \bar{\lambda}=\frac{\sqrt{(\rho+\delta)^{2}+r^{2}\pi}-(\rho+\delta)}{r^{2}/2}.
  24. U t * = u * ( X t ) = r λ ¯ 1 - X t 2 = { > u ¯ if X t < x ¯ , = u ¯ if X t = x ¯ , < u ¯ if X t > x ¯ , U^{*}_{t}=u^{*}(X_{t})=\frac{r\bar{\lambda}\sqrt{1-\ X_{t}}}{2}=\begin{cases}{% }>\bar{u}&\,\text{if }X_{t}<\bar{x},\\ {}=\bar{u}&\,\text{if }X_{t}=\bar{x},\\ {}<\bar{u}&\,\text{if }X_{t}>\bar{x},\end{cases}
  25. x ¯ = r 2 λ ¯ / 2 r 2 λ ¯ / 2 + δ \bar{x}=\frac{r^{2}\bar{\lambda}/2}{r^{2}\bar{\lambda}/2+\delta}
  26. u ¯ = r λ ¯ 1 - x ¯ 2 . \bar{u}=\frac{r\bar{\lambda}\sqrt{1-\bar{x}}}{2}.

Seven-dimensional_space.html

  1. S 6 = { x 7 : x = r } . S^{6}=\left\{x\in\mathbb{R}^{7}:\|x\|=r\right\}.
  2. V 7 = 16 π 3 105 r 7 V_{7}\,=\frac{16\pi^{3}}{105}\,r^{7}

Severe_plastic_deformation.html

  1. d m i n b = A 3 ( e - β Q 4 R T ) ( D p 0 G b 2 ν 0 k T ) 0.25 ( γ G b ) 0.5 ( G H ) 1.25 \frac{d_{min}}{b}=A_{3}\left(e^{-\tfrac{\beta Q}{4RT}}\right){\left(\frac{D_{p% 0}Gb^{2}}{\nu_{0}kT}\right)}^{0.25}{\left(\frac{\gamma}{Gb}\right)}^{0.5}{% \left(\frac{G}{H}\right)}^{1.25}

Shahn_Majid.html

  1. q q

Shanks_transformation.html

  1. { a m } m \left\{a_{m}\right\}_{m\in\mathbb{N}}
  2. A = m = 0 a m A=\sum_{m=0}^{\infty}a_{m}\,
  3. A n A_{n}
  4. A n = m = 0 n a m A_{n}=\sum_{m=0}^{n}a_{m}\,
  5. { A n } n \left\{A_{n}\right\}_{n\in\mathbb{N}}
  6. A n A_{n}
  7. A A
  8. n . n\to\infty.
  9. S ( A n ) S(A_{n})
  10. A n A_{n}
  11. S ( A n ) = A n + 1 A n - 1 - A n 2 A n + 1 - 2 A n + A n - 1 S(A_{n})=\frac{A_{n+1}\,A_{n-1}\,-\,A_{n}^{2}}{A_{n+1}-2A_{n}+A_{n-1}}
  12. S ( A n ) S(A_{n})
  13. A n . A_{n}.
  14. S 2 ( A n ) = S ( S ( A n ) ) , S^{2}(A_{n})=S(S(A_{n})),
  15. S 3 ( A n ) = S ( S ( S ( A n ) ) ) , S^{3}(A_{n})=S(S(S(A_{n}))),
  16. 4 k = 0 ( - 1 ) k 1 2 k + 1 = 4 ( 1 - 1 3 + 1 5 - 1 7 + ) 4\sum_{k=0}^{\infty}(-1)^{k}\frac{1}{2k+1}=4\left(1-\frac{1}{3}+\frac{1}{5}-% \frac{1}{7}+\cdots\right)
  17. A 6 A_{6}
  18. A n A_{n}
  19. S ( A n ) S(A_{n})
  20. S 2 ( A n ) S^{2}(A_{n})
  21. S 3 ( A n ) S^{3}(A_{n})
  22. n n
  23. n n
  24. A n A_{n}
  25. S ( A n ) S(A_{n})
  26. S 2 ( A n ) S^{2}(A_{n})
  27. S 3 ( A n ) S^{3}(A_{n})
  28. S ( A 1 ) S(A_{1})
  29. A 24 . A_{24}.
  30. S 3 ( A 3 ) S^{3}(A_{3})
  31. A 0 A_{0}
  32. A 6 . A_{6}.
  33. A n A_{n}
  34. n n
  35. A n A_{n}
  36. A n = A + α q n , A_{n}=A+\alpha q^{n},\,
  37. | q | < 1 |q|<1
  38. A A
  39. n . n\to\infty.
  40. n - 1 , n-1,
  41. n n
  42. n + 1 n+1
  43. A n - 1 = A + α q n - 1 , A n = A + α q n and A n + 1 = A + α q n + 1 . A_{n-1}=A+\alpha q^{n-1}\quad,\qquad A_{n}=A+\alpha q^{n}\qquad\,\text{and}% \qquad A_{n+1}=A+\alpha q^{n+1}.
  44. A , A,
  45. α \alpha
  46. q . q.
  47. A A
  48. A = A n + 1 A n - 1 - A n 2 A n + 1 - 2 A n + A n - 1 . A=\frac{A_{n+1}\,A_{n-1}\,-\,A_{n}^{2}}{A_{n+1}-2A_{n}+A_{n-1}}.
  49. A n = A A_{n}=A
  50. n . n.
  51. S k ( A n ) = | A n - k A n - 1 A n Δ A n - k Δ A n - 1 Δ A n Δ A n - k + 1 Δ A n Δ A n + 1 Δ A n - 1 Δ A n + k - 2 Δ A n + k - 1 | | 1 1 1 Δ A n - k Δ A n - 1 Δ A n Δ A n - k + 1 Δ A n Δ A n + 1 Δ A n - 1 Δ A n + k - 2 Δ A n + k - 1 | , S_{k}(A_{n})=\frac{\begin{vmatrix}A_{n-k}&\cdots&A_{n-1}&A_{n}\\ \Delta A_{n-k}&\cdots&\Delta A_{n-1}&\Delta A_{n}\\ \Delta A_{n-k+1}&\cdots&\Delta A_{n}&\Delta A_{n+1}\\ \vdots&&\vdots&\vdots\\ \Delta A_{n-1}&\cdots&\Delta A_{n+k-2}&\Delta A_{n+k-1}\\ \end{vmatrix}}{\begin{vmatrix}1&\cdots&1&1\\ \Delta A_{n-k}&\cdots&\Delta A_{n-1}&\Delta A_{n}\\ \Delta A_{n-k+1}&\cdots&\Delta A_{n}&\Delta A_{n+1}\\ \vdots&&\vdots&\vdots\\ \Delta A_{n-1}&\cdots&\Delta A_{n+k-2}&\Delta A_{n+k-1}\\ \end{vmatrix}},
  52. Δ A p = A p + 1 - A p . \Delta A_{p}=A_{p+1}-A_{p}.
  53. A n A_{n}
  54. k k
  55. A n = A + p = 1 k α p q p n . A_{n}=A+\sum_{p=1}^{k}\alpha_{p}q_{p}^{n}.
  56. 2 k + 1 2k+1
  57. A n - k , A n - k + 1 , , A n + k A_{n-k},A_{n-k+1},\ldots,A_{n+k}
  58. A , A,
  59. S 1 ( A n ) = S ( A n ) . S_{1}(A_{n})=S(A_{n}).

Shannon_multigraph.html

  1. S h ( n ) Sh(n)
  2. n 2 \left\lfloor\frac{n}{2}\right\rfloor
  3. n 2 \left\lfloor\frac{n}{2}\right\rfloor
  4. n + 1 2 \left\lfloor\frac{n+1}{2}\right\rfloor
  5. n n
  6. n + 1 2 \left\lfloor\frac{n+1}{2}\right\rfloor
  7. Δ \Delta
  8. 3 2 Δ \frac{3}{2}\Delta
  9. Δ \Delta
  10. Δ / 2 \Delta/2
  11. Δ \Delta
  12. 3 2 Δ \frac{3}{2}\Delta
  13. 3 2 Δ \frac{3}{2}\Delta
  14. Δ \Delta
  15. μ \mu
  16. Δ + μ \Delta+\mu

Shear_wave_splitting.html

  1. x i [ c i j k l U k x l ] = ρ 2 U j t 2 \frac{\partial}{\partial x_{i}}\left[c_{ijkl}\frac{\partial U_{k}}{\partial x_% {l}}\right]=\rho\frac{\partial^{2}U_{j}}{\partial t^{2}}
  2. ρ \rho
  3. U j U_{j}
  4. c i j k l c_{ijkl}
  5. U k ( x i , t ) = n = 0 U k ( n ) ( x i ) f n ( t - τ ( x i ) ) U_{k}\left(x_{i},t\right)=\sum_{n=0}^{\infty}U_{k}^{\left(}n\right)\left(x_{i}% \right)f_{n}\left(t-\tau\left(x_{i}\right)\right)
  6. f n ( ϑ ) f_{n}\left(\vartheta\right)
  7. d f n + 1 ( ϑ ) / d ϑ = f n ( ϑ ) df_{n+1}\left(\vartheta\right)/d\vartheta=f_{n}\left(\vartheta\right)
  8. N ( U ( n ) ) - M ( U ( n - 1 ) ) + L ( U ( n - 2 ) ) = 0 N\left(U^{\left(}n\right)\right)-M\left(U^{\left(}{n-1}\right)\right)+L\left(U% ^{\left(}{n-2}\right)\right)=0
  9. { N j ( U ( n ) ) = Γ j k U k ( n ) - U j ( n ) M j ( U ( n ) ) = p i a i j k l U k ( n ) x l + ρ - 1 x i ( ρ a i j k l p l U k ( n ) ) L j ( U ( n ) ) = ρ - 1 x i ( ρ a i j k l U k ( n ) x l ) \begin{cases}N_{j}\left(U^{\left(}n\right)\right)=\Gamma_{jk}U_{k}^{\left(}n% \right)-U_{j}^{\left(}n\right)\\ M_{j}\left(U^{\left(}n\right)\right)=p_{i}~{}a_{ijkl}\frac{\partial U_{k}^{% \left(}n\right)}{\partial x_{l}}+\rho^{-1}\frac{\partial}{\partial x_{i}}\left% (\rho~{}a_{ijkl}~{}p_{l}U_{k}^{\left(}n\right)\right)\\ L_{j}\left(U^{\left(}n\right)\right)=\rho^{-1}\frac{\partial}{\partial x_{i}}% \left(\rho~{}a_{ijkl}\frac{\partial U_{k}^{\left(}n\right)}{\partial x_{l}}% \right)\end{cases}
  10. Γ j k = p i p l a i j k l , a i j k l = c i j k l / ρ , p i = τ x i \Gamma_{jk}=p_{i}~{}p_{l}~{}a_{ijkl},\quad a_{ijkl}=c_{ijkl}/\rho,\quad p_{i}=% \frac{\partial\tau}{\partial x_{i}}
  11. n = 0 n=0
  12. U ( - 1 ) = U ( - 2 ) = 0 U^{\left(}-1\right)=U^{\left(}-2\right)=0
  13. N j ( U ( 0 ) ) = Γ j k U k ( 0 ) - U j ( 0 ) = = ( Γ j k - δ j k ) U k ( 0 ) = = 0 N_{j}\left(U^{\left(}0\right)\right)=\Gamma_{jk}U_{k}^{\left(}0\right)-U_{j}^{% \left(}0\right)==\left(\Gamma_{jk}-\delta_{jk}\right)U_{k}^{\left(}0\right)==0
  14. Γ j k \Gamma_{jk}
  15. D e t ( Γ j k - G δ j k ) = 0 Det\left(\Gamma_{jk}-G\delta_{jk}\right)=0
  16. P , Q P,Q
  17. R R
  18. Γ j k \Gamma_{jk}
  19. Γ j k \Gamma_{jk}
  20. g 1 , g 2 , g 3 g_{1},~{}g_{2},~{}g_{3}
  21. G 1 , G 2 , G_{1},~{}G_{2},
  22. G 3 ~{}G_{3}
  23. G 1 = α 2 p i p i G_{1}=\alpha^{2}p_{i}p_{i}
  24. G 2 = G 3 = β 2 p i p i G_{2}=G_{3}=\beta^{2}p_{i}p_{i}
  25. G 2 G 3 G_{2}\neq G_{3}
  26. w ( ω ) w\left(\omega\right)
  27. p ^ \hat{p}
  28. f ^ \hat{f}
  29. s ^ \hat{s}
  30. δ t \delta t
  31. ϕ \phi
  32. 4 m s / k m 4ms/km
  33. 10 m s / k m 10ms/km
  34. 10 m s / k m 10ms/km
  35. 2 m s / k m / y e a r 2ms/km/year

Shields_parameter.html

  1. τ \tau_{\ast}
  2. θ \theta
  3. τ = θ = τ ( ρ s - ρ ) g D , \tau_{\ast}=\theta=\frac{\tau}{(\rho_{s}-\rho)gD},
  4. τ \tau
  5. ρ s \rho_{s}
  6. ρ \rho
  7. g g
  8. D D

Shintani_zeta_function.html

  1. n 1 , , n m 0 1 L 1 s 1 L k s k , \sum_{n_{1},\dots,n_{m}\geq 0}\frac{1}{L_{1}^{s_{1}}\cdots L_{k}^{s_{k}}},

Shock_polar.html

  1. ( φ , p ) (\varphi,p)
  2. θ \theta
  3. μ = sin - 1 ( 1 / M ) \mu=\sin^{-1}(1/M)
  4. M M
  5. sin - 1 ( 1 / M ) θ π / 2 \sin^{-1}(1/M)\leq\theta\leq\pi/2
  6. p p 0 = 1 + 2 γ γ + 1 ( M 2 sin 2 θ - 1 ) \frac{p}{p_{0}}=1+\frac{2\gamma}{\gamma+1}(M^{2}\sin^{2}\theta-1)
  7. θ \theta
  8. φ \varphi
  9. tan φ = 2 cot θ M 2 sin 2 θ - 1 2 + M 2 ( γ + cos 2 θ ) . \tan\varphi=2\cot\theta\frac{M^{2}\sin^{2}\theta-1}{2+M^{2}(\gamma+\cos 2% \theta)}.
  10. γ \gamma
  11. φ \varphi

Sholl_analysis.html

  1. log 10 ( N S ) = - k r + m \log_{10}\left(\frac{N}{S}\right)=-k\cdot r+m
  2. log 10 ( N S ) = - k log 10 ( r ) + m \log_{10}\left(\frac{N}{S}\right)=-k\cdot\log_{10}(r)+m
  3. N ( r ) = a 0 + a 1 * r + a 2 * r 2 + + a t * r t . \ N(r)\ =a_{0}+a_{1}*r+a_{2}*r^{2}+...+a_{t}*r^{t}.

Shoshana_Kamin.html

  1. t u = Δ x u m , m > 1 , \partial_{t}u=\Delta_{x}u^{m},\,\,m>1,\,

Shriek_map.html

  1. f ! f_{!}
  2. f ! , f^{!},
  3. f * f_{*}
  4. f * , f^{*},
  5. f ! f_{!}
  6. F E B , F\to E\to B,
  7. H * ( E ) H * ( B ) H_{*}(E)\to H_{*}(B)
  8. H * ( B ) H * ( E ) . H^{*}(B)\to H^{*}(E).

Sievert_integral.html

  1. F ( x , θ ) = 0 θ e - x sec ( φ ) d φ . F(x,\theta)=\int_{0}^{\theta}{e^{-x\sec(\varphi)}}\,d{\varphi}.\,

Sigma_heat.html

  1. S S
  2. S = 0.24 BTU lb F t + W ( 0.45 BTU lb F t + 1061 BTU lb ) S=0.24\mathrm{\tfrac{BTU}{lb\cdot^{\circ}F}\;}t+W\;(0.45\mathrm{\tfrac{BTU}{lb% \cdot^{\circ}F}\;}t+1061\mathrm{\tfrac{BTU}{lb}})
  3. S S
  4. t t
  5. W W
  6. S = 17.86 kJ kg + 1.005 kJ kg K t + W ( 2501 kJ kg + 1.884 kJ kg K t ) S=17.86\mathrm{\tfrac{kJ}{kg}}+1.005\mathrm{\tfrac{kJ}{kg\cdot K}}t+W\;(2501% \mathrm{\tfrac{kJ}{kg}}+1.884\mathrm{\tfrac{kJ}{kg\cdot K}}t)
  7. S S
  8. t t
  9. W W
  10. h = S + 1 BTU lb W t h=S+1\mathrm{\tfrac{BTU}{lb}\;}Wt^{\prime}
  11. h h
  12. S S
  13. W W
  14. t t^{\prime}

Signal_averaging.html

  1. S N = n S n σ 2 = n S σ \frac{S}{N}=\frac{nS}{\sqrt{n\sigma^{2}}}=\sqrt{n}\frac{S}{\sigma}

Signorini_problem.html

  1. \scriptstylesymbol u ( s y m b o l x ) = ( u 1 ( s y m b o l x ) , u 2 ( s y m b o l x ) , u 3 ( s y m b o l x ) ) \scriptstylesymbol{u}(symbol{x})=\left(u_{1}(symbol{x}),u_{2}(symbol{x}),u_{3}% (symbol{x})\right)
  2. A A
  3. A \scriptstyle\partial A
  4. n n
  5. Σ \Sigma
  6. \scriptstylesymbol f ( s y m b o l x ) = ( f 1 ( s y m b o l x ) , f 2 ( s y m b o l x ) , f 3 ( s y m b o l x ) ) \scriptstylesymbol{f}(symbol{x})=\left(f_{1}(symbol{x}),f_{2}(symbol{x}),f_{3}% (symbol{x})\right)
  7. \scriptstylesymbol g ( s y m b o l x ) = ( g 1 ( s y m b o l x ) , g 2 ( s y m b o l x ) , g 3 ( s y m b o l x ) ) \scriptstylesymbol{g}(symbol{x})=\left(g_{1}(symbol{x}),g_{2}(symbol{x}),g_{3}% (symbol{x})\right)
  8. A Σ \scriptstyle\partial A\setminus\Sigma
  9. A A
  10. Σ \Sigma
  11. σ i k x k - f i = 0 for i = 1 , 2 , 3 \qquad\frac{\partial\sigma_{ik}}{\partial x_{k}}-f_{i}=0\qquad\,\text{for }i=1% ,2,3
  12. A Σ \scriptstyle\partial A\setminus\Sigma
  13. σ i k n k - g i = 0 for i = 1 , 2 , 3 \qquad\sigma_{ik}n_{k}-g_{i}=0\qquad\,\text{for }i=1,2,3
  14. Σ \Sigma
  15. \scriptstylesymbol σ = s y m b o l σ ( s y m b o l u ) \scriptstylesymbol{\sigma}=symbol{\sigma}(symbol{u})
  16. \scriptstylesymbol τ = ( τ 1 , τ 2 , τ 3 ) \scriptstylesymbol{\tau}=(\tau_{1},\tau_{2},\tau_{3})
  17. Σ \Sigma
  18. { u i n i = 0 σ i k n i n k 0 σ i k n i τ k = 0 \quad\begin{cases}u_{i}n_{i}&=0\\ \sigma_{ik}n_{i}n_{k}&\geq 0\\ \sigma_{ik}n_{i}\tau_{k}&=0\end{cases}
  19. { u i n i > 0 σ i k n i n k = 0 σ i k n i τ k = 0 \begin{cases}u_{i}n_{i}&>0\\ \sigma_{ik}n_{i}n_{k}&=0\\ \sigma_{ik}n_{i}\tau_{k}&=0\end{cases}
  20. n n
  21. n n
  22. τ \tau
  23. Σ \Sigma
  24. + 1 \scriptstyle+1
  25. - 1 \scriptstyle-1
  26. Σ \Sigma
  27. u u
  28. n n
  29. n n
  30. u u
  31. n n
  32. n n
  33. Σ \Sigma
  34. Σ \Sigma
  35. W ( s y m b o l ε ) = a i k j h ( s y m b o l x ) ε i k ε i k W(symbol{\varepsilon})=a_{ikjh}(symbol{x})\varepsilon_{ik}\varepsilon_{ik}
  36. \scriptstylesymbol a ( s y m b o l x ) = ( a i k j h ( s y m b o l x ) ) \scriptstylesymbol{a}(symbol{x})=\left(a_{ikjh}(symbol{x})\right)
  37. \scriptstylesymbol ε = s y m b o l ε ( s y m b o l u ) = ( ε i k ( s y m b o l u ) ) = ( 1 2 ( u i x k + u k x i ) ) \scriptstylesymbol{\varepsilon}=symbol{\varepsilon}(symbol{u})=\left(% \varepsilon_{ik}(symbol{u})\right)=\left(\frac{1}{2}\left(\frac{\partial u_{i}% }{\partial x_{k}}+\frac{\partial u_{k}}{\partial x_{i}}\right)\right)
  38. σ i k = - W ε i k for i , k = 1 , 2 , 3 \sigma_{ik}=-\frac{\partial W}{\partial\varepsilon_{ik}}\qquad\,\text{for }i,k% =1,2,3
  39. I ( s y m b o l u ) = A W ( s y m b o l x , s y m b o l ε ) d x - A u i f i d x - A Σ u i g i d σ I(symbol{u})=\int_{A}W(symbol{x},symbol{\varepsilon})\mathrm{d}x-\int_{A}u_{i}% f_{i}\mathrm{d}x-\int_{\partial A\setminus\Sigma}u_{i}g_{i}\mathrm{d}\sigma
  40. u u
  41. 𝒰 Σ \scriptstyle\mathcal{U}_{\Sigma}
  42. u u
  43. I ( u ) I(u)
  44. C 1 C^{1}
  45. A ¯ \scriptstyle\bar{A}
  46. A A
  47. \scriptstylesymbol u 𝒰 Σ \scriptstylesymbol{u}\in\mathcal{U}_{\Sigma}
  48. d d t I ( s y m b o l u + t s y m b o l v ) | t = 0 = - A σ i k ( s y m b o l u ) ε i k ( s y m b o l v ) d x - A v i f i d x - A Σ v i g i d σ 0 s y m b o l v 𝒰 Σ \left.\frac{\mathrm{d}}{\mathrm{d}t}I(symbol{u}+tsymbol{v})\right|_{t=0}=-\int% _{A}\sigma_{ik}(symbol{u})\varepsilon_{ik}(symbol{v})\mathrm{d}x-\int_{A}v_{i}% f_{i}\mathrm{d}x-\int_{\partial A\setminus\Sigma}\!\!\!\!\!v_{i}g_{i}\mathrm{d% }\sigma\geq 0\qquad\forall symbol{v}\in\mathcal{U}_{\Sigma}
  49. B ( s y m b o l u , s y m b o l v ) = - A σ i k ( s y m b o l u ) ε i k ( s y m b o l v ) d x s y m b o l u , s y m b o l v 𝒰 Σ B(symbol{u},symbol{v})=-\int_{A}\sigma_{ik}(symbol{u})\varepsilon_{ik}(symbol{% v})\mathrm{d}x\qquad symbol{u},symbol{v}\in\mathcal{U}_{\Sigma}
  50. F ( s y m b o l v ) = A v i f i d x + A Σ v i g i d σ s y m b o l v 𝒰 Σ F(symbol{v})=\int_{A}v_{i}f_{i}\mathrm{d}x+\int_{\partial A\setminus\Sigma}\!% \!\!\!\!v_{i}g_{i}\mathrm{d}\sigma\qquad symbol{v}\in\mathcal{U}_{\Sigma}
  51. B ( s y m b o l u , s y m b o l v ) - F ( s y m b o l v ) 0 s y m b o l v 𝒰 Σ B(symbol{u},symbol{v})-F(symbol{v})\geq 0\qquad\forall symbol{v}\in\mathcal{U}% _{\Sigma}

SigSpec.html

  1. K K
  2. ( t k , x k ) (t_{k},x_{k})
  3. α 0 \alpha_{0}
  4. β 0 \beta_{0}
  5. θ 0 \theta_{0}
  6. tan 2 θ 0 = K k = 0 K - 1 sin 2 ω t k - 2 ( k = 0 K - 1 cos ω t k ) ( k = 0 K - 1 sin ω t k ) K k = 0 K - 1 cos 2 ω t k - ( k = 0 K - 1 cos ω t k ) 2 + ( k = 0 K - 1 sin ω t k ) 2 , \tan 2\theta_{0}=\frac{K\sum_{k=0}^{K-1}\sin 2\omega t_{k}-2\left(\sum_{k=0}^{% K-1}\cos\omega t_{k}\right)\left(\sum_{k=0}^{K-1}\sin\omega t_{k}\right)}{K% \sum_{k=0}^{K-1}\cos 2\omega t_{k}-\big(\sum_{k=0}^{K-1}\cos\omega t_{k}\big)^% {2}+\big(\sum_{k=0}^{K-1}\sin\omega t_{k}\big)^{2}},
  7. α 0 = 2 K 2 ( K k = 0 K - 1 cos 2 ( ω t k - θ 0 ) - [ l = 0 K - 1 cos ( ω t k - θ 0 ) ] 2 ) , \alpha_{0}=\sqrt{\frac{2}{K^{2}}\left(K\sum_{k=0}^{K-1}\cos^{2}\left(\omega t_% {k}-\theta_{0}\right)-\left[\sum_{l=0}^{K-1}\cos\left(\omega t_{k}-\theta_{0}% \right)\right]^{2}\right)},
  8. β 0 = 2 K 2 ( K k = 0 K - 1 sin 2 ( ω t k - θ 0 ) - [ l = 0 K - 1 sin ( ω t k - θ 0 ) ] 2 ) . \beta_{0}=\sqrt{\frac{2}{K^{2}}\left(K\sum_{k=0}^{K-1}\sin^{2}\left(\omega t_{% k}-\theta_{0}\right)-\left[\sum_{l=0}^{K-1}\sin\left(\omega t_{k}-\theta_{0}% \right)\right]^{2}\right)}.
  9. θ \theta
  10. tan θ = k = 0 K - 1 sin ω t k k = 0 K - 1 cos ω t k , \tan\theta=\frac{\sum_{k=0}^{K-1}\sin\omega t_{k}}{\sum_{k=0}^{K-1}\cos\omega t% _{k}},
  11. ϕ ( A ) = K A sock 2 < x 2 > exp ( - K A 2 4 < x 2 > sock ) , \phi(A)=\frac{KA\cdot\operatorname{sock}}{2<x^{2}>}\exp\left(-\frac{KA^{2}}{4<% x^{2}>}\cdot\operatorname{sock}\right),
  12. sock ( ω , θ ) = [ cos 2 ( θ - θ 0 ) α 0 2 + sin 2 ( θ - θ 0 ) β 0 2 ] \operatorname{sock}(\omega,\theta)=\left[\frac{\cos^{2}\left(\theta-\theta_{0}% \right)}{\alpha_{0}^{2}}+\frac{\sin^{2}\left(\theta-\theta_{0}\right)}{\beta_{% 0}^{2}}\right]
  13. < x 2 > <x^{2}>
  14. x k x_{k}
  15. A A
  16. Φ FA ( A ) = exp ( - K A 2 4 < x 2 > sock ) . \Phi_{\operatorname{FA}}(A)=\exp\left(-\frac{KA^{2}}{4<x^{2}>}\cdot% \operatorname{sock}\right).
  17. sig ( A ) = K A 2 log e 4 < x 2 > sock . \operatorname{sig}(A)=\frac{KA^{2}\log e}{4<x^{2}>}\cdot\operatorname{sock}.
  18. A A

Silhouette_(clustering).html

  1. k k
  2. i i
  3. a ( i ) a(i)
  4. i i
  5. a ( i ) a(i)
  6. i i
  7. i i
  8. c c
  9. i i
  10. c c
  11. b ( i ) b(i)
  12. i i
  13. i i
  14. i i
  15. i i
  16. s ( i ) = b ( i ) - a ( i ) max { a ( i ) , b ( i ) } s(i)=\frac{b(i)-a(i)}{\max\{a(i),b(i)\}}
  17. s ( i ) = { 1 - a ( i ) / b ( i ) , if a ( i ) < b ( i ) 0 , if a ( i ) = b ( i ) b ( i ) / a ( i ) - 1 , if a ( i ) > b ( i ) s(i)=\begin{cases}1-a(i)/b(i),&\mbox{if }~{}a(i)<b(i)\\ 0,&\mbox{if }~{}a(i)=b(i)\\ b(i)/a(i)-1,&\mbox{if }~{}a(i)>b(i)\\ \end{cases}
  18. - 1 s ( i ) 1 -1\leq s(i)\leq 1
  19. s ( i ) s(i)
  20. a ( i ) b ( i ) a(i)\ll b(i)
  21. a ( i ) a(i)
  22. i i
  23. b ( i ) b(i)
  24. i i
  25. s ( i ) s(i)
  26. s ( i ) s(i)
  27. i i
  28. s ( i ) s(i)
  29. s ( i ) s(i)
  30. s ( i ) s(i)
  31. k k

Similarity_solution.html

  1. ν \nu
  2. t = 0 t=0
  3. U U
  4. x x
  5. x - y x-y
  6. u = U u=U
  7. y = 0 y=0
  8. u 0 u\rightarrow 0
  9. y y\rightarrow\infty
  10. ρ ( u t + u . u ) = - p + μ 2 u \rho\left(\dfrac{\partial\vec{u}}{\partial t}+\vec{u}.\nabla\vec{u}\right)=-% \nabla p+\mu\nabla^{2}\vec{u}
  11. y y
  12. x x
  13. p y = 0 \dfrac{\partial p}{\partial y}=0
  14. x x
  15. u t = ν y 2 u \dfrac{\partial\vec{u}}{\partial t}=\nu\partial^{2}_{y}\vec{u}
  16. U t ν U y 2 \frac{U}{t}\sim\nu\frac{U}{y^{2}}
  17. y y
  18. y ( ν t ) 1 / 2 y\sim(\nu t)^{1/2}
  19. f f
  20. η \eta
  21. u = U f ( η y ( ν t ) 1 / 2 ) u=Uf\left(\eta\equiv\dfrac{y}{(\nu t)^{1/2}}\right)
  22. - η f / 2 = f ′′ -\eta f^{\prime}/2=f^{\prime\prime}
  23. f = 1 - e r f ( η / 2 ) f=1-erf(\eta/2)
  24. u = U ( 1 - e r f ( - y / ( 4 ν t ) 1 / 2 ) ) u=U\left(1-erf\left(-y/(4\nu t)^{1/2}\right)\right)

Simon–Glatzel_equation.html

  1. T M = T R e f ( P M - P R e f a + 1 ) 1 c T_{M}=T_{Ref}\left(\frac{P_{M}-P_{Ref}}{a}+1\right)^{\frac{1}{c}}

Simple_random_sample.html

  1. P \displaystyle P
  2. P = 1 - ( 1 - 1 N ) n = 1 - ( 999 1000 ) 100 = 0.0952 9.5 % P=1-\left(1-\frac{1}{N}\right)^{n}=1-\left(\frac{999}{1000}\right)^{100}=0.095% 2\dots\approx 9.5\%

Simplicial_map.html

  1. f : | K | | L | f:|K|\rightarrow|L|
  2. st ( v ) \,\text{st}(v)
  3. f : K L f_{\triangle}:K\rightarrow L
  4. f ( st ( v ) ) st ( f ( v ) ) f(\,\text{st}(v))\subseteq\,\text{st}(f_{\triangle}(v))
  5. f f

Simplicity_theory.html

  1. U = C exp - C obs . U=C\text{exp}-C\text{obs}.
  2. C exp C\text{exp}
  3. U U
  4. P P
  5. P = 2 - U . P=2^{-U}.

Simulation_algorithms_for_atomic_DEVS.html

  1. t s [ 0 , ] t_{s}\in[0,\infty]
  2. t e [ 0 , ) t_{e}\in[0,\infty)
  3. t l [ 0 , ) t_{l}\in[0,\infty)
  4. t n [ 0 , ] t_{n}\in[0,\infty]
  5. t e = t - t l \,t_{e}=t-t_{l}
  6. t s = t n - t l \,t_{s}=t_{n}-t_{l}
  7. t [ 0 , ) t\in[0,\infty)
  8. t r = t s - t e \,t_{r}=t_{s}-t_{e}
  9. t r = t n - t \,t_{r}=t_{n}-t
  10. t r [ 0 , ] t_{r}\in[0,\infty]
  11. t l t_{l}
  12. t n t_{n}
  13. A = ( X , Y , S , t a , δ e x t , δ i n t , λ ) A=(X,Y,S,ta,\delta_{ext},\delta_{int},\lambda)
  14. t t
  15. t l t ; t_{l}\leftarrow t;
  16. t n t l + t a ( s ) ; t_{n}\leftarrow t_{l}+ta(s);
  17. t t
  18. t t n t\neq t_{n}
  19. y λ ( s ) ; y\leftarrow\lambda(s);
  20. y , t y,t
  21. s δ i n t ( s ) s\leftarrow\delta_{int}(s)
  22. t l t ; t_{l}\leftarrow t;
  23. t n t l + t a ( s ) ; t_{n}\leftarrow t_{l}+ta(s);
  24. δ e x t \delta_{ext}
  25. t l t_{l}
  26. t t
  27. t e t_{e}
  28. t e = t - t l t_{e}=t-t_{l}
  29. x X x\in X
  30. t t
  31. ( t l t (t_{l}\leq t
  32. t t n ) t\leq t_{n})
  33. s δ e x t ( s , t - t l , x ) s\leftarrow\delta_{ext}(s,t-t_{l},x)
  34. t l t ; t_{l}\leftarrow t;
  35. t n t l + t a ( s ) ; t_{n}\leftarrow t_{l}+ta(s);
  36. b b
  37. δ e x t \delta_{ext}
  38. t l t_{l}
  39. t n t_{n}
  40. t e t_{e}
  41. t n t_{n}
  42. b = 1 b=1
  43. b = 0 b=0
  44. x X x\in X
  45. t t
  46. ( t l t (t_{l}\leq t
  47. t t n ) t\leq t_{n})
  48. ( s , b ) δ e x t ( s , t - t l , x ) (s,b)\leftarrow\delta_{ext}(s,t-t_{l},x)
  49. b = 1 b=1
  50. t l t ; t_{l}\leftarrow t;
  51. t n t l + t a ( s ) ; t_{n}\leftarrow t_{l}+ta(s);

Simulation_algorithms_for_coupled_DEVS.html

  1. t s [ 0 , ] t_{s}\in[0,\infty]
  2. t e [ 0 , ) t_{e}\in[0,\infty)
  3. t l [ 0 , ) t_{l}\in[0,\infty)
  4. t n [ 0 , ] t_{n}\in[0,\infty]
  5. t e = t - t l \,t_{e}=t-t_{l}
  6. t s = t n - t l \,t_{s}=t_{n}-t_{l}
  7. t [ 0 , ) t\in[0,\infty)
  8. t r = t s - t e \,t_{r}=t_{s}-t_{e}
  9. t r = t n - t \,t_{r}=t_{n}-t
  10. t r [ 0 , ] t_{r}\in[0,\infty]
  11. t l t_{l}
  12. t n t_{n}
  13. N = ( X , Y , D , { M i } , C x x , C y x , C y y , S e l e c t ) N=(X,Y,D,\{M_{i}\},C_{xx},C_{yx},C_{yy},Select)
  14. i D i\in D
  15. i i
  16. t l max { t l i : i D } t_{l}\leftarrow\max\{t_{li}:i\in D\}
  17. t n min { t n i : i D } t_{n}\leftarrow\min\{t_{ni}:i\in D\}
  18. t t n t\neq t_{n}
  19. i * S e l e c t ( { i D : t n i = t n } ) ; i^{*}\leftarrow Select(\{i\in D:t_{ni}=t_{n}\});
  20. i * i^{*}
  21. t l max { t l i : i D } t_{l}\leftarrow\max\{t_{li}:i\in D\}
  22. t n min { t n i : i D } t_{n}\leftarrow\min\{t_{ni}:i\in D\}
  23. x X x\in X
  24. ( t l t (t_{l}\leq t
  25. t t n ) t\leq t_{n})
  26. ( x , x i ) C x x (x,x_{i})\in C_{xx}
  27. x i x_{i}
  28. i i
  29. t l max { t l i : i D } t_{l}\leftarrow\max\{t_{li}:i\in D\}
  30. t n min { t n i : i D } t_{n}\leftarrow\min\{t_{ni}:i\in D\}
  31. y i Y i y_{i}\in Y_{i}
  32. ( y i , x i ) C y x (y_{i},x_{i})\in C_{yx}
  33. x i x_{i}
  34. i i
  35. C y y ( y i ) ϕ C_{yy}(y_{i})\neq\phi
  36. C y y ( y i ) C_{yy}(y_{i})
  37. t l max { t l i : i D } t_{l}\leftarrow\max\{t_{li}:i\in D\}
  38. t n min { t n i : i D } t_{n}\leftarrow\min\{t_{ni}:i\in D\}

Simultaneous_perturbation_stochastic_approximation.html

  1. u * u^{*}
  2. J ( u ) J(u)
  3. u * = arg min u U J ( u ) . u^{*}=\arg\min_{u\in U}J(u).
  4. u n + 1 = u n - a n g ^ n ( u n ) , u_{n+1}=u_{n}-a_{n}\hat{g}_{n}(u_{n}),
  5. u n = ( ( u n ) 1 , ( u n ) 2 , , ( u n ) p ) T u_{n}=((u_{n})_{1},(u_{n})_{2},\ldots,(u_{n})_{p})^{T}
  6. n t h n^{th}
  7. g ^ n ( u n ) \hat{g}_{n}(u_{n})
  8. g ( u ) = u J ( u ) g(u)=\frac{\partial}{\partial u}J(u)
  9. u n {u_{n}}
  10. { a n } \{a_{n}\}
  11. u n u_{n}
  12. i t h i^{th}
  13. ( g n ^ ( u n ) ) i = J ( u n + c n e i ) - J ( u n - c n e i ) 2 c n , (\hat{g_{n}}(u_{n}))_{i}=\frac{J(u_{n}+c_{n}e_{i})-J(u_{n}-c_{n}e_{i})}{2c_{n}},
  14. e i e_{i}
  15. i t h i^{th}
  16. c n c_{n}
  17. g n g_{n}
  18. Δ n \Delta_{n}
  19. i t h i^{th}
  20. ( g n ^ ( u n ) ) i = J ( u n + c n Δ n ) - J ( u n - c n Δ n ) 2 c n ( Δ n ) i . (\hat{g_{n}}(u_{n}))_{i}=\frac{J(u_{n}+c_{n}\Delta_{n})-J(u_{n}-c_{n}\Delta_{n% })}{2c_{n}(\Delta_{n})_{i}}.
  21. g n g_{n}
  22. b n = E [ g ^ n | u n ] - J ( u n ) b_{n}=E[\hat{g}_{n}|u_{n}]-\nabla J(u_{n})
  23. g ^ n \hat{g}_{n}
  24. { ( Δ n ) i } \{(\Delta_{n})_{i}\}
  25. E ( | ( Δ n ) i | - 1 ) E(|(\Delta_{n})_{i}|^{-1})
  26. b n b_{n}
  27. Δ n \Delta_{n}
  28. E [ ( g ^ n ) i ] E[(\hat{g}_{n})_{i}]
  29. J ( u n + c n Δ n ) i J(u_{n}+c_{n}\Delta_{n})_{i}
  30. J ( u n - c n Δ n ) i J(u_{n}-c_{n}\Delta_{n})_{i}
  31. { ( Δ n ) i } \{(\Delta_{n})_{i}\}
  32. E [ ( g ^ n ) i ] = ( g n ) i + O ( c n 2 ) E[(\hat{g}_{n})_{i}]=(g_{n})_{i}+O(c_{n}^{2})
  33. c n c_{n}
  34. u t u_{t}
  35. J ( u ) J(u)
  36. J ( u ) J(u)
  37. a k a_{k}
  38. c k c_{k}
  39. Δ k i \Delta_{ki}
  40. a t a_{t}
  41. a t a_{t}
  42. t = 1 a t = \sum_{t=1}^{\infty}a_{t}=\infty
  43. a t = a t ; a_{t}=\frac{a}{t};
  44. c t = c t γ c_{t}=\frac{c}{t^{\gamma}}
  45. γ [ 1 6 , 1 2 ] \gamma\in\left[\frac{1}{6},\frac{1}{2}\right]
  46. t = 1 ( a t c t ) 2 < \sum_{t=1}^{\infty}(\frac{a_{t}}{c_{t}})^{2}<\infty
  47. Δ t i \Delta_{ti}
  48. Δ k i < a 1 < \Delta_{ki}<a_{1}<\infty
  49. Δ t i \Delta_{ti}
  50. Δ k i \Delta_{ki}
  51. | J ( 3 ) ( u ) | < a 3 < |J^{(3)}(u)|<a_{3}<\infty
  52. J \nabla J
  53. u ˙ = g ( u ) \dot{u}=g(u)
  54. u k u_{k}

SINADR.html

  1. SINADR = P signal P quantizationError + P randomNoise + P distortion \mathrm{SINADR}=\frac{P_{\mathrm{signal}}}{P_{\mathrm{quantizationError}}+P_{% \mathrm{randomNoise}}+P_{\mathrm{distortion}}}
  2. P P
  3. SINADR = E N O B 6.02 + 1.76 \mathrm{SINADR}=ENOB\cdot 6.02+1.76

Sinc_numerical_methods.html

  1. C ( f , h ) ( x ) = k = - f ( k h ) sinc ( x h - k ) C(f,h)(x)=\sum_{k=-\infty}^{\infty}f(kh)\,\textrm{sinc}\left(\dfrac{x}{h}-k\right)
  2. sinc ( x ) = sin ( π x ) π x \textrm{sinc}(x)=\frac{\sin(\pi x)}{\pi x}
  3. C M , N ( f , h ) ( x ) = k = - M N f ( k h ) sinc ( x h - k ) C_{M,N}(f,h)(x)=\displaystyle\sum_{k=-M}^{N}f(kh)\,\textrm{sinc}\left(\dfrac{x% }{h}-k\right)
  4. O ( e - c n ) O\left(e^{-c\sqrt{n}}\right)
  5. O ( e - k n ln n ) O\left(e^{-\frac{kn}{\ln n}}\right)

Single-precision_floating-point_format.html

  1. e 127 {}^{e−127}
  2. = ( - 1 ) sign ( 1. b 22 b 21 b 0 ) 2 × 2 e - 127 =(-1)\text{sign}(1.b_{22}b_{21}...b_{0})_{2}\times 2^{e-127}
  3. value = ( - 1 ) sign ( 1 + i = 1 23 b 23 - i 2 - i ) × 2 ( e - 127 ) \,\text{value}=(-1)\text{sign}\left(1+\sum_{i=1}^{23}b_{23-i}2^{-i}\right)% \times 2^{(e-127)}
  4. sign = 0 \,\text{sign}=0
  5. 1 + i = 1 23 b 23 - i 2 - i = 1 + 2 - 2 = 1.25 1+\sum_{i=1}^{23}b_{23-i}2^{-i}=1+2^{-2}=1.25
  6. 2 ( e - 127 ) = 2 124 - 127 = 2 - 3 2^{(e-127)}=2^{124-127}=2^{-3}
  7. value = 1.25 × 2 - 3 = 0.15625 \,\text{value}=1.25\times 2^{-3}=0.15625
  8. ( 1. x 1 x 2 x 23 ) 2 × 2 e (1.x_{1}x_{2}...x_{23})_{2}\times 2^{e}
  9. ( 1.100011 ) 2 × 2 3 (1.100011)_{2}\times 2^{3}
  10. ( 12.375 ) 10 = ( 1.100011 ) 2 × 2 3 (12.375)_{10}=(1.100011)_{2}\times 2^{3}
  11. ( 1 ) 10 = ( 1.0 ) 2 × 2 0 (1)_{10}=(1.0)_{2}\times 2^{0}
  12. ( 0.25 ) 10 = ( 1.0 ) 2 × 2 - 2 (0.25)_{10}=(1.0)_{2}\times 2^{-2}
  13. 0.375 = ( 1.1 ) 2 × 2 - 2 0.375={(1.1)_{2}}\times 2^{-2}
  14. ( 1.1 ) 2 × 2 - 2 {(1.1)_{2}}\times 2^{-2}
  15. n = ( - 1 ) s × ( 1 + m * 2 - 23 ) × 2 x - 127 n=(-1)^{s}\times(1+m*2^{-23})\times 2^{x-127}
  16. s s
  17. x x
  18. m m

Single_peaked_preferences.html

  1. { x 1 , , x N } \{x_{1},\ldots,x_{N}\}
  2. \succsim
  3. x * { x 1 , , x N } x^{*}\in\{x_{1},\ldots,x_{N}\}
  4. x m < x n x * x n x m x_{m}<x_{n}\leq x^{*}\Rightarrow x_{n}\succ x_{m}
  5. x m > x n x * x n x m x_{m}>x_{n}\geq x^{*}\Rightarrow x_{n}\succ x_{m}
  6. x * x^{*}
  7. x * x^{*}
  8. { x 1 , x 2 , , x n } \{x_{1},x_{2},\ldots,x_{n}\}
  9. x i x_{i}
  10. i i
  11. x i x_{i}

Six-dimensional_space.html

  1. S 5 = { x 6 : x = r } . S^{5}=\left\{x\in\mathbb{R}^{6}:\|x\|=r\right\}.
  2. V 6 = π 3 r 6 6 V_{6}=\frac{\pi^{3}r^{6}}{6}
  3. S 6 = { x 7 : x = r } . S^{6}=\left\{x\in\mathbb{R}^{7}:\|x\|=r\right\}.
  4. V 7 = 16 π 3 r 7 105 V_{7}=\frac{16\pi^{3}r^{7}}{105}
  5. 𝐅 = 𝐉 \partial\mathbf{F}=\mathbf{J}\,
  6. 𝐅 \mathbf{F}
  7. 𝐉 \mathbf{J}
  8. 2 4 {2}_{4}
  9. 𝐁 = B 12 𝐞 12 + B 13 𝐞 13 + B 14 𝐞 14 + B 23 𝐞 23 + B 24 𝐞 24 + B 34 𝐞 34 \mathbf{B}=B_{12}\mathbf{e}_{12}+B_{13}\mathbf{e}_{13}+B_{14}\mathbf{e}_{14}+B% _{23}\mathbf{e}_{23}+B_{24}\mathbf{e}_{24}+B_{34}\mathbf{e}_{34}
  10. 𝐚 𝐛 = a 1 b 1 + a 2 b 2 + a 3 b 3 + a 4 b 4 + a 5 b 5 + a 6 b 6 . \mathbf{a}\cdot\mathbf{b}=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}+a_{4}b_{4}+a_{5}b_{% 5}+a_{6}b_{6}.
  11. | 𝐚 | = 𝐚 𝐚 = a 1 2 + a 2 2 + a 3 2 + a 4 2 + a 5 2 + a 6 2 . \left|\mathbf{a}\right|=\sqrt{\mathbf{a}\cdot\mathbf{a}}=\sqrt{{a_{1}}^{2}+{a_% {2}}^{2}+{a_{3}}^{2}+{a_{4}}^{2}+{a_{5}}^{2}+{a_{6}}^{2}}.
  12. 1 + 1 + 1 + 1 + 1 + 1 = 6 = 2.4495 , \sqrt{1+1+1+1+1+1}=\sqrt{6}=2.4495,
  13. C H C\otimes H

Six_factor_formula.html

  1. k = η f p ε P F N L P T N L k=\eta fp\varepsilon P_{FNL}P_{TNL}
  2. η \eta
  3. η = ν σ f F σ a F \eta=\frac{\nu\sigma_{f}^{F}}{\sigma_{a}^{F}}
  4. f f
  5. f = Σ a F Σ a f=\frac{\Sigma_{a}^{F}}{\Sigma_{a}}
  6. p p
  7. p exp ( - i = 1 N N i I r , A , i ( ξ ¯ Σ p ) m o d ) p\approx\mathrm{exp}\left(-\frac{\sum\limits_{i=1}^{N}N_{i}I_{r,A,i}}{\left(% \overline{\xi}\Sigma_{p}\right)_{mod}}\right)
  8. ε \varepsilon
  9. total number of fission neutrons number of fission neutrons from just thermal fissions \tfrac{\mbox{total number of fission neutrons}~{}}{\mbox{number of fission % neutrons from just thermal fissions}~{}}
  10. ε 1 + 1 - p p u f ν f P F A F f ν t P T A F P T N L \varepsilon\approx 1+\frac{1-p}{p}\frac{u_{f}\nu_{f}P_{FAF}}{f\nu_{t}P_{TAF}P_% {TNL}}
  11. P F N L P_{FNL}
  12. P F N L exp ( - B g 2 τ t h ) P_{FNL}\approx\mathrm{exp}\left(-{B_{g}}^{2}\tau_{th}\right)
  13. P T N L P_{TNL}
  14. P T N L 1 1 + L t h 2 B g 2 P_{TNL}\approx\frac{1}{1+{L_{th}}^{2}{B_{g}}^{2}}
  15. ν \nu
  16. ν f \nu_{f}
  17. ν t \nu_{t}
  18. σ f F \sigma_{f}^{F}
  19. σ a F \sigma_{a}^{F}
  20. Σ a F \Sigma_{a}^{F}
  21. Σ a \Sigma_{a}
  22. N i N_{i}
  23. I r , A , i I_{r,A,i}
  24. I r , A , i = E t h E 0 d E Σ p m o d Σ t ( E ) σ a i ( E ) E I_{r,A,i}=\int_{E_{th}}^{E_{0}}dE^{\prime}\frac{\Sigma_{p}^{mod}}{\Sigma_{t}(E% ^{\prime})}\frac{\sigma_{a}^{i}(E^{\prime})}{E^{\prime}}
  25. ξ ¯ \overline{\xi}
  26. u f u_{f}
  27. P F A F P_{FAF}
  28. P T A F P_{TAF}
  29. B g 2 {B_{g}}^{2}
  30. L t h 2 {L_{th}}^{2}
  31. L t h 2 = D Σ a , t h {L_{th}}^{2}=\frac{D}{\Sigma_{a,th}}
  32. τ t h \tau_{th}
  33. τ = E t h E d E ′′ 1 E ′′ D ( E ′′ ) ξ ¯ [ D ( E ′′ ) B g 2 + Σ t ( E ) ] \tau=\int_{E_{th}}^{E^{\prime}}dE^{\prime\prime}\frac{1}{E^{\prime\prime}}% \frac{D(E^{\prime\prime})}{\overline{\xi}\left[D(E^{\prime\prime}){B_{g}}^{2}+% \Sigma_{t}(E^{\prime})\right]}
  34. τ t h \tau_{th}
  35. τ \tau
  36. E E^{\prime}
  37. k = number of neutrons in one generation number of neutrons in preceding generation k=\frac{\mbox{number of neutrons in one generation}~{}}{\mbox{number of % neutrons in preceding generation}~{}}

Size_theory.html

  1. k \mathbb{R}^{k}
  2. k \mathbb{R}^{k}
  3. 0
  4. ( N , ψ ) ( x ¯ , y ¯ ) \ell_{(N,\psi)}(\bar{x},\bar{y})
  5. ( M , φ ) ( x ~ , y ~ ) \ell_{(M,\varphi)}(\tilde{x},\tilde{y})
  6. d ( ( M , φ ) , ( N , ψ ) ) d((M,\varphi),(N,\psi))
  7. ( M , φ ) , ( N , ψ ) (M,\varphi),\ (N,\psi)
  8. If ( N , ψ ) ( x ¯ , y ¯ ) > ( M , φ ) ( x ~ , y ~ ) then d ( ( M , φ ) , ( N , ψ ) ) min { x ~ - x ¯ , y ¯ - y ~ } . \,\text{If }\ell_{(N,\psi)}(\bar{x},\bar{y})>\ell_{(M,\varphi)}(\tilde{x},% \tilde{y})\,\text{ then }d((M,\varphi),(N,\psi))\geq\min\{\tilde{x}-\bar{x},% \bar{y}-\tilde{y}\}.

Skew_coordinates.html

  1. g i j = 𝐞 i 𝐞 j g_{ij}=\mathbf{e}_{i}\cdot\mathbf{e}_{j}
  2. g i j g_{ij}
  3. 𝐞 i \mathbf{e}_{i}
  4. ϕ \phi
  5. ϕ \phi
  6. 𝐞 1 \mathbf{e}_{1}
  7. 𝐞 2 \mathbf{e}_{2}
  8. 𝐞 3 \mathbf{e}_{3}
  9. x x
  10. y y
  11. z z
  12. g 11 = g 22 = g 33 = 1 ; g 12 = g 23 = 0 ; g 13 = cos ( π 2 - ϕ ) = sin ( ϕ ) g_{11}=g_{22}=g_{33}=1\quad;\quad g_{12}=g_{23}=0\quad;\quad g_{13}=\cos\left(% \frac{\pi}{2}-\phi\right)=\sin(\phi)
  13. g = 𝐞 1 ( 𝐞 2 × 𝐞 3 ) = cos ( ϕ ) \sqrt{g}=\mathbf{e}_{1}\cdot(\mathbf{e}_{2}\times\mathbf{e}_{3})=\cos(\phi)
  14. 𝐞 1 = 𝐞 2 × 𝐞 3 g = 𝐞 2 × 𝐞 3 cos ( ϕ ) \mathbf{e}^{1}=\frac{\mathbf{e}_{2}\times\mathbf{e}_{3}}{\sqrt{g}}=\frac{% \mathbf{e}_{2}\times\mathbf{e}_{3}}{\cos(\phi)}
  15. 𝐞 2 = 𝐞 3 × 𝐞 1 g = 𝐞 2 \mathbf{e}^{2}=\frac{\mathbf{e}_{3}\times\mathbf{e}_{1}}{\sqrt{g}}=\mathbf{e}_% {2}
  16. 𝐞 3 = 𝐞 1 × 𝐞 2 g = 𝐞 1 × 𝐞 2 cos ( ϕ ) \mathbf{e}^{3}=\frac{\mathbf{e}_{1}\times\mathbf{e}_{2}}{\sqrt{g}}=\frac{% \mathbf{e}_{1}\times\mathbf{e}_{2}}{\cos(\phi)}
  17. 𝐚 = i a i 𝐞 i and 𝐛 = i b i 𝐞 i \mathbf{a}=\sum_{i}a^{i}\mathbf{e}_{i}\quad\mbox{and}~{}\quad\mathbf{b}=\sum_{% i}b^{i}\mathbf{e}_{i}
  18. a i = j a j g i j a^{i}=\sum_{j}a_{j}g^{ij}\,
  19. a 1 = a 1 - sin ( ϕ ) a 3 cos ( ϕ ) 2 , a^{1}=\frac{a_{1}-\sin(\phi)a_{3}}{\cos(\phi)^{2}},\,
  20. a 2 = a 2 , a^{2}=a_{2},\,
  21. a 3 = - sin ( ϕ ) a 1 + a 3 cos ( ϕ ) 2 . a^{3}=\frac{-\sin(\phi)a_{1}+a_{3}}{\cos(\phi)^{2}}.\,
  22. 𝐚 𝐛 = i a i b i = a 1 b 1 + a 2 b 2 + a 3 b 3 + sin ( ϕ ) ( a 1 b 3 + a 3 b 1 ) . \mathbf{a}\cdot\mathbf{b}=\sum_{i}a^{i}b_{i}=a^{1}b^{1}+a^{2}b^{2}+a^{3}b^{3}+% \sin(\phi)(a^{1}b^{3}+a^{3}b^{1}).
  23. f = i 𝐞 i f q i = f x 𝐞 1 + f y 𝐞 2 + f z 𝐞 3 \nabla f=\sum_{i}\mathbf{e}^{i}\frac{\partial f}{\partial q^{i}}=\frac{% \partial f}{\partial x}\mathbf{e}^{1}+\frac{\partial f}{\partial y}\mathbf{e}^% {2}+\frac{\partial f}{\partial z}\mathbf{e}^{3}
  24. q i q_{i}
  25. f = f x - sin ( ϕ ) f z cos ( ϕ ) 2 𝐞 1 + f y 𝐞 2 + - sin ( ϕ ) f x + f z cos ( ϕ ) 2 𝐞 3 . \nabla f=\frac{\frac{\partial f}{\partial x}-\sin(\phi)\frac{\partial f}{% \partial z}}{\cos(\phi)^{2}}\mathbf{e}_{1}+\frac{\partial f}{\partial y}% \mathbf{e}_{2}+\frac{-\sin(\phi)\frac{\partial f}{\partial x}+\frac{\partial f% }{\partial z}}{\cos(\phi)^{2}}\mathbf{e}_{3}.
  26. 𝐚 \mathbf{a}
  27. 𝐚 = 1 g i q i ( g a i ) = a 1 x + a 2 y + a 3 z . \nabla\cdot\mathbf{a}=\frac{1}{\sqrt{g}}\sum_{i}\frac{\partial}{\partial q^{i}% }\left(\sqrt{g}a^{i}\right)=\frac{\partial a^{1}}{\partial x}+\frac{\partial a% ^{2}}{\partial y}+\frac{\partial a^{3}}{\partial z}.
  28. 𝐀 \mathbf{A}
  29. 𝐀 = 1 g i , j q i ( g a i j 𝐞 j ) = i , j 𝐞 j a i j q i . \nabla\cdot\mathbf{A}=\frac{1}{\sqrt{g}}\sum_{i,j}\frac{\partial}{\partial q^{% i}}\left(\sqrt{g}a^{ij}\mathbf{e}_{j}\right)=\sum_{i,j}\mathbf{e}_{j}\frac{% \partial a^{ij}}{\partial q^{i}}.
  30. 2 f = f = 1 cos ( ϕ ) 2 ( 2 f x 2 + 2 f z 2 - 2 sin ( ϕ ) 2 f x z ) + 2 f y 2 \nabla^{2}f=\nabla\cdot\nabla f=\frac{1}{\cos(\phi)^{2}}\left(\frac{\partial^{% 2}f}{\partial x^{2}}+\frac{\partial^{2}f}{\partial z^{2}}-2\sin(\phi)\frac{% \partial^{2}f}{\partial x\partial z}\right)+\frac{\partial^{2}f}{\partial y^{2}}
  31. ( 𝐚 ) = ( i a i e i ) ( i q i 𝐞 i ) = ( i a i q i ) (\mathbf{a}\cdot\nabla)=\left(\sum_{i}a^{i}e_{i}\right)\cdot\left(\sum_{i}% \frac{\partial}{\partial q^{i}}\mathbf{e}^{i}\right)=\left(\sum_{i}a^{i}\frac{% \partial}{\partial q^{i}}\right)
  32. × 𝐚 = i , j , k 𝐞 k ϵ i j k a i q i = \nabla\times\mathbf{a}=\sum_{i,j,k}\mathbf{e}_{k}\epsilon^{ijk}\frac{\partial a% _{i}}{\partial q^{i}}=
  33. 1 cos ( ϕ ) ( ( sin ( ϕ ) a 1 y + a 3 y - a 2 z ) 𝐞 1 + ( a 1 z + sin ( ϕ ) ( a 3 z - a 1 x ) - a 3 x ) 𝐞 2 + ( a 2 x - a 1 y - sin ( ϕ ) a 3 y ) 𝐞 3 ) . \frac{1}{\cos(\phi)}\left(\left(\sin(\phi)\frac{\partial a^{1}}{\partial y}+% \frac{\partial a^{3}}{\partial y}-\frac{\partial a^{2}}{\partial z}\right)% \mathbf{e}_{1}+\left(\frac{\partial a^{1}}{\partial z}+\sin(\phi)\left(\frac{% \partial a^{3}}{\partial z}-\frac{\partial a^{1}}{\partial x}\right)-\frac{% \partial a^{3}}{\partial x}\right)\mathbf{e}_{2}+\left(\frac{\partial a^{2}}{% \partial x}-\frac{\partial a^{1}}{\partial y}-\sin(\phi)\frac{\partial a^{3}}{% \partial y}\right)\mathbf{e}_{3}\right).

Slab_pull.html

  1. F s p = K × Δ ρ × L × A F_{sp}=K\times\Delta\rho\times L\times\sqrt{A}
  2. 4.2 < v a r > g 4.2<var>g

Slash_distribution.html

  1. f ( x ) = φ ( 0 ) - φ ( x ) x 2 . f(x)=\frac{\varphi(0)-\varphi(x)}{x^{2}}.
  2. lim x 0 f ( x ) = φ ( 0 ) 2 = 1 2 2 π \lim_{x\to 0}f(x)=\frac{\varphi(0)}{2}=\frac{1}{2\sqrt{2\pi}}
  3. { 2 π x f ( x ) + 2 π ( x 2 + 2 ) f ( x ) - 2 = 0 , f ( 1 ) = 1 2 π - 1 2 e π } \left\{\begin{array}[]{l}2\sqrt{\pi}xf^{\prime}(x)+2\sqrt{\pi}\left(x^{2}+2% \right)f(x)-\sqrt{2}=0,\\ f(1)=\frac{1}{\sqrt{2\pi}}-\frac{1}{\sqrt{2e\pi}}\end{array}\right\}

Slater_integrals.html

  1. c c
  2. c k ( , m , , m ) = d 2 Ω Y m ( Ω ) * Y m ( Ω ) Y k m - m ( Ω ) c^{k}(\ell,m,\ell^{\prime},m^{\prime})=\int d^{2}\Omega\ Y_{\ell}^{m}(\Omega)^% {*}Y_{\ell^{\prime}}^{m^{\prime}}(\Omega)Y_{k}^{m-m^{\prime}}(\Omega)
  3. Y m Y m = ′′ A ^ ′′ ( , m , , m , ) Y ′′ m + m , Y_{\ell}^{m}Y_{\ell^{\prime}}^{m^{\prime}}=\sum_{\ell^{\prime\prime}}\hat{A}^{% \ell^{\prime\prime}}(\ell,m,\ell^{\prime},m^{\prime},)Y_{\ell^{\prime\prime}}^% {m+m^{\prime}},
  4. Y * Y^{*}
  5. Y m Y m Y L - M d 2 Ω = ( - 1 ) m + m A ^ L ( , m , , m ) = ( - 1 ) m c L ( , - m , , m ) . \int Y_{\ell}^{m}Y_{\ell^{\prime}}^{m^{\prime}}Y_{L}^{-M}d^{2}\Omega=(-1)^{m+m% ^{\prime}}\hat{A}^{L}(\ell,m,\ell^{\prime},m^{\prime})=(-1)^{m}c^{L}(\ell,-m,% \ell^{\prime},m^{\prime}).
  6. Y m Y m = ′′ ( - 1 ) m c ′′ ( , - m , , m , ) Y ′′ m + m , Y_{\ell}^{m}Y_{\ell^{\prime}}^{m^{\prime}}=\sum_{\ell^{\prime\prime}}(-1)^{m^{% \prime}}c^{\ell^{\prime\prime}}(\ell,-m,\ell^{\prime},m^{\prime},)Y_{\ell^{% \prime\prime}}^{m+m^{\prime}},
  7. c k ( , m , , m ) \displaystyle c^{k}(\ell,m,\ell^{\prime},m^{\prime})

Sliced_inverse_regression.html

  1. x ¯ \underline{x}
  2. E ( x ¯ | y ) E(\underline{x}\,|\,y)
  3. Y \,Y
  4. X \R p X\in\R^{p}
  5. Y = f ( β 1 X , , β k X , ε ) ( 1 ) Y=f(\beta_{1}^{\top}X,\ldots,\beta_{k}^{\top}X,\varepsilon)\quad\quad\quad% \quad\quad(1)
  6. β 1 , , β k \beta_{1},\ldots,\beta_{k}
  7. k \,k
  8. p \,p
  9. f \;f
  10. \R k + 1 \R^{k+1}
  11. k \,k
  12. ε \varepsilon
  13. E [ ε | X ] = 0 E[\varepsilon|X]=0
  14. σ 2 \sigma^{2}
  15. Y \,Y
  16. X \R p X\in\R^{p}
  17. k \,k
  18. p \,p
  19. k \,k
  20. ( 1 ) \,(1)
  21. Y \,Y
  22. X \,X
  23. X \,X
  24. k \,k
  25. ( β 1 X , , β k X ) (\beta_{1}^{\top}X,\ldots,\beta_{k}^{\top}X)
  26. X \,X
  27. Y \,Y
  28. β i s \,\beta_{i}^{\prime}s
  29. k \,k
  30. \R n \R^{n}
  31. U \R n U\in\R^{n}
  32. a ¯ , b ¯ U a ¯ + b ¯ U \underline{a},\underline{b}\in U\Rightarrow\underline{a}+\underline{b}\in U
  33. a ¯ U , λ \R λ a ¯ U \underline{a}\in U,\lambda\in\R\Rightarrow\lambda\underline{a}\in U
  34. a ¯ 1 , , a ¯ r \R n \underline{a}_{1},\ldots,\underline{a}_{r}\in\R^{n}
  35. V := L ( a ¯ 1 , , a ¯ r ) V:=L(\underline{a}_{1},\ldots,\underline{a}_{r})
  36. a ¯ 1 , , a ¯ r \underline{a}_{1},\ldots,\underline{a}_{r}
  37. V \,V
  38. V \,V
  39. B = { b ¯ 1 , , b ¯ r } B=\{\underline{b}_{1},\ldots,\underline{b}_{r}\}
  40. V \,V
  41. V \,V
  42. V := L ( b ¯ 1 , , b ¯ r ) V:=L(\underline{b}_{1},\ldots,\underline{b}_{r})
  43. V ( \R n ) \,V(\in\R^{n})
  44. V \,V
  45. n \,n
  46. \R n \R^{n}
  47. \R n \R^{n}
  48. k \,k
  49. k \,k
  50. k \,k
  51. [ 0 , 1 ] [0,1]
  52. 10 10
  53. E [ Y | X = x ] \,E[Y|X=x]
  54. \R p \R^{p}
  55. E [ X | Y = y ] \,E[X|Y=y]
  56. \R p \R^{p}
  57. p \,p
  58. E [ E [ X | Y ] ] = E [ X ] \,E[E[X|Y]]=E[X]
  59. E [ X | Y = y ] - E [ X ] \,E[X|Y=y]-E[X]
  60. p \,p
  61. \R p \R^{p}
  62. k \,k
  63. Σ x x β i s \,\Sigma_{xx}\beta_{i}\,^{\prime}s
  64. Y \,Y
  65. H \,H
  66. m ^ h \,\hat{m}_{h}
  67. m ( y ) \,m(y)
  68. y i s \,y_{i}\,^{\prime}s
  69. k \,k
  70. Σ x x β i s \,\Sigma_{xx}\beta_{i}\,^{\prime}s
  71. b ¯ \R p : E [ b X | β 1 X = β 1 x , , β k X = β k x ) = c 0 + i = 1 k c i β i x \forall\,\underline{b}\in\R^{p}:\,E[b^{\top}X|\beta_{1}^{\top}X=\beta_{1}^{% \top}x,\ldots,\beta_{k}^{\top}X=\beta_{k}^{\top}x)=c_{0}+\sum_{i=1}^{k}c_{i}% \beta_{i}^{\top}x
  72. β 1 X , , β k X \beta_{1}X,\ldots,\beta_{k}X
  73. c 0 , , c K c_{0},\ldots,c_{K}
  74. X \,X
  75. ( 1 ) \,(1)
  76. E [ X | Y = y ] - E [ X ] \,E[X|Y=y]-E[X]
  77. Σ x x β k ( k = 1 , , K ) \,\Sigma_{xx}\beta_{k}(k=1,\ldots,K)
  78. Σ x x = C o v ( X ) \,\Sigma_{xx}=Cov(X)
  79. m ^ h s \,\hat{m}_{h}\,^{\prime}s
  80. X \,X
  81. Z = Σ x x - 1 / 2 { X - E ( X ) } \,Z=\Sigma_{xx}^{-1/2}\{X-E(X)\}
  82. m 1 ( y ) = E [ Z | Y = y ] \,m_{1}(y)=E[Z|Y=y]
  83. ( η 1 , , η k ) \,(\eta_{1},\ldots,\eta_{k})
  84. η i = Σ x x 1 / 2 β i \,\eta_{i}=\Sigma^{1/2}_{xx}\beta_{i}
  85. η i s \,\eta_{i}\,^{\prime}s
  86. c o v [ E [ Z | Y ] ] \,cov[E[Z|Y]]
  87. η i s \,\eta_{i}\,^{\prime}s
  88. η k ( k = 1 , , K ) \,\eta_{k}(k=1,\ldots,K)
  89. K \,K
  90. C o v { m 1 ( y ) } \,Cov\{m_{1}(y)\}
  91. V ^ = n - 1 i = 1 S n s z ¯ s z ¯ s \hat{V}=n^{-1}\sum_{i=1}^{S}n_{s}\bar{z}_{s}\bar{z}_{s}^{\top}
  92. λ ^ i \hat{\lambda}_{i}
  93. η ^ i \hat{\eta}_{i}
  94. V ^ \hat{V}
  95. C o v ( X | Y ) \,Cov(X|Y)
  96. Σ x x \,\Sigma_{xx}
  97. X \,X
  98. X \,X
  99. Z = Σ x x - 1 / 2 { X - E ( X ) } \,Z=\Sigma_{xx}^{-1/2}\{X-E(X)\}
  100. ( 1 ) \,(1)
  101. Y = f ( η 1 Z , , η k Z , ε ) Y=f(\eta_{1}^{\top}Z,\ldots,\eta_{k}^{\top}Z,\varepsilon)
  102. η k = β k Σ x x 1 / 2 k \,\eta_{k}=\beta_{k}\Sigma_{xx}^{1/2}\quad\forall\;k
  103. E [ Z ] = 0 \,E[Z]=0
  104. C o v ( Z ) = I \,Cov(Z)=I
  105. y i \,y_{i}
  106. S \,S
  107. H s ( s = 1 , , S ) . n s \,H_{s}(s=1,\ldots,S).\;n_{s}
  108. I H s \,I_{H_{s}}
  109. n s = i = 1 n I H s ( y i ) n_{s}=\sum_{i=1}^{n}I_{H_{s}}(y_{i})
  110. z i \,z_{i}
  111. m ^ 1 \,\hat{m}_{1}
  112. m 1 \,m_{1}
  113. z ¯ s = n s - 1 i = 1 n z i I H s ( y i ) \,\bar{z}_{s}=n_{s}^{-1}\sum_{i=1}^{n}z_{i}I_{H_{s}}(y_{i})
  114. C o v { m 1 ( y ) } \,Cov\{m_{1}(y)\}
  115. V ^ = n - 1 i = 1 S n s z ¯ s z ¯ s \,\hat{V}=n^{-1}\sum_{i=1}^{S}n_{s}\bar{z}_{s}\bar{z}_{s}^{\top}
  116. λ ^ i \,\hat{\lambda}_{i}
  117. η ^ i \,\hat{\eta}_{i}
  118. V ^ \,\hat{V}
  119. β ^ i = Σ ^ x x - 1 / 2 η ^ i \,\hat{\beta}_{i}=\hat{\Sigma}_{xx}^{-1/2}\hat{\eta}_{i}

Slope_stability_analysis.html

  1. τ = σ tan ϕ + c \tau=\sigma^{\prime}\tan\phi^{\prime}+c^{\prime}
  2. τ \tau
  3. σ = σ - u \sigma^{\prime}=\sigma-u
  4. σ \sigma
  5. u u
  6. ϕ \phi^{\prime}
  7. c c^{\prime}
  8. τ = c \tau=c^{\prime}
  9. E r , E l E_{r},E_{l}
  10. S r , S l S_{r},S_{l}
  11. F v = 0 = W - N cos α - T sin α F h = 0 = k W + N sin α - T cos α \begin{aligned}\displaystyle\sum F_{v}=0&\displaystyle=W-N\cos\alpha-T\sin% \alpha\\ \displaystyle\sum F_{h}=0&\displaystyle=kW+N\sin\alpha-T\cos\alpha\end{aligned}
  12. k k
  13. N N
  14. N = W cos α - k W sin α . N=W\cos\alpha-kW\sin\alpha\,.
  15. M = 0 = j ( W j x j - T j R j - N j f j - k W j e j ) \sum M=0=\sum_{j}(W_{j}x_{j}-T_{j}R_{j}-N_{j}f_{j}-kW_{j}e_{j})
  16. j j
  17. x j , R j , f j , e j x_{j},R_{j},f_{j},e_{j}
  18. j T j R j = j [ W j x j - ( W j cos α j - k W j sin α j ) f j - k W j e j ] \sum_{j}T_{j}R_{j}=\sum_{j}[W_{j}x_{j}-(W_{j}\cos\alpha_{j}-kW_{j}\sin\alpha_{% j})f_{j}-kW_{j}e_{j}]
  19. j τ l j R j = l j R j σ j tan ϕ + l j R j c = R j ( N j - u j l j ) tan ϕ + l j R j c \sum_{j}\tau l_{j}R_{j}=l_{j}R_{j}\sigma_{j}^{\prime}\tan\phi^{\prime}+l_{j}R_% {j}c^{\prime}=R_{j}(N_{j}-u_{j}l_{j})\tan\phi^{\prime}+l_{j}R_{j}c^{\prime}
  20. u j u_{j}
  21. Factor of safety = j τ l j R j j T j R j . \,\text{Factor of safety}=\frac{\sum_{j}\tau l_{j}R_{j}}{\sum_{j}T_{j}R_{j}}\,.
  22. F = j [ c l j + ( W j - u j l j ) tan ϕ ] ψ j j W j sin α j F=\cfrac{\sum_{j}\cfrac{\left[c^{\prime}l_{j}+(W_{j}-u_{j}l_{j})\tan\phi^{% \prime}\right]}{\psi_{j}}}{\sum_{j}W_{j}\sin\alpha_{j}}
  23. ψ j = cos α j + sin α j tan ϕ F \psi_{j}=\cos\alpha_{j}+\frac{\sin\alpha_{j}\tan\phi^{\prime}}{F}
  24. j j
  25. c c^{\prime}
  26. ϕ \phi^{\prime}
  27. l l
  28. W W
  29. u u
  30. F F

Slow-growing_hierarchy.html

  1. g k + 1 ( n ) = g k ( n ) + 1 g_{k+1}(n)=g_{k}(n)+1
  2. g α ( n ) = g α [ n ] ( n ) g_{\alpha}(n)=g_{\alpha[n]}(n)
  3. ω 2 \omega^{2}
  4. Π 1 1 \Pi^{1}_{1}

Slow_manifold.html

  1. d x d t = f ( x ) \frac{d{\vec{x}}}{dt}=\vec{f}(\vec{x})
  2. x ( t ) \vec{x}(t)
  3. x * \vec{x}^{*}
  4. d x d t = A x where A = d f d x ( x * ) . \frac{d{\vec{x}}}{dt}=A\vec{x}\quad\,\text{where }A=\frac{d\vec{f}}{d\vec{x}}(% \vec{x}^{*}).
  5. A A
  6. λ \lambda
  7. λ \lambda
  8. λ = 0 \lambda=0
  9. x ( t ) x(t)
  10. y ( t ) y(t)
  11. d x d t = - x y and d y d t = - y + x 2 - 2 y 2 \frac{dx}{dt}=-xy\quad\,\text{ and }\quad\frac{dy}{dt}=-y+x^{2}-2y^{2}
  12. y = x 2 y=x^{2}
  13. d x / d t = - x 3 dx/dt=-x^{3}
  14. y > - 1 / 2 y>-1/2
  15. d U d t = - V W + b V Z , d V d t = U W - b U Z , d W d t = - U V , d X d t = - Z , d Z d t = X + b U V . \begin{aligned}\displaystyle\frac{dU}{dt}&\displaystyle=-VW+bVZ,\\ \displaystyle\frac{dV}{dt}&\displaystyle=UW-bUZ,\\ \displaystyle\frac{dW}{dt}&\displaystyle=-UV,\\ \displaystyle\frac{dX}{dt}&\displaystyle=-Z,\\ \displaystyle\frac{dZ}{dt}&\displaystyle=X+bUV.\end{aligned}
  16. ± i \pm i
  17. X X
  18. Z Z
  19. u ( x , t ) u(x,t)
  20. u t = u 2 u x 2 on the domain - 1 < x < 1 \frac{\partial u}{\partial t}=u\frac{\partial^{2}u}{\partial x^{2}}\quad\,% \text{on the domain }-1<x<1
  21. 2 b u ± ( 1 - b ) u x = 0 on x = ± 1. 2bu\pm(1-b)\frac{\partial u}{\partial x}=0\quad\,\text{on }x=\pm 1.
  22. b b
  23. b = 0 b=0
  24. b = 1 b=1
  25. b b
  26. b t = 0 \frac{\partial b}{\partial t}=0
  27. ( b , u ( x ) ) (b,u(x))
  28. b = 0 b=0
  29. u = u=
  30. u = a u=a
  31. a > 0 a>0
  32. - π 2 a / 4 -\pi^{2}a/4
  33. u ( x , t ) = a ( t ) ( 1 - b x 2 ) u(x,t)=a(t)(1-bx^{2})
  34. a a
  35. d a d t = - 2 a 2 b and d b d t = 0. \frac{da}{dt}=-2a^{2}b\quad\,\text{and }\frac{db}{dt}=0.
  36. a a
  37. b b
  38. a a
  39. b b
  40. b b
  41. b b
  42. ϵ \epsilon
  43. W ( t ) W(t)
  44. d x = ε y d t and d y = - y d t + d W . dx=\varepsilon y\,dt\quad\,\text{and}\quad dy=-y\,dt+dW\,.
  45. y y
  46. y = - t exp ( s - t ) d W ( s ) y=\int_{-\infty}^{t}\exp(s-t)\,dW(s)
  47. x ( t ) x(t)
  48. exp ( s - t ) \exp(s-t)
  49. ( X ( t ) , Y ( t ) ) (X(t),Y(t))
  50. y = Y + - t exp ( s - t ) d W ( s ) and x = X - ε Y - ε - t exp ( s - t ) d W ( s ) y=Y+\int_{-\infty}^{t}\exp(s-t)\,dW(s)\quad\,\text{and}\quad x=X-\varepsilon Y% -\varepsilon\int_{-\infty}^{t}\exp(s-t)\,dW(s)
  51. d X = ε d W and d Y = - Y d t . dX=\varepsilon\,dW\quad\,\text{and}\quad dY=-Y\,dt.
  52. Y ( t ) 0 Y(t)\to 0
  53. X ( t ) = ϵ W ( t ) X(t)=\epsilon W(t)
  54. Y = 0 Y=0

Slowed_rotor.html

  1. μ \mu
  2. μ = V u = V Ω R \mu=\frac{V}{u}=\frac{V}{\Omega\cdot R}

Slowly_varying_envelope_approximation.html

  1. 2 E - μ 0 ε 0 2 E t 2 = 0. \nabla^{2}E-\mu_{0}\,\varepsilon_{0}\,\frac{\partial^{2}E}{\partial t^{2}}=0.
  2. E ( 𝐫 , t ) = { E 0 ( 𝐫 , t ) e i ( 𝐤 0 𝐫 - ω 0 t ) } , E(\mathbf{r},t)=\Re\left\{E_{0}(\mathbf{r},t)\,e^{i\,(\mathbf{k}_{0}\,\cdot\,% \mathbf{r}-\omega_{0}\,t)}\right\},
  3. { } \scriptstyle\Re\{\cdot\}
  4. | 2 E 0 k o 2 | | k 0 E 0 k o | \displaystyle\left|\frac{\partial^{2}E_{0}}{\partial{k_{o}}^{2}}\right|\ll% \left|k_{0}\,\frac{\partial E_{0}}{\partial k_{o}}\right|
  5. | 2 E 0 t 2 | | ω 0 E 0 t | , \displaystyle\left|\frac{\partial^{2}E_{0}}{\partial t^{2}}\right|\ll\left|% \omega_{0}\,\frac{\partial E_{0}}{\partial t}\right|,
  6. k 0 = | 𝐤 0 | . k_{0}=|\mathbf{k}_{0}|.
  7. 2 i 𝐤 0 E 0 + 2 i ω 0 μ 0 ε 0 E 0 t - ( k 0 2 - ω 0 2 μ 0 ε 0 ) E 0 = 0. 2\,i\,\mathbf{k}_{0}\,\cdot\nabla E_{0}+2\,i\,\omega_{0}\,\mu_{0}\,\varepsilon% _{0}\,\frac{\partial E_{0}}{\partial t}-\left(k_{0}^{2}-\omega_{0}^{2}\,\mu_{0% }\,\varepsilon_{0}\right)\,E_{0}=0.
  8. k 0 2 - ω 0 2 μ 0 ε 0 = 0. k_{0}^{2}-\omega_{0}^{2}\,\mu_{0}\,\varepsilon_{0}=0.\,
  9. 𝐤 0 E 0 + ω 0 μ 0 ε 0 E 0 t = 0. \mathbf{k}_{0}\cdot\nabla E_{0}+\omega_{0}\,\mu_{0}\,\varepsilon_{0}\,\frac{% \partial E_{0}}{\partial t}=0.
  10. Δ = 2 / x 2 + 2 / y 2 \scriptstyle\Delta_{\perp}=\partial^{2}/\partial x^{2}+\partial^{2}/\partial y% ^{2}
  11. k 0 E 0 z + ω 0 μ 0 ε 0 E 0 t - 1 2 i Δ E 0 = 0. k_{0}\frac{\partial E_{0}}{\partial z}+\omega_{0}\,\mu_{0}\,\varepsilon_{0}\,% \frac{\partial E_{0}}{\partial t}-\tfrac{1}{2}\,i\,\Delta_{\perp}E_{0}=0.

Slowness_(seismology).html

  1. Γ = i = 1 N s i Δ x i \Gamma=\sum_{i=1}^{N}s_{i}\Delta x_{i}

Small_control_property.html

  1. x ˙ = f ( x , u ) \dot{x}=f(x,u)
  2. ε > 0 \varepsilon>0
  3. δ > 0 \delta>0
  4. x < δ \|x\|<\delta
  5. u < ε \|u\|<\varepsilon

SMASH_(hash).html

  1. m 1 , m 2 , , m t m_{1},m_{2},...,m_{t}
  2. θ G F ( 2 n ) \theta\in GF(2^{n})
  3. h 0 = f ( i v ) i v h_{0}=f(iv)\oplus iv
  4. h i = h ( h i - 1 , m i ) = f ( h i 1 m i ) m i θ m i h_{i}=h(h_{i-1},m_{i})=f(h_{i_{1}}\oplus m_{i})\oplus m_{i}\oplus\theta m_{i}
  5. h t + 1 = f ( h t ) h t h_{t+1}=f(h_{t})\oplus h_{t}

Smooth_functor.html

  1. F : Hom 𝐕𝐞𝐜𝐭 ( T , U ) Hom 𝐕𝐞𝐜𝐭 ( F ( T ) , F ( U ) ) , F:\mathrm{Hom}_{\mathbf{Vect}}(T,U)\rightarrow\mathrm{Hom}_{\mathbf{Vect}}(F(T% ),F(U)),

Smoothed_octagon.html

  1. 8 - 4 2 - ln 2 2 2 - 1 0.902414 . \frac{8-4\sqrt{2}-\ln{2}}{2\sqrt{2}-1}\approx 0.902414\,.
  2. π 12 0.906899. \frac{\pi}{\sqrt{12}}\approx 0.906899.
  3. 4 + 4 2 5 + 4 2 0.906163 , \frac{4+4\sqrt{2}}{5+4\sqrt{2}}\approx 0.906163,
  4. 2 \sqrt{2}
  5. ( 2 + 2 , 0 ) (2+\sqrt{2},0)
  6. ( 2 , 0 ) (2,0)
  7. = 2 - 1 \ell=\sqrt{2}-1
  8. m = 1 2 4 m=\sqrt[4]{\frac{1}{2}}
  9. 2 x 2 - y 2 = m 2 \ell^{2}x^{2}-y^{2}=m^{2}
  10. { x = m cosh t y = m sinh t - < t < \begin{cases}x=\frac{m}{\ell}\cosh{t}\\ y=m\sinh{t}\end{cases}-\infty<t<\infty
  11. - ln 2 4 < t < ln 2 4 -\frac{\ln{2}}{4}<t<\frac{\ln{2}}{4}
  12. y = ± ( 2 + 1 ) ( x - 2 ) y=\pm\left(\sqrt{2}+1\right)\left(x-2\right)
  13. y = ± x . y=\pm\ell x.

Smoothness_(probability_theory).html

  1. d 0 | t | - β φ X ( t ) d 1 | t | - β as t d_{0}|t|^{-\beta}\leq\varphi_{X}(t)\leq d_{1}|t|^{-\beta}\quad\,\text{as }t\to\infty
  2. d 0 | t | β 0 exp ( - | t | β / γ ) φ X ( t ) d 1 | t | β 1 exp ( - | t | β / γ ) as t d_{0}|t|^{\beta_{0}}\exp\big(-|t|^{\beta}/\gamma\big)\leq\varphi_{X}(t)\leq d_% {1}|t|^{\beta_{1}}\exp\big(-|t|^{\beta}/\gamma\big)\quad\,\text{as }t\to\infty

Smoothstep.html

  1. smoothstep ( t ) = 3 t 2 - 2 t 3 \operatorname{smoothstep}(t)=3t^{2}-2t^{3}
  2. smootherstep ( t ) = 6 t 5 - 15 t 4 + 10 t 3 \operatorname{smootherstep}(t)=6t^{5}-15t^{4}+10t^{3}
  3. f ( t ) \displaystyle f(t)
  4. f ( 0 ) \displaystyle f(0)
  5. f ( 0 ) \displaystyle f^{\prime}(0)
  6. a 0 = 0 , a 1 = 0 , a 2 = 3 , a 3 = - 2 a_{0}=0,\;\;\;\;\;\;a_{1}=0,\;\;\;\;\;\;a_{2}=3,\;\;\;\;\;\;a_{3}=-2
  7. f ( t ) = - 2 t 3 + 3 t 2 f(t)=-2t^{3}+3t^{2}
  8. f ( t ) \displaystyle f(t)
  9. f ( 0 ) \displaystyle f(0)
  10. f ( 0 ) \displaystyle f^{\prime}(0)
  11. f ′′ ( 0 ) \displaystyle f^{\prime\prime}(0)
  12. a 0 = 0 , a 1 = 0 , a 2 = 0 , a 3 = 10 , a 4 = - 15 , a 5 = 6 a_{0}=0,\;\;\;\;\;\;a_{1}=0,\;\;\;\;\;\;a_{2}=0,\;\;\;\;\;\;a_{3}=10,\;\;\;\;% \;\;a_{4}=-15,\;\;\;\;\;\;a_{5}=6
  13. f ( t ) = 6 t 5 - 15 t 4 + 10 t 3 f(t)=6t^{5}-15t^{4}+10t^{3}

Snellius–Pothenot_problem.html

  1. α \alpha
  2. β \beta
  3. x + y = 2 π - α - β - C x+y=2\pi-\alpha-\beta-C
  4. π \pi
  5. AC sin x sin α = PC = BC sin y sin β . \frac{\rm{AC}\sin x}{\sin\alpha}=\rm{PC}=\frac{\rm{BC}\sin y}{\sin\beta}.
  6. ϕ \phi
  7. tan ϕ = BC sin α AC sin β . \tan\phi=\frac{\rm{BC}\sin\alpha}{\rm{AC}\sin\beta}.
  8. sin x sin y = tan ϕ . \frac{\sin x}{\sin y}=\tan\phi.
  9. tan ( π 4 - ϕ ) = 1 - tan ϕ tan ϕ + 1 \tan\left(\frac{\pi}{4}-\phi\right)=\frac{1-\tan\phi}{\tan\phi+1}
  10. tan [ ( x - y ) / 2 ] tan [ ( x + y ) / 2 ] = sin x - sin y sin x + sin y \frac{\tan[(x-y)/2]}{\tan[(x+y)/2]}=\frac{\sin x-\sin y}{\sin x+\sin y}
  11. tan 1 2 ( x - y ) = tan 1 2 ( α + β + C ) tan ( π 4 - ϕ ) . \tan\frac{1}{2}(x-y)=\tan\frac{1}{2}(\alpha+\beta+C)\tan\left(\frac{\pi}{4}-% \phi\right).
  12. α \alpha
  13. β \beta
  14. ϕ = atan2 ( BC sin α , AC sin β ) \phi=\operatorname{atan2}(\rm{BC}\sin\alpha,\rm{AC}\sin\beta)
  15. K = 2 π - α - β - C . K=2\pi-\alpha-\beta-C.
  16. W = 2 * atan [ tan ( π / 4 - ϕ ) tan ( 1 2 ( α + β + C ) ) ] . W=2*\operatorname{atan}[\tan(\pi/4-\phi)\tan(\frac{1}{2}(\alpha+\beta+C))].
  17. x = ( K + W ) / 2 x=(K+W)/2
  18. y = ( K - W ) / 2. y=(K-W)/2.
  19. | β | > | α | |\beta|>|\alpha|
  20. PC = BC sin y sin β \rm{PC}=\frac{\rm{BC}\sin y}{\sin\beta}
  21. PC = AC sin x sin α . \rm{PC}=\frac{\rm{AC}\sin x}{\sin\alpha}.
  22. PA = sqrt ( AC 2 + PC 2 - 2 * AC * PC * cos ( π - α - x ) ) . \rm{PA}=\operatorname{sqrt}(\rm{AC}^{2}+\rm{PC}^{2}-2*\rm{AC}*\rm{PC}*cos(\pi-% \alpha-x)).
  23. PB = sqrt ( BC 2 + PC 2 - 2 * BC * PC * cos ( π - β - y ) ) . \rm{PB}=\operatorname{sqrt}(\rm{BC}^{2}+\rm{PC}^{2}-2*\rm{BC}*\rm{PC}*\cos(\pi% -\beta-y)).
  24. α \alpha
  25. 2 α 2\alpha
  26. β \beta
  27. M O = A C 2 tan α MO=\frac{AC}{2\tan\alpha}
  28. β \beta
  29. s ( x ) = sin 2 ( x ) s(x)=\sin^{2}(x)
  30. A ( x , y , z ) = ( x + y + z ) 2 - 2 ( x 2 + y 2 + z 2 ) A(x,y,z)=(x+y+z)^{2}-2(x^{2}+y^{2}+z^{2})
  31. r 1 = s ( β ) r_{1}=s(\beta)
  32. r 2 = s ( α ) r_{2}=s(\alpha)
  33. r 3 = s ( α + β ) r_{3}=s(\alpha+\beta)
  34. Q 1 = B C 2 Q_{1}=BC^{2}
  35. Q 2 = A C 2 Q_{2}=AC^{2}
  36. Q 3 = A B 2 Q_{3}=AB^{2}
  37. R 1 = r 2 Q 3 / r 3 R_{1}=r_{2}Q_{3}/r_{3}
  38. R 2 = r 1 Q 3 / r 3 R_{2}=r_{1}Q_{3}/r_{3}
  39. C 0 = ( ( Q 1 + Q 2 + Q 3 ) ( r 1 + r 2 + r 3 ) - 2 ( Q 1 r 1 + Q 2 r 2 + Q 3 r 3 ) ) / ( 2 r 3 ) C_{0}=((Q_{1}+Q_{2}+Q_{3})(r_{1}+r_{2}+r_{3})-2(Q_{1}r_{1}+Q_{2}r_{2}+Q_{3}r_{% 3}))/(2r_{3})
  40. D 0 = r 1 r 2 A ( Q 1 , Q 2 , Q 3 ) / r 3 D_{0}=r_{1}r_{2}A(Q_{1},Q_{2},Q_{3})/r_{3}
  41. R 3 R_{3}
  42. ( R 3 - C 0 ) 2 = D 0 (R_{3}-C_{0})^{2}=D_{0}
  43. v 1 = 1 - ( R 1 + R 3 - Q 2 ) 2 / ( 4 R 1 R 3 ) v_{1}=1-(R_{1}+R_{3}-Q_{2})^{2}/(4R_{1}R_{3})
  44. v 2 = 1 - ( R 2 + R 3 - Q 1 ) 2 / ( 4 R 2 R 3 ) v_{2}=1-(R_{2}+R_{3}-Q_{1})^{2}/(4R_{2}R_{3})
  45. A P 2 = v 1 R 1 / r 2 = v 1 Q 3 / r 3 AP^{2}=v_{1}R_{1}/r_{2}=v_{1}Q_{3}/r_{3}
  46. B P 2 = v 2 R 2 / r 1 = v 2 Q 3 / r 3 BP^{2}=v2R_{2}/r_{1}=v_{2}Q_{3}/r_{3}
  47. α + β + C = k π , ( k = 1 , 2 , ) \alpha+\beta+C=k\pi,(k=1,2,\cdots)

Snub_octahedron.html

  1. s { 3 , 4 } s\begin{Bmatrix}3,4\end{Bmatrix}
  2. s { 4 3 } s\begin{Bmatrix}4\\ 3\end{Bmatrix}

Software_testability.html

  1. F S F_{S}
  2. I k I_{k}
  3. O k O_{k}
  4. F S : I O F_{S}:I\to O
  5. ( I k , O k ) (I_{k},O_{k})
  6. Σ \Sigma
  7. I t I_{t}
  8. O t O_{t}
  9. τ = ( I t , O t ) \tau=(I_{t},O_{t})
  10. τ Σ \tau\in\Sigma
  11. τ Σ \tau\not\in\Sigma
  12. τ \tau
  13. Σ \Sigma
  14. 1 Σ 1_{\Sigma}
  15. Σ \Sigma

Solar_eclipse_of_March_9,_1997.html

  1. × \times

Solar_micro-inverter.html

  1. V o c V_{oc}

Solid-state_dye_lasers.html

  1. Δ ν \Delta\nu
  2. Δ λ \Delta\lambda

Solinas_prime.html

  1. 2 a ± 2 b ± 1 2^{a}\pm 2^{b}\pm 1
  2. 0 < b < a 0<b<a

Soliton_distribution.html

  1. p ( 1 ) = 1 N , p(1)=\frac{1}{N},
  2. p ( k ) = 1 k ( k - 1 ) ( k = 2 , 3 , , N ) . p(k)=\frac{1}{k(k-1)}\qquad(k=2,3,\dots,N).\,
  3. t ( i ) = 1 i M , ( i = 1 , 2 , , M - 1 ) , t(i)=\frac{1}{iM},\qquad\qquad(i=1,2,\dots,M-1),\,
  4. t ( i ) = ln ( R / δ ) M , ( i = M ) , t(i)=\frac{\ln(R/\delta)}{M},\qquad(i=M),\,
  5. t ( i ) = 0 , ( i = M + 1 , , N ) . t(i)=0,\qquad\qquad(i=M+1,\dots,N).\,

Solomon_Mikhlin.html

  1. s y m b o l 𝒜 ( ω ) s y m b o l u = Δ 2 s y m b o l u + ω ( \cdotsymbol u ) symbol{\mathcal{A}}(\omega)symbol{u}=\Delta_{2}symbol{u}+\omega\nabla\left(% \nabla\cdotsymbol{u}\right)
  2. u u
  3. Δ 2 \scriptstyle\Delta_{2}
  4. \scriptstyle\nabla
  5. \scriptstyle\nabla\cdot
  6. ω \omega
  7. A u = v ( s y m b o l x ) = n f ( s y m b o l x , s y m b o l θ ) r n u ( s y m b o l y ) d s y m b o l y Au=v(symbol{x})=\int_{\mathbb{R}^{n}}\frac{f(symbol{x},symbol{\theta})}{r^{n}}% u(symbol{y})\mathrm{d}symbol{y}
  8. x x
  9. r r
  10. y - x y-x
  11. \scriptstylesymbol θ = s y m b o l y - s y m b o l x r \scriptstylesymbol{\theta}=\frac{symbol{y}-symbol{x}}{r}
  12. n = 2 n=2
  13. n = 3 n=3
  14. y y
  15. x x
  16. 𝐲 - 𝐱 \mathbf{y-x}
  17. 𝐱 \mathbf{x}
  18. 𝐲 \mathbf{y}
  19. 𝐱 \mathbf{x}

Solution_of_triangles.html

  1. a , b , c ~{}a,b,c
  2. α , β , γ ~{}\alpha,\beta,\gamma
  3. a 2 = b 2 + c 2 - 2 b c cos α a^{2}=b^{2}+c^{2}-2bc\cos\alpha
  4. b 2 = a 2 + c 2 - 2 a c cos β b^{2}=a^{2}+c^{2}-2ac\cos\beta
  5. c 2 = a 2 + b 2 - 2 a b cos γ c^{2}=a^{2}+b^{2}-2ab\cos\gamma
  6. a sin α = b sin β = c sin γ \frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}
  7. α + β + γ = 180 \alpha+\beta+\gamma=180^{\circ}
  8. a - b a + b = tan [ 1 2 ( α - β ) ] tan [ 1 2 ( α + β ) ] . \frac{a-b}{a+b}=\frac{\mathrm{tan}[\frac{1}{2}(\alpha-\beta)]}{\mathrm{tan}[% \frac{1}{2}(\alpha+\beta)]}.
  9. sin β = 0.5 \sin\beta=0.5
  10. β \beta
  11. 30 30^{\circ}
  12. 150 150^{\circ}
  13. 0 0^{\circ}
  14. 180 180^{\circ}
  15. a , b , c a,b,c
  16. α , β \alpha,\beta
  17. α = arccos b 2 + c 2 - a 2 2 b c \alpha=\arccos\frac{b^{2}+c^{2}-a^{2}}{2bc}
  18. β = arccos a 2 + c 2 - b 2 2 a c . \beta=\arccos\frac{a^{2}+c^{2}-b^{2}}{2ac}.
  19. γ = 180 - α - β \gamma=180^{\circ}-\alpha-\beta
  20. β \beta
  21. a , b a,b
  22. γ \gamma
  23. c = a 2 + b 2 - 2 a b cos γ . c=\sqrt{a^{2}+b^{2}-2ab\cos\gamma}.
  24. α = arccos b 2 + c 2 - a 2 2 b c . \alpha=\arccos\frac{b^{2}+c^{2}-a^{2}}{2bc}.
  25. β = 180 - α - γ . \beta=180^{\circ}-\alpha-\gamma.
  26. b , c b,c
  27. β \beta
  28. γ \gamma
  29. sin γ = c b sin β . \sin\gamma=\frac{c}{b}\sin\beta.
  30. D = c b sin β ~{}D=\frac{c}{b}\sin\beta
  31. D > 1 D>1
  32. b b
  33. β 90 \beta\geqslant 90^{\circ}
  34. b c . b\leqslant c.
  35. D = 1 D=1
  36. γ = 90 \gamma=90^{\circ}
  37. D < 1 D<1
  38. b < c b<c
  39. γ \gamma
  40. γ = arcsin D ~{}\gamma=\arcsin D
  41. γ = 180 - γ ~{}\gamma^{\prime}=180^{\circ}-\gamma
  42. C C
  43. b b
  44. γ \gamma
  45. C C^{\prime}
  46. b b^{\prime}
  47. γ \gamma^{\prime}
  48. b c b\geqslant c
  49. β γ \beta\geqslant\gamma
  50. γ ~{}\gamma
  51. γ = arcsin D ~{}\gamma=\arcsin D
  52. γ \gamma
  53. α = 180 - β - γ \alpha=180^{\circ}-\beta-\gamma
  54. a = b sin α sin β a=b\ \frac{\sin\alpha}{\sin\beta}
  55. c c
  56. α , β \alpha,\beta
  57. γ = 180 - α - β . ~{}\gamma=180^{\circ}-\alpha-\beta.
  58. a = c sin α sin γ ; b = c sin β sin γ . a=c\ \frac{\sin\alpha}{\sin\gamma};\quad b=c\ \frac{\sin\beta}{\sin\gamma}.
  59. a , b , c a,b,c
  60. α + β + γ \alpha+\beta+\gamma
  61. tan c 2 cos α - β 2 = tan a + b 2 cos α + β 2 \tan\frac{c}{2}\cos\frac{\alpha-\beta}{2}=\tan\frac{a+b}{2}\cos\frac{\alpha+% \beta}{2}
  62. tan c 2 sin α - β 2 = tan a - b 2 sin α + β 2 \tan\frac{c}{2}\sin\frac{\alpha-\beta}{2}=\tan\frac{a-b}{2}\sin\frac{\alpha+% \beta}{2}
  63. cot γ 2 cos a - b 2 = tan α + β 2 cos a + b 2 \cot\frac{\gamma}{2}\cos\frac{a-b}{2}=\tan\frac{\alpha+\beta}{2}\cos\frac{a+b}% {2}
  64. cot γ 2 sin a - b 2 = tan α - β 2 sin a + b 2 . \cot\frac{\gamma}{2}\sin\frac{a-b}{2}=\tan\frac{\alpha-\beta}{2}\sin\frac{a+b}% {2}.
  65. a , b , c a,b,c
  66. α = arccos ( cos a - cos b cos c sin b sin c ) , \alpha=\arccos\left(\frac{\cos a-\cos b\ \cos c}{\sin b\ \sin c}\right),
  67. β = arccos ( cos b - cos c cos a sin c sin a ) , \beta=\arccos\left(\frac{\cos b-\cos c\ \cos a}{\sin c\ \sin a}\right),
  68. γ = arccos ( cos c - cos a cos b sin a sin b ) , \gamma=\arccos\left(\frac{\cos c-\cos a\ \cos b}{\sin a\ \sin b}\right),
  69. a , b a,b
  70. γ \gamma
  71. c c
  72. c = arccos ( cos a cos b + sin a sin b cos γ ) c=\arccos\left(\cos a\cos b+\sin a\sin b\cos\gamma\right)
  73. α , β \alpha,\beta
  74. α = arctan 2 sin a tan ( γ 2 ) sin ( b + a ) + cot ( γ 2 ) sin ( b - a ) \alpha=\arctan\ \frac{2\sin a}{\tan(\frac{\gamma}{2})\sin(b+a)+\cot(\frac{% \gamma}{2})\sin(b-a)}
  75. β = arctan 2 sin b tan ( γ 2 ) sin ( a + b ) + cot ( γ 2 ) sin ( a - b ) . \beta=\arctan\ \frac{2\sin b}{\tan(\frac{\gamma}{2})\sin(a+b)+\cot(\frac{% \gamma}{2})\sin(a-b)}.
  76. c = arctan ( sin a cos b - cos a sin b cos γ ) 2 + ( sin b sin γ ) 2 cos a cos b + sin a sin b cos γ , α = arctan sin a sin γ sin b cos a - cos b sin a cos γ , β = arctan sin b sin γ sin a cos b - cos a sin b cos γ , \begin{aligned}\displaystyle c&\displaystyle=\arctan\frac{\sqrt{(\sin a\cos b-% \cos a\sin b\cos\gamma)^{2}+(\sin b\sin\gamma)^{2}}}{\cos a\cos b+\sin a\sin b% \cos\gamma},\\ \displaystyle\alpha&\displaystyle=\arctan\frac{\sin a\sin\gamma}{\sin b\cos a-% \cos b\sin a\cos\gamma},\\ \displaystyle\beta&\displaystyle=\arctan\frac{\sin b\sin\gamma}{\sin a\cos b-% \cos a\sin b\cos\gamma},\end{aligned}
  77. b , c b,c
  78. β \beta
  79. b > arcsin ( sin c sin β ) b>\arcsin(\sin c\,\sin\beta)
  80. γ \gamma
  81. γ = arcsin ( sin c sin β sin b ) \gamma=\arcsin\left(\frac{\sin c\,\sin\beta}{\sin b}\right)
  82. b < c b<c
  83. γ \gamma
  84. 180 - γ ~{}180^{\circ}-\gamma
  85. a = 2 arctan { tan ( 1 2 ( b - c ) ) sin ( 1 2 ( β + γ ) ) sin ( 1 2 ( β - γ ) ) } , a=2\arctan\left\{\tan\left(\frac{1}{2}(b-c)\right)\frac{\sin\left(\frac{1}{2}(% \beta+\gamma)\right)}{\sin\left(\frac{1}{2}(\beta-\gamma)\right)}\right\},
  86. α = 2 \arccot { tan ( 1 2 ( β - γ ) ) sin ( 1 2 ( b + c ) ) sin ( 1 2 ( b - c ) ) } . \alpha=2\arccot\left\{\tan\left(\frac{1}{2}(\beta-\gamma)\right)\frac{\sin% \left(\frac{1}{2}(b+c)\right)}{\sin\left(\frac{1}{2}(b-c)\right)}\right\}.
  87. c c
  88. α , β \alpha,\beta
  89. γ \gamma
  90. γ = arccos ( sin α sin β cos c - cos α cos β ) , \gamma=\arccos(\sin\alpha\sin\beta\cos c-\cos\alpha\cos\beta),\,
  91. γ \gamma
  92. a = arccos ( cos α + cos β cos γ sin β sin γ ) a=\arccos\left(\frac{\cos\alpha+\cos\beta\cos\gamma}{\sin\beta\sin\gamma}\right)
  93. b = arccos ( cos β + cos γ cos α sin γ sin α ) b=\arccos\left(\frac{\cos\beta+\cos\gamma\cos\alpha}{\sin\gamma\sin\alpha}\right)
  94. a = arctan { 2 sin α cot ( c / 2 ) sin ( β + α ) + tan ( c / 2 ) sin ( β - α ) } , a=\arctan\left\{\frac{2\sin\alpha}{\cot(c/2)\sin(\beta+\alpha)+\tan(c/2)\sin(% \beta-\alpha)}\right\},
  95. b = arctan { 2 sin β cot ( c / 2 ) sin ( α + β ) + tan ( c / 2 ) sin ( α - β ) } , b=\arctan\left\{\frac{2\sin\beta}{\cot(c/2)\sin(\alpha+\beta)+\tan(c/2)\sin(% \alpha-\beta)}\right\},
  96. a a
  97. α , β \alpha,\beta
  98. b b
  99. b = arcsin ( sin a sin β sin α ) , b=\arcsin\left(\frac{\sin a\,\sin\beta}{\sin\alpha}\right),
  100. a a
  101. α > β \alpha>\beta
  102. b = π - arcsin ( sin a sin β sin α ) b=\pi-\arcsin\left(\frac{\sin a\,\sin\beta}{\sin\alpha}\right)
  103. c = 2 arctan { tan ( 1 2 ( a - b ) ) sin ( 1 2 ( α + β ) ) sin ( 1 2 ( α - β ) ) } , c=2\arctan\left\{\tan\left(\frac{1}{2}(a-b)\right)\frac{\sin\left(\frac{1}{2}(% \alpha+\beta)\right)}{\sin\left(\frac{1}{2}(\alpha-\beta)\right)}\right\},
  104. γ = 2 \arccot { tan ( 1 2 ( α - β ) ) sin ( 1 2 ( a + b ) ) sin ( 1 2 ( a - b ) ) } , \gamma=2\arccot\left\{\tan\left(\frac{1}{2}(\alpha-\beta)\right)\frac{\sin% \left(\frac{1}{2}(a+b)\right)}{\sin\left(\frac{1}{2}(a-b)\right)}\right\},
  105. α , β , γ \alpha,\beta,\gamma
  106. a = arccos ( cos α + cos β cos γ sin β sin γ ) , a=\arccos\left(\frac{\cos\alpha+\cos\beta\cos\gamma}{\sin\beta\sin\gamma}% \right),
  107. b = arccos ( cos β + cos γ cos α sin γ sin α ) , b=\arccos\left(\frac{\cos\beta+\cos\gamma\cos\alpha}{\sin\gamma\sin\alpha}% \right),
  108. c = arccos ( cos γ + cos α cos β sin α sin β ) . c=\arccos\left(\frac{\cos\gamma+\cos\alpha\cos\beta}{\sin\alpha\sin\beta}% \right).
  109. C C
  110. sin a = sin c sin A \sin a=\sin c\cdot\sin A
  111. tan a = sin b tan A \tan a=\sin b\cdot\tan A
  112. cos c = cos a cos b \cos c=\cos a\cdot\cos b
  113. tan b = tan c cos A \tan b=\tan c\cdot\cos A
  114. cos A = cos a sin B \cos A=\cos a\cdot\sin B
  115. cos c = cot A cot B \cos c=\cot A\cdot\cot B
  116. d d
  117. l l
  118. α , β \alpha,\beta
  119. d = sin α sin β sin ( α + β ) l = tan α tan β tan α + tan β l d=\frac{\sin\alpha\,\sin\beta}{\sin(\alpha+\beta)}\,l=\frac{\tan\alpha\,\tan% \beta}{\tan\alpha+\tan\beta}\,l
  120. α , β \alpha,\beta
  121. h h
  122. α , β \alpha,\beta
  123. l l
  124. h = sin α sin β sin ( β - α ) l = tan α tan β tan β - tan α l h=\frac{\sin\alpha\,\sin\beta}{\sin(\beta-\alpha)}\,l=\frac{\tan\alpha\,\tan% \beta}{\tan\beta-\tan\alpha}\,l
  125. λ A , \lambda_{\mathrm{A}},
  126. L A L_{\mathrm{A}}
  127. λ B , \lambda_{\mathrm{B}},
  128. L B L_{\mathrm{B}}
  129. A B C ABC
  130. C C
  131. a = 90 o - λ B a=90^{\mathrm{o}}-\lambda_{\mathrm{B}}\,
  132. b = 90 o - λ A b=90^{\mathrm{o}}-\lambda_{\mathrm{A}}\,
  133. γ = L A - L B \gamma=L_{\mathrm{A}}-L_{\mathrm{B}}\,
  134. AB = R arccos { sin λ A sin λ B + cos λ A cos λ B cos ( L A - L B ) } , \mathrm{AB}=R\arccos\left\{\sin\lambda_{\mathrm{A}}\,\sin\lambda_{\mathrm{B}}+% \cos\lambda_{\mathrm{A}}\,\cos\lambda_{\mathrm{B}}\,\cos\left(L_{\mathrm{A}}-L% _{\mathrm{B}}\right)\right\},
  135. R R

Solvency_ratio.html

  1. n e t . a s s e t s ÷ n e t . p r e m i u m . w r i t t e n net.assets\div net.premium.written

Solvent_effects.html

  1. 𝐊 T = [ c i s - e n o l ] [ d i k e t o ] {\mathbf{K}}_{\mathrm{T}}=\frac{[cis-enol]}{[diketo]}

Sombrero_function.html

  1. somb ( ρ ) = 2 J 1 ( π ρ ) π ρ \operatorname{somb}(\rho)=\frac{2J_{1}(\pi\rho)}{\pi\rho}
  2. 2 2
  3. s o m b ( 0 ) = 1 somb(0)=1
  4. π π
  5. somb ( ρ ) = 2 J 1 ( ρ ) ρ \operatorname{somb}^{\prime}(\rho)=\frac{2J_{1}(\rho)}{\rho}

Sommerfeld_expansion.html

  1. β \beta
  2. β \beta
  3. - H ( ε ) e β ( ε - μ ) + 1 d ε = - μ H ( ε ) d ε + π 2 6 ( 1 β ) 2 H ( μ ) + O ( 1 β μ ) 4 \int_{-\infty}^{\infty}\frac{H(\varepsilon)}{e^{\beta(\varepsilon-\mu)}+1}\,% \mathrm{d}\varepsilon=\int_{-\infty}^{\mu}H(\varepsilon)\,\mathrm{d}% \varepsilon+\frac{\pi^{2}}{6}\left(\frac{1}{\beta}\right)^{2}H^{\prime}(\mu)+O% \left(\frac{1}{\beta\mu}\right)^{4}
  4. H ( μ ) H^{\prime}(\mu)
  5. H ( ε ) H(\varepsilon)
  6. ε = μ \varepsilon=\mu
  7. O ( x n ) O(x^{n})
  8. x n x^{n}
  9. H ( ε ) H(\varepsilon)
  10. ε - \varepsilon\rightarrow-\infty
  11. ε \varepsilon
  12. ε \varepsilon\rightarrow\infty
  13. H ( ε ) H(\varepsilon)
  14. β \beta
  15. μ \mu
  16. β \beta
  17. τ 2 \tau^{2}
  18. β - 1 = τ = k B T \beta^{-1}=\tau=k_{B}T
  19. τ x = ε - μ \tau x=\varepsilon-\mu
  20. I = - H ( ε ) e β ( ε - μ ) + 1 d ε = τ - H ( μ + τ x ) e x + 1 d x , I=\int_{-\infty}^{\infty}\frac{H(\varepsilon)}{e^{\beta(\varepsilon-\mu)}+1}\,% \mathrm{d}\varepsilon=\tau\int_{-\infty}^{\infty}\frac{H(\mu+\tau x)}{e^{x}+1}% \,\mathrm{d}x\,,
  21. I = I 1 + I 2 I=I_{1}+I_{2}
  22. I 1 I_{1}
  23. x - x x\rightarrow-x
  24. I = τ - 0 H ( μ + τ x ) e x + 1 d x I 1 + τ 0 H ( μ + τ x ) e x + 1 d x I 2 . I=\underbrace{\tau\int_{-\infty}^{0}\frac{H(\mu+\tau x)}{e^{x}+1}\,\mathrm{d}x% }_{I_{1}}+\underbrace{\tau\int_{0}^{\infty}\frac{H(\mu+\tau x)}{e^{x}+1}\,% \mathrm{d}x}_{I_{2}}\,.
  25. I 1 = τ - 0 H ( μ + τ x ) e x + 1 d x = τ 0 H ( μ - τ x ) e - x + 1 d x I_{1}=\tau\int_{-\infty}^{0}\frac{H(\mu+\tau x)}{e^{x}+1}\,\mathrm{d}x=\tau% \int_{0}^{\infty}\frac{H(\mu-\tau x)}{e^{-x}+1}\,\mathrm{d}x\,
  26. I 1 I_{1}
  27. 1 e - x + 1 = 1 - 1 e x + 1 , \frac{1}{e^{-x}+1}=1-\frac{1}{e^{x}+1}\,,
  28. I 1 = τ 0 H ( μ - τ x ) d x - τ 0 H ( μ - τ x ) e x + 1 d x I_{1}=\tau\int_{0}^{\infty}H(\mu-\tau x)\,\mathrm{d}x-\tau\int_{0}^{\infty}% \frac{H(\mu-\tau x)}{e^{x}+1}\,\mathrm{d}x\,
  29. - τ d x = d ε -\tau\mathrm{d}x=\mathrm{d}\varepsilon
  30. I 1 I_{1}
  31. I = I 1 + I 2 I=I_{1}+I_{2}
  32. I = - μ H ( ε ) d ε + τ 0 H ( μ + τ x ) - H ( μ - τ x ) e x + 1 d x I=\int_{-\infty}^{\mu}H(\varepsilon)\,\mathrm{d}\varepsilon+\tau\int_{0}^{% \infty}\frac{H(\mu+\tau x)-H(\mu-\tau x)}{e^{x}+1}\,\mathrm{d}x\,
  33. τ \tau
  34. H ( ε ) H(\varepsilon)
  35. Δ H = H ( μ + τ x ) - H ( μ - τ x ) 2 τ x H ( μ ) + , \Delta H=H(\mu+\tau x)-H(\mu-\tau x)\approx 2\tau xH^{\prime}(\mu)+\cdots\,,
  36. I = - μ H ( ε ) d ε + 2 τ 2 H ( μ ) 0 x d x e x + 1 I=\int_{-\infty}^{\mu}H(\varepsilon)\,\mathrm{d}\varepsilon+2\tau^{2}H^{\prime% }(\mu)\int_{0}^{\infty}\frac{x\mathrm{d}x}{e^{x}+1}\,
  37. 0 x d x e x + 1 = π 2 12 \int_{0}^{\infty}\frac{x\mathrm{d}x}{e^{x}+1}=\frac{\pi^{2}}{12}
  38. I = - H ( ε ) e β ( ε - μ ) + 1 d ε - μ H ( ε ) d ε + π 2 6 β 2 H ( μ ) I=\int_{-\infty}^{\infty}\frac{H(\varepsilon)}{e^{\beta(\varepsilon-\mu)}+1}\,% \mathrm{d}\varepsilon\approx\int_{-\infty}^{\mu}H(\varepsilon)\,\mathrm{d}% \varepsilon+\frac{\pi^{2}}{6\beta^{2}}H^{\prime}(\mu)\,

Somos_sequence.html

  1. ( a 0 , a 1 , a 2 , ) (a_{0},a_{1},a_{2},\ldots)
  2. a n a n - k = a n - 1 a n - k + 1 + a n - 2 a n - k + 2 + + a n - ( k - 1 ) / 2 a n - ( k + 1 ) / 2 a_{n}a_{n-k}=a_{n-1}a_{n-k+1}+a_{n-2}a_{n-k+2}+\cdots+a_{n-(k-1)/2}a_{n-(k+1)/2}
  3. a n a n - k = a n - 1 a n - k + 1 + a n - 2 a n - k + 2 + + ( a n - k / 2 ) 2 a_{n}a_{n-k}=a_{n-1}a_{n-k+1}+a_{n-2}a_{n-k+2}+\cdots+(a_{n-k/2})^{2}
  4. a n a n - 4 = a n - 1 a n - 3 + a n - 2 2 a_{n}a_{n-4}=a_{n-1}a_{n-3}+a_{n-2}^{2}
  5. a n a n - 5 = a n - 1 a n - 4 + a n - 2 a n - 3 . a_{n}a_{n-5}=a_{n-1}a_{n-4}+a_{n-2}a_{n-3}\,.
  6. a n = a n - 1 a n - 3 + a n - 2 2 a n - 4 a_{n}=\frac{a_{n-1}a_{n-3}+a_{n-2}^{2}}{a_{n-4}}
  7. a n = a n - 1 a n - 4 + a n - 2 a n - 3 a n - 5 . a_{n}=\frac{a_{n-1}a_{n-4}+a_{n-2}a_{n-3}}{a_{n-5}}.

Sorption_calorimetry.html

  1. n w = P v a p d t H w v a p n_{w}=\frac{\int{P^{vap}dt}}{H^{vap}_{w}}
  2. a w = 1 - P v a p P m a x v a p a_{w}=1-\frac{P^{vap}}{P^{vap}_{max}}

Soundness_(interactive_proof).html

  1. y L y\not\in L
  2. ( 𝒫 ~ ) (\tilde{\mathcal{P}})
  3. y L y\not\in L
  4. Pr [ ( , ( accept ) ) ( 𝒫 ~ ) ( y ) ( 𝒱 ) ( y ) ] < 2 - 80 . \Pr[(\perp,(\,\text{accept}))\leftarrow(\tilde{\mathcal{P}})(y)\leftrightarrow% (\mathcal{V})(y)]<2^{-80}.
  5. 1 / poly ( | y | ) \leq 1/\mathrm{poly}(|y|)
  6. \ell
  7. ϵ \epsilon
  8. ϵ \epsilon^{\ell}

Southampton_BASIC_System.html

  1. x x
  2. sin x \sin x
  3. cos x \cos x
  4. arctan x \arctan x
  5. x \sqrt{x}
  6. log x \log x
  7. e x e^{x}
  8. x x
  9. x x
  10. - x -x
  11. x x
  12. x x
  13. ( 1 1 2 1 0 2 0 2 1 ) \begin{pmatrix}1&1&2\\ 1&0&2\\ 0&2&1\end{pmatrix}
  14. ( 0 0 1 0 1 0 1 0 0 ) \begin{pmatrix}0&0&1\\ 0&1&0\\ 1&0&0\end{pmatrix}
  15. ( 2 1 1 2 0 1 1 2 0 ) \begin{pmatrix}2&1&1\\ 2&0&1\\ 1&2&0\end{pmatrix}
  16. ( 2 2 1 1 - 1 0 4 - 3 - 2 ) \begin{pmatrix}2&2&1\\ 1&-1&0\\ 4&-3&-2\end{pmatrix}

Space-filling_tree.html

  1. T s q u a r e T_{square}
  2. T s q u a r e T_{square}
  3. [ 0 , 1 ] 2 2 [0,1]^{2}\subset\Re^{2}
  4. 3 \Re^{3}
  5. n \Re^{n}
  6. 3 \Re^{3}

Space_time_(chemical_engineering).html

  1. τ = N A O 1 ( - r A ) V R d f A \tau=N_{AO}\int\frac{1}{(-r_{A})V_{R}}\,df_{A}
  2. τ = C A O 1 ( - r A ) d f A \tau=C_{AO}\int\frac{1}{(-r_{A})}\,df_{A}
  3. τ = C A i n - C A o u t ( - r A F ) \tau=\frac{C_{Ain}-C_{Aout}}{(-r_{AF})}
  4. τ = C A O ( R + 1 ) 1 ( - r A ) d f A \tau=C_{AO}(R+1)\int\frac{1}{(-r_{A})}\,df_{A}
  5. - r A -r_{A}
  6. V R = V R i n i t i a l ( 1 - δ A f A ) V_{R}=V_{Rinitial}(1-\delta_{A}f_{A})
  7. τ = N A O 1 ( - r A ) V R ( 1 - δ A f A ) d f A \tau=N_{AO}\int\frac{1}{(-r_{A})V_{R}(1-\delta_{A}f_{A})}\,df_{A}
  8. τ = C A O 1 ( - r A ) ( 1 - δ A f A ) d f A \tau=C_{AO}\int\frac{1}{(-r_{A})(1-\delta_{A}f_{A})}\,df_{A}
  9. τ = C A i n - C A o u t ( - r A F ) ( 1 - δ A f A ) \tau=\frac{C_{Ain}-C_{Aout}}{(-r_{AF})(1-\delta_{A}f_{A})}

Spaghettification.html

  1. F = μ l m 4 r 3 F=\frac{\mu lm}{4r^{3}}
  2. μ \mu

Sparse_matrix_converter.html

  1. ϕ 1 \phi_{1}
  2. ( 0... π / 6 ) (0...\pi/6)
  3. ϕ 2 \phi_{2}
  4. ( 0... π / 6 ) (0...\pi/6)
  5. i ¯ \bar{i}
  6. τ a c \tau_{ac}
  7. τ a b \tau_{ab}
  8. s A , s B s_{A},s_{B}
  9. s C s_{C}
  10. u a c , u a b , i A u_{ac},u_{ab},i_{A}
  11. i C i_{C}

Sparse_ruler.html

  1. L L
  2. m m
  3. a 1 , a 2 , , a m a_{1},a_{2},...,a_{m}
  4. 0 = a 1 < a 2 < < a m = L 0=a_{1}<a_{2}<...<a_{m}=L
  5. a 1 a_{1}
  6. a m a_{m}
  7. K K
  8. 0 K L 0<=K<=L
  9. a i a_{i}
  10. a j a_{j}
  11. a j - a i = K a_{j}-a_{i}=K
  12. L L
  13. m - 1 m-1
  14. L + 1 L+1
  15. m m
  16. m ( m - 1 ) / 2 m(m-1)/2
  17. L L
  18. m m
  19. a j - a i a_{j}-a_{i}
  20. m m
  21. m m
  22. m ( m - 1 ) / 2 m(m-1)/2
  23. { 0 , 2 , 7 , 14 , 15 , 18 , 24 } \{0,2,7,14,15,18,24\}
  24. { 0 , 2 , 7 , 13 , 16 , 17 , 25 } \{0,2,7,13,16,17,25\}
  25. { 0 , 5 , 7 , 13 , 16 , 17 , 31 } \{0,5,7,13,16,17,31\}
  26. { 0 , 6 , 10 , 15 , 17 , 18 , 31 } \{0,6,10,15,17,18,31\}

Specht's_theorem.html

  1. W ( x , y ) = x m 1 y n 1 x m 2 y n 2 x m p , W(x,y)=x^{m_{1}}y^{n_{1}}x^{m_{2}}y^{n_{2}}\cdots x^{m_{p}},\,
  2. m 1 + n 1 + m 2 + n 2 + + m p . m_{1}+n_{1}+m_{2}+n_{2}+\cdots+m_{p}.\,
  3. tr A = tr B , tr A 2 = tr B 2 , and tr A A * = tr B B * . \operatorname{tr}\,A=\operatorname{tr}\,B,\quad\operatorname{tr}\,A^{2}=% \operatorname{tr}\,B^{2},\quad\,\text{and}\quad\operatorname{tr}\,AA^{*}=% \operatorname{tr}\,BB^{*}.
  4. tr A = tr B , tr A 2 = tr B 2 , tr A A * = tr B B * , tr A 3 = tr B 3 , \displaystyle\operatorname{tr}\,A=\operatorname{tr}\,B,\quad\operatorname{tr}% \,A^{2}=\operatorname{tr}\,B^{2},\quad\operatorname{tr}\,AA^{*}=\operatorname{% tr}\,BB^{*},\quad\operatorname{tr}\,A^{3}=\operatorname{tr}\,B^{3},
  5. n 2 n 2 n - 1 + 1 4 + n 2 - 2. n\sqrt{\frac{2n^{2}}{n-1}+\frac{1}{4}}+\frac{n}{2}-2.

Special_classes_of_semigroups.html

  1. \emptyset

Special_conformal_transformation.html

  1. U ( q , 1 ) ( 1 0 t 1 ) = U ( q + t , 1 ) . U(q,1)\begin{pmatrix}1&0\\ t&1\end{pmatrix}=U(q+t,1).
  2. ( 0 1 1 0 ) ( 1 0 t 1 ) ( 0 1 1 0 ) = ( 1 t 0 1 ) , \begin{pmatrix}0&1\\ 1&0\end{pmatrix}\begin{pmatrix}1&0\\ t&1\end{pmatrix}\begin{pmatrix}0&1\\ 1&0\end{pmatrix}=\begin{pmatrix}1&t\\ 0&1\end{pmatrix},
  3. x μ = x μ - b μ x 2 1 - 2 b x + b 2 x 2 . x^{\prime\mu}=\frac{x^{\mu}-b^{\mu}x^{2}}{1-2b\cdot x+b^{2}x^{2}}\,.
  4. x μ x 2 = x μ x 2 - b μ . \frac{x^{\prime\mu}}{x^{\prime 2}}=\frac{x^{\mu}}{x^{2}}-b^{\mu}\,.
  5. K μ = - i ( 2 x μ x ν ν - x 2 μ ) . K_{\mu}=-i(2x_{\mu}x^{\nu}\partial_{\nu}-x^{2}\partial_{\mu})\,.

Special_functions.html

  1. J n ( x ) , ~{}J_{n}(x),~{}
  2. besselj ( n , x ) , ~{}{\rm besselj}(n,x),~{}
  3. BesselJ [ n , x ] . ~{}{\rm BesselJ}[n,x].~{}
  4. cos 3 ( x ) ~{}\cos^{3}(x)~{}
  5. ( cos ( x ) ) 3 ~{}(\cos(x))^{3}~{}
  6. cos 2 ( x ) ~{}\cos^{2}(x)~{}
  7. ( cos ( x ) ) 2 ~{}(\cos(x))^{2}~{}
  8. cos ( cos ( x ) ) ~{}\cos(\cos(x))~{}
  9. cos - 1 ( x ) ~{}\cos^{-1}(x)~{}
  10. arccos ( x ) ~{}\arccos(x)~{}
  11. ( cos ( x ) ) - 1 ~{}(\cos(x))^{-1}~{}

Special_values_of_L-functions.html

  1. 1 - 1 3 + 1 5 - 1 7 + 1 9 - = π 4 , 1\,-\,\frac{1}{3}\,+\,\frac{1}{5}\,-\,\frac{1}{7}\,+\,\frac{1}{9}\,-\,\cdots\;% =\;\frac{\pi}{4},\!

Specific_ion_interaction_theory.html

  1. K = { H A } { A - } { H + } K=\frac{\{HA\}}{\{A^{-}\}\{H^{+}\}}
  2. K = [ H A ] [ H + ] [ A - ] × γ H A γ H + γ A - K=\frac{[HA]}{[H^{+}][A^{-}]}\times\frac{\gamma_{HA}}{\gamma_{H^{+}}\gamma_{A^% {-}}}
  3. log K = log K 0 + log γ H A - log γ H + - log γ A - \log K=\log K^{0}+\log\gamma_{HA}-\log\gamma_{H^{+}}-\log\gamma_{A^{-}}
  4. log γ j = - z j 2 0.51 I 1 + 1.5 I + k ϵ j k m k \log\gamma_{j}=-z_{j}^{2}\frac{0.51\sqrt{I}}{1+1.5\sqrt{I}}+\sum_{k}\epsilon_{% jk}m_{k}
  5. log y j = - z j 2 0.51 I 1 + 1.5 I + k b j k c k \log y_{j}=-z_{j}^{2}\frac{0.51\sqrt{I}}{1+1.5\sqrt{I}}+\sum_{k}b_{jk}c_{k}
  6. log γ = k m I \log\gamma=k_{m}I\,

Specific_radiative_intensity.html

  1. 𝐱 \mathbf{x}
  2. t t
  3. ν ν
  4. 𝐫 \mathbf{r}
  5. d E dE
  6. ( ν , ν + d ν ) (ν,ν+dν)
  7. d t dt
  8. 𝐫 \mathbf{r}
  9. d E dE
  10. 𝐫 \mathbf{r}
  11. 𝐫 \mathbf{r}
  12. 𝐫 \mathbf{r}
  13. 𝐫 \mathbf{r}
  14. 𝐫 \mathbf{r}
  15. d E dE
  16. d E dE
  17. r r
  18. r r
  19. 𝐫 \mathbf{r}
  20. r r
  21. 𝐫 \mathbf{r}
  22. d A 1 d A 2 cos θ 1 cos θ 2 r 2 \frac{\mbox{d}~{}A_{1}\ \,\mbox{d}~{}A_{2}\ \cos{\theta_{1}}\ \cos{\theta_{2}}% }{r^{2}}
  23. = d A < s u b > 2 c o s < v a r > θ < / v a r > 2 d < v a r > Ω < / v a r > 2 =dA<sub>2cos<var>θ</var>_{2}d<var>Ω</var>_{2}

Specker_sequence.html

  1. q n = i = 0 n 2 - a i - 1 . q_{n}=\sum_{i=0}^{n}2^{-a_{i}-1}.
  2. i = 0 2 - i - 1 = 1. \sum_{i=0}^{\infty}2^{-i-1}=1.

Spectral_signal-to-noise_ratio.html

  1. S S N R ( r ) = r i R | k i F r i , k | 2 K K - 1 r i R k i | F r i , k - F ¯ r i | 2 - 1 SSNR(r)=\frac{\displaystyle\sum_{r_{i}\in R}\left|\sum_{k_{i}}{F_{r_{i},k}}% \right|^{2}}{\displaystyle\frac{K}{K-1}\sum_{r_{i}\in R}\sum_{k_{i}}{\left|{F_% {r_{i},k}-\bar{F}_{r_{i}}}\right|^{2}}}-1
  2. F r i , k F_{r_{i},k}
  3. r i r_{i}
  4. R R
  5. F R C = S S N R S S N R + 1 FRC=\frac{SSNR}{SSNR+1}

Speed_skydiving.html

  1. v t = 2 m g ρ A C d v_{\mathrm{t}}=\sqrt{\frac{2mg}{\rho AC_{\mathrm{d}}}}
  2. m m
  3. g g
  4. C < s u b > d C<sub>d

Sperm_donation.html

  1. N = V s × c × r s n r N=\frac{V_{s}\times c\times r_{s}}{n_{r}}
  2. N = V s × f c × c × r s n r N=\frac{V_{s}\times f_{c}\times c\times r_{s}}{n_{r}}

Spherical_image.html

  1. α \alpha
  2. α = 1 \|\alpha^{\prime}\|=1
  3. T T
  4. α \alpha
  5. σ = T \sigma=T
  6. α \alpha
  7. σ \sigma
  8. σ = T = 1 \|\sigma\|=\|T\|=1

Spherical_measure.html

  1. σ n = 1 H n ( 𝐒 n ) H n . \sigma^{n}=\frac{1}{H^{n}(\mathbf{S}^{n})}H^{n}.
  2. σ n ( A ) := 1 α ( n + 1 ) λ n + 1 ( { t x x A , t [ 0 , 1 ] } ) , \sigma^{n}(A):=\frac{1}{\alpha(n+1)}\lambda^{n+1}(\{tx\mid x\in A,t\in[0,1]\}),
  3. α ( m ) := λ m ( 𝐁 1 m ( 0 ) ) . \alpha(m):=\lambda^{m}(\mathbf{B}_{1}^{m}(0)).
  4. θ n ( { g O ( n ) g ( x ) A } ) = σ n - 1 ( A ) . \theta^{n}(\{g\in\mathrm{O}(n)\mid g(x)\in A\})=\sigma^{n-1}(A).
  5. σ n ( A r ) σ n ( B r ) , \sigma^{n}(A_{r})\geq\sigma^{n}(B_{r}),
  6. A r := { x 𝐒 n ρ n ( x , A ) r } . A_{r}:=\{x\in\mathbf{S}^{n}\mid\rho_{n}(x,A)\leq r\}.
  7. σ n ( A r ) 1 - π 8 exp ( - ( n - 1 ) r 2 2 ) . \sigma^{n}(A_{r})\geq 1-\sqrt{\frac{\pi}{8}}\,\exp\left(-\frac{(n-1)r^{2}}{2}% \right).

Spherical_segment.html

  1. V = π h 6 ( 3 r 1 2 + 3 r 2 2 + h 2 ) V=\frac{\pi h}{6}(3r_{1}^{2}+3r_{2}^{2}+h^{2})\,
  2. A = 2 π R h A=2\pi Rh\,

Spherical_wedge.html

  1. 4 3 π r 3 \tfrac{4}{3}\pi r^{3}
  2. V = α 2 π 4 3 π r 3 = 2 3 α r 3 . V=\frac{\alpha}{2\pi}\cdot\frac{4}{3}\pi r^{3}=\frac{2}{3}\alpha r^{3}.
  3. 4 π r 2 4\pi r^{2}
  4. A = α 2 π 4 π r 2 = 2 α r 2 A=\frac{\alpha}{2\pi}\cdot 4\pi r^{2}=2\alpha r^{2}
  5. V s V_{s}
  6. V w V_{w}
  7. V w V s = α 2 π \frac{V_{w}}{V_{s}}=\frac{\alpha}{2\pi}
  8. S l S s = α 2 π \frac{S_{l}}{S_{s}}=\frac{\alpha}{2\pi}

Spherium.html

  1. R R
  2. H ^ = - 1 2 2 - 2 2 2 + 1 u \hat{H}=-\frac{\nabla_{1}^{2}}{2}-\frac{\nabla_{2}^{2}}{2}+\frac{1}{u}
  3. u u
  4. S ( u ) S(u)
  5. ( u 2 4 R 2 - 1 ) d 2 S ( u ) d u 2 + ( 3 u 4 R 2 - 1 u ) d S ( u ) d u + 1 u S ( u ) = E S ( u ) \left(\frac{u^{2}}{4R^{2}}-1\right)\frac{d^{2}S(u)}{du^{2}}+\left(\frac{3u}{4R% ^{2}}-\frac{1}{u}\right)\frac{dS(u)}{du}+\frac{1}{u}S(u)=ES(u)
  6. x = u / 2 R x=u/2R
  7. x = - 1 , 0 , + 1 x=-1,0,+1
  8. S ( u ) = k = 0 s k u k S(u)=\sum_{k=0}^{\infty}s_{k}\,u^{k}
  9. s k + 2 = s k + 1 + [ k ( k + 2 ) 1 4 R 2 - E ] s k ( k + 2 ) 2 s_{k+2}=\frac{s_{k+1}+\left[k(k+2)\frac{1}{4R^{2}}-E\right]s_{k}}{(k+2)^{2}}
  10. s 0 = s 1 = 1 s_{0}=s_{1}=1
  11. S ( 0 ) S ( 0 ) = 1 \frac{S^{\prime}(0)}{S(0)}=1
  12. S n , m ( u ) = k = 0 n s k u k S_{n,m}(u)=\sum_{k=0}^{n}s_{k}\,u^{k}
  13. m m
  14. 0
  15. 2 R 2R
  16. s n + 1 = s n + 2 = 0 s_{n+1}=s_{n+2}=0
  17. E n , m E_{n,m}
  18. s n + 1 = 0 s_{n+1}=0
  19. deg s n + 1 = ( n + 1 ) / 2 \deg s_{n+1}=\lfloor(n+1)/2\rfloor
  20. R n , m R_{n,m}
  21. R n , m 2 E n , m = n 2 ( n 2 + 1 ) R_{n,m}^{2}E_{n,m}=\frac{n}{2}\left(\frac{n}{2}+1\right)
  22. S n , m ( u ) S_{n,m}(u)
  23. m m
  24. R n , m R_{n,m}
  25. E HF = 1 / R E_{\rm HF}=1/R
  26. R = 3 / 2 R=\sqrt{3}/2
  27. E corr = 1 - 2 / 3 - 0.1547 E_{\rm corr}=1-2/\sqrt{3}\approx-0.1547
  28. - 0.0467 -0.0467
  29. - 0.0497 -0.0497
  30. - .0476 -.0476

Spike-triggered_covariance.html

  1. 𝐱 𝐢 \mathbf{x_{i}}
  2. i i
  3. y i y_{i}
  4. E [ 𝐱 ] = 0 E[\mathbf{x}]=0
  5. STC = 1 n s - 1 i = 1 T y i ( 𝐱 𝐢 - S T A ) ( 𝐱 𝐢 - S T A ) T , \mathrm{STC}=\tfrac{1}{n_{s}-1}\sum_{i=1}^{T}y_{i}(\mathbf{x_{i}}-STA)(\mathbf% {x_{i}}-STA)^{T},
  6. n s = y i n_{s}=\sum y_{i}
  7. C = 1 n p - 1 i = 1 T 𝐱 𝐢 𝐱 𝐢 𝐓 , \mathrm{C}=\tfrac{1}{n_{p}-1}\sum_{i=1}^{T}\mathbf{x_{i}}\mathbf{x_{i}^{T}},
  8. n p n_{p}
  9. 𝐱 𝐢 \mathbf{x_{i}}
  10. ( S T C - C ) (STC-C)

SPIKE_algorithm.html

  1. n × n n\times n
  2. n n
  3. n × s n\times s
  4. s s
  5. [ s y m b o l A 1 s y m b o l B 1 s y m b o l C 2 s y m b o l A 2 s y m b o l B 2 s y m b o l C p - 1 s y m b o l A p - 1 s y m b o l B p - 1 s y m b o l C p s y m b o l A p ] [ s y m b o l X 1 s y m b o l X 2 s y m b o l X p - 1 s y m b o l X p ] = [ s y m b o l F 1 s y m b o l F 2 s y m b o l F p - 1 s y m b o l F p ] . \begin{bmatrix}symbol{A}_{1}&symbol{B}_{1}\\ symbol{C}_{2}&symbol{A}_{2}&symbol{B}_{2}\\ &\ddots&\ddots&\ddots\\ &&symbol{C}_{p-1}&symbol{A}_{p-1}&symbol{B}_{p-1}\\ &&&symbol{C}_{p}&symbol{A}_{p}\end{bmatrix}\begin{bmatrix}symbol{X}_{1}\\ symbol{X}_{2}\\ \vdots\\ symbol{X}_{p-1}\\ symbol{X}_{p}\end{bmatrix}=\begin{bmatrix}symbol{F}_{1}\\ symbol{F}_{2}\\ \vdots\\ symbol{F}_{p-1}\\ symbol{F}_{p}\end{bmatrix}.
  6. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  7. [ s y m b o l I s y m b o l V 1 s y m b o l W 2 s y m b o l I s y m b o l V 2 s y m b o l W p - 1 s y m b o l I s y m b o l V p - 1 s y m b o l W p s y m b o l I ] [ s y m b o l X 1 s y m b o l X 2 s y m b o l X p - 1 s y m b o l X p ] = [ s y m b o l G 1 s y m b o l G 2 s y m b o l G p - 1 s y m b o l G p ] , \begin{bmatrix}symbol{I}&symbol{V}_{1}\\ symbol{W}_{2}&symbol{I}&symbol{V}_{2}\\ &\ddots&\ddots&\ddots\\ &&symbol{W}_{p-1}&symbol{I}&symbol{V}_{p-1}\\ &&&symbol{W}_{p}&symbol{I}\end{bmatrix}\begin{bmatrix}symbol{X}_{1}\\ symbol{X}_{2}\\ \vdots\\ symbol{X}_{p-1}\\ symbol{X}_{p}\end{bmatrix}=\begin{bmatrix}symbol{G}_{1}\\ symbol{G}_{2}\\ \vdots\\ symbol{G}_{p-1}\\ symbol{G}_{p}\end{bmatrix},
  8. p p
  9. j = 1 j=1
  10. j = p j=p
  11. m m
  12. m m
  13. n n
  14. [ s y m b o l V j ( t ) s y m b o l V j s y m b o l V j ( b ) ] \begin{bmatrix}symbol{V}_{j}^{(t)}\\ symbol{V}_{j}^{\prime}\\ symbol{V}_{j}^{(b)}\end{bmatrix}
  15. [ s y m b o l W j ( t ) s y m b o l W j s y m b o l W j ( b ) ] \begin{bmatrix}symbol{W}_{j}^{(t)}\\ symbol{W}_{j}^{\prime}\\ symbol{W}_{j}^{(b)}\\ \end{bmatrix}
  16. m × m m\times m
  17. [ s y m b o l X j ( t ) s y m b o l X j s y m b o l X j ( b ) ] \begin{bmatrix}symbol{X}_{j}^{(t)}\\ symbol{X}_{j}^{\prime}\\ symbol{X}_{j}^{(b)}\end{bmatrix}
  18. [ s y m b o l G j ( t ) s y m b o l G j s y m b o l G j ( b ) ] . \begin{bmatrix}symbol{G}_{j}^{(t)}\\ symbol{G}_{j}^{\prime}\\ symbol{G}_{j}^{(b)}\\ \end{bmatrix}.
  19. m m
  20. n n
  21. [ s y m b o l I m s y m b o l 0 s y m b o l V 1 ( t ) s y m b o l 0 s y m b o l I m s y m b o l V 1 ( b ) s y m b o l 0 s y m b o l 0 s y m b o l W 2 ( t ) s y m b o l I m s y m b o l 0 s y m b o l V 2 ( t ) s y m b o l W 2 ( b ) s y m b o l 0 s y m b o l I m s y m b o l V 2 ( b ) s y m b o l 0 s y m b o l 0 s y m b o l W p - 1 ( t ) s y m b o l I m s y m b o l 0 s y m b o l V p - 1 ( t ) s y m b o l W p - 1 ( b ) s y m b o l 0 s y m b o l I m s y m b o l V p - 1 ( b ) s y m b o l 0 s y m b o l 0 s y m b o l W p ( t ) s y m b o l I m s y m b o l 0 s y m b o l W p ( b ) s y m b o l 0 s y m b o l I m ] [ s y m b o l X 1 ( t ) s y m b o l X 1 ( b ) s y m b o l X 2 ( t ) s y m b o l X 2 ( b ) s y m b o l X p - 1 ( t ) s y m b o l X p - 1 ( b ) s y m b o l X p ( t ) s y m b o l X p ( b ) ] = [ s y m b o l G 1 ( t ) s y m b o l G 1 ( b ) s y m b o l G 2 ( t ) s y m b o l G 2 ( b ) s y m b o l G p - 1 ( t ) s y m b o l G p - 1 ( b ) s y m b o l G p ( t ) s y m b o l G p ( b ) ] , \begin{bmatrix}symbol{I}_{m}&symbol{0}&symbol{V}_{1}^{(t)}\\ symbol{0}&symbol{I}_{m}&symbol{V}_{1}^{(b)}&symbol{0}\\ symbol{0}&symbol{W}_{2}^{(t)}&symbol{I}_{m}&symbol{0}&symbol{V}_{2}^{(t)}\\ &symbol{W}_{2}^{(b)}&symbol{0}&symbol{I}_{m}&symbol{V}_{2}^{(b)}&symbol{0}\\ &&\ddots&\ddots&\ddots&\ddots&\ddots\\ &&&symbol{0}&symbol{W}_{p-1}^{(t)}&symbol{I}_{m}&symbol{0}&symbol{V}_{p-1}^{(t% )}\\ &&&&symbol{W}_{p-1}^{(b)}&symbol{0}&symbol{I}_{m}&symbol{V}_{p-1}^{(b)}&symbol% {0}\\ &&&&&symbol{0}&symbol{W}_{p}^{(t)}&symbol{I}_{m}&symbol{0}\\ &&&&&&symbol{W}_{p}^{(b)}&symbol{0}&symbol{I}_{m}\end{bmatrix}\begin{bmatrix}% symbol{X}_{1}^{(t)}\\ symbol{X}_{1}^{(b)}\\ symbol{X}_{2}^{(t)}\\ symbol{X}_{2}^{(b)}\\ \vdots\\ symbol{X}_{p-1}^{(t)}\\ symbol{X}_{p-1}^{(b)}\\ symbol{X}_{p}^{(t)}\\ symbol{X}_{p}^{(b)}\end{bmatrix}=\begin{bmatrix}symbol{G}_{1}^{(t)}\\ symbol{G}_{1}^{(b)}\\ symbol{G}_{2}^{(t)}\\ symbol{G}_{2}^{(b)}\\ \vdots\\ symbol{G}_{p-1}^{(t)}\\ symbol{G}_{p-1}^{(b)}\\ symbol{G}_{p}^{(t)}\\ symbol{G}_{p}^{(b)}\end{bmatrix}\,\text{,}
  22. { s y m b o l X 1 = s y m b o l G 1 - s y m b o l V 1 s y m b o l X 2 ( t ) , s y m b o l X j = s y m b o l G j - s y m b o l V j s y m b o l X j + 1 ( t ) - s y m b o l W j s y m b o l X j - 1 ( b ) , j = 2 , , p - 1 , s y m b o l X p = s y m b o l G p - s y m b o l W p s y m b o l X p - 1 ( b ) . \begin{cases}symbol{X}_{1}^{\prime}=symbol{G}_{1}^{\prime}-symbol{V}_{1}^{% \prime}symbol{X}_{2}^{(t)}\,\text{,}\\ symbol{X}_{j}^{\prime}=symbol{G}_{j}^{\prime}-symbol{V}_{j}^{\prime}symbol{X}_% {j+1}^{(t)}-symbol{W}_{j}^{\prime}symbol{X}_{j-1}^{(b)}\,\text{,}&j=2,\ldots,p% -1\,\text{,}\\ symbol{X}_{p}^{\prime}=symbol{G}_{p}^{\prime}-symbol{W}_{p}symbol{X}_{p-1}^{(b% )}\,\text{.}\end{cases}
  23. pivot = { pivot + ϵ \lVertsymbol A j 1 if pivot 0 , pivot - ϵ \lVertsymbol A j 1 if pivot < 0 \mathrm{pivot}=\begin{cases}\mathrm{pivot}+\epsilon\lVertsymbol{A}_{j}\rVert_{% 1}&\,\text{if }\mathrm{pivot}\geq 0\,\text{,}\\ \mathrm{pivot}-\epsilon\lVertsymbol{A}_{j}\rVert_{1}&\,\text{if }\mathrm{pivot% }<0\end{cases}
  24. [ s y m b o l I m s y m b o l 0 s y m b o l V 1 ( t ) s y m b o l 0 s y m b o l I m s y m b o l V 1 ( b ) s y m b o l 0 s y m b o l 0 s y m b o l W 2 ( t ) s y m b o l I m s y m b o l 0 s y m b o l W 2 ( b ) s y m b o l 0 s y m b o l I m ] [ s y m b o l X 1 ( t ) s y m b o l X 1 ( b ) s y m b o l X 2 ( t ) s y m b o l X 2 ( b ) ] = [ s y m b o l G 1 ( t ) s y m b o l G 1 ( b ) s y m b o l G 2 ( t ) s y m b o l G 2 ( b ) ] . \begin{bmatrix}symbol{I}_{m}&symbol{0}&symbol{V}_{1}^{(t)}\\ symbol{0}&symbol{I}_{m}&symbol{V}_{1}^{(b)}&symbol{0}\\ symbol{0}&symbol{W}_{2}^{(t)}&symbol{I}_{m}&symbol{0}\\ &symbol{W}_{2}^{(b)}&symbol{0}&symbol{I}_{m}\end{bmatrix}\begin{bmatrix}symbol% {X}_{1}^{(t)}\\ symbol{X}_{1}^{(b)}\\ symbol{X}_{2}^{(t)}\\ symbol{X}_{2}^{(b)}\end{bmatrix}=\begin{bmatrix}symbol{G}_{1}^{(t)}\\ symbol{G}_{1}^{(b)}\\ symbol{G}_{2}^{(t)}\\ symbol{G}_{2}^{(b)}\end{bmatrix}\,\text{.}
  25. [ s y m b o l I m s y m b o l V 1 ( b ) s y m b o l W 2 ( t ) s y m b o l I m ] [ s y m b o l X 1 ( b ) s y m b o l X 2 ( t ) ] = [ s y m b o l G 1 ( b ) s y m b o l G 2 ( t ) ] , \begin{bmatrix}symbol{I}_{m}&symbol{V}_{1}^{(b)}\\ symbol{W}_{2}^{(t)}&symbol{I}_{m}\end{bmatrix}\begin{bmatrix}symbol{X}_{1}^{(b% )}\\ symbol{X}_{2}^{(t)}\end{bmatrix}=\begin{bmatrix}symbol{G}_{1}^{(b)}\\ symbol{G}_{2}^{(t)}\end{bmatrix}\,\text{,}
  26. [ s y m b o l I m s y m b o l V 1 ( b ) s y m b o l W 2 ( t ) s y m b o l I m ] = [ s y m b o l I m s y m b o l W 2 ( t ) s y m b o l I m ] [ s y m b o l I m s y m b o l V 1 ( b ) s y m b o l I m - s y m b o l W 2 ( t ) s y m b o l V 1 ( b ) ] . \begin{bmatrix}symbol{I}_{m}&symbol{V}_{1}^{(b)}\\ symbol{W}_{2}^{(t)}&symbol{I}_{m}\end{bmatrix}=\begin{bmatrix}symbol{I}_{m}\\ symbol{W}_{2}^{(t)}&symbol{I}_{m}\end{bmatrix}\begin{bmatrix}symbol{I}_{m}&% symbol{V}_{1}^{(b)}\\ &symbol{I}_{m}-symbol{W}_{2}^{(t)}symbol{V}_{1}^{(b)}\end{bmatrix}\,\text{.}
  27. s y m b o l D ~ k [ 1 ] = [ s y m b o l I m s y m b o l 0 s y m b o l V 2 k - 1 ( t ) s y m b o l 0 s y m b o l I m s y m b o l V 2 k - 1 ( b ) s y m b o l 0 s y m b o l 0 s y m b o l W 2 k ( t ) s y m b o l I m s y m b o l 0 s y m b o l W 2 k ( b ) s y m b o l 0 s y m b o l I m ] symbol{\tilde{D}}_{k}^{[1]}=\begin{bmatrix}symbol{I}_{m}&symbol{0}&symbol{V}_{% 2k-1}^{(t)}\\ symbol{0}&symbol{I}_{m}&symbol{V}_{2k-1}^{(b)}&symbol{0}\\ symbol{0}&symbol{W}_{2k}^{(t)}&symbol{I}_{m}&symbol{0}\\ &symbol{W}_{2k}^{(b)}&symbol{0}&symbol{I}_{m}\end{bmatrix}
  28. [ s y m b o l I 3 m s y m b o l 0 s y m b o l V 1 [ 2 ] ( t ) s y m b o l 0 s y m b o l I m s y m b o l V 1 [ 2 ] ( b ) s y m b o l 0 s y m b o l 0 s y m b o l W 2 [ 2 ] ( t ) s y m b o l I m s y m b o l 0 s y m b o l V 2 [ 2 ] ( t ) s y m b o l W 2 [ 2 ] ( b ) s y m b o l 0 s y m b o l I 3 m s y m b o l V 2 [ 2 ] ( b ) s y m b o l 0 s y m b o l 0 s y m b o l W p / 2 - 1 [ 2 ] ( t ) s y m b o l I 3 m s y m b o l 0 s y m b o l V p / 2 - 1 [ 2 ] ( t ) s y m b o l W p / 2 - 1 [ 2 ] ( b ) s y m b o l 0 s y m b o l I m s y m b o l V p / 2 - 1 [ 2 ] ( b ) s y m b o l 0 s y m b o l 0 s y m b o l W p / 2 [ 2 ] ( t ) s y m b o l I m s y m b o l 0 s y m b o l W p / 2 [ 2 ] ( b ) s y m b o l 0 s y m b o l I 3 m ] . \begin{bmatrix}symbol{I}_{3m}&symbol{0}&symbol{V}_{1}^{[2](t)}\\ symbol{0}&symbol{I}_{m}&symbol{V}_{1}^{[2](b)}&symbol{0}\\ symbol{0}&symbol{W}_{2}^{[2](t)}&symbol{I}_{m}&symbol{0}&symbol{V}_{2}^{[2](t)% }\\ &symbol{W}_{2}^{[2](b)}&symbol{0}&symbol{I}_{3m}&symbol{V}_{2}^{[2](b)}&symbol% {0}\\ &&\ddots&\ddots&\ddots&\ddots&\ddots\\ &&&symbol{0}&symbol{W}_{p/2-1}^{[2](t)}&symbol{I}_{3m}&symbol{0}&symbol{V}_{p/% 2-1}^{[2](t)}\\ &&&&symbol{W}_{p/2-1}^{[2](b)}&symbol{0}&symbol{I}_{m}&symbol{V}_{p/2-1}^{[2](% b)}&symbol{0}\\ &&&&&symbol{0}&symbol{W}_{p/2}^{[2](t)}&symbol{I}_{m}&symbol{0}\\ &&&&&&symbol{W}_{p/2}^{[2](b)}&symbol{0}&symbol{I}_{3m}\end{bmatrix}\,\text{.}
  29. [ s y m b o l I m s y m b o l 0 s y m b o l V 1 ( t ) s y m b o l 0 s y m b o l I m s y m b o l V 1 ( b ) s y m b o l 0 s y m b o l 0 s y m b o l W 2 ( t ) s y m b o l I m s y m b o l 0 s y m b o l V 2 ( t ) s y m b o l W 2 ( b ) s y m b o l 0 s y m b o l I m s y m b o l V 2 ( b ) s y m b o l 0 s y m b o l 0 s y m b o l W p - 1 ( t ) s y m b o l I m s y m b o l 0 s y m b o l V p - 1 ( t ) s y m b o l W p - 1 ( b ) s y m b o l 0 s y m b o l I m s y m b o l V p - 1 ( b ) s y m b o l 0 s y m b o l 0 s y m b o l W p ( t ) s y m b o l I m s y m b o l 0 s y m b o l W p ( b ) s y m b o l 0 s y m b o l I m ] [ s y m b o l X 1 ( t ) s y m b o l X 1 ( b ) s y m b o l X 2 ( t ) s y m b o l X 2 ( b ) s y m b o l X p - 1 ( t ) s y m b o l X p - 1 ( b ) s y m b o l X p ( t ) s y m b o l X p ( b ) ] = [ s y m b o l G 1 ( t ) s y m b o l G 1 ( b ) s y m b o l G 2 ( t ) s y m b o l G 2 ( b ) s y m b o l G p - 1 ( t ) s y m b o l G p - 1 ( b ) s y m b o l G p ( t ) s y m b o l G p ( b ) ] , \begin{bmatrix}symbol{I}_{m}&symbol{0}&symbol{V}_{1}^{(t)}\\ symbol{0}&symbol{I}_{m}&symbol{V}_{1}^{(b)}&symbol{0}\\ symbol{0}&symbol{W}_{2}^{(t)}&symbol{I}_{m}&symbol{0}&symbol{V}_{2}^{(t)}\\ &symbol{W}_{2}^{(b)}&symbol{0}&symbol{I}_{m}&symbol{V}_{2}^{(b)}&symbol{0}\\ &&\ddots&\ddots&\ddots&\ddots&\ddots\\ &&&symbol{0}&symbol{W}_{p-1}^{(t)}&symbol{I}_{m}&symbol{0}&symbol{V}_{p-1}^{(t% )}\\ &&&&symbol{W}_{p-1}^{(b)}&symbol{0}&symbol{I}_{m}&symbol{V}_{p-1}^{(b)}&symbol% {0}\\ &&&&&symbol{0}&symbol{W}_{p}^{(t)}&symbol{I}_{m}&symbol{0}\\ &&&&&&symbol{W}_{p}^{(b)}&symbol{0}&symbol{I}_{m}\end{bmatrix}\begin{bmatrix}% symbol{X}_{1}^{(t)}\\ symbol{X}_{1}^{(b)}\\ symbol{X}_{2}^{(t)}\\ symbol{X}_{2}^{(b)}\\ \vdots\\ symbol{X}_{p-1}^{(t)}\\ symbol{X}_{p-1}^{(b)}\\ symbol{X}_{p}^{(t)}\\ symbol{X}_{p}^{(b)}\end{bmatrix}=\begin{bmatrix}symbol{G}_{1}^{(t)}\\ symbol{G}_{1}^{(b)}\\ symbol{G}_{2}^{(t)}\\ symbol{G}_{2}^{(b)}\\ \vdots\\ symbol{G}_{p-1}^{(t)}\\ symbol{G}_{p-1}^{(b)}\\ symbol{G}_{p}^{(t)}\\ symbol{G}_{p}^{(b)}\end{bmatrix}\,\text{,}
  30. [ s y m b o l I m s y m b o l I m s y m b o l V 1 ( b ) s y m b o l W 2 ( t ) s y m b o l I m s y m b o l I m s y m b o l V 2 ( b ) s y m b o l W p - 1 ( t ) s y m b o l I m s y m b o l I m s y m b o l V p - 1 ( b ) s y m b o l W p ( t ) s y m b o l I m s y m b o l I m ] [ s y m b o l X 1 ( t ) s y m b o l X 1 ( b ) s y m b o l X 2 ( t ) s y m b o l X 2 ( b ) s y m b o l X p - 1 ( t ) s y m b o l X p - 1 ( b ) s y m b o l X p ( t ) s y m b o l X p ( b ) ] = [ s y m b o l G 1 ( t ) s y m b o l G 1 ( b ) s y m b o l G 2 ( t ) s y m b o l G 2 ( b ) s y m b o l G p - 1 ( t ) s y m b o l G p - 1 ( b ) s y m b o l G p ( t ) s y m b o l G p ( b ) ] \begin{bmatrix}symbol{I}_{m}\\ &symbol{I}_{m}&symbol{V}_{1}^{(b)}\\ &symbol{W}_{2}^{(t)}&symbol{I}_{m}\\ &&&symbol{I}_{m}&symbol{V}_{2}^{(b)}\\ &&&\ddots&\ddots&\ddots\\ &&&&symbol{W}_{p-1}^{(t)}&symbol{I}_{m}\\ &&&&&&symbol{I}_{m}&symbol{V}_{p-1}^{(b)}\\ &&&&&&symbol{W}_{p}^{(t)}&symbol{I}_{m}\\ &&&&&&&&symbol{I}_{m}\end{bmatrix}\begin{bmatrix}symbol{X}_{1}^{(t)}\\ symbol{X}_{1}^{(b)}\\ symbol{X}_{2}^{(t)}\\ symbol{X}_{2}^{(b)}\\ \vdots\\ symbol{X}_{p-1}^{(t)}\\ symbol{X}_{p-1}^{(b)}\\ symbol{X}_{p}^{(t)}\\ symbol{X}_{p}^{(b)}\end{bmatrix}=\begin{bmatrix}symbol{G}_{1}^{(t)}\\ symbol{G}_{1}^{(b)}\\ symbol{G}_{2}^{(t)}\\ symbol{G}_{2}^{(b)}\\ \vdots\\ symbol{G}_{p-1}^{(t)}\\ symbol{G}_{p-1}^{(b)}\\ symbol{G}_{p}^{(t)}\\ symbol{G}_{p}^{(b)}\end{bmatrix}
  31. < 𝐯𝐚𝐫 > 𝐀𝐱 = < 𝐯𝐚𝐫 > 𝐛 < / 𝐯𝐚𝐫 > \mathbf{<var>Ax}=\mathbf{<var>b</var>}

Spin_crossover.html

  1. T 1 2 P = Δ V Δ S H L \frac{\partial{{T}_{{}^{1}\!\!\diagup\!\!{}_{2}\;}}}{\partial P}=\frac{\,\text% { }\!\!\Delta\!\!\,\text{ V}}{\,\text{ }\!\!\Delta\!\!\,\text{ }{{S}_{HL}}}
  2. T 1 2 P \frac{\partial{{T}_{{}^{1}\!\!\diagup\!\!{}_{2}\;}}}{\partial P}

Spin_stiffness.html

  1. ρ s = 2 θ 2 E 0 ( θ ) N | θ = 0 \rho_{s}=\cfrac{\partial^{2}}{\partial\theta^{2}}\cfrac{E_{0}(\theta)}{N}|_{% \theta=0}
  2. E 0 E_{0}
  3. θ \theta
  4. H Heisenberg = - J < i , j > [ S i z S j z + 1 2 ( S i + S j - + S i - S j + ) ] H_{\mathrm{Heisenberg}}=-J\sum_{<i,j>}\left[S_{i}^{z}S_{j}^{z}+\cfrac{1}{2}(S_% {i}^{+}S_{j}^{-}+S_{i}^{-}S_{j}^{+})\right]
  5. S i + S i + e i θ i S_{i}^{+}\longrightarrow S_{i}^{+}e^{i\theta_{i}}
  6. S i - S i - e - i θ i S_{i}^{-}\longrightarrow S_{i}^{-}e^{-i\theta_{i}}
  7. H ( θ i j ) = - J < i , j > [ S i z S j z + 1 2 ( S i + e i θ i S j - e - i θ j + S i - e - i θ i S j + e i θ j ) ] H(\theta_{ij})=-J\sum_{<i,j>}\left[S_{i}^{z}S_{j}^{z}+\cfrac{1}{2}(S_{i}^{+}e^% {i\theta_{i}}S_{j}^{-}e^{-i\theta_{j}}+S_{i}^{-}e^{-i\theta_{i}}S_{j}^{+}e^{i% \theta_{j}})\right]
  8. H H Heisenberg - J < i j > [ θ i j J i j ( s ) - 1 2 θ i j 2 T i j ( s ) ] H\approx H_{\mathrm{Heisenberg}}-J\sum_{<ij>}\left[\theta_{ij}J_{ij}^{(s)}-% \cfrac{1}{2}\theta_{ij}^{2}T_{ij}^{(s)}\right]
  9. J i j s = i 2 ( S i + S j - - S i - S j + ) J_{ij}^{s}=\cfrac{i}{2}(S_{i}^{+}S_{j}^{-}-S_{i}^{-}S_{j}^{+})
  10. T i j = 1 2 ( S i + S j - + S i - S j + ) T_{ij}=\cfrac{1}{2}(S_{i}^{+}S_{j}^{-}+S_{i}^{-}S_{j}^{+})
  11. E ( θ ) - E ( 0 ) = N ρ s θ x 2 E(\theta)-E(0)=N\rho_{s}\theta_{x}^{2}
  12. ρ s = 1 N [ 1 2 T x + ν 0 | 0 | j x ( s ) | ν | 2 E ν - E 0 ] \rho_{s}=\cfrac{1}{N}\left[\cfrac{1}{2}\langle T_{x}\rangle+\sum_{\nu\neq 0}% \cfrac{|\langle 0|j_{x}^{(s)}|\nu\rangle|^{2}}{E_{\nu}-E_{0}}\right]

Split_Lie_algebra.html

  1. ( 𝔤 , 𝔥 ) (\mathfrak{g},\mathfrak{h})
  2. 𝔤 \mathfrak{g}
  3. 𝔥 < 𝔤 \mathfrak{h}<\mathfrak{g}
  4. x 𝔥 x\in\mathfrak{h}
  5. ad 𝔤 x \operatorname{ad}_{\mathfrak{g}}x
  6. A n , 𝔰 𝔩 n + 1 ( 𝐂 ) : 𝔰 𝔩 n + 1 ( 𝐑 ) A_{n},\mathfrak{sl}_{n+1}(\mathbf{C}):\mathfrak{sl}_{n+1}(\mathbf{R})
  7. B n , 𝔰 𝔬 2 n + 1 ( 𝐂 ) : 𝔰 𝔬 n , n + 1 ( 𝐑 ) B_{n},\mathfrak{so}_{2n+1}(\mathbf{C}):\mathfrak{so}_{n,n+1}(\mathbf{R})
  8. C n , 𝔰 𝔭 n ( 𝐂 ) : 𝔰 𝔭 n ( 𝐑 ) C_{n},\mathfrak{sp}_{n}(\mathbf{C}):\mathfrak{sp}_{n}(\mathbf{R})
  9. D n , 𝔰 𝔬 2 n ( 𝐂 ) : 𝔰 𝔬 n , n ( 𝐑 ) D_{n},\mathfrak{so}_{2n}(\mathbf{C}):\mathfrak{so}_{n,n}(\mathbf{R})
  10. E 6 , E 7 , E 8 , F 4 , G 2 E_{6},E_{7},E_{8},F_{4},G_{2}

SPOJ.html

  1. 80 40 + number _ of _ people _ who _ have _ solved _ it \tfrac{80}{40+\mathrm{number\_of\_people\_who\_have\_solved\_it}}

Sponsored_search_auction.html

Square_knot_(mathematics).html

  1. Δ ( t ) = ( t - 1 + t - 1 ) 2 , \Delta(t)=(t-1+t^{-1})^{2},\,
  2. ( z ) = ( z 2 + 1 ) 2 . \nabla(z)=(z^{2}+1)^{2}.
  3. V ( q ) = ( q - 1 + q - 3 - q - 4 ) ( q + q 3 - q 4 ) = - q 3 + q 2 - q + 3 - q - 1 + q - 2 - q - 3 . V(q)=(q^{-1}+q^{-3}-q^{-4})(q+q^{3}-q^{4})=-q^{3}+q^{2}-q+3-q^{-1}+q^{-2}-q^{-% 3}.\,
  4. x , y , z x y x = y x y , x z x = z x z . \langle x,y,z\mid xyx=yxy,xzx=zxz\rangle.\,

Square_root_biased_sampling.html

  1. n n
  2. n \sqrt{n}

Stably_finite_ring.html

  1. R R

Standard_sea_level.html

  1. ρ \rho
  2. γ \gamma
  3. μ \mu

Standard_translation.html

  1. S T x ( p ) P ( x ) ST_{x}(p)\equiv P(x)
  2. p p
  3. p p
  4. x x
  5. S T x ( ) ST_{x}(\top)\equiv\top
  6. S T x ( ) ST_{x}(\bot)\equiv\bot
  7. S T x ( ¬ φ ) ¬ S T x ( φ ) ST_{x}(\neg\varphi)\equiv\neg ST_{x}(\varphi)
  8. S T x ( φ ψ ) S T x ( φ ) S T x ( ψ ) ST_{x}(\varphi\wedge\psi)\equiv ST_{x}(\varphi)\wedge ST_{x}(\psi)
  9. S T x ( φ ψ ) S T x ( φ ) S T x ( ψ ) ST_{x}(\varphi\vee\psi)\equiv ST_{x}(\varphi)\vee ST_{x}(\psi)
  10. S T x ( φ ψ ) S T x ( φ ) S T x ( ψ ) ST_{x}(\varphi\rightarrow\psi)\equiv ST_{x}(\varphi)\rightarrow ST_{x}(\psi)
  11. S T x ( m φ ) y ( R m ( x , y ) S T y ( φ ) ) ST_{x}(\Diamond_{m}\varphi)\equiv\exists y(R_{m}(x,y)\wedge ST_{y}(\varphi))
  12. S T x ( m φ ) y ( R m ( x , y ) S T y ( φ ) ) ST_{x}(\Box_{m}\varphi)\equiv\forall y(R_{m}(x,y)\rightarrow ST_{y}(\varphi))
  13. x x
  14. x x
  15. m m
  16. \Box
  17. \Diamond
  18. R R
  19. a \Box_{a}
  20. a \Diamond_{a}
  21. R a R_{a}
  22. b \Box_{b}
  23. b \Diamond_{b}
  24. R b R_{b}
  25. p \Box\Box p
  26. y ( R ( x , y ) S T y ( p ) ) \forall y(R(x,y)\rightarrow ST_{y}(\Box p))
  27. x x
  28. y y
  29. p \Box p
  30. y ( R ( x , y ) ( z ( R ( y , z ) P ( z ) ) ) ) \forall y(R(x,y)\rightarrow(\forall z(R(y,z)\rightarrow P(z))))
  31. p \Box\Box p
  32. x x
  33. y y
  34. x x
  35. z z
  36. y y
  37. P P
  38. z z
  39. x x
  40. R ( x , y ) R(x,y)
  41. x x
  42. x x
  43. y y
  44. y y
  45. z z

Standardized_uptake_value.html

  1. S U V ( t ) = c ( t ) 𝑖𝑛𝑗𝑒𝑐𝑡𝑒𝑑 𝑎𝑐𝑡𝑖𝑣𝑖𝑡𝑦 ( t ) / 𝑏𝑜𝑑𝑦 𝑤𝑒𝑖𝑔ℎ𝑡 SUV(t)=\frac{c(t)}{\mathit{injected\ activity}(t)\quad/\quad\mathit{body\ % weight}}

Stanley–Reisner_ring.html

  1. I Δ = ( x i 1 x i r : { i 1 , , i r } Δ ) , k [ Δ ] = k [ x 1 , , x n ] / I Δ . I_{\Delta}=(x_{i_{1}}\ldots x_{i_{r}}:\{i_{1},\ldots,i_{r}\}\notin\Delta),% \quad k[\Delta]=k[x_{1},\ldots,x_{n}]/I_{\Delta}.
  2. k [ Δ ] = σ Δ k [ Δ ] σ , k[\Delta]=\bigoplus_{\sigma\in\Delta}k[\Delta]_{\sigma},
  3. H ( k [ Δ ] ; x 1 , , x n ) = σ Δ i σ x i 1 - x i . H(k[\Delta];x_{1},\ldots,x_{n})=\sum_{\sigma\in\Delta}\prod_{i\in\sigma}\frac{% x_{i}}{1-x_{i}}.
  4. H ( k [ Δ ] ; t , , t ) = 1 ( 1 - t ) n i = 0 d f i - 1 t i ( 1 - t ) n - i , H(k[\Delta];t,\ldots,t)=\frac{1}{(1-t)^{n}}\sum_{i=0}^{d}f_{i-1}t^{i}(1-t)^{n-% i},
  5. H ( k [ Δ ] ; t , , t ) = h 0 + h 1 t + + h d t d ( 1 - t ) d H(k[\Delta];t,\ldots,t)=\frac{h_{0}+h_{1}t+\cdots+h_{d}t^{d}}{(1-t)^{d}}
  6. k [ Δ ] = k [ x 1 , , x n ] k[\Delta]=k[x_{1},\ldots,x_{n}]
  7. I Δ = { x i x j : 1 i < j n } I_{\Delta}=\{x_{i}x_{j}:1\leq i<j\leq n\}
  8. k [ Δ ] = k 1 i n x i k [ x i ] . k[\Delta]=k\oplus\bigoplus_{1\leq i\leq n}x_{i}k[x_{i}].
  9. k [ Δ ] = k 0 r d i 0 < < i r x i 0 x i r k [ x i 0 , , x i r ] . k[\Delta]=k\oplus\bigoplus_{0\leq r\leq d}\bigoplus_{i_{0}<\ldots<i_{r}}x_{i_{% 0}}\ldots x_{i_{r}}k[x_{i_{0}},\ldots,x_{i_{r}}].
  10. k [ Δ ] k [ Δ ] k k [ Δ ′′ ] . k[\Delta]\simeq k[\Delta^{\prime}]\otimes_{k}k[\Delta^{\prime\prime}].
  11. H ~ i ( link Δ ( σ ) ; k ) = 0 for all i < dim link Δ ( σ ) . \tilde{H}_{i}(\operatorname{link}_{\Delta}(\sigma);k)=0\quad\,\text{for all}% \quad i<\dim\operatorname{link}_{\Delta}(\sigma).
  12. For all p | Δ | and for all i < dim | Δ | = d - 1 , H ~ i ( | Δ | ; k ) = H i ( | Δ | , | Δ | - p ; k ) = 0. \,\text{For all }p\in|\Delta|\,\text{ and for all }i<\dim|\Delta|=d-1,\quad% \tilde{H}_{i}(\operatorname{|}\Delta|;k)=H_{i}(\operatorname{|}\Delta|,% \operatorname{|}\Delta|-p;k)=0.

Star_of_David_theorem.html

  1. gcd { ( n - 1 k - 1 ) , ( n k + 1 ) , ( n + 1 k ) } \displaystyle{}\quad\gcd\left\{{\left({{n-1}\atop{k-1}}\right)},{\left({{n}% \atop{k+1}}\right)},{\left({{n+1}\atop{k}}\right)}\right\}

StarTram.html

  1. F = ( μ I 1 I 2 l ) / ( 2 π r ) F=(\mu I_{1}I_{2}l)/(2\pi r)
  2. μ = μ 0 μ r \mu=\mu_{0}\mu_{r}
  3. I 1 , I 2 I_{1},I_{2}
  4. l l
  5. r r
  6. μ r \mu_{r}
  7. I 1 I_{1}
  8. I 2 I_{2}

Statistical_association_football_predictions.html

  1. y 1 = 3 - 1 y_{1}=3-1
  2. y 2 = 2 - 1 y_{2}=2-1
  3. y 3 = 1 - 4 y_{3}=1-4
  4. y 4 = 3 - 1 y_{4}=3-1
  5. y 5 = 2 - 0 y_{5}=2-0
  6. r A r_{A}
  7. r B r_{B}
  8. r C r_{C}
  9. r D r_{D}
  10. y 1 = r A - r B + ε 1 y_{1}=r_{A}-r_{B}+\varepsilon_{1}
  11. y 1 y_{1}
  12. ε 1 \varepsilon_{1}
  13. y 1 = r A - r B + ε 1 y 2 = r C - r D + ε 2 y 5 = r B - r C + ε 5 \begin{matrix}y_{1}=r_{A}-r_{B}+\varepsilon_{1}\\ y_{2}=r_{C}-r_{D}+\varepsilon_{2}\\ ...\\ y_{5}=r_{B}-r_{C}+\varepsilon_{5}\\ \end{matrix}
  14. 𝐲 = 𝐗𝐫 + 𝐞 \mathbf{y}=\mathbf{Xr}+\mathbf{e}
  15. 𝐲 = [ 2 1 - 3 2 2 ] , 𝐗 = [ 1 - 1 0 0 0 0 1 - 1 0 - 1 0 1 1 0 0 - 1 0 1 - 1 0 ] , 𝐫 = [ r A r B r C r D ] , 𝐞 = [ ε 1 ε 2 ε 3 ε 4 ε 5 ] \begin{matrix}\mathbf{y}=\left[\begin{matrix}2\\ 1\\ -3\\ 2\\ 2\\ \end{matrix}\right],&\mathbf{X}=\left[\begin{matrix}1&-1&0&0\\ 0&0&1&-1\\ 0&-1&0&1\\ 1&0&0&-1\\ 0&1&-1&0\\ \end{matrix}\right],&\mathbf{r}=\left[\begin{matrix}r_{A}\\ r_{B}\\ r_{C}\\ r_{D}\\ \end{matrix}\right],&\mathbf{e}=\left[\begin{matrix}\varepsilon_{1}\\ \varepsilon_{2}\\ \varepsilon_{3}\\ \varepsilon_{4}\\ \varepsilon_{5}\\ \end{matrix}\right]\\ \end{matrix}
  16. 𝐗 T 𝐗 \mathbf{X}^{T}\mathbf{X}
  17. 𝐫 = ( 𝐗 T 𝐗 ) - 1 𝐗 T 𝐲 \mathbf{r}=\left(\mathbf{X}^{T}\mathbf{X}\right)^{-1}\mathbf{X}^{T}\mathbf{y}
  18. 𝐫 = 𝐗 + 𝐲 \mathbf{r}=\mathbf{X}^{+}\mathbf{y}
  19. 𝐫 = [ 1.625 , 0.75 , - 0.875 , - 1.5 ] T . \mathbf{r}=[1.625,\ 0.75,\ -0.875,\ -1.5]^{T}.
  20. X i , j X_{i,j}
  21. Y i , j Y_{i,j}
  22. X i , j \displaystyle X_{i,j}
  23. X i , j X_{i,j}
  24. Y i , j Y_{i,j}
  25. λ \lambda
  26. μ \mu
  27. P ( X i , j = x , Y i , j = y ) = λ x exp ( - λ ) x ! μ y exp ( - μ ) y ! P\left(X_{i,j}=x,Y_{i,j}=y\right)=\frac{\lambda^{x}\exp(-\lambda)}{x!}\frac{% \mu^{y}\exp(-\mu)}{y!}
  28. λ \lambda
  29. μ \mu
  30. log ( λ ) = c λ + a i - d j + h \log\left(\lambda\right)=c^{\lambda}+a_{i}-d_{j}+h
  31. log ( μ ) = c μ + a j - d i \log\left(\mu\right)=c^{\mu}+a_{j}-d_{i}
  32. a i , d i , h > 0 a_{i},d_{i},h>0
  33. c λ c^{\lambda}
  34. c μ c^{\mu}
  35. λ \lambda
  36. μ \mu
  37. L ( a i , d i , h ; i = 1 , . . C ) = - log n = 1 N λ n x n exp ( - λ n ) x n ! μ n y n exp ( - μ n ) y n ! \displaystyle L(a_{i},d_{i},h;\ i=1,..C)=-\log\prod\limits_{n=1}^{N}{\frac{% \lambda_{n}^{x_{n}}\exp(-\lambda_{n})}{x_{n}!}\frac{\mu_{n}^{y_{n}}\exp(-\mu_{% n})}{y_{n}!}}
  38. x n x_{n}
  39. y n y_{n}
  40. ( a i , d i ) \left(a_{i},d_{i}\right)
  41. ( h ) \left(h\right)
  42. min a i , d i , h L ( a i , d i , h , i = 1 , . . C ) \underset{a_{i},d_{i},h}{\mathop{\min}}\,L(a_{i},d_{i},h,i=1,..C)
  43. λ \lambda
  44. μ \mu
  45. log ( λ ) = c λ + a i - d j - γ Δ i , j \displaystyle\log\left(\lambda\right)=c^{\lambda}+a_{i}-d_{j}-\gamma\cdot% \Delta_{i,j}
  46. Δ i , j = ( a i - d j ) + ( d i - a j ) 2 \Delta_{i,j}=\frac{\left(a_{i}-d_{j}\right)+\left(d_{i}-a_{j}\right)}{2}
  47. γ > 0 \gamma>0
  48. ( a ) \left(a\right)
  49. B a , A ( t ) B_{a,A}\left(t\right)
  50. t 1 > t 0 t_{1}>t_{0}
  51. a A t 1 = a A t 0 + ( B a , A ( t 1 / τ ) - B a , A ( t 0 / τ ) ) σ a , A 1 - γ ( 1 - γ / 2 ) a_{A}^{t_{1}}=a_{A}^{t_{0}}+\left(B_{a,A}\left(t_{1}/\tau\right)-B_{a,A}\left(% t_{0}/\tau\right)\right)\cdot\frac{\sigma_{a,A}}{\sqrt{1-\gamma\left(1-{\gamma% }/{2}\;\right)}}
  52. τ \tau
  53. σ a , A 2 \sigma_{a,A}^{2}
  54. a A t 1 | a A t 0 N ( a A t 0 , t 1 - t 0 τ σ a , A 2 ) {a_{A}^{t_{1}}}|{a_{A}^{t_{0}}}\;\sim N\left(a_{A}^{t_{0}},\ \frac{t_{1}-t_{0}% }{\tau}\sigma_{a,A}^{2}\right)
  55. t 0 t_{0}
  56. t 0 t_{0}
  57. t 1 t_{1}
  58. P ( a i , d i , γ , τ ; A , B , C ) = P ( λ A , t 0 ) P ( λ B , t 0 ) P ( λ C , t 0 ) \displaystyle P(a_{i},d_{i},\gamma,\,\tau;\ A,B,C)=P\left(\lambda_{A},t_{0}% \right)\cdot P\left(\lambda_{B},t_{0}\right)\cdot P\left(\lambda_{C},t_{0}\right)

Stellar_pulsations.html

  1. κ \kappa
  2. κ \kappa
  3. κ \kappa
  4. κ \kappa
  5. d A 1 / d t = κ 1 A 1 + ( Q 11 A 1 2 + Q 12 A 2 2 ) A 1 dA_{1}/dt=\kappa_{1}A_{1}+(Q_{11}A_{1}^{2}+Q_{12}A_{2}^{2})A_{1}
  6. d A 2 / d t = κ 2 A 2 + ( Q 21 A 1 2 + Q 22 A 2 2 ) A 2 dA_{2}/dt=\kappa_{2}A_{2}+(Q_{21}A_{1}^{2}+Q_{22}A_{2}^{2})A_{2}
  7. \neq
  8. \neq
  9. \neq
  10. \neq
  11. κ \kappa
  12. κ \kappa

Stericated_5-simplexes.html

  1. ( ± 1 , 0 , 0 , 0 , 0 ) \left(\pm 1,\ 0,\ 0,\ 0,\ 0\right)
  2. ( 0 , ± 1 , 0 , 0 , 0 ) \left(0,\ \pm 1,\ 0,\ 0,\ 0\right)
  3. ( 0 , 0 , ± 1 , 0 , 0 ) \left(0,\ 0,\ \pm 1,\ 0,\ 0\right)
  4. ( ± 1 / 2 , 0 , ± 1 / 2 , - 1 / 8 , - 3 / 8 ) \left(\pm 1/2,\ 0,\ \pm 1/2,\ -\sqrt{1/8},\ -\sqrt{3/8}\right)
  5. ( ± 1 / 2 , 0 , ± 1 / 2 , 1 / 8 , 3 / 8 ) \left(\pm 1/2,\ 0,\ \pm 1/2,\ \sqrt{1/8},\ \sqrt{3/8}\right)
  6. ( 0 , ± 1 / 2 , ± 1 / 2 , - 1 / 8 , 3 / 8 ) \left(0,\ \pm 1/2,\ \pm 1/2,\ -\sqrt{1/8},\ \sqrt{3/8}\right)
  7. ( 0 , ± 1 / 2 , ± 1 / 2 , 1 / 8 , - 3 / 8 ) \left(0,\ \pm 1/2,\ \pm 1/2,\ \sqrt{1/8},\ -\sqrt{3/8}\right)
  8. ( ± 1 / 2 , ± 1 / 2 , 0 , ± 1 / 2 , 0 ) \left(\pm 1/2,\ \pm 1/2,\ 0,\ \pm\sqrt{1/2},\ 0\right)
  9. I ~ 1 {\tilde{I}}_{1}
  10. A ~ 2 {\tilde{A}}_{2}
  11. A ~ 3 {\tilde{A}}_{3}
  12. A ~ 4 {\tilde{A}}_{4}
  13. A ~ 5 {\tilde{A}}_{5}

Sterling_ratio.html

  1. S R = C o m p o u n d R O R A B S ( A v g . A n n u a l D D - 10 % ) SR=\frac{CompoundROR}{ABS(Avg.AnnualDD-10\%)}
  2. S R = A n n u a l P o r t f o l i o R e t u r n - A n n u a l R i s k - F r e e R a t e A v e r a g e L a r g e s t D r a w d o w n SR=\frac{Annual\ Portfolio\ Return-Annual\ Risk\operatorname{-}Free\ Rate}{% Average\ Largest\ Drawdown}

Stevedore_knot_(mathematics).html

  1. Δ ( t ) = - 2 t + 5 - 2 t - 1 , \Delta(t)=-2t+5-2t^{-1},\,
  2. ( z ) = 1 - 2 z 2 , \nabla(z)=1-2z^{2},\,
  3. V ( q ) = q 2 - q + 2 - 2 q - 1 + q - 2 - q - 3 + q - 4 . V(q)=q^{2}-q+2-2q^{-1}+q^{-2}-q^{-3}+q^{-4}.\,

Stieltjes–Wigert_polynomials.html

  1. w ( x ) = k π x - 1 / 2 exp ( - k 2 log 2 x ) w(x)=\frac{k}{\sqrt{\pi}}x^{-1/2}\exp(-k^{2}\log^{2}x)
  2. S n ( x ; q ) = 1 ( q ; q ) n ) ϕ 1 1 ( q - n , 0 ; q , - q n + 1 x ) \displaystyle S_{n}(x;q)=\frac{1}{(q;q)_{n})}{}_{1}\phi_{1}(q^{-n},0;q,-q^{n+1% }x)
  3. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  4. 1 ( - x , - q x - 1 ; q ) \frac{1}{(-x,-qx^{-1};q)_{\infty}}
  5. k π x - 1 / 2 exp ( - k 2 log 2 x ) \frac{k}{\sqrt{\pi}}x^{-1/2}\exp(-k^{2}\log^{2}x)

Stiffness_matrix.html

  1. - 2 u = f -\nabla^{2}u=f
  2. u u h = u 1 φ 1 + + u n φ n . u\approx u^{h}=u_{1}\varphi_{1}+\cdots+u_{n}\varphi_{n}.
  3. Ω φ i f d x = - Ω φ i 2 u h d x = - j ( Ω φ i 2 φ j d x ) u j = j ( Ω φ i φ j d x ) u j . \int_{\Omega}\varphi_{i}\cdot f\,dx=-\int_{\Omega}\varphi_{i}\nabla^{2}u^{h}\,% dx=-\sum_{j}\left(\int_{\Omega}\varphi_{i}\nabla^{2}\varphi_{j}\,dx\right)\,u_% {j}=\sum_{j}\left(\int_{\Omega}\nabla\varphi_{i}\cdot\nabla\varphi_{j}\,dx% \right)u_{j}.
  4. A i j = Ω φ i φ j d x . A_{ij}=\int_{\Omega}\nabla\varphi_{i}\cdot\nabla\varphi_{j}\,dx.
  5. - k , l x k ( a k l u x l ) = f -\sum_{k,l}\frac{\partial}{\partial x_{k}}\left(a^{kl}\frac{\partial u}{% \partial x_{l}}\right)=f
  6. - k , l ν k a k l u x l = c ( u - g ) , -\sum_{k,l}\nu_{k}a^{kl}\frac{\partial u}{\partial x_{l}}=c(u-g),
  7. j ( k , l Ω a k l φ i x k φ j x l d x + Ω c φ i φ j d s ) u j = Ω φ i f d x + Ω c φ i g d s , \sum_{j}\left(\sum_{k,l}\int_{\Omega}a^{kl}\frac{\partial\varphi_{i}}{\partial x% _{k}}\frac{\partial\varphi_{j}}{\partial x_{l}}dx+\int_{\partial\Omega}c% \varphi_{i}\varphi_{j}\,ds\right)u_{j}=\int_{\Omega}\varphi_{i}f\,dx+\int_{% \partial\Omega}c\varphi_{i}g\,ds,
  8. B [ u , v ] = ( f , v ) B[u,v]=(f,v)
  9. A i j = B [ φ j , φ i ] . A_{ij}=B[\varphi_{j},\varphi_{i}].
  10. A i j [ k ] = T k φ i φ j d x . A^{[k]}_{ij}=\int_{T_{k}}\nabla\varphi_{i}\cdot\nabla\varphi_{j}\,dx.
  11. D = [ x 3 - x 2 x 1 - x 3 x 2 - x 1 y 3 - y 2 y 1 - y 3 y 2 - y 1 ] . D=\left[\begin{matrix}x_{3}-x_{2}&x_{1}-x_{3}&x_{2}-x_{1}\\ y_{3}-y_{2}&y_{1}-y_{3}&y_{2}-y_{1}\end{matrix}\right].
  12. A [ k ] = D D / ( 4 area ( T ) ) A^{[k]}=D^{\top}D/(4\cdot\mathrm{area}(T))

Stochastic_computing.html

  1. p , q [ 0 , 1 ] p,q\in[0,1]
  2. p × q p\times q
  3. p p
  4. q q
  5. a i a_{i}
  6. b i b_{i}
  7. a i b i a_{i}\land b_{i}
  8. p q pq
  9. p q pq
  10. p p
  11. q q
  12. a i b i a_{i}\land b_{i}
  13. p p
  14. q q
  15. n n
  16. n 2 n^{2}
  17. 2 n 2n
  18. 2 n 2n
  19. 2 - 2 n 2^{-2n}
  20. 2 - 2 n 2^{-2n}
  21. O ( 2 4 n ) O(2^{4n})
  22. p 2 p^{2}
  23. p p
  24. a i a i = a i a_{i}\land a_{i}=a_{i}
  25. p × p = p p\times p=p
  26. p = p=
  27. f ( p , q ) p q / ( p q + ( 1 - p ) ( 1 - q ) ) f(p,q)\rightarrow pq/(pq+(1-p)(1-q))
  28. f ( p , q ) ( p + q ) / 2 f(p,q)\rightarrow(p+q)/2
  29. s s
  30. O ( 1 / s ) O(1/\sqrt{s})
  31. 1 / s 1/s

Stokes_operator.html

  1. P σ P_{\sigma}
  2. A A
  3. A := - P σ , A:=-P_{\sigma}\triangle,
  4. 2 \triangle\equiv\nabla^{2}
  5. A A
  6. 𝒟 ( A ) = H 2 V \mathcal{D}(A)=H^{2}\cap V
  7. V = { u ( H 0 1 ( Ω ) ) n | div u = 0 } V=\{\vec{u}\in(H^{1}_{0}(\Omega))^{n}|\operatorname{div}\,\vec{u}=0\}
  8. Ω \Omega
  9. n \mathbb{R}^{n}
  10. H 2 ( Ω ) H^{2}(\Omega)
  11. H 0 1 ( Ω ) H^{1}_{0}(\Omega)
  12. u \vec{u}
  13. Ω \Omega
  14. C 2 C^{2}
  15. A A
  16. L 2 L^{2}
  17. { w k } k = 1 \{w_{k}\}_{k=1}^{\infty}
  18. { λ k } k = 1 \{\lambda_{k}\}_{k=1}^{\infty}
  19. 0 < λ 1 < λ 2 λ 3 λ k 0<\lambda_{1}<\lambda_{2}\leq\lambda_{3}\cdots\leq\lambda_{k}\leq\cdots
  20. λ k \lambda_{k}\rightarrow\infty
  21. k k\rightarrow\infty
  22. α > 0 \alpha>0
  23. A α A^{\alpha}
  24. u 𝒟 ( A ) \vec{u}\in\mathcal{D}(A)
  25. A α u = k = 1 λ k α u k w k A^{\alpha}\vec{u}=\sum_{k=1}^{\infty}\lambda_{k}^{\alpha}u_{k}\vec{w_{k}}
  26. u k := ( u , w k ) u_{k}:=(\vec{u},\vec{w_{k}})
  27. ( , ) (\cdot,\cdot)
  28. L 2 ( Ω ) L^{2}(\Omega)
  29. A - 1 A^{-1}
  30. H := { u ( L 2 ( Ω ) ) n | div u = 0 and γ ( u ) = 0 } H:=\{\vec{u}\in(L^{2}(\Omega))^{n}|\operatorname{div}\,\vec{u}=0\,\text{ and }% \gamma(\vec{u})=0\}
  31. γ \gamma
  32. A - 1 : H V A^{-1}:H\rightarrow V

Stoletov_curve.html

  1. χ \chi

Stoner_criterion.html

  1. E ( k ) = ϵ ( k ) - I N - N N , E ( k ) = ϵ ( k ) + I N - N N , E_{\uparrow}(k)=\epsilon(k)-I\frac{N_{\uparrow}-N_{\downarrow}}{N},\qquad E_{% \downarrow}(k)=\epsilon(k)+I\frac{N_{\uparrow}-N_{\downarrow}}{N},
  2. N / N N_{\uparrow}/N
  3. N / N N_{\downarrow}/N
  4. ϵ ( k ) \epsilon(k)
  5. N + N N_{\uparrow}+N_{\downarrow}
  6. E ( k ) , E ( k ) E_{\uparrow}(k),E_{\downarrow}(k)
  7. P = ( N - N ) / N P=(N_{\uparrow}-N_{\downarrow})/N
  8. I D ( E F ) > 1 ID(E_{F})>1
  9. D ( E F ) D(E_{F})
  10. n i \langle n_{i}\rangle
  11. n i - n i n_{i}-\langle n_{i}\rangle
  12. H = U i n i , n i , + n i , n i , - n i , n i , + i , σ ϵ i n i , σ . H=U\sum_{i}n_{i,\uparrow}\langle n_{i,\downarrow}\rangle+n_{i,\downarrow}% \langle n_{i,\uparrow}\rangle-\langle n_{i,\uparrow}\rangle\langle n_{i,% \downarrow}\rangle+\sum_{i,\sigma}\epsilon_{i}n_{i,\sigma}.
  13. D ( E F ) U > 1. D(E_{F})U>1.
  14. N N_{\uparrow}
  15. ϵ ( k ) \epsilon(k)
  16. ϵ i \epsilon_{i}
  17. D ( E ) = i δ ( E - ϵ i ) D(E)=\sum_{i}\delta(E-\epsilon_{i})

Stoney_units.html

  1. c = G = k e = e = 1. c=G=k\text{e}=e=1.
  2. = 1 α 137.035999679 , \hbar=\frac{1}{\alpha}\approx 137.035999679,
  3. 1 / 16 {1}/{16}

Storage_efficiency.html

  1. storage efficiency = effective capacity + free capacity raw capacity . \,\text{storage efficiency}=\frac{\,\text{effective capacity}+\,\text{free % capacity}}{\,\text{raw capacity}}.

Strafing_(gaming).html

  1. 2 \sqrt{2}
  2. 3 \sqrt{3}

Stream_power.html

  1. Ω = ρ g Q S \Omega=\rho gQS
  2. Δ P E Δ t = m g Δ z Δ t \frac{\Delta PE}{\Delta t}=mg\frac{\Delta z}{\Delta t}
  3. Δ z / Δ t {\Delta z}/{\Delta t}
  4. u z u_{z}
  5. α \alpha
  6. Δ z Δ t = u z = u sin ( α ) u S \frac{\Delta z}{\Delta t}=u_{z}=u\sin(\alpha)\approx uS
  7. u u
  8. sin ( α ) tan ( α ) = S \sin(\alpha)\approx\tan(\alpha)=S
  9. Δ P E Δ t = m g u S \frac{\Delta PE}{\Delta t}=mguS
  10. Ω = Δ P E Δ t \Omega=\frac{\Delta PE}{\Delta t}
  11. m = ρ L b h m=\rho Lbh
  12. L L
  13. b b
  14. h h
  15. Q = u b h Q=ubh
  16. A A
  17. Ω = ρ g Q \cancelto 1 L S \Omega=\rho gQ\cancelto{1}{L}S
  18. ω = ρ g Q S b \omega=\frac{\rho gQS}{b}

Streaming_algorithm.html

  1. 𝐚 = ( a 1 , , a n ) \mathbf{a}=(a_{1},\dots,a_{n})
  2. 𝟎 \mathbf{0}
  3. 𝐚 \mathbf{a}
  4. 𝐚 \mathbf{a}
  5. i , c \langle i,c\rangle
  6. a i a_{i}
  7. c c
  8. c = 1 c=1
  9. i , c \langle i,c\rangle
  10. a i a_{i}
  11. c c
  12. a i a_{i}
  13. ( ϵ , δ ) (\epsilon,\delta)
  14. ϵ \epsilon
  15. 1 - δ 1-\delta
  16. k k
  17. 𝐚 \mathbf{a}
  18. F k ( 𝐚 ) = i = 1 n a i k F_{k}(\mathbf{a})=\sum_{i=1}^{n}a_{i}^{k}
  19. F 1 F_{1}
  20. F 2 F_{2}
  21. F F_{\infty}
  22. F 0 F_{0}
  23. 𝐚 \mathbf{a}
  24. F k ( 𝐚 ) = i = 1 n a i m log a i m F_{k}(\mathbf{a})=\sum_{i=1}^{n}\frac{a_{i}}{m}\log{\frac{a_{i}}{m}}
  25. m = i = 1 n a i m=\sum_{i=1}^{n}a_{i}

Strength_of_a_graph.html

  1. σ ( G ) \sigma(G)
  2. Π \Pi
  3. V V
  4. π \partial\pi
  5. π Π \pi\in\Pi
  6. σ ( G ) = min π Π | π | | π | - 1 \displaystyle\sigma(G)=\min_{\pi\in\Pi}\frac{|\partial\pi|}{|\pi|-1}
  7. 𝒯 \mathcal{T}
  8. σ ( G ) = max { T 𝒯 λ T : T 𝒯 λ T 0 and e E T e λ T 1 } . \sigma(G)=\max\left\{\sum_{T\in\mathcal{T}}\lambda_{T}\ :\ \forall T\in{% \mathcal{T}}\ \lambda_{T}\geq 0\mbox{ and }~{}\forall e\in E\ \sum_{T\ni e}% \lambda_{T}\leq 1\right\}.
  9. σ ( G ) = min { e E y e : e E y e 0 and T 𝒯 e E y e 1 } . \sigma(G)=\min\left\{\sum_{e\in E}y_{e}\ :\ \forall e\in E\ y_{e}\geq 0\mbox{ % and }~{}\forall T\in{\mathcal{T}}\ \sum_{e\in E}y_{e}\geq 1\right\}.
  10. O ( min ( m , n 2 / 3 ) m n log ( n 2 / m + 2 ) ) O(\min(\sqrt{m},n^{2/3})mn\log(n^{2}/m+2))
  11. π = { V 1 , , V k } \pi=\{V_{1},\dots,V_{k}\}
  12. i { 1 , , k } i\in\{1,\dots,k\}
  13. G i = G / V i G_{i}=G/V_{i}
  14. V i V_{i}
  15. σ ( G k ) σ ( G ) \sigma(G_{k})\geq\sigma(G)
  16. σ ( G ) \lfloor\sigma(G)\rfloor

Stress–strength_analysis.html

  1. ( μ x ) \left(\mu_{x}\right)
  2. ( s x ) \left(s_{x}\right)
  3. ( μ y ) \left(\mu_{y}\right)
  4. ( s y ) \left(s_{y}\right)
  5. ( Z ) \left(Z\right)
  6. R = 1 - P ( Z ) R=1-P(Z)
  7. Z = - μ x - μ y s x 2 + s y 2 Z=-\frac{\mu_{x}-\mu_{y}}{\sqrt{s_{x}^{2}+s_{y}^{2}}}

String_bending.html

  1. ν = 1 2 L T + cos θ ( T - E A ) μ o \nu=\frac{1}{2L}\sqrt{\frac{T+\cos\theta(T-EA)}{\mu_{o}}}
  2. θ \theta
  3. μ o \mu_{o}
  4. θ \theta
  5. F B = ( T + E A ( 1 - cos θ cos θ ) ) sin θ . F_{B}=\left(T+EA\left(\frac{1-\cos\theta}{\cos\theta}\right)\right)\sin\theta.

Structural_engineering_theory.html

  1. F = 0 \sum\vec{F}=0
  2. M = 0 \sum\vec{M}=0
  3. b b
  4. j j
  5. r r
  6. r + b = 2 j r+b=2j
  7. 𝐅 = k 𝐱 \vec{\mathbf{F}}=k\vec{\mathbf{x}}
  8. E I d 4 u d x 4 = w ( x ) . EI\frac{d^{4}u}{dx^{4}}=w(x).\,
  9. u u
  10. w ( x ) w(x)
  11. E E
  12. I I
  13. u \textstyle{u}\,
  14. u x \textstyle{\frac{\partial u}{\partial x}}\,
  15. E I 2 u x 2 \textstyle{EI\frac{\partial^{2}u}{\partial x^{2}}}\,
  16. - x ( E I 2 u x 2 ) \textstyle{-\frac{\partial}{\partial x}\left(EI\frac{\partial^{2}u}{\partial x% ^{2}}\right)}\,
  17. σ = M y I = E y 2 u x 2 \sigma=\frac{My}{I}=Ey\frac{\partial^{2}u}{\partial x^{2}}\,
  18. σ \sigma
  19. M M
  20. y y
  21. I I
  22. F = π 2 E I ( K l ) 2 F=\frac{\pi^{2}EI}{(Kl)^{2}}
  23. F F
  24. E E
  25. I I
  26. l l
  27. K K
  28. K K
  29. K K
  30. K K\approx
  31. K K
  32. σ = π 2 E ( K l r ) 2 \sigma=\frac{\pi^{2}E}{(\frac{Kl}{r})^{2}}
  33. σ \sigma
  34. r r

Sturm_series.html

  1. p 0 p_{0}
  2. p 1 p_{1}
  3. p 0 p_{0}
  4. p 1 p_{1}
  5. p i := p i + 1 q i + 1 - p i + 2 for i 0. p_{i}:=p_{i+1}q_{i+1}-p_{i+2}\,\text{ for }i\geq 0.
  6. p i + 2 p_{i+2}
  7. p 0 , p 1 , , p k p_{0},p_{1},\dots,p_{k}
  8. P P
  9. λ \lambda
  10. P ( λ ) = a 0 λ k + a 1 λ k - 1 + + a k - 1 λ + a k P(\lambda)=a_{0}\lambda^{k}+a_{1}\lambda^{k-1}+\cdots+a_{k-1}\lambda+a_{k}
  11. a i a_{i}
  12. i i
  13. { 1 , , k } \{1,\dots,k\}
  14. ( Z ) \mathbb{R}(Z)
  15. Z Z
  16. P ( ı μ ) P(\imath\mu)
  17. ı k \imath^{k}
  18. ı \imath
  19. - 1 \sqrt{-1}
  20. p 0 ( μ ) \displaystyle p_{0}(\mu)
  21. p i ( μ ) = c i , 0 μ k - i + c i , 1 μ k - i - 2 + c i , 2 μ k - i - 4 + p_{i}(\mu)=c_{i,0}\mu^{k-i}+c_{i,1}\mu^{k-i-2}+c_{i,2}\mu^{k-i-4}+\cdots
  22. q i q_{i}
  23. ( c i - 1 , 0 / c i , 0 ) μ (c_{i-1,0}/c_{i,0})\mu
  24. c i , 0 0 c_{i,0}\neq 0
  25. p i p_{i}
  26. c i , j c_{i,j}
  27. c i + 1 , j = c i , j + 1 c i - 1 , 0 c i , 0 - c i - 1 , j + 1 = 1 c i , 0 det ( c i - 1 , 0 c i - 1 , j + 1 c i , 0 c i , j + 1 ) . c_{i+1,j}=c_{i,j+1}\frac{c_{i-1,0}}{c_{i,0}}-c_{i-1,j+1}=\frac{1}{c_{i,0}}\det% \begin{pmatrix}c_{i-1,0}&c_{i-1,j+1}\\ c_{i,0}&c_{i,j+1}\end{pmatrix}.
  28. c i , 0 = 0 c_{i,0}=0
  29. i i
  30. q i q_{i}
  31. p i p_{i}
  32. p h p_{h}
  33. h < k h<k

SUBCLU.html

  1. S S
  2. T S T\subseteq S
  3. C D B C\subseteq DB
  4. S S
  5. T S T\subseteq S
  6. T T
  7. C C
  8. S S
  9. T S T\subseteq S
  10. k + 1 k+1
  11. k k
  12. k - 1 k-1
  13. k k
  14. k + 1 k+1
  15. ϵ \epsilon\!\,
  16. M i n P t s MinPts
  17. S U B C L U ( D B , e p s , M i n P t s ) \!\,SUBCLU(DB,eps,MinPts)
  18. S 1 := S_{1}:=\emptyset
  19. C 1 := C_{1}:=\emptyset
  20. f o r e a c h a A t t r i b u t e s for\,each\,a\in Attributes
  21. C { a } = D B S C A N ( D B , { a } , e p s , M i n P t s ) C^{\{a\}}=DBSCAN(DB,\{a\},eps,MinPts)\!\,
  22. i f ( C { a } ) if(C^{\{a\}}\neq\emptyset)
  23. S 1 := S 1 { a } S_{1}:=S_{1}\cup\{a\}
  24. C 1 := C 1 C { a } C_{1}:=C_{1}\cup C^{\{a\}}
  25. e n d i f end\,if
  26. e n d f o r end\,for
  27. k + 1 k+1
  28. k k
  29. k := 1 k:=1\!\,
  30. w h i l e ( C k ) while(C_{k}\neq\emptyset)
  31. C a n d S k + 1 := G e n e r a t e C a n d i d a t e S u b s p a c e s ( S k ) CandS_{k+1}:=GenerateCandidateSubspaces(S_{k})\!\,
  32. f o r e a c h c a n d C a n d S k + 1 for\,each\,cand\in CandS_{k+1}
  33. b e s t S u b s p a c e := min s S k s c a n d C i C s | C i | bestSubspace:=\min_{s\in S_{k}\wedge s\subset cand}\sum_{C_{i}\in C^{s}}|C_{i}|
  34. C c a n d := C^{cand}:=\emptyset
  35. f o r e a c h c l u s t e r c l C b e s t S u b s p a c e for\,each\,cluster\,cl\in C^{bestSubspace}
  36. C c a n d := C c a n d D B S C A N ( c l , c a n d , e p s , M i n P t s ) C^{cand}:=C^{cand}\cup DBSCAN(cl,cand,eps,MinPts)
  37. i f ( C c a n d ) if\,(C^{cand}\neq\emptyset)
  38. S k + 1 := S k + 1 c a n d S_{k+1}:=S_{k+1}\cup cand
  39. C k + 1 := C k + 1 C c a n d C_{k+1}:=C_{k+1}\cup C^{cand}
  40. e n d i f end\,if
  41. e n d f o r end\,for
  42. e n d f o r end\,for
  43. k := k + 1 k:=k+1\!\,
  44. e n d w h i l e end\,while
  45. e n d end\!\,
  46. S k S_{k}
  47. k k
  48. C k C_{k}
  49. b e s t S u b s p a c e bestSubspace
  50. k k
  51. k + 1 k+1
  52. k k
  53. G e n e r a t e C a n d i d a t e S u b s p a c e s ( S k ) \,\!GenerateCandidateSubspaces(S_{k})
  54. C a n d S k + 1 := CandS_{k+1}:=\emptyset
  55. f o r e a c h s 1 S k for\,each\,s_{1}\in S_{k}
  56. f o r e a c h s 2 S k for\,each\,s_{2}\in S_{k}
  57. i f ( s 1 a n d s 2 𝑑𝑖𝑓𝑓𝑒𝑟 i n e x a c t e l y o n e a t t r i b u t e ) if\,(s_{1}\,and\,s_{2}\,\,\mathit{differ}\,\,in\,\,exactely\,\,one\,\,attribute)
  58. C a n d S k + 1 := C a n d S k + 1 { s 1 s 2 } CandS_{k+1}:=CandS_{k+1}\cup\{s_{1}\cup s_{2}\}
  59. e n d i f end\,if
  60. e n d f o r end\,for
  61. e n d f o r end\,for
  62. f o r e a c h c a n d C a n d S k + 1 for\,each\,cand\in CandS_{k+1}
  63. f o r e a c h k - e l e m e n t s c a n d for\,each\,k-element\,s\subset cand
  64. i f ( s S k ) if\,(s\not\in S_{k})
  65. C a n d S k + 1 = C a n d S k + 1 { c a n d } CandS_{k+1}=CandS_{k+1}\setminus\{cand\}
  66. e n d i f end\,if
  67. e n d f o r end\,for
  68. e n d f o r end\,for
  69. e n d end\,\!

Subindependence.html

  1. φ X + Y ( t ) = φ X ( t ) φ Y ( t ) . \varphi_{X+Y}(t)=\varphi_{X}(t)\cdot\varphi_{Y}(t).\,

Subliminal_channel.html

  1. k k
  2. m m^{\prime}
  3. p = 2347 p=2347
  4. q = 23 q=23
  5. g = 266 g=266
  6. x = 1468 x=1468
  7. y = g x y=g^{x}
  8. p = 2100 p=2100
  9. m = 1337 m=1337
  10. H ( m ) H(m)
  11. h = m h=m
  12. q = 1337 q=1337
  13. 107 = 53 107=53
  14. k = ? k=?
  15. m = 17 m^{\prime}=17
  16. m - 1 = 19 m^{\prime-1}=19
  17. 23 23
  18. r = ( g k r=(g^{k}
  19. p ) p)
  20. q = ( 266 17 q=(266^{17}
  21. 2347 ) 2347)
  22. 23 = 12 23=12
  23. s = k - 1 * ( h + x * r ) s=k^{-1}*(h+x*r)
  24. q = 19 * ( 53 + 1468 * 12 ) q=19*(53+1468*12)
  25. 23 = 3 23=3
  26. ( 1337 ; 12 , 3 ) (1337;12,3)
  27. ( m ; r , s ) = ( 1337 ; 12 , 3 ) (m;r,s)=(1337;12,3)
  28. h = H ( m ) h=H(m)
  29. q = 1337 q=1337
  30. 107 = 53 107=53
  31. w = s - 1 w=s^{-1}
  32. q = 8 q=8
  33. u 1 = ( h * w ) u_{1}=(h*w)
  34. q = 53 * 8 q=53*8
  35. 23 = 10 23=10
  36. u 2 = ( r * w ) u_{2}=(r*w)
  37. q = 12 * 8 q=12*8
  38. 23 = 4 23=4
  39. v = ( g u 1 * y u 2 v=(g^{u_{1}}*y^{u_{2}}
  40. p ) p)
  41. q = ( 266 10 * 2100 4 q=(266^{10}*2100^{4}
  42. 2347 ) 2347)
  43. 23 = 12 23=12
  44. v = r v=r
  45. m = 8 * ( 53 + 1468 * 12 ) m^{\prime}=8*(53+1468*12)
  46. 23 = 17 23=17
  47. s s
  48. s = m - 1 * ( h + x r ) s=m^{\prime-1}*(h+xr)
  49. q q
  50. s * m = h + x r s*m^{\prime}=h+xr
  51. q q
  52. m = s - 1 * ( h + x r ) m^{\prime}=s^{-1}*(h+xr)
  53. q q

Submerged_specific_gravity.html

  1. SSG = ρ o - ρ f ρ f \,\text{SSG}=\frac{\rho_{o}-\rho_{f}}{\rho_{f}}
  2. SSG \,\text{SSG}
  3. ρ o \rho_{o}
  4. ρ f \rho_{f}

Subthreshold_slope.html

  1. S s - t h = l n ( 10 ) k T q ( 1 + C d C o x ) S_{s-th}=ln(10){kT\over q}(1+{C_{d}\over C_{ox}})
  2. C d C_{d}
  3. C o x C_{ox}
  4. k T q {kT\over q}
  5. C o x \textstyle{C_{ox}}\rightarrow\infty

Succinct_game.html

  1. n n
  2. s s
  3. n s n ns^{n}
  4. d d
  5. n s d + 1 ns^{d+1}
  6. d d
  7. n n
  8. s s
  9. s ( n + s - 2 s - 1 ) s{\textstyle\left({{n+s-2}\atop{s-1}}\right)}
  10. O ( log n / log log n ) O(\log n/\log\log n)
  11. s n ( n + s - 2 s - 1 ) sn{\textstyle\left({{n+s-2}\atop{s-1}}\right)}
  12. O ( n 2 * s 2 ) O(n^{2}*s^{2})
  13. Σ 2 P \Sigma_{2}^{\rm P}
  14. n s n ns^{n}
  15. n s d + 1 ns^{d+1}
  16. s ( n + s - 1 s - 1 ) s{\textstyle\left({{n+s-1}\atop{s-1}}\right)}
  17. s n ( n + s - 1 s - 1 ) sn{\textstyle\left({{n+s-1}\atop{s-1}}\right)}
  18. n 2 * s 2 n^{2}*s^{2}
  19. Σ 2 P \Sigma_{2}^{\rm P}

Sum_of_radicals.html

  1. Σ i = 1 n k i x i r i , \Sigma_{i=1}^{n}k_{i}\sqrt[r_{i}]{x_{i}},
  2. n , r i n,r_{i}
  3. k i , x i k_{i},x_{i}

Summation_equation.html

  1. x ( t ) = f ( t ) + i = m n k ( t , s , x ( s ) ) x(t)=f(t)+\sum_{i=m}^{n}k(t,s,x(s))

Sun-Ni_law.html

  1. W * \textstyle W^{*}
  2. f \textstyle f
  3. ( 1 - f ) \textstyle(1-f)
  4. y = g ( x ) \textstyle y=g(x)
  5. W = g ( M ) \textstyle W=g(M)
  6. W * = g ( m M ) \textstyle W^{*}=g(m\cdot M)
  7. M \textstyle M
  8. W * = g ( m g - 1 ( W ) ) W^{*}=g(m\cdot g^{-1}(W))
  9. ( 1 - f ) W + f g ( m g - 1 ( W ) ) ( 1 - f ) W + f g ( m g - 1 ( W ) ) m \frac{(1-f)W+f\cdot g(m\cdot g^{-1}(W))}{(1-f)W+\frac{f\cdot g(m\cdot g^{-1}(W% ))}{m}}
  10. g ( x ) = a x b \textstyle g(x)=ax^{b}
  11. g ( m x ) = a ( m x ) b = m b a x b = m b g ( x ) = g ¯ ( m ) g ( x ) g(mx)=a(mx)^{b}=m^{b}\cdot ax^{b}=m^{b}g(x)=\bar{g}(m)g(x)
  12. g ¯ ( m ) \textstyle\bar{g}(m)
  13. ( 1 - f ) W + f g ¯ ( m ) W ( 1 - f ) W + f g ¯ ( m ) W m = ( 1 - f ) + f g ¯ ( m ) ( 1 - f ) + f g ¯ ( m ) m \frac{(1-f)W+f\cdot\bar{g}(m)W}{(1-f)W+\frac{f\cdot\bar{g}(m)W}{m}}=\frac{(1-f% )+f\cdot\bar{g}(m)}{(1-f)+\frac{f\cdot\bar{g}(m)}{m}}
  14. g ¯ ( m ) \textstyle\bar{g}(m)
  15. g ¯ ( m ) = 1 \textstyle\bar{g}(m)=1
  16. g ¯ ( m ) = m \textstyle\bar{g}(m)=m
  17. g ¯ ( m ) \textstyle\bar{g}(m)
  18. S p e e d u p memory-bounded = ( 1 - f ) + f G ( n ) ( 1 - f ) + f G ( n ) n Speedup\text{memory-bounded}=\frac{(1-f)+f\cdot G(n)}{(1-f)+\frac{f\cdot G(n)}% {n}}
  19. x = 3 N 2 \textstyle x=3N^{2}
  20. 2 N 3 2N^{3}
  21. g ( x ) = 2 ( x / 3 ) 3 / 2 = 2 3 3 / 2 x 3 / 2 g(x)=2(x/3)^{3/2}=\frac{2}{3^{3/2}}x^{3/2}
  22. g ¯ ( x ) = x 3 / 2 \bar{g}(x)=x^{3/2}
  23. ( 1 - f ) + f g ¯ ( m ) ( 1 - f ) + f g ¯ ( m ) m = ( 1 - f ) + f m 3 / 2 ( 1 - f ) + f m 1 / 2 \frac{(1-f)+f\cdot\bar{g}(m)}{(1-f)+\frac{f\cdot\bar{g}(m)}{m}}=\frac{(1-f)+f% \cdot m^{3/2}}{(1-f)+f\cdot m^{1/2}}
  24. O ( n 3 ) O(n^{3})
  25. O ( n 2 ) O(n^{2})

Sun-Yung_Alice_Chang.html

  1. S 2 S^{2}
  2. S 2 S^{2}
  3. H p H^{p}
  4. H 1 H^{1}

Super-resolution_microscopy.html

  1. Ω \Omega
  2. Δ r Δ 1 + I max / I s \Delta r\approx\frac{\Delta}{\sqrt{1+I_{\max}/I_{s}}}
  3. I s I_{s}
  4. I s I_{s}
  5. Δ loc Δ N \Delta\mathrm{loc}\approx\frac{\Delta}{\sqrt{N}}

Super_Bloch_oscillations.html

  1. k m a x k_{max}
  2. F 0 F_{0}
  3. + k m a x / m +\hbar k_{max}/m
  4. - k m a x / m -\hbar k_{max}/m
  5. | k m a x / m | |\hbar k_{max}/m|
  6. F 0 F_{0}
  7. F ( t ) = F 0 + Δ F sin ( ω t + φ ) F(t)=F_{0}+\Delta F\sin(\omega t+\varphi)
  8. ω \omega
  9. ω B \omega_{B}
  10. ω - ω B \omega-\omega_{B}

Superconducting_tunnel_junction.html

  1. e V eV
  2. f f
  3. n h f / ( 2 e ) nhf/(2e)
  4. h h
  5. e e
  6. n n
  7. n h f / e nhf/e
  8. A l 2 O 3 Al_{2}O_{3}
  9. n f / K J nf/K_{J}
  10. n n
  11. f f
  12. K J = 483597.9 G H z / V K_{J}=483597.9GHz/V
  13. 2 e / h 2e/h

Superfunction.html

  1. S ( z ; x ) = f ( f ( f ( x ) ) ) z evaluations of the function f . S(z;x)=\underbrace{f\Big(f\big(\dots f(x)\dots\big)\Big)}_{z\,\text{ % evaluations of the function }f}.
  2. exp \sqrt{\exp}~{}
  3. ! \sqrt{!~{}}~{}
  4. exp \sqrt{\exp}
  5. ! ~{}\sqrt{!~{}}~{}~{}
  6. φ \varphi
  7. φ ( φ ( u ) ) = exp ( u ) \varphi(\varphi(u))=\exp(u)
  8. 𝒳 \mathcal{X}
  9. 𝒳 ( exp ( u ) ) = 𝒳 ( u ) + 1. \mathcal{X}(\exp(u))=\mathcal{X}(u)+1.
  10. 𝒳 ( S ( z ; u ) ) = 𝒳 ( u ) + z \mathcal{X}(S(z;u))=\mathcal{X}(u)+z
  11. S ( z ; u ) = 𝒳 - 1 ( z + 𝒳 ( u ) ) S(z;u)=\mathcal{X}^{-1}(z+\mathcal{X}(u))
  12. S ( 0 ; x ) = x S(0;x)=x
  13. exp \sqrt{\exp}
  14. ! \sqrt{!~{}}~{}
  15. S ( z + 1 ; x ) = f ( S ( z ; x ) ) z : z > 0 S(z+1;x)=f(S(z;x))~{}~{}~{}~{}~{}~{}~{}~{}\forall z\in\mathbb{N}:z>0
  16. S ( 1 ) = f ( x ) . S(1)=f(x).
  17. S ( 0 ) = x , S(0)=x~{},
  18. S ( - 1 ) = f - 1 ( x ) , S(-1)=f^{-1}(x),
  19. S ( - 2 ) = f - 2 ( x ) S(-2)=f^{-2}(x)
  20. x x
  21. b b
  22. f = exp b . f=\exp_{b}.
  23. S ( - 1 ) = log b 1 = 0 , S(-1)=\log_{b}1=0,
  24. S ( - 2 ) = log b 0 S(-2)=\log_{b}0
  25. a ~{}a~{}
  26. b ~{}b~{}
  27. a ~{}a~{}
  28. D D\subseteq\mathbb{C}
  29. a a
  30. b b
  31. D D
  32. S S
  33. D D
  34. S ( z + 1 ) = f ( S ( z ) ) z D : z + 1 D S(z\!+\!1)=f(S(z))~{}\forall z\in D:z\!+\!1\in D
  35. S ( a ) = b . S(a)=b.
  36. f f
  37. ( a d ) (a\mapsto d)
  38. S S
  39. ( a d ) (a\mapsto d)
  40. G G
  41. G ( z ) = S ( z + μ ( z ) ) G(z)=S(z+\mu(z))
  42. μ \mu
  43. μ ( a ) = 0 \mu(a)=0
  44. H H
  45. ( C , 0 1 ) (C,0\mapsto 1)
  46. exp b \exp_{b}
  47. b > 1 b>1
  48. C = { z : ( z ) > - 2 } C=\{z\in\mathbb{C}~{}:~{}\Re(z)>-2\}
  49. b > exp ( 1 / e ) b>\exp(1/\mathrm{e})
  50. c c
  51. add c \mathrm{add}_{c}
  52. add c ( x ) = c + x , x \mathrm{add}_{c}(x)=c+x,~{}~{}~{}~{}\forall x\in\mathbb{C}
  53. mul c \mathrm{mul_{c}}
  54. mul c ( x ) = c x , x \mathrm{mul_{c}}(x)=c\cdot x,~{}~{}~{}~{}\forall x\in\mathbb{C}
  55. S ( z ; x ) = x + mul c ( z ) S(z;x)=x+\mathrm{mul_{c}}(z)
  56. add c ~{}\mathrm{add_{c}}~{}
  57. exp c \exp_{c}
  58. c c
  59. mul c \mathrm{mul}_{c}
  60. f ( x ) = 2 x 2 - 1 f(x)=2x^{2}-1
  61. S ( z ; x ) = cos ( 2 z arccos ( x ) ) S(z;x)=\cos(2^{z}\arccos(x))
  62. ( , 0 1 ) (\mathbb{C},~{}0\!\rightarrow\!1)
  63. S ( z + 1 ; x ) = cos ( 2 2 z arccos ( x ) ) = 2 cos ( 2 z arccos ( x ) ) 2 - 1 = f ( S ( z ; x ) ) S(z+1;x)=\cos(2\cdot 2^{z}\arccos(x))=2\cos(2^{z}\arccos(x))^{2}-1=f(S(z;x))
  64. S ( 0 ; x ) = x . S(0;x)=x~{}.
  65. S S
  66. T = 2 π ln ( 2 ) i 9.0647202836543876194 i T=\frac{2\pi}{\ln(2)}i\approx 9.0647202836543876194\!~{}i
  67. lim z - S ( z ) = 1. \lim_{z\rightarrow-\infty}S(z)=1.
  68. f ( x ) = 2 x 1 - x 2 f(x)=2x\sqrt{1-x^{2}}
  69. S ( z ; x ) = sin ( 2 z arcsin ( x ) ) . S(z;x)=\sin(2^{z}\arcsin(x)).
  70. f ( x ) = 2 x 1 - x 2 x D f(x)=\frac{2x}{1-x^{2}}~{}~{}~{}~{}~{}\forall x\in D~{}
  71. D = \ { - 1 , 1 } . ~{}D=\mathbb{C}\backslash\{-1,1\}.
  72. S ( z ; x ) = tan ( 2 z arctan ( x ) ) S(z;x)=\tan(2^{z}\arctan(x))
  73. tan ( 2 α ) = 2 tan ( α ) 1 - tan ( α ) 2 α \ { α : cos ( α ) = 0 | | sin ( α ) = ± cos ( α ) } . \tan(2\alpha)=\frac{2\tan(\alpha)}{1-\tan(\alpha)^{2}}~{}~{}\forall\alpha\in% \mathbb{C}\backslash\{\alpha\in\mathbb{C}:\cos(\alpha)=0||\sin(\alpha)=\pm\cos% (\alpha)\}.
  74. b > 1 b>1
  75. f ( u ) = exp b ( u ) f(u)=\exp_{b}(u)
  76. C = { z : ( u ) > - 2 } C=\{z\in\mathbb{C}:\Re(u)>-2\}
  77. tet b \mathrm{tet}_{b}
  78. ( C , 0 1 ) (C,~{}0\!\rightarrow\!1)
  79. exp b \exp_{b}
  80. 𝒳 ( exp ( u ) ) = 𝒳 ( u ) + 1. \mathcal{X}(\exp(u))=\mathcal{X}(u)+1.
  81. 𝒳 ( S ( z ; u ) ) = 𝒳 ( u ) + z . \mathcal{X}(S(z;u))=\mathcal{X}(u)+z.
  82. S ( z ; u ) = 𝒳 - 1 ( z + 𝒳 ( u ) ) , S(z;u)=\mathcal{X}^{-1}(z+\mathcal{X}(u)),
  83. exp 1 = exp \exp^{1}\!=\!\exp
  84. exp - 1 = ln \exp^{\!-1}\!=\!\ln
  85. exp z \exp^{z}
  86. z = 2 , 1 , 0.9 , 0.5 , 0.1 , - 0.1 , - 0.5 , - 0.9 , - 1 , - 2 z=2,1,0.9,0.5,0.1,-0.1,-0.5,-0.9,-1,-2
  87. S S
  88. A A
  89. L L
  90. L * L^{*}
  91. z 0 z\!\geq\!0
  92. exp z ( x ) \exp^{z}~{}(x)
  93. | ( z ) | < ( L ) 1.3 |\Im(z)|<\Im(L)\approx 1.3
  94. H H
  95. h = H h=\sqrt{H}
  96. H H

Supergeometry.html

  1. 2 \mathbb{Z}_{2}
  2. 2 \mathbb{Z}_{2}
  3. H H^{\infty}
  4. G G^{\infty}
  5. G H GH^{\infty}
  6. G G
  7. G G

Superheterodyne_transmitter.html

  1. f ( t ) f(t)
  2. ω R \omega_{R}
  3. ω I \omega_{I}
  4. ω s \omega_{s}
  5. RF = ( 1 + f ( t ) ) sin ( ω R t ) \mbox{RF}~{}=(1+f(t))\cdot\sin(\omega_{R}t)
  6. IF = ( 1 + f ( t ) ) sin ( ω I t ) \mbox{IF}~{}=(1+f(t))\cdot\sin(\omega_{I}t)
  7. SC = sin ( ω s t ) \mbox{SC}~{}=\sin(\omega_{s}t)
  8. IF SC = ( 1 + f ( t ) ) sin ( ω I t ) sin ( ω s t ) \mbox{IF}~{}\cdot\mbox{SC}~{}=(1+f(t))\cdot\sin(\omega_{I}t)\cdot\sin(\omega_{% s}t)
  9. IF SC = 1 2 ( 1 + f ( t ) ) ( cos ( ω s t - ω I t ) + cos ( ω s t + ω I t ) ) \mbox{IF}~{}\cdot\mbox{SC}~{}=\frac{1}{2}(1+f(t))\cdot(\cos(\omega_{s}t-\omega% _{I}t)+\cos(\omega_{s}t+\omega_{I}t))
  10. RF = 1 2 ( 1 + f ( t ) ) cos ( ω s t - ω I t ) \mbox{RF}~{}=\frac{1}{2}(1+f(t))\cdot\cos(\omega_{s}t-\omega_{I}t)
  11. ω s - ω I \omega_{s}-\omega_{I}
  12. ω R = ω s - ω I \omega_{R}=\omega_{s}-\omega_{I}
  13. RF = ( 1 + f ( t ) ) sin ( ω R t ) \mbox{RF}~{}=(1+f(t))\cdot\sin(\omega_{R}t)

Superintegrable_Hamiltonian_system.html

  1. F : Z N = F ( Z ) F:Z\to N=F(Z)
  2. N k N\subset\mathbb{R}^{k}
  3. s i j s_{ij}
  4. N N
  5. { F i , F j } = s i j F \{F_{i},F_{j}\}=s_{ij}\circ F
  6. s i j s_{ij}
  7. m = 2 n - k m=2n-k
  8. N N
  9. k = n k=n
  10. F F
  11. T m T^{m}
  12. M M
  13. U U
  14. M M
  15. ( I A , p i , q i , ϕ A ) (I_{A},p_{i},q^{i},\phi^{A})
  16. A = 1 , , m A=1,\ldots,m
  17. i = 1 , , n - m i=1,\ldots,n-m
  18. ( ϕ A ) (\phi^{A})
  19. T m T^{m}
  20. U U
  21. I A I_{A}
  22. F ( U ) F(U)
  23. T m - r × r T^{m-r}\times\mathbb{R}^{r}

Superluminescent_diode.html

  1. P o u t = h c ν Π R s p exp [ ( g - α ) L ] - 1 g - α P_{out}=\frac{h}{c}\cdot\nu\cdot\Pi\cdot R_{sp}\frac{\exp[(g-\alpha)L]-1}{g-\alpha}
  2. λ \lambda
  3. λ \lambda
  4. ω \omega
  5. R I N ( ω ) = < I n 2 ( ω ) Align g t ; / < I 0 2 Align g t ; RIN(\omega)=<I_{n}^{2}(\omega)&gt;/<I_{0}^{2}&gt;
  6. L c = λ 2 / B W L_{c}=\lambda^{2}/BW
  7. λ \lambda

Supersingular_prime_(for_an_elliptic_curve).html

  1. X ln X \frac{\sqrt{X}}{\ln X}
  2. 𝔭 \mathfrak{p}
  3. 𝔭 \mathfrak{p}

Suresh_P._Sethi.html

  1. ( s , S ) (s,S)
  2. ( s , S ) (s,S)

Surface_wave_inversion.html

  1. V group = V phase - λ δ V phase δ λ V_{\mathrm{group}}=V_{\mathrm{phase}}-\lambda\frac{\delta V_{\mathrm{phase}}}{% \delta\lambda}
  2. U ( x , ω ) = u ( x , t ) e - i ω t d t U(x,\omega)=\int u(x,t)e^{-i\omega t}\,dt
  3. U ( x , ω ) = P ( x , ω ) A ( x , ω ) U(x,\omega)=P(x,\omega)A(x,\omega)
  4. U ( x , ω ) = e - i Φ x A ( x , ω ) U(x,\omega)=e^{-i\Phi x}A(x,\omega)
  5. V ( ω , Φ ) = e i Φ x U ( x , ω ) | U ( x , ω ) | d x V(\omega,\Phi)=\int e^{i\Phi x}\frac{U(x,\omega)}{|U(x,\omega)|}\,dx
  6. M i s f i t = i - σ ( x d i - x c i ) 2 σ i 2 n F Misfit=\sqrt{\sum_{i-\sigma}\frac{(x_{di}-x_{ci})^{2}}{\sigma^{2}_{i}n_{F}}}

Susskind–Glogower_operator.html

  1. V = 1 a a a V=\frac{1}{\sqrt{aa^{\dagger}}}a
  2. V = a 1 a a V^{\dagger}=a^{\dagger}\frac{1}{\sqrt{aa^{\dagger}}}
  3. [ V , V ] = | 0 0 | [V,V^{\dagger}]=|0\rangle\langle 0|
  4. | 0 |0\rangle
  5. V a a V = a a + 1 Va^{\dagger}aV^{\dagger}=a^{\dagger}a+1
  6. a a a^{\dagger}a
  7. exp ( i p x o ) x exp ( - i p x o ) = x + x 0 \exp\left(i\frac{px_{o}}{\hbar}\right)x\exp\left(-i\frac{px_{o}}{\hbar}\right)% =x+x_{0}

Suzuki-Kasami_algorithm.html

  1. n n
  2. 1 , , n 1,...,n
  3. i i
  4. R N i [ n ] RN_{i}[n]
  5. R N i [ j ] RN_{i}[j]
  6. j j
  7. L N [ n ] LN[n]
  8. L N [ j ] LN[j]
  9. j j
  10. i i
  11. R N i [ i ] RN_{i}[i]
  12. i i
  13. L N [ i ] LN[i]
  14. R N i [ i ] RN_{i}[i]
  15. R N i [ i ] RN_{i}[i]
  16. k k
  17. Q Q
  18. k k
  19. Q Q
  20. R N i [ k ] = L N [ k ] + 1 RN_{i}[k]=LN[k]+1
  21. k k
  22. Q Q
  23. j j
  24. Q Q
  25. j j
  26. i i
  27. j j
  28. s s
  29. R N i [ j ] RN_{i}[j]
  30. m a x ( R N i [ j ] , s ) max(RN_{i}[j],s)
  31. s < R N i [ j ] s<RN_{i}[j]
  32. i i
  33. R N i [ j ] = = L N [ j ] + 1 RN_{i}[j]==LN[j]+1
  34. j j
  35. N - 1 N-1
  36. 1 1

Sverdrup_wave.html

  1. c = ( g h ) 1 / 2 c=(gh)^{1/2}
  2. c g = ( g h ) 1 / 2 c_{\mathrm{g}}=(gh)^{1/2}
  3. c = c g c=c_{\mathrm{g}}

Swain–Lupton_equation.html

  1. log K X K H = σ ρ \log\frac{K_{X}}{K_{H}}=\sigma\rho
  2. σ ρ = f F + r R \sigma\rho=fF+rR
  3. σ X = c 1 σ 1 X + c 2 σ 2 X \sigma_{X}=c_{1}\sigma_{1X}+c_{2}\sigma_{2X}
  4. Z = a X + b Y + i Z=aX+bY+i
  5. % r = 100 r f + r \%r=100\frac{r}{f+r}
  6. ρ σ X - ρ σ H f = F X + ( % r ) R X 100 - % r \frac{\rho\sigma_{X}-\rho\sigma_{H}}{f}=F_{X}+\frac{(\%r)R_{X}}{100-\%r}

SWIFFT.html

  1. α \alpha
  2. M M
  3. m n mn
  4. M M
  5. m m
  6. p i p_{i}
  7. R R
  8. p i p_{i}
  9. a i a_{i}
  10. p i p_{i}
  11. a i a_{i}
  12. i i
  13. m m
  14. f i f_{i}
  15. < 2 n <2n
  16. f = i = 1 m ( f i ) f=\sum_{i=1}^{m}(f_{i})
  17. p p
  18. α n + 1 \alpha^{n}+1
  19. f f
  20. n log ( p ) n\log(p)
  21. p n \mathbb{Z}^{n}_{p}
  22. p n = 257 64 p^{n}=257^{64}
  23. p n \mathbb{Z}^{n}_{p}
  24. R R
  25. n n
  26. m > 0 m>0
  27. p > 0 p>0
  28. R R
  29. R = p [ α ] / ( α n + 1 ) R=\mathbb{Z}_{p}[\alpha]/(\alpha^{n}+1)
  30. α \alpha
  31. p p
  32. α n + 1 \alpha^{n}+1
  33. R R
  34. < n <n
  35. Z p Z_{p}
  36. m m
  37. a 1 , , a m R a_{1},\ldots,a_{m}\in R
  38. R R
  39. i = 1 m ( a i x i ) \sum_{i=1}^{m}(a_{i}\cdot x_{i})
  40. x 1 , , x m R x_{1},\ldots,x_{m}\in R
  41. m n mn
  42. a i x i a_{i}\cdot x_{i}
  43. { f * g } = { f } { g } \mathcal{F}\{f*g\}=\mathcal{F}\{f\}\cdot\mathcal{F}\{g\}
  44. \mathcal{F}
  45. \cdot
  46. * *
  47. O ( n log ( n ) ) O(n\log(n))
  48. < 2 n <2n
  49. p \mathbb{Z}_{p}
  50. p \mathbb{Z}_{p}
  51. p p
  52. 2 n 2n
  53. p - 1 p-1
  54. log ( p ) \log(p)
  55. 2 106 2^{106}
  56. x , x * x,x*
  57. f f
  58. f ( x ) = f ( x * ) f(x)=f(x*)
  59. f f
  60. x x
  61. f ( x ) f(x)
  62. f f
  63. x 1 x_{1}
  64. x 2 x_{2}
  65. x 1 + x 2 x_{1}+x_{2}
  66. f ( x 1 ) + f ( x 2 ) = f ( x 1 + x 2 ) f(x_{1})+f(x_{2})=f(x_{1}+x_{2})
  67. f f
  68. f f
  69. T T
  70. p p
  71. f 2 f_{2}
  72. p [ α ] / ( α n + 1 ) \mathbb{Z}_{p}[\alpha]/(\alpha^{n}+1)
  73. T 2 T_{2}
  74. T T
  75. p p
  76. p [ α ] / ( α n + 1 ) \mathbb{Z}_{p}[\alpha]/(\alpha^{n}+1)
  77. n n

Swing_(United_Kingdom).html

  1. 1 2 ( 10 % + 5 % ) = + 7.5 % \frac{1}{2}\left(10\%+5\%\right)=+7.5\%
  2. 45 % ( 40 % + 45 % ) - 35 % ( 35 % + 45 % ) = 52.94 % - 43.75 % = + 9.19 % 45\%\left(40\%+45\%\right)-35\%\left(35\%+45\%\right)=52.94\%-43.75\%=+9.19\%
  3. 40 % ( 40 % + 45 % ) - 45 % ( 35 % + 45 % ) = 47.06 % - 56.25 % = - 9.19 % 40\%\left(40\%+45\%\right)-45\%\left(35\%+45\%\right)=47.06\%-56.25\%=-9.19\%

Switched-mode_power_supply_applications.html

  1. 2 \scriptstyle\sqrt{2}

Symbiocom.html

  1. G 1 G_{1}

Symbolic_circuit_analysis.html

  1. s \mathit{s}\,
  2. 𝐱 \mathbf{x}
  3. T ( s , 𝐱 ) = N ( s , 𝐱 ) D ( s , 𝐱 ) T(s,\mathbf{x})=\frac{N(s,\mathbf{x})}{D(s,\mathbf{x})}
  4. N ( s , 𝐱 ) N(s,\mathbf{x})
  5. D ( s , 𝐱 ) D(s,\mathbf{x})
  6. s \mathit{s}\,
  7. T ( s , 𝐱 ) = i = 0 n a i ( 𝐱 ) s i i = 0 m b i ( 𝐱 ) s i = K i = 1 n ( s - z i ( 𝐱 ) ) i = 1 m ( s - p i ( 𝐱 ) ) T(s,\mathbf{x})=\frac{\displaystyle\sum_{i=0}^{n}a_{i}(\mathbf{x})s^{i}}{% \displaystyle\sum_{i=0}^{m}b_{i}(\mathbf{x})s^{i}}=K\frac{\displaystyle\prod_{% i=1}^{n}(s-z_{i}(\mathbf{x}))}{\displaystyle\prod_{i=1}^{m}(s-p_{i}(\mathbf{x}% ))}
  8. z i ( 𝐱 ) z_{i}(\mathbf{x})
  9. p i ( 𝐱 ) p_{i}(\mathbf{x})
  10. m n m\geqslant n
  11. a i ( 𝐱 ) a_{i}(\mathbf{x})
  12. b i ( 𝐱 ) b_{i}(\mathbf{x})
  13. T v ( s ) = V o u t ( s ) / V i n ( s ) {T_{v}(s)=V_{out}(s)/V_{in}(s)}\,
  14. s \mathit{s}\,
  15. R i = i , C i = 0.01 i R_{i}=i,C_{i}=0.01i\,
  16. T ( s ) = 3.48 s 13.2 s 2 + 1.32 s + 0.33 T(s)=\frac{3.48s}{13.2s^{2}+1.32s+0.33}
  17. s \mathit{s}\,
  18. T ( s , 𝐱 ) = 1.74 C 2 s 6.6 C 1 C 2 s 2 + 0.66 C 2 s + 0.33 𝐱 = [ C 1 C 2 ] \begin{aligned}\displaystyle T(s,\mathbf{x})&\displaystyle=\frac{1.74C_{2}s}{6% .6C_{1}C_{2}s^{2}+0.66C_{2}s+0.33}\\ \displaystyle\mathbf{x}&\displaystyle=[C_{1}~{}C_{2}]\end{aligned}
  19. s \mathit{s}\,
  20. G i = 1 / R i G_{i}=1/R_{i}\,
  21. T ( s , 𝐱 ) = G 4 G 6 G 8 C 2 s G 6 G 11 C 1 C 2 s 2 + G 1 G 6 G 11 C 2 s + G 2 G 3 G 5 G 11 𝐱 = [ C 1 C 2 G 1 G 2 G 3 G 4 G 5 G 6 G 8 G 11 ] \begin{aligned}\displaystyle T(s,\mathbf{x})&\displaystyle=\frac{G_{4}G_{6}G_{% 8}C_{2}s}{G_{6}G_{11}C_{1}C_{2}s^{2}+G_{1}G_{6}G_{11}C_{2}s+G_{2}G_{3}G_{5}G_{% 11}}\\ \displaystyle\mathbf{x}&\displaystyle=[C_{1}~{}C_{2}~{}G_{1}~{}G_{2}~{}G_{3}~{% }G_{4}~{}G_{5}~{}G_{6}~{}G_{8}~{}G_{11}]\end{aligned}
  22. T v ( s ) T_{v}(s)\,
  23. N ( s , 𝐱 ) N(s,\mathbf{x})
  24. D ( s , 𝐱 ) D(s,\mathbf{x})

Symbolic_computation.html

  1. a + b a+b
  2. a a
  3. b b
  4. a a
  5. b b
  6. x x
  7. a x a^{x}
  8. x a x - 1 0 + a x ( 1 log a + x 0 a ) . x\cdot a^{x-1}\cdot 0+a^{x}\cdot\left(1\cdot\log a+x\cdot\frac{0}{a}\right).
  9. E E 0 E−E→0
  10. s i n ( 0 ) 0 sin(0)→0
  11. a + b + c a+b+c
  12. " + " ( a , b , c ) "+"(a,b,c)
  13. a + ( b + c ) a+(b+c)
  14. ( a + b ) + c (a+b)+c
  15. " + " ( a , b , c ) "+"(a,b,c)
  16. a + b + c a+b+c
  17. a b + c a−b+c
  18. E −E
  19. E F E−F
  20. E / F E/F
  21. ( 1 ) E (−1)⋅E
  22. E + ( 1 ) F E+(−1)⋅F
  23. ( x + 1 ) 4 x 4 + 4 x 3 + 6 x 2 + 4 x + 1 (x+1)^{4}\rightarrow x^{4}+4x^{3}+6x^{2}+4x+1
  24. ( x - 1 ) ( x 4 + x 3 + x 2 + x + 1 ) x 5 - 1. (x-1)(x^{4}+x^{3}+x^{2}+x+1)\rightarrow x^{5}-1.
  25. ( sin ( x + y ) 2 + log ( z 2 - 5 ) ) 3 (\sin(x+y)^{2}+\log(z^{2}-5))^{3}
  26. sin ( x + y ) \sin(x+y)
  27. log ( z 2 - 5 ) \log(z^{2}-5)
  28. ( x + y ) 2 = x 2 + 2 x y + y 2 . (x+y)^{2}=x^{2}+2xy+y^{2}.

SymbolicC++.html

  1. ( cos θ sin θ - sin θ cos θ ) \begin{pmatrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{pmatrix}

Symmetric_closure.html

  1. S = R { ( x , y ) : ( y , x ) R } . S=R\cup\left\{(x,y):(y,x)\in R\right\}.\,

Symmetric_function.html

  1. f ( x 1 , x 2 , x 3 ) = ( x - x 1 ) ( x - x 2 ) ( x - x 3 ) f(x_{1},x_{2},x_{3})=(x-x_{1})(x-x_{2})(x-x_{3})
  2. f ( x 1 , x 2 , , x n ) = f ( x 2 , x 1 , , x n ) = f ( x 3 , x 1 , , x n , x n - 1 ) f(x_{1},x_{2},...,x_{n})=f(x_{2},x_{1},...,x_{n})=f(x_{3},x_{1},...,x_{n},x_{n% -1})
  3. ( x - x 1 ) ( x - x 2 ) ( x - x 3 ) = ( x - x 2 ) ( x - x 1 ) ( x - x 3 ) = ( x - x 3 ) ( x - x 1 ) ( x - x 2 ) (x-x_{1})(x-x_{2})(x-x_{3})=(x-x_{2})(x-x_{1})(x-x_{3})=(x-x_{3})(x-x_{1})(x-x% _{2})
  4. x 1 , x 2 , x 3 x_{1},x_{2},x_{3}
  5. f ( x , y ) = x 2 + y 2 - r 2 f(x,y)=x^{2}+y^{2}-r^{2}
  6. f ( y , x ) = y 2 + x 2 - r 2 f(y,x)=y^{2}+x^{2}-r^{2}
  7. f ( x , y ) = a x 2 + b y 2 - r 2 f(x,y)=ax^{2}+by^{2}-r^{2}
  8. f ( y , x ) = a y 2 + b x 2 - r 2 . f(y,x)=ay^{2}+bx^{2}-r^{2}.

Symmetric_mean_absolute_percentage_error.html

  1. SMAPE = 1 n t = 1 n | F t - A t | ( | A t | + | F t | ) / 2 \mbox{SMAPE}~{}=\frac{1}{n}\sum_{t=1}^{n}\frac{\left|F_{t}-A_{t}\right|}{(|A_{% t}|+|F_{t}|)/2}
  2. SMAPE = 1 n t = 1 n | F t - A t | ( A t + F t ) / 2 \mbox{SMAPE}~{}=\frac{1}{n}\sum_{t=1}^{n}\frac{\left|F_{t}-A_{t}\right|}{(A_{t% }+F_{t})/2}
  3. A t + F t < 0 A_{t}+F_{t}<0
  4. A t + F t = 0 A_{t}+F_{t}=0
  5. SMAPE = 1 n t = 1 n | F t - A t | A t + F t \mbox{SMAPE}~{}=\frac{1}{n}\sum_{t=1}^{n}\frac{\left|F_{t}-A_{t}\right|}{A_{t}% +F_{t}}
  6. SMAPE = t = 1 n | F t - A t | t = 1 n ( A t + F t ) \mbox{SMAPE}~{}=\frac{\sum_{t=1}^{n}\left|F_{t}-A_{t}\right|}{\sum_{t=1}^{n}(A% _{t}+F_{t})}

Symmetric_set.html

  1. S = S - 1 S=S^{-1}
  2. S - 1 = { x - 1 : x S } S^{-1}=\{x^{-1}:x\in S\}
  3. x - 1 S x^{-1}\in S
  4. x S x\in S
  5. S = - S = { - x : x S } S=-S=\{-x:x\in S\}
  6. ( - k , k ) (-k,k)
  7. k > 0 k>0
  8. { - 1 , 1 } \{-1,1\}
  9. S S - 1 SS^{-1}
  10. S - 1 S S^{-1}S

Symmetric_tensor.html

  1. T ( v 1 , v 2 , , v r ) = T ( v σ 1 , v σ 2 , , v σ r ) T(v_{1},v_{2},\dots,v_{r})=T(v_{\sigma 1},v_{\sigma 2},\dots,v_{\sigma r})
  2. T i 1 i 2 i r = T i σ 1 i σ 2 i σ r . T_{i_{1}i_{2}\dots i_{r}}=T_{i_{\sigma 1}i_{\sigma 2}\dots i_{\sigma r}}.
  3. T V k T\in V^{\otimes k}
  4. τ σ T = T \tau_{\sigma}T=T\,
  5. T = i 1 , , i k = 1 N T i 1 i 2 i k e i 1 e i 2 e i k T=\sum_{i_{1},\dots,i_{k}=1}^{N}T_{i_{1}i_{2}\dots i_{k}}e^{i_{1}}\otimes e^{i% _{2}}\otimes\cdots\otimes e^{i_{k}}
  6. T i 1 i 2 i k T_{i_{1}i_{2}\dots i_{k}}
  7. T i σ 1 i σ 2 i σ k = T i 1 i 2 i k T_{i_{\sigma 1}i_{\sigma 2}\dots i_{\sigma k}}=T_{i_{1}i_{2}\dots i_{k}}
  8. dim Sym k ( V ) = ( N + k - 1 k ) . \dim\,\operatorname{Sym}^{k}(V)={N+k-1\choose k}.
  9. Sym ( V ) = k = 0 Sym k ( V ) . \operatorname{Sym}(V)=\bigoplus_{k=0}^{\infty}\operatorname{Sym}^{k}(V).
  10. g μ ν g_{\mu\nu}
  11. G μ ν G_{\mu\nu}
  12. R μ ν R_{\mu\nu}
  13. V V
  14. k k
  15. T T
  16. Sym T = 1 k ! σ 𝔖 k τ σ T , \operatorname{Sym}\,T=\frac{1}{k!}\sum_{\sigma\in\mathfrak{S}_{k}}\tau_{\sigma% }T,
  17. T = T i 1 i 2 i k e i 1 e i 2 e i k , T=T_{i_{1}i_{2}\dots i_{k}}e^{i_{1}}\otimes e^{i_{2}}\otimes\cdots\otimes e^{i% _{k}},
  18. Sym T = 1 k ! σ 𝔖 k T i σ 1 i σ 2 i σ k e i 1 e i 2 e i k . \operatorname{Sym}\,T=\frac{1}{k!}\sum_{\sigma\in\mathfrak{S}_{k}}T_{i_{\sigma 1% }i_{\sigma 2}\dots i_{\sigma k}}e^{i_{1}}\otimes e^{i_{2}}\otimes\cdots\otimes e% ^{i_{k}}.
  19. T ( i 1 i 2 i k ) = 1 k ! σ 𝔖 k T i σ 1 i σ 2 i σ k T_{(i_{1}i_{2}\dots i_{k})}=\frac{1}{k!}\sum_{\sigma\in\mathfrak{S}_{k}}T_{i_{% \sigma 1}i_{\sigma 2}\dots i_{\sigma k}}
  20. T = v 1 v 2 v r T=v_{1}\otimes v_{2}\otimes\cdots\otimes v_{r}
  21. v 1 v 2 v r := 1 r ! σ 𝔖 r v σ 1 v σ 2 v σ r . v_{1}\odot v_{2}\odot\cdots\odot v_{r}:=\frac{1}{r!}\sum_{\sigma\in\mathfrak{S% }_{r}}v_{\sigma 1}\otimes v_{\sigma 2}\otimes\cdots\otimes v_{\sigma r}.
  22. \odot
  23. T 1 T 2 = Sym ( T 1 T 2 ) ( Sym k 1 + k 2 ( V ) ) . T_{1}\odot T_{2}=\operatorname{Sym}(T_{1}\otimes T_{2})\quad\left(\in% \operatorname{Sym}^{k_{1}+k_{2}}(V)\right).
  24. \odot
  25. v k = v v v k times = v v v k times = v k . v^{\odot k}=\underbrace{v\odot v\odot\cdots\odot v}_{k\,\text{ times}}=% \underbrace{v\otimes v\otimes\cdots\otimes v}_{k\,\text{ times}}=v^{\otimes k}.
  26. \odot
  27. v k = v v v k times = v v v k times . v^{k}=\underbrace{v\,v\,\cdots\,v}_{k\,\text{ times}}=\underbrace{v\odot v% \odot\cdots\odot v}_{k\,\text{ times}}.
  28. T = i = 1 r λ i v i v i . T=\sum_{i=1}^{r}\lambda_{i}\,v_{i}\otimes v_{i}.
  29. T = i = 1 r λ i v i k T=\sum_{i=1}^{r}\lambda_{i}\,v_{i}^{\otimes k}

Symmetry-preserving_filter.html

  1. G G
  2. φ g , ψ g , ρ g \varphi_{g},\psi_{g},\rho_{g}
  3. g G g\in G
  4. x ˙ = f ( x , u ) y = h ( x , u ) \begin{aligned}\displaystyle\dot{x}&\displaystyle=f(x,u)\\ \displaystyle y&\displaystyle=h(x,u)\end{aligned}
  5. φ g , ψ g , ρ g \varphi_{g},\psi_{g},\rho_{g}
  6. X ˙ = f ( X , U ) Y = h ( X , U ) , \begin{aligned}\displaystyle\dot{X}&\displaystyle=f(X,U)\\ \displaystyle Y&\displaystyle=h(X,U),\end{aligned}
  7. ( X , U , Y ) = ( φ g ( x ) , ψ g ( u ) , ρ g ( y ) ) (X,U,Y)=(\varphi_{g}(x),\psi_{g}(u),\rho_{g}(y))
  8. x ^ ˙ = F ( x ^ , u , y ) \dot{\hat{x}}=F(\hat{x},u,y)
  9. F ( x , u , h ( x , u ) ) = f ( x , u ) F(x,u,h(x,u))=f(x,u)
  10. x ^ ˙ = f ( x ^ , u ) + C \dot{\hat{x}}=f(\hat{x},u)+C
  11. C C
  12. 0
  13. y ^ = y \hat{y}=y
  14. X ^ ˙ = F ( X ^ , U , Y ) \dot{\hat{X}}=F(\hat{X},U,Y)
  15. x ^ ˙ = f ( x ^ , u ) + W ( x ^ ) L ( I ( x ^ , u ) , E ( x ^ , u , y ) ) E ( x ^ , u , y ) , \dot{\hat{x}}=f(\hat{x},u)+W(\hat{x})L\Bigl(I(\hat{x},u),E(\hat{x},u,y)\Bigr)E% (\hat{x},u,y),
  16. E ( x ^ , u , y ) E(\hat{x},u,y)
  17. y ^ - y \hat{y}-y
  18. W ( x ^ ) = ( w 1 ( x ^ ) , . . , w n ( x ^ ) ) W(\hat{x})=\bigl(w_{1}(\hat{x}),..,w_{n}(\hat{x})\bigr)
  19. I ( x ^ , u ) I(\hat{x},u)
  20. L ( I , E ) L(I,E)
  21. E ( x ^ , u , y ) , W ( x ^ ) , I ( x ^ , u ) E(\hat{x},u,y),W(\hat{x}),I(\hat{x},u)
  22. η ( x ^ , x ) \eta(\hat{x},x)
  23. x ^ - x \hat{x}-x
  24. I ( x ^ , u ) I(\hat{x},u)
  25. η ˙ = Υ ( η , I ( x ^ , u ) ) \dot{\eta}=\Upsilon\bigl(\eta,I(\hat{x},u)\bigr)
  26. L ( I , E ) L(I,E)