wpmath0000006_11

Residence_time_distribution.html

  1. E ( t ) E(t)
  2. E ( t ) E(t)
  3. 0 E ( t ) d t = 1 \int_{0}^{\infty}E(t)\,dt=1
  4. t t
  5. E ( t ) d t E(t)dt
  6. t 1 t_{1}
  7. 0 t 1 E ( t ) d t \int_{0}^{t_{1}}E(t)\,dt
  8. t 1 t_{1}
  9. t 1 E ( t ) d t = 1 - 0 t 1 E ( t ) d t \int_{t_{1}}^{\infty}E(t)\,dt=1-\int_{0}^{t_{1}}E(t)\,dt
  10. t ¯ = 0 t E ( t ) d t \bar{t}=\int_{0}^{\infty}t\cdot E(t)\,dt
  11. t ¯ \bar{t}
  12. τ \tau
  13. τ = V v \tau=\frac{V}{v}
  14. E ( t ) E(t)
  15. σ 2 \sigma^{2}
  16. σ 2 = 0 ( t - t ¯ ) 2 E ( t ) d t \sigma^{2}=\int_{0}^{\infty}(t-\bar{t})^{2}\cdot E(t)\,dt
  17. I ( t ) I(t)
  18. E ( t ) E(t)
  19. t t
  20. I ( t ) d t I(t)dt
  21. E ( t ) E(t)
  22. I ( t ) I(t)
  23. I ( t ) = 1 τ ( 1 - 0 t E ( t ) d t ) E ( t ) = - τ d I ( t ) d t I(t)=\frac{1}{\tau}\left(1-\int_{0}^{t}E(t)\ dt\right)\qquad E(t)=-\tau\frac{% dI(t)}{dt}
  24. E ( t ) E(t)
  25. M M
  26. V V
  27. τ \tau
  28. C ( t ) C(t)
  29. E ( t ) = C ( t ) 0 C ( t ) d t E(t)=\frac{C(t)}{\int_{0}^{\infty}C(t)\ dt}
  30. C 0 C_{0}
  31. C 0 C_{0}
  32. F ( t ) F(t)
  33. F ( t ) = C ( t ) C 0 F(t)=\frac{C(t)}{C_{0}}
  34. F ( t ) = 0 t E ( t ) d t E ( t ) = d F ( t ) d t F(t)=\int_{0}^{t}E(t)\,dt\qquad E(t)=\frac{dF(t)}{dt}
  35. F ( t ) F(t)
  36. t ¯ = 0 t E ( t ) d t = 0 1 t d F ( t ) = - 1 0 t d ( 1 - F ( t ) ) = 0 ( 1 - F ( t ) ) d t \overline{t}=\int_{0}^{\infty}t\cdot E(t)\,dt=\int_{0}^{1}t\,dF(t)=-\int_{1}^{% 0}t\,d(1-F(t))=\int_{0}^{\infty}(1-F(t))\,dt
  37. σ 2 = 0 ( t - t ¯ ) 2 E ( t ) d t = 0 1 ( t - t ¯ ) 2 d F ( t ) = 0 1 t 2 d F ( t ) - t ¯ 2 = 2 0 t ( 1 - F ( t ) ) d t - t ¯ 2 \sigma^{2}=\int_{0}^{\infty}(t-\overline{t})^{2}\cdot E(t)\,dt=\int_{0}^{1}(t-% \overline{t})^{2}\,dF(t)=\int_{0}^{1}t^{2}\,dF(t)-\overline{t}^{2}=2\int_{0}^{% \infty}t(1-F(t))\,dt-\overline{t}^{2}
  38. t t
  39. t + τ t+\tau
  40. τ \tau
  41. τ \tau
  42. E ( t ) = δ ( t - τ ) E(t)=\delta(t-\tau)\,
  43. E ( t ) E(t)
  44. τ \tau
  45. E ( t ) = 1 τ e - t / τ E(t)=\frac{1}{\tau}e^{-t/\tau}\,
  46. E ( t ) E(t)

Residual_gas_analyzer.html

  1. 10 - 14 10^{-14}
  2. 10 - 5 10^{-5}
  3. 10 - 4 10^{-4}
  4. 10 - 14 10^{-14}
  5. H 2 H_{2}
  6. 10 - 4 10^{-4}
  7. 10 - 3 10^{-3}
  8. 10 - 2 10^{-2}
  9. 10 - 11 10^{-11}
  10. 10 - 4 10^{-4}
  11. 10 - 2 10^{-2}

Residual_sum_of_squares.html

  1. R S S = i = 1 n ( y i - f ( x i ) ) 2 , RSS=\sum_{i=1}^{n}(y_{i}-f(x_{i}))^{2},
  2. f ( x i ) f(x_{i})
  3. y i ^ \hat{y_{i}}
  4. y i = a + b x i + ε i y_{i}=a+bx_{i}+\varepsilon_{i}\,
  5. R S S = i = 1 n ( ε i ) 2 = i = 1 n ( y i - ( α + β x i ) ) 2 , RSS=\sum_{i=1}^{n}(\varepsilon_{i})^{2}=\sum_{i=1}^{n}(y_{i}-(\alpha+\beta x_{% i}))^{2},
  6. α \alpha
  7. a a
  8. β \beta
  9. y = X β + e y=X\beta+e
  10. β \beta
  11. β \beta
  12. β ^ = ( X T X ) - 1 X T y . \hat{\beta}=(X^{T}X)^{-1}X^{T}y.
  13. e ^ \hat{e}
  14. y - X β ^ = y - X ( X T X ) - 1 X T y y-X\hat{\beta}=y-X(X^{T}X)^{-1}X^{T}y
  15. e ^ T e ^ \hat{e}^{T}\hat{e}
  16. R S S = y T y - y T X ( X T X ) - 1 X T y = y T [ I - X ( X T X ) - 1 X T ] y = y T [ I - H ] y RSS=y^{T}y-y^{T}X(X^{T}X)^{-1}X^{T}y=y^{T}[I-X(X^{T}X)^{-1}X^{T}]y=y^{T}[I-H]y

Residuated_lattice.html

  1. A / B := { r R r B A } A/B:=\{r\in R\mid rB\subseteq A\}\,
  2. B A := { r R B r A } B\setminus A:=\{r\in R\mid Br\subseteq A\}\,
  3. \infty

Resolution_(logic).html

  1. ¬ c \lnot c
  2. c c
  3. a 1 a i - 1 c a i + 1 a n , b 1 b j - 1 ¬ c b j + 1 b m a 1 a i - 1 a i + 1 a n b 1 b j - 1 b j + 1 b m \frac{a_{1}\lor\ldots\vee a_{i-1}\lor c\lor a_{i+1}\vee\ldots\lor a_{n},\quad b% _{1}\lor\ldots\vee b_{j-1}\lor\lnot c\lor b_{j+1}\vee\ldots\lor b_{m}}{a_{1}% \lor\ldots\lor a_{i-1}\lor a_{i+1}\lor\ldots\lor a_{n}\lor b_{1}\lor\ldots\lor b% _{j-1}\lor b_{j+1}\lor\ldots\lor b_{m}}
  4. a a
  5. b b
  6. c c
  7. p q , p q \frac{p\rightarrow q,p}{q}
  8. ¬ p q , p q \frac{\lnot p\lor q,p}{q}
  9. ( A 1 A 2 ) ( B 1 B 2 B 3 ) ( C 1 ) (A_{1}\lor A_{2})\land(B_{1}\lor B_{2}\lor B_{3})\land(C_{1})
  10. S = { A 1 A 2 , B 1 B 2 B 3 , C 1 } S=\{A_{1}\lor A_{2},B_{1}\lor B_{2}\lor B_{3},C_{1}\}
  11. a b , ¬ a c b c \frac{a\vee b,\quad\neg a\vee c}{b\vee c}
  12. a a
  13. a b a\vee b
  14. b b
  15. a a
  16. ¬ a c \neg a\vee c
  17. c c
  18. a a
  19. b c b\vee c
  20. Γ 1 { L 1 } Γ 2 { L 2 } ( Γ 1 Γ 2 ) ϕ ϕ \frac{\Gamma_{1}\cup\left\{L_{1}\right\}\,\,\,\,\Gamma_{2}\cup\left\{L_{2}% \right\}}{(\Gamma_{1}\cup\Gamma_{2})\phi}\phi
  21. ϕ \phi
  22. L 1 L_{1}
  23. L 2 ¯ \overline{L_{2}}
  24. Γ 1 \Gamma_{1}
  25. Γ 2 \Gamma_{2}
  26. P ( x ) , Q ( x ) P(x),Q(x)
  27. ¬ P ( b ) \neg P(b)
  28. [ b / x ] [b/x]
  29. P ( x ) , Q ( x ) ¬ P ( b ) Q ( b ) [ b / x ] \frac{P(x),Q(x)\,\,\,\,\neg P(b)}{Q(b)}[b/x]
  30. P ( x ) , Q ( x ) P(x),Q(x)
  31. ¬ P ( b ) \neg P(b)
  32. Q ( b ) Q(b)
  33. P ( x ) P(x)
  34. ¬ P ( b ) \neg P(b)
  35. P P
  36. [ b / x ] [b/x]
  37. x . P ( x ) Q ( x ) \forall x.P(x)\Rightarrow Q(x)
  38. P ( a ) P(a)
  39. Q ( a ) Q(a)
  40. ¬ P ( x ) Q ( x ) \neg P(x)\vee Q(x)
  41. P ( a ) P(a)
  42. Q ( a ) Q(a)
  43. \mapsto

Resonant_trans-Neptunian_object.html

  1. p λ - q λ N \rm p\cdot\lambda-\rm q\cdot\lambda_{\rm N}
  2. ϕ = p λ - q λ N - m ϖ - n Ω - r ϖ N - s Ω N \phi=\rm p\cdot\lambda-\rm q\cdot\lambda_{\rm N}-\rm m\cdot\varpi-\rm n\cdot% \Omega-\rm r\cdot\varpi_{\rm N}-\rm s\cdot\Omega_{\rm N}
  3. ϖ \varpi
  4. Ω \Omega
  5. ϕ \phi
  6. ϕ = 3 λ - 2 λ N - ϖ \phi=\rm 3\cdot\lambda-\rm 2\cdot\lambda_{\rm N}-\varpi
  7. ϕ = p λ - q λ N - ( p - q ) ϖ \phi=\rm p\cdot\lambda-\rm q\cdot\lambda_{\rm N}-(\rm p-\rm q)\cdot\varpi
  8. ϕ \phi\,
  9. λ = ϖ \lambda=\varpi
  10. ϕ = q ( ϖ - λ N ) \phi=q\cdot(\varpi-\lambda_{\rm N})
  11. ϕ \phi\,
  12. ϕ \phi\,

Resource_holding_potential.html

  1. C a b s C_{abs}

Respiratory_alkalosis.html

  1. HCO 3 - + H + H 2 CO 3 CO 2 + H 2 O \rm HCO_{3}^{-}+H^{+}\rightarrow H_{2}CO_{3}\rightarrow CO_{2}+H_{2}O

Respiratory_physiology.html

  1. P = P e l + P r e + P i n P = E V + R V ˙ + I V ¨ \begin{aligned}\displaystyle P&\displaystyle=P_{el}+P_{re}+P_{in}\\ \displaystyle P&\displaystyle=EV+R\dot{V}+I\ddot{V}\end{aligned}

Resultant.html

  1. ( x , y ) : P ( x ) = Q ( y ) = 0 ( x - y ) \prod_{(x,y):\,P(x)=Q(y)=0}(x-y)
  2. P = P + R Q P^{\prime}=P+RQ
  3. deg P = deg P \deg P^{\prime}=\deg P
  4. res ( P , Q ) = res ( P , Q ) \mathrm{res}(P^{\prime},Q)=\mathrm{res}(P,Q)
  5. res ( X , Y ) = det ( a 00 a 01 a 10 a 11 ) deg P res ( P , Q ) \mathrm{res}(X,Y)=\det{\begin{pmatrix}a_{00}&a_{01}\\ a_{10}&a_{11}\end{pmatrix}}^{\deg P}\cdot\mathrm{res}(P,Q)
  6. res ( P ( - z ) , Q ( z ) ) = res ( Q ( - z ) , P ( z ) ) \mathrm{res}(P(-z),Q(z))=\mathrm{res}(Q(-z),P(z))
  7. p deg ( Q ) P ( x ) = 0 Q ( x ) , p^{\deg(Q)}\prod_{P(x)=0}Q(x),
  8. P ( x ) = Q ( y ) = 0 P(x)=Q(y)=0
  9. z = x + y z=x+y
  10. P ( x ) P(x)
  11. Q ( z - x ) Q(z-x)
  12. t = x y t=xy
  13. P ( x ) P(x)
  14. x n Q ( t / x ) x^{n}Q(t/x)
  15. 1 / y 1/y
  16. y n Q ( 1 / y ) y^{n}Q(1/y)
  17. f ( x , y ) = 0 f(x,y)=0
  18. g ( x , y ) = 0 g(x,y)=0
  19. 𝔸 k 2 \mathbb{A}^{2}_{k}
  20. f f
  21. g g
  22. x x
  23. k [ y ] k[y]
  24. f f
  25. g g
  26. y y
  27. y y
  28. x x
  29. p p

Retarded_time.html

  1. t = t - | 𝐫 - 𝐫 | c t^{\prime}=t-\frac{|\mathbf{r}-\mathbf{r}^{\prime}|}{c}
  2. c = | 𝐫 - 𝐫 | t - t c=\frac{|\mathbf{r}-\mathbf{r}^{\prime}|}{t-t^{\prime}}
  3. t a = t + | 𝐫 - 𝐫 | c t_{a}=t+\frac{|\mathbf{r}-\mathbf{r}^{\prime}|}{c}

Reticulocyte_index.html

  1. R e t i c I n d e x = R e t i c C o u n t * H e m a t o c r i t N o r m a l H e m a t o c r i t ReticIndex=ReticCount*{Hematocrit\over NormalHematocrit}
  2. R P I = R e t i c I n d e x M a t u r a t i o n C o r r e c t i o n RPI={ReticIndex\over MaturationCorrection}
  3. = 5 * 25 45 2 = ={{5*{25\over 45}}\over 2}=
  4. R P I = R e t i c C o u n t * H e m o g l o b i n ( o b s e r v e d ) N o r m a l H e m o g l o b i n * 0.5 RPI=ReticCount*{Hemoglobin(observed)\over NormalHemoglobin}*0.5

Return_on_capital_employed.html

  1. ROCE = Earning Before Interest and Tax (EBIT) Capital Employed \mbox{ROCE}~{}=\frac{\mbox{Earning Before Interest and Tax (EBIT)}~{}}{\mbox{% Capital Employed}~{}}

Reversible_dynamics.html

  1. U - t = π U t π U_{-t}=\pi\,U_{t}\,\pi
  2. p ( x t , x t + τ 1 , x t + τ 2 . . x t + τ k ) = p ( x t , x t - τ 1 , x t - τ 2 . . x t - τ k ) p(x_{t},x_{t+\tau_{1}},x_{t+\tau_{2}}..x_{t+\tau_{k}})=p(x_{t^{\prime}},x_{t^{% \prime}-\tau_{1}},x_{t^{\prime}-\tau_{2}}..x_{t^{\prime}-\tau_{k}})
  3. p ( x t = i , x t + 1 = j ) = p ( x t = j , x t + 1 = i ) p(x_{t}=i,x_{t+1}=j)=\,p(x_{t}=j,x_{t+1}=i)

Rho_meson.html

  1. u u ¯ - d d ¯ 2 \mathrm{\tfrac{u\bar{u}-d\bar{d}}{\sqrt{2}}}\,
  2. ħ / Γ {ħ}/{Γ}

Ricci_decomposition.html

  1. R a b c d = S a b c d + E a b c d + C a b c d . R_{abcd}=\,S_{abcd}+E_{abcd}+C_{abcd}.
  2. S a b c d S_{abcd}
  3. E a b c d E_{abcd}
  4. C a b c d C_{abcd}
  5. n > 2 n>2
  6. S a b c d = R n ( n - 1 ) H a b c d S_{abcd}=\frac{R}{n\,(n-1)}\,H_{abcd}
  7. R = R m m R={R^{m}}_{m}
  8. R a b = R c a c b R_{ab}={R^{c}}_{acb}
  9. g a b g_{ab}
  10. H a b c d = g a c g d b - g a d g c b = 2 g a [ c g d ] b . H_{abcd}=g_{ac}\,g_{db}-g_{ad}\,g_{cb}=2g_{a[c}\,g_{d]b}.
  11. E a b c d = 1 n - 2 ( g a c S b d - g a d S b c + g b d S a c - g b c S a d ) = 2 n - 2 ( g a [ c S d ] b - g b [ c S d ] a ) E_{abcd}=\frac{1}{n-2}\,\left(g_{ac}\,S_{bd}-g_{ad}\,S_{bc}+g_{bd}\,S_{ac}-g_{% bc}\,S_{ad}\right)=\frac{2}{n-2}\,\left(g_{a[c}\,S_{d]b}-g_{b[c}\,S_{d]a}\right)
  12. S a b = R a b - 1 n g a b R S_{ab}=R_{ab}-\frac{1}{n}\,g_{ab}\,R
  13. g a b g_{ab}
  14. C a b c d C_{abcd}
  15. n = 4 n=4
  16. G a b = R a b - 1 / 2 g a b R G_{ab}=R_{ab}-1/2\,g_{ab}R
  17. S a b = R a b - 1 4 g a b R = G a b - 1 4 g a b G S_{ab}=R_{ab}-\frac{1}{4}\,g_{ab}\,R=G_{ab}-\frac{1}{4}\,g_{ab}\,G
  18. R a b c d , C a b c d R_{abcd},\,C_{abcd}
  19. S a b , E a b c d S_{ab},\,E_{abcd}
  20. R ( x , y , z , w ) = - R ( y , x , z , w ) = - R ( x , y , w , z ) R(x,y,z,w)=-R(y,x,z,w)=-R(x,y,w,z)\,
  21. R ( x , y , z , w ) = R ( z , w , x , y ) , R(x,y,z,w)=R(z,w,x,y),\,
  22. b ( R ) ( x , y , z , w ) = R ( x , y , z , w ) + R ( y , z , x , w ) + R ( z , x , y , w ) . b(R)(x,y,z,w)=R(x,y,z,w)+R(y,z,x,w)+R(z,x,y,w).\,
  23. c : S 2 Λ 2 V S 2 V c:S^{2}\Lambda^{2}V\to S^{2}V
  24. c ( R ) ( x , y ) = tr R ( x , , y , ) . c(R)(x,y)=\operatorname{tr}R(x,\cdot,y,\cdot).
  25. ( h k ) ( x , y , z , w ) = h ( x , z ) k ( y , w ) + h ( y , w ) k ( x , z ) - h ( x , w ) k ( y , z ) - h ( y , z ) k ( x , w ) (h{~{}\wedge\!\!\!\!\!\!\bigcirc~{}}k)(x,y,z,w)=h(x,z)k(y,w)+h(y,w)k(x,z)-h(x,% w)k(y,z)-h(y,z)k(x,w)
  26. 𝐒 V = g g \mathbf{S}V=\mathbb{R}g{~{}\wedge\!\!\!\!\!\!\bigcirc~{}}g
  27. \mathbb{R}
  28. 𝐄 V = g S 0 2 V \mathbf{E}V=g{~{}\wedge\!\!\!\!\!\!\bigcirc~{}}S^{2}_{0}V
  29. 𝐂 V = ker c ker b . \mathbf{C}V=\ker c\cap\ker b.
  30. R = S + E + C R=S+E+C
  31. | R | 2 = | S | 2 + | E | 2 + | C | 2 . |R|^{2}=|S|^{2}+|E|^{2}+|C|^{2}.
  32. G a b = 8 π T a b G_{ab}=8\pi\,T_{ab}
  33. T a b T_{ab}

Richardson–Lucy_deconvolution.html

  1. d i = j p i j u j d_{i}=\sum_{j}p_{ij}u_{j}\,
  2. p i j p_{ij}
  3. j j
  4. i i
  5. u j u_{j}
  6. j j
  7. d i d_{i}
  8. i i
  9. u j u_{j}
  10. u j u_{j}
  11. d i d_{i}
  12. p i j p_{ij}
  13. u j u_{j}
  14. u j ( t + 1 ) = u j ( t ) i d i c i p i j u_{j}^{(t+1)}=u_{j}^{(t)}\sum_{i}\frac{d_{i}}{c_{i}}p_{ij}
  15. c i = j p i j u j ( t ) . c_{i}=\sum_{j}p_{ij}u_{j}^{(t)}.
  16. u j u_{j}
  17. u ( t + 1 ) = u ( t ) ( d u ( t ) p p ^ ) u^{(t+1)}=u^{(t)}\cdot\left(\frac{d}{u^{(t)}\otimes p}\otimes\hat{p}\right)
  18. p ^ \hat{p}
  19. p ^ n m = p ( i - n ) ( j - m ) , 0 n , m i , j \hat{p}_{nm}=p_{(i-n)(j-m)},0\leq n,m\leq i,j
  20. p i j p_{ij}

Riemann's_differential_equation.html

  1. \infty
  2. z s z_{s}
  3. x s f ( x ) x^{s}f(x)
  4. x = z - z s x=z-z_{s}
  5. f f
  6. f ( 0 ) 0 f(0)\neq 0
  7. s s
  8. z s z_{s}
  9. α + α + β + β + γ + γ = 1. \alpha+\alpha^{\prime}+\beta+\beta^{\prime}+\gamma+\gamma^{\prime}=1.
  10. d 2 w d z 2 + [ 1 - α - α z - a + 1 - β - β z - b + 1 - γ - γ z - c ] d w d z \frac{d^{2}w}{dz^{2}}+\left[\frac{1-\alpha-\alpha^{\prime}}{z-a}+\frac{1-\beta% -\beta^{\prime}}{z-b}+\frac{1-\gamma-\gamma^{\prime}}{z-c}\right]\frac{dw}{dz}
  11. + [ α α ( a - b ) ( a - c ) z - a + β β ( b - c ) ( b - a ) z - b + γ γ ( c - a ) ( c - b ) z - c ] w ( z - a ) ( z - b ) ( z - c ) = 0. +\left[\frac{\alpha\alpha^{\prime}(a-b)(a-c)}{z-a}+\frac{\beta\beta^{\prime}(b% -c)(b-a)}{z-b}+\frac{\gamma\gamma^{\prime}(c-a)(c-b)}{z-c}\right]\frac{w}{(z-a% )(z-b)(z-c)}=0.
  12. a a
  13. b b
  14. c c
  15. α ; α α;α′
  16. β ; β β;β′
  17. γ ; γ γ;γ′
  18. α + α + β + β + γ + γ = 1. \alpha+\alpha^{\prime}+\beta+\beta^{\prime}+\gamma+\gamma^{\prime}=1.
  19. w ( z ) = P { a b c α β γ z α β γ } w(z)=P\left\{\begin{matrix}a&b&c&\\ \alpha&\beta&\gamma&z\\ \alpha^{\prime}&\beta^{\prime}&\gamma^{\prime}&\end{matrix}\right\}
  20. F 1 2 ( a , b ; c ; z ) = P { 0 1 0 a 0 z 1 - c b c - a - b } \;{}_{2}F_{1}(a,b;c;z)=P\left\{\begin{matrix}0&\infty&1&\\ 0&a&0&z\\ 1-c&b&c-a-b&\end{matrix}\right\}
  21. P { a b c α β γ z α β γ } = ( z - a z - b ) α ( z - c z - b ) γ P { 0 1 0 α + β + γ 0 ( z - a ) ( c - b ) ( z - b ) ( c - a ) α - α α + β + γ γ - γ } P\left\{\begin{matrix}a&b&c&\\ \alpha&\beta&\gamma&z\\ \alpha^{\prime}&\beta^{\prime}&\gamma^{\prime}&\end{matrix}\right\}=\left(% \frac{z-a}{z-b}\right)^{\alpha}\left(\frac{z-c}{z-b}\right)^{\gamma}P\left\{% \begin{matrix}0&\infty&1&\\ 0&\alpha+\beta+\gamma&0&\;\frac{(z-a)(c-b)}{(z-b)(c-a)}\\ \alpha^{\prime}-\alpha&\alpha+\beta^{\prime}+\gamma&\gamma^{\prime}-\gamma&% \end{matrix}\right\}
  22. w ( z ) = ( z - a z - b ) α ( z - c z - b ) 2 γ F 1 ( α + β + γ , α + β + γ ; 1 + α - α ; ( z - a ) ( c - b ) ( z - b ) ( c - a ) ) w(z)=\left(\frac{z-a}{z-b}\right)^{\alpha}\left(\frac{z-c}{z-b}\right)^{\gamma% }\;_{2}F_{1}\left(\alpha+\beta+\gamma,\alpha+\beta^{\prime}+\gamma;1+\alpha-% \alpha^{\prime};\frac{(z-a)(c-b)}{(z-b)(c-a)}\right)
  23. G L ( 2 , 𝐂 ) GL(2,\mathbf{C})
  24. A A
  25. B B
  26. C C
  27. D D
  28. A D B C 0 AD−BC≠0
  29. u = A z + B C z + D and η = A a + B C a + D u=\frac{Az+B}{Cz+D}\quad\,\text{ and }\quad\eta=\frac{Aa+B}{Ca+D}
  30. ζ = A b + B C b + D and θ = A c + B C c + D \zeta=\frac{Ab+B}{Cb+D}\quad\,\text{ and }\quad\theta=\frac{Ac+B}{Cc+D}
  31. P { a b c α β γ z α β γ } = P { η ζ θ α β γ u α β γ } P\left\{\begin{matrix}a&b&c&\\ \alpha&\beta&\gamma&z\\ \alpha^{\prime}&\beta^{\prime}&\gamma^{\prime}&\end{matrix}\right\}=P\left\{% \begin{matrix}\eta&\zeta&\theta&\\ \alpha&\beta&\gamma&u\\ \alpha^{\prime}&\beta^{\prime}&\gamma^{\prime}&\end{matrix}\right\}

Riemannian_submersion.html

  1. f : M N f:M\to N
  2. d f : ker ( d f ) T N df:\mathrm{ker}(df)^{\perp}\rightarrow TN
  3. G G
  4. ( M , g ) (M,g)
  5. π : M N \pi:M\rightarrow N
  6. N = M / G N=M/G
  7. S 3 2 S^{3}\subset\mathbb{C}^{2}
  8. K N ( X , Y ) = K M ( X ~ , Y ~ ) + 3 4 | [ X ~ , Y ~ ] V | 2 K_{N}(X,Y)=K_{M}(\tilde{X},\tilde{Y})+\tfrac{3}{4}|[\tilde{X},\tilde{Y}]^{V}|^% {2}
  9. X , Y X,Y
  10. N N
  11. X ~ , Y ~ \tilde{X},\tilde{Y}
  12. M M
  13. [ * , * ] [*,*]
  14. Z V Z^{V}
  15. Z Z
  16. N N
  17. M M

Riemann–Roch_theorem_for_smooth_manifolds.html

  1. ch ( f K * ( x ) ) = f H * ( ch ( x ) e d ( v f ) / 2 A ^ ( v f ) ) , \mathrm{ch}(f_{K*}(x))=f_{H*}(\mathrm{ch}(x)e^{d(v_{f})/2}\hat{A}(v_{f})),
  2. u : H * ( B ( N ) , S ( N ) ) H * ( Y , Y - B ( N ) ) H * ( Y ) u:H^{*}(B(N),S(N))\to H^{*}(Y,Y-B(N))\to H^{*}(Y)
  3. v : K ( B ( N ) , S ( N ) ) K ( Y , Y - B ( N ) ) K ( Y ) v:K(B(N),S(N))\to K(Y,Y-B(N))\to K(Y)
  4. i : X Y × S 2 n i:X\to Y\times S^{2n}
  5. p : Y × S 2 n Y . p:Y\times S^{2n}\to Y.

Riemann–Roch_theorem_for_surfaces.html

  1. χ ( D ) = χ ( 0 ) + 1 2 D . ( D - K ) \chi(D)=\chi(0)+\tfrac{1}{2}D.(D-K)\,
  2. χ = c 1 2 + c 2 12 = ( K . K ) + e 12 \chi=\frac{c_{1}^{2}+c_{2}}{12}=\frac{(K.K)+e}{12}
  3. 1 + c 1 ( X ) / 2 + ( c 1 ( X ) 2 + c 2 ( X ) ) / 12 1+c_{1}(X)/2+(c_{1}(X)^{2}+c_{2}(X))/12
  4. 1 + c 1 ( L ) + c 1 ( L ) 2 / 2 1+c_{1}(L)+c_{1}(L)^{2}/2
  5. χ ( D ) \displaystyle\chi(D)
  6. χ ( 0 ) = 1 12 ( c 1 ( X ) 2 + c 2 ( X ) ) \chi(0)=\frac{1}{12}\left(c_{1}(X)^{2}+c_{2}(X)\right)
  7. χ ( D ) = χ ( 0 ) + 1 2 ( D . D - D . K ) \chi(D)=\chi(0)+\frac{1}{2}(D.D-D.K)
  8. r n - π + p a + 1 - i r\geq n-\pi+p_{a}+1-i

Rietveld_refinement.html

  1. Q = 4 π sin ( θ ) λ . Q=\frac{4\pi\sin\left(\theta\right)}{\lambda}.
  2. y i = I k exp [ - 4 ln ( 2 ) H k 2 ( 2 θ i - 2 θ k ) 2 ] y_{i}=I_{k}\exp\left[\frac{-4\ln\left(2\right)}{H_{k}^{2}}\left(2\theta_{i}-2% \theta_{k}\right)^{2}\right]
  3. A s = 1 - [ s P ( 2 θ i - 2 θ k ) 2 tan θ k ] A_{s}=1-\left[\frac{sP\left(2\theta_{i}-2\theta_{k}\right)^{2}}{\tan\theta_{k}% }\right]
  4. H k 2 = U tan 2 θ k + V tan θ k + W H_{k}^{2}=U\tan^{2}\theta_{k}+V\tan\theta_{k}+W
  5. I c o r r = I o b s exp ( - G α 2 ) I_{corr}=I_{obs}\exp\left(-G\alpha^{2}\right)
  6. M = i W i { y i o b s - 1 c y i c a l c } 2 M=\sum_{i}W_{i}\left\{y_{i}^{obs}-\frac{1}{c}y_{i}^{calc}\right\}^{2}
  7. y c a l c = c y o b s y^{calc}=cy^{obs}

Rijndael_key_schedule.html

  1. 2 = 00000010 = 0 x 7 + 0 x 6 + 0 x 5 + 0 x 4 + 0 x 3 + 0 x 2 + 1 x + 0 = x 2=00000010=0x^{7}+0x^{6}+0x^{5}+0x^{4}+0x^{3}+0x^{2}+1x+0=x
  2. rcon ( i ) = x ( i - 1 ) \textrm{rcon}(i)=x^{(i-1)}
  3. 𝔽 2 8 \mathbb{F}_{2^{8}}
  4. rcon ( i ) = x ( i - 1 ) mod x 8 + x 4 + x 3 + x + 1 \textrm{rcon}(i)=x^{(i-1)}\mod x^{8}+x^{4}+x^{3}+x+1
  5. 𝔽 2 [ x ] \mathbb{F}_{2}[x]

Rijndael_mix_columns.html

  1. x 4 + 1 x^{4}+1
  2. c ( x ) = 3 x 3 + x 2 + x + 2 c(x)=3x^{3}+x^{2}+x+2
  3. c - 1 ( x ) = 11 x 3 + 13 x 2 + 9 x + 14 c^{-1}(x)=11x^{3}+13x^{2}+9x+14
  4. [ b 0 b 1 b 2 b 3 ] = [ 2 3 1 1 1 2 3 1 1 1 2 3 3 1 1 2 ] [ a 0 a 1 a 2 a 3 ] \begin{bmatrix}b_{0}\\ b_{1}\\ b_{2}\\ b_{3}\end{bmatrix}=\begin{bmatrix}2&3&1&1\\ 1&2&3&1\\ 1&1&2&3\\ 3&1&1&2\end{bmatrix}\begin{bmatrix}a_{0}\\ a_{1}\\ a_{2}\\ a_{3}\end{bmatrix}
  5. b 0 = 2 a 0 + 3 a 1 + 1 a 2 + 1 a 3 b_{0}=2a_{0}+3a_{1}+1a_{2}+1a_{3}
  6. b 1 = 1 a 0 + 2 a 1 + 3 a 2 + 1 a 3 b_{1}=1a_{0}+2a_{1}+3a_{2}+1a_{3}
  7. b 2 = 1 a 0 + 1 a 1 + 2 a 2 + 3 a 3 b_{2}=1a_{0}+1a_{1}+2a_{2}+3a_{3}
  8. b 3 = 3 a 0 + 1 a 1 + 1 a 2 + 2 a 3 b_{3}=3a_{0}+1a_{1}+1a_{2}+2a_{3}
  9. [ r 0 r 1 r 2 r 3 ] = [ 14 11 13 9 9 14 11 13 13 9 14 11 11 13 9 14 ] [ a 0 a 1 a 2 a 3 ] \begin{bmatrix}r_{0}\\ r_{1}\\ r_{2}\\ r_{3}\end{bmatrix}=\begin{bmatrix}14&11&13&9\\ 9&14&11&13\\ 13&9&14&11\\ 11&13&9&14\end{bmatrix}\begin{bmatrix}a_{0}\\ a_{1}\\ a_{2}\\ a_{3}\end{bmatrix}
  10. r 0 = 14 a 0 + 11 a 1 + 13 a 2 + 9 a 3 r_{0}=14a_{0}+11a_{1}+13a_{2}+9a_{3}
  11. r 1 = 9 a 0 + 14 a 1 + 11 a 2 + 13 a 3 r_{1}=9a_{0}+14a_{1}+11a_{2}+13a_{3}
  12. r 2 = 13 a 0 + 9 a 1 + 14 a 2 + 11 a 3 r_{2}=13a_{0}+9a_{1}+14a_{2}+11a_{3}
  13. r 3 = 11 a 0 + 13 a 1 + 9 a 2 + 14 a 3 r_{3}=11a_{0}+13a_{1}+9a_{2}+14a_{3}

Ring_of_sets.html

  1. \mathcal{R}
  2. A A
  3. B B
  4. A , B A,B\in\mathcal{R}
  5. A B A\cap B\in\mathcal{R}
  6. A , B A,B\in\mathcal{R}
  7. A B . A\cup B\in\mathcal{R}.
  8. \mathcal{R}
  9. A A
  10. B B
  11. A , B A,B\in\mathcal{R}
  12. A B A\setminus B\in\mathcal{R}
  13. A , B A,B\in\mathcal{R}
  14. A B . A\cup B\in\mathcal{R}.
  15. \mathcal{R}
  16. \mathcal{R}
  17. A B = ( A B ) ( B A ) A\,\triangle\,B=(A\setminus B)\cup(B\setminus A)
  18. A B = A ( A B ) . A\cap B=A\setminus(A\setminus B).
  19. \mathcal{R}
  20. A B = ( A B ) ( A B ) A\cup B=(A\,\triangle\,B)\,\triangle\,(A\cap B)
  21. A B = A ( A B ) . A\setminus B=A\,\triangle\,(A\cap B).
  22. 𝒮 \mathcal{S}
  23. 𝒮 , \emptyset\in\mathcal{S},
  24. A , B 𝒮 A,B\in\mathcal{S}
  25. A B 𝒮 , A\cap B\in\mathcal{S},
  26. A , B 𝒮 A,B\in\mathcal{S}
  27. A B = i = 1 n C i A\setminus B=\bigcup_{i=1}^{n}C_{i}
  28. C 1 , , C n 𝒮 . C_{1},\dots,C_{n}\in\mathcal{S}.

Risk_reversal.html

  1. R 25 = σ c a l l , 25 - σ p u t , 25 R_{25}=\sigma_{call,25}-\sigma_{put,25}

Robert_M._Solovay.html

  1. λ \lambda
  2. 2 λ = λ + 2^{\lambda}=\lambda^{+}
  3. κ \kappa
  4. S κ S\subseteq\kappa
  5. S S
  6. κ \kappa
  7. ( A A ) A \Box(\Box A\to A)\to\Box A

Roborovski_hamster.html

  1. N = ( 1 , 422 ln ( M / gramm ) - 1 , 780 ) gramm N=(1{,}422\cdot\ln(M/\,\text{gramm})-1{,}780)\,\text{ gramm}

Robust_regression.html

  1. x ( x - μ ) / σ x\leftarrow(x-\mu)/\sigma
  2. 1 - ε 1-\varepsilon
  3. σ 2 \sigma^{2}
  4. ε \varepsilon
  5. ε \varepsilon
  6. c σ 2 c\sigma^{2}
  7. c > 1 c>1
  8. e i ( 1 - ε ) N ( 0 , σ 2 ) + ε N ( 0 , c σ 2 ) . e_{i}\sim(1-\varepsilon)N(0,\sigma^{2})+\varepsilon N(0,c\sigma^{2}).
  9. ε < 0.1 \varepsilon<0.1
  10. ε \varepsilon
  11. ε \varepsilon

Robust_statistics.html

  1. n n
  2. ( X 1 , , X n ) (X_{1},\dots,X_{n})
  3. x 1 , , x n x_{1},\dots,x_{n}
  4. X n ¯ := X 1 + + X n n \overline{X_{n}}:=\frac{X_{1}+\cdots+X_{n}}{n}
  5. x ¯ \overline{x}
  6. x 1 , , x n x_{1},\dots,x_{n}
  7. ( Ω , 𝒜 , P ) (\Omega,\mathcal{A},P)
  8. ( 𝒳 , Σ ) (\mathcal{X},\Sigma)
  9. Θ \Theta
  10. p * p\in\mathbb{N}^{*}
  11. ( Γ , S ) (\Gamma,S)
  12. ( Ω , 𝒜 , P ) (\Omega,\mathcal{A},P)
  13. ( 𝒳 , Σ ) = ( , ) (\mathcal{X},\Sigma)=(\mathbb{R},\mathcal{B})
  14. Θ = × + \Theta=\mathbb{R}\times\mathbb{R}^{+}
  15. ( Γ , S ) = ( , ) (\Gamma,S)=(\mathbb{R},\mathcal{B})
  16. n * n\in\mathbb{N}^{*}
  17. X 1 , , X n : ( Ω , 𝒜 ) ( 𝒳 , Σ ) X_{1},\dots,X_{n}:(\Omega,\mathcal{A})\rightarrow(\mathcal{X},\Sigma)
  18. ( x 1 , , x n ) (x_{1},\dots,x_{n})
  19. T n : ( 𝒳 n , Σ n ) ( Γ , S ) T_{n}:(\mathcal{X}^{n},\Sigma^{n})\rightarrow(\Gamma,S)
  20. i { 1 , , n } i\in\{1,\dots,n\}
  21. E I F i EIF_{i}
  22. i i
  23. E I F i : x 𝒳 n * ( T n ( x 1 , , x i - 1 , x , x i + 1 , , x n ) - T n ( x 1 , , x i - 1 , x i , x i + 1 , , x n ) ) EIF_{i}:x\in\mathcal{X}\mapsto n*(T_{n}(x_{1},\dots,x_{i-1},x,x_{i+1},\dots,x_% {n})-T_{n}(x_{1},\dots,x_{i-1},x_{i},x_{i+1},\dots,x_{n}))
  24. E I F i : x Γ EIF_{i}:x\in\Gamma
  25. x x
  26. A A
  27. Σ \Sigma
  28. θ Θ \theta\in\Theta
  29. F F
  30. A A
  31. T : A Γ T:A\rightarrow\Gamma
  32. ( T n ) n (T_{n})_{n\in\mathbb{N}}
  33. θ Θ , T ( F θ ) = θ \forall\theta\in\Theta,T(F_{\theta})=\theta
  34. F F
  35. G G
  36. A A
  37. F F
  38. G G
  39. d T G - F ( F ) = lim t 0 + T ( t G + ( 1 - t ) F ) - T ( F ) t dT_{G-F}(F)=\lim_{t\rightarrow 0^{+}}\frac{T(tG+(1-t)F)-T(F)}{t}
  40. T T
  41. F F
  42. G - F G-F
  43. x 𝒳 x\in\mathcal{X}
  44. Δ x \Delta_{x}
  45. { x } \{x\}
  46. G = Δ x G=\Delta_{x}
  47. I F ( x ; T ; F ) := lim t 0 + T ( t Δ x + ( 1 - t ) F ) - T ( F ) t . IF(x;T;F):=\lim_{t\rightarrow 0^{+}}\frac{T(t\Delta_{x}+(1-t)F)-T(F)}{t}.
  48. x x
  49. t t
  50. ρ * \rho^{*}
  51. γ * \gamma^{*}
  52. λ * \lambda^{*}
  53. ρ * := inf r > 0 { r : I F ( x ; T ; F ) = 0 , | x | > r } \rho^{*}:=\inf_{r>0}\{r:IF(x;T;F)=0,|x|>r\}
  54. γ * ( T ; F ) := sup x 𝒳 | I F ( x ; T ; F ) | \gamma^{*}(T;F):=\sup_{x\in\mathcal{X}}|IF(x;T;F)|
  55. λ * ( T ; F ) := sup ( x , y ) 𝒳 2 x y I F ( y ; T ; F ) - I F ( x ; T ; F ) y - x \lambda^{*}(T;F):=\sup_{(x,y)\in\mathcal{X}^{2}\atop x\neq y}\left\|\frac{IF(y% ;T;F)-IF(x;T;F)}{y-x}\right\|
  56. x x
  57. y y
  58. y y
  59. x x
  60. i = 1 n f ( x i ) \prod_{i=1}^{n}f(x_{i})
  61. i = 1 n - log f ( x i ) \sum_{i=1}^{n}-\log f(x_{i})
  62. i = 1 n ρ ( x i ) \sum_{i=1}^{n}\rho(x_{i})
  63. ρ \rho
  64. i = 1 n ρ ( x i ) \sum_{i=1}^{n}\rho(x_{i})
  65. ρ \rho
  66. i = 1 n ψ ( x i ) = 0 \sum_{i=1}^{n}\psi(x_{i})=0
  67. ψ ( x ) = d ρ ( x ) d x \psi(x)=\frac{d\rho(x)}{dx}
  68. ρ \rho
  69. ρ \rho
  70. ψ \psi
  71. ρ \rho
  72. ψ \psi
  73. ρ ( x ) \rho(x)
  74. ρ \rho
  75. T T
  76. ψ \psi
  77. ψ \psi
  78. I F ( x ; T , F ) = M - 1 ψ ( x , T ( F ) ) IF(x;T,F)=M^{-1}\psi(x,T(F))
  79. p × p p\times p
  80. M = - 𝒳 ( ψ ( x , θ ) θ ) T ( F ) d F ( x ) M=-\int_{\mathcal{X}}\left(\frac{\partial\psi(x,\theta)}{\partial\theta}\right% )_{T(F)}dF(x)
  81. ψ \psi
  82. ψ \psi
  83. ν \nu
  84. ψ ( x ) = x x 2 + ν \psi(x)=\frac{x}{x^{2}+\nu}
  85. ν = 1 \nu=1
  86. ν \nu
  87. ν \nu
  88. ν \nu
  89. ψ \psi
  90. ν \nu
  91. μ ^ = 27.40 , σ ^ = 3.81 , ν ^ = 2.13. \hat{\mu}=27.40,\hat{\sigma}=3.81,\hat{\nu}=2.13.
  92. ν = 4 \nu=4
  93. μ ^ = 27.49 , σ ^ = 4.51. \hat{\mu}=27.49,\hat{\sigma}=4.51.

Rogers–Ramanujan_identities.html

  1. G ( q ) = n = 0 q n 2 ( q ; q ) n = 1 ( q ; q 5 ) ( q 4 ; q 5 ) = 1 + q + q 2 + q 3 + 2 q 4 + 2 q 5 + 3 q 6 + G(q)=\sum_{n=0}^{\infty}\frac{q^{n^{2}}}{(q;q)_{n}}=\frac{1}{(q;q^{5})_{\infty% }(q^{4};q^{5})_{\infty}}=1+q+q^{2}+q^{3}+2q^{4}+2q^{5}+3q^{6}+\cdots\,
  2. H ( q ) = n = 0 q n 2 + n ( q ; q ) n = 1 ( q 2 ; q 5 ) ( q 3 ; q 5 ) = 1 + q 2 + q 3 + q 4 + q 5 + 2 q 6 + H(q)=\sum_{n=0}^{\infty}\frac{q^{n^{2}+n}}{(q;q)_{n}}=\frac{1}{(q^{2};q^{5})_{% \infty}(q^{3};q^{5})_{\infty}}=1+q^{2}+q^{3}+q^{4}+q^{5}+2q^{6}+\cdots\,
  3. ( ; ) n (\cdot;\cdot)_{n}
  4. q n 2 ( q ; q ) n \frac{q^{n^{2}}}{(q;q)_{n}}
  5. n n
  6. 1 ( q ; q 5 ) ( q 4 ; q 5 ) \frac{1}{(q;q^{5})_{\infty}(q^{4};q^{5})_{\infty}}
  7. q n 2 + n ( q ; q ) n \frac{q^{n^{2}+n}}{(q;q)_{n}}
  8. n n
  9. 1 ( q 2 ; q 5 ) ( q 3 ; q 5 ) \frac{1}{(q^{2};q^{5})_{\infty}(q^{3};q^{5})_{\infty}}
  10. n n
  11. n n
  12. n n
  13. n n
  14. n n
  15. 1 + q 1 + q 2 1 + q 3 1 + = G ( q ) H ( q ) . 1+\frac{q}{1+\frac{q^{2}}{1+\frac{q^{3}}{1+\cdots}}}=\frac{G(q)}{H(q)}.

Rolling_(metalworking).html

  1. t 0 t_{0}
  2. t f t_{f}
  3. d d
  4. d = t 0 - t f d=t_{0}-t_{f}
  5. R R
  6. f f
  7. d m a x = f 2 R d_{max}=f^{2}R

Rossby_radius_of_deformation.html

  1. L R ( g D ) 1 / 2 f L_{R}\equiv\frac{(gD)^{1/2}}{f}
  2. g \,g
  3. D \,D
  4. f \,f
  5. L R , n N H n π f 0 L_{R,n}\equiv\frac{NH}{n\pi f_{0}}
  6. N \,N
  7. H \,H

Rosser's_theorem.html

  1. p n > n ln n . p_{n}>n\cdot\ln n.
  2. p n > n ( ln n + ln ( ln n ) - 1 ) . p_{n}>n\cdot(\ln n+\ln(\ln n)-1).

Rotating_wave_approximation.html

  1. ω L + ω 0 \omega_{L}+\omega_{0}
  2. ω L - ω 0 \omega_{L}-\omega_{0}
  3. ω L \omega_{L}
  4. ω 0 \omega_{0}
  5. | g |\,\text{g}\rangle
  6. | e |\,\text{e}\rangle
  7. ω 0 \hbar\omega_{0}
  8. ω 0 \omega_{0}
  9. H 0 = ω 0 | e e | H_{0}=\hbar\omega_{0}|\,\text{e}\rangle\langle\,\text{e}|
  10. ω L \omega_{L}
  11. E ( t ) = E 0 e - i ω L t + E 0 * e i ω L t \vec{E}(t)=\vec{E}_{0}e^{-i\omega_{L}t}+\vec{E}_{0}^{*}e^{i\omega_{L}t}
  12. H 1 = - d E H_{1}=-\vec{d}\cdot\vec{E}
  13. d \vec{d}
  14. H = H 0 + H 1 . H=H_{0}+H_{1}.
  15. e | d | e = g | d | g = 0. \langle\,\text{e}|\vec{d}|\,\text{e}\rangle=\langle\,\text{g}|\vec{d}|\,\text{% g}\rangle=0.
  16. d eg := e | d | g \vec{d}_{\,\text{eg}}:=\langle\,\text{e}|\vec{d}|\,\text{g}\rangle
  17. d = d eg | e g | + d eg * | g e | \vec{d}=\vec{d}_{\,\text{eg}}|\,\text{e}\rangle\langle\,\text{g}|+\vec{d}_{\,% \text{eg}}^{*}|\,\text{g}\rangle\langle\,\text{e}|
  18. * {}^{*}
  19. H 1 = - ( Ω e - i ω L t + Ω ~ e i ω L t ) | e g | - ( Ω ~ * e - i ω L t + Ω * e i ω L t ) | g e | H_{1}=-\hbar\left(\Omega e^{-i\omega_{L}t}+\tilde{\Omega}e^{i\omega_{L}t}% \right)|\,\text{e}\rangle\langle\,\text{g}|-\hbar\left(\tilde{\Omega}^{*}e^{-i% \omega_{L}t}+\Omega^{*}e^{i\omega_{L}t}\right)|\,\text{g}\rangle\langle\,\text% {e}|
  20. Ω = - 1 d eg E 0 \Omega=\hbar^{-1}\vec{d}\text{eg}\cdot\vec{E}_{0}
  21. Ω ~ := - 1 d eg E 0 * \tilde{\Omega}:=\hbar^{-1}\vec{d}\text{eg}\cdot\vec{E}_{0}^{*}
  22. Ω ~ \tilde{\Omega}
  23. H 1 , I H_{1,I}
  24. H 1 , I = - ( Ω e - i Δ t + Ω ~ e i ( ω L + ω 0 ) t ) | e g | - ( Ω ~ * e - i ( ω L + ω 0 ) t + Ω * e i Δ t ) | g e | , H_{1,I}=-\hbar\left(\Omega e^{-i\Delta t}+\tilde{\Omega}e^{i(\omega_{L}+\omega% _{0})t}\right)|\,\text{e}\rangle\langle\,\text{g}|-\hbar\left(\tilde{\Omega}^{% *}e^{-i(\omega_{L}+\omega_{0})t}+\Omega^{*}e^{i\Delta t}\right)|\,\text{g}% \rangle\langle\,\text{e}|,
  25. Δ := ω L - ω 0 \Delta:=\omega_{L}-\omega_{0}
  26. Δ ω L + ω 0 \Delta\ll\omega_{L}+\omega_{0}
  27. Ω ~ \tilde{\Omega}
  28. Ω ~ * \tilde{\Omega}^{*}
  29. H 1 , I RWA = - Ω e - i Δ t | e g | - Ω * e i Δ t | g e | . H_{1,I}^{\,\text{RWA}}=-\hbar\Omega e^{-i\Delta t}|\,\text{e}\rangle\langle\,% \text{g}|-\hbar\Omega^{*}e^{i\Delta t}|\,\text{g}\rangle\langle\,\text{e}|.
  30. H RWA = ω 0 | e e | - Ω e - i ω L t | e g | - Ω * e i ω L t | g e | . H\text{RWA}=\hbar\omega_{0}|\,\text{e}\rangle\langle\,\text{e}|-\hbar\Omega e^% {-i\omega_{L}t}|\,\text{e}\rangle\langle\,\text{g}|-\hbar\Omega^{*}e^{i\omega_% {L}t}|\,\text{g}\rangle\langle\,\text{e}|.
  31. H 1 \displaystyle H_{1}
  32. H 1 , I H_{1,I}
  33. U = e i H 0 t / = e i ω 0 t | e e | = | g g | + e i ω 0 t | e e | U=e^{iH_{0}t/\hbar}=e^{i\omega_{0}t|\,\text{e}\rangle\langle\,\text{e}|}=|\,% \text{g}\rangle\langle\,\text{g}|+e^{i\omega_{0}t}|\,\text{e}\rangle\langle\,% \text{e}|
  34. | g |\,\text{g}\rangle
  35. | e |\,\text{e}\rangle
  36. H 1 , I \displaystyle H_{1,I}
  37. H 1 , I RWA H_{1,I}^{\,\text{RWA}}
  38. H 1 RWA \displaystyle H_{1}^{\,\text{RWA}}
  39. H RWA = H 0 + H 1 RWA = ω 0 | e e | - Ω e - i ω L t | e g | - Ω * e i ω L t | g e | . H\text{RWA}=H_{0}+H_{1}^{\,\text{RWA}}=\hbar\omega_{0}|\,\text{e}\rangle% \langle\,\text{e}|-\hbar\Omega e^{-i\omega_{L}t}|\,\text{e}\rangle\langle\,% \text{g}|-\hbar\Omega^{*}e^{i\omega_{L}t}|\,\text{g}\rangle\langle\,\text{e}|.

Rotation_around_a_fixed_axis.html

  1. F net = M a cm F_{\mathrm{net}}=Ma_{\mathrm{cm}}\;\!
  2. r r
  3. s s
  4. θ \theta
  5. θ = s r \theta=\frac{s}{r}
  6. 1 rev = 360 = 2 π rad , and 1\mathrm{\ rev}=360^{\circ}=2\pi\mathrm{\ rad}\mathrm{,and}
  7. 1 rad = 180 / π 57.3 . 1\mathrm{\ rad}=180^{\circ}/{\pi}\approx 57.3^{\circ}.
  8. Δ θ = θ 2 - θ 1 , \Delta\theta=\theta_{2}-\theta_{1},\!
  9. Δ θ \Delta\theta
  10. θ 1 \theta_{1}
  11. θ 2 \theta_{2}
  12. ω \omega
  13. ω ¯ = Δ θ Δ t = θ 2 - θ 1 t 2 - t 1 . \overline{\omega}=\frac{\Delta\theta}{\Delta t}=\frac{\theta_{2}-\theta_{1}}{t% _{2}-t_{1}}.
  14. ω ( t ) = d θ d t . \omega(t)=\frac{d\theta}{dt}.
  15. v = d s d t v=\frac{ds}{dt}
  16. ω = d θ d t = v r , \omega=\frac{d\theta}{dt}=\frac{v}{r},
  17. v v
  18. ω = 2 π f \omega={2\pi f}\!
  19. α ¯ \overline{\alpha}
  20. α ¯ = Δ ω Δ t = ω 2 - ω 1 t 2 - t 1 . \overline{\alpha}=\frac{\Delta\omega}{\Delta t}=\frac{\omega_{2}-\omega_{1}}{t% _{2}-t_{1}}.
  21. α ( t ) = d ω d t = d 2 θ d t 2 . \alpha(t)=\frac{d\omega}{dt}=\frac{d^{2}\theta}{dt^{2}}.
  22. a = r α , a=r\alpha,\!
  23. a R = v 2 r = ω 2 r a_{\mathrm{R}}=\frac{v^{2}}{r}=\omega^{2}r\!
  24. T = I α T=I\alpha
  25. θ \theta
  26. ω i \omega_{i}
  27. ω f \omega_{f}
  28. α \alpha
  29. t t
  30. ω f = ω i + α t \omega_{f}=\omega_{i}+\alpha t\;\!
  31. θ = ω i t + 1 2 α t 2 \theta=\omega_{i}t+\begin{matrix}\frac{1}{2}\end{matrix}\alpha t^{2}
  32. ω f 2 = ω i 2 + 2 α θ \omega_{f}^{2}=\omega_{i}^{2}+2\alpha\theta
  33. θ = 1 2 ( ω f + ω i ) t \theta=\tfrac{1}{2}\left(\omega_{f}+\omega_{i}\right)t
  34. m m
  35. r r
  36. I = m r 2 . I=mr^{2}.
  37. s y m b o l τ symbol{\tau}
  38. s y m b o l τ = 𝐫 × 𝐅 , symbol{\tau}=\mathbf{r}\times\mathbf{F},
  39. s y m b o l τ = I s y m b o l α , symbol{\tau}=Isymbol{\alpha},
  40. W = τ θ . W=\tau\theta.\!
  41. P = τ ω . P=\tau\omega.\!
  42. 𝐋 = 𝐫 × 𝐩 . \mathbf{L}=\mathbf{r}\times\mathbf{p}.
  43. 𝐋 = I s y m b o l ω , \mathbf{L}=Isymbol{\omega},
  44. s y m b o l τ = d 𝐋 d t , symbol{\tau}=\frac{d\mathbf{L}}{dt},
  45. K rot = 1 2 I ω 2 , K_{\mathrm{rot}}=\tfrac{1}{2}I\omega^{2},
  46. 1 / 2 {1}/{2}
  47. Δ θ \Delta\theta

Rotation_system.html

  1. ( σ , θ ) (\sigma,\theta)
  2. g = 1 - 1 2 ( | Z ( σ ) | - | Z ( θ ) | + | Z ( σ θ ) | ) g=1-\frac{1}{2}(|Z(\sigma)|-|Z(\theta)|+|Z(\sigma\theta)|)
  3. Z ( ϕ ) Z(\phi)
  4. ϕ \phi

Rotations_in_4-dimensional_Euclidean_space.html

  1. [ 0 , π ] [0,\pi]
  2. α \alpha
  3. α \alpha
  4. < α <\alpha
  5. α \alpha
  6. α \alpha
  7. ( + α , + α ) (+\alpha,+\alpha)
  8. ( - α , - α ) (-\alpha,-\alpha)
  9. ( + α , - α ) (+\alpha,-\alpha)
  10. ( - α , + α ) (-\alpha,+\alpha)
  11. α = 0 \alpha=0
  12. α = π \alpha=\pi
  13. α = 0 \alpha=0
  14. α = π \alpha=\pi
  15. A = ( a 00 a 01 a 02 a 03 a 10 a 11 a 12 a 13 a 20 a 21 a 22 a 23 a 30 a 31 a 32 a 33 ) A=\begin{pmatrix}a_{00}&a_{01}&a_{02}&a_{03}\\ a_{10}&a_{11}&a_{12}&a_{13}\\ a_{20}&a_{21}&a_{22}&a_{23}\\ a_{30}&a_{31}&a_{32}&a_{33}\\ \end{pmatrix}
  16. M = 1 4 ( a 00 + a 11 + a 22 + a 33 + a 10 - a 01 - a 32 + a 23 + a 20 + a 31 - a 02 - a 13 + a 30 - a 21 + a 12 - a 03 a 10 - a 01 + a 32 - a 23 - a 00 - a 11 + a 22 + a 33 + a 30 - a 21 - a 12 + a 03 - a 20 - a 31 - a 02 - a 13 a 20 - a 31 - a 02 + a 13 - a 30 - a 21 - a 12 - a 03 - a 00 + a 11 - a 22 + a 33 + a 10 + a 01 - a 32 - a 23 a 30 + a 21 - a 12 - a 03 + a 20 - a 31 + a 02 - a 13 - a 10 - a 01 - a 32 - a 23 - a 00 + a 11 + a 22 - a 33 ) M=\frac{1}{4}\begin{pmatrix}a_{00}+a_{11}+a_{22}+a_{33}&+a_{10}-a_{01}-a_{32}+% a_{23}&+a_{20}+a_{31}-a_{02}-a_{13}&+a_{30}-a_{21}+a_{12}-a_{03}\\ a_{10}-a_{01}+a_{32}-a_{23}&-a_{00}-a_{11}+a_{22}+a_{33}&+a_{30}-a_{21}-a_{12}% +a_{03}&-a_{20}-a_{31}-a_{02}-a_{13}\\ a_{20}-a_{31}-a_{02}+a_{13}&-a_{30}-a_{21}-a_{12}-a_{03}&-a_{00}+a_{11}-a_{22}% +a_{33}&+a_{10}+a_{01}-a_{32}-a_{23}\\ a_{30}+a_{21}-a_{12}-a_{03}&+a_{20}-a_{31}+a_{02}-a_{13}&-a_{10}-a_{01}-a_{32}% -a_{23}&-a_{00}+a_{11}+a_{22}-a_{33}\end{pmatrix}
  17. M = ( a p a q a r a s b p b q b r b s c p c q c r c s d p d q d r d s ) M=\begin{pmatrix}ap&aq&ar&as\\ bp&bq&br&bs\\ cp&cq&cr&cs\\ dp&dq&dr&ds\end{pmatrix}
  18. ( a p ) 2 + + ( d s ) 2 = (ap)^{2}+\cdots+(ds)^{2}=
  19. ( a 2 + b 2 + c 2 + d 2 ) ( p 2 + q 2 + r 2 + s 2 ) = 1 (a^{2}+b^{2}+c^{2}+d^{2})(p^{2}+q^{2}+r^{2}+s^{2})=1
  20. a 2 + b 2 + c 2 + d 2 = 1 a^{2}+b^{2}+c^{2}+d^{2}=1
  21. p 2 + q 2 + r 2 + s 2 = 1 p^{2}+q^{2}+r^{2}+s^{2}=1
  22. A = ( a p - b q - c r - d s - a q - b p + c s - d r - a r - b s - c p + d q - a s + b r - c q - d p b p + a q - d r + c s - b q + a p + d s + c r - b r + a s - d p - c q - b s - a r - d q + c p c p + d q + a r - b s - c q + d p - a s - b r - c r + d s + a p + b q - c s - d r + a q - b p d p - c q + b r + a s - d q - c p - b s + a r - d r - c s + b p - a q - d s + c r + b q + a p ) A=\begin{pmatrix}ap-bq-cr-ds&-aq-bp+cs-dr&-ar-bs-cp+dq&-as+br-cq-dp\\ bp+aq-dr+cs&-bq+ap+ds+cr&-br+as-dp-cq&-bs-ar-dq+cp\\ cp+dq+ar-bs&-cq+dp-as-br&-cr+ds+ap+bq&-cs-dr+aq-bp\\ dp-cq+br+as&-dq-cp-bs+ar&-dr-cs+bp-aq&-ds+cr+bq+ap\end{pmatrix}
  23. = ( a - b - c - d b a - d c c d a - b d - c b a ) ( p - q - r - s q p s - r r - s p q s r - q p ) . =\begin{pmatrix}a&-b&-c&-d\\ b&\;\,\,a&-d&\;\,\,c\\ c&\;\,\,d&\;\,\,a&-b\\ d&-c&\;\,\,b&\;\,\,a\end{pmatrix}\cdot\begin{pmatrix}p&-q&-r&-s\\ q&\;\,\,p&\;\,\,s&-r\\ r&-s&\;\,\,p&\;\,\,q\\ s&\;\,\,r&-q&\;\,\,p\end{pmatrix}.
  24. ( u x y z ) = ( a - b - c - d b a - d c c d a - b d - c b a ) ( u x y z ) \begin{pmatrix}u^{\prime}\\ x^{\prime}\\ y^{\prime}\\ z^{\prime}\end{pmatrix}=\begin{pmatrix}a&-b&-c&-d\\ b&\;\,\,a&-d&\;\,\,c\\ c&\;\,\,d&\;\,\,a&-b\\ d&-c&\;\,\,b&\;\,\,a\end{pmatrix}\cdot\begin{pmatrix}u\\ x\\ y\\ z\end{pmatrix}
  25. ( u x y z ) = ( p - q - r - s q p s - r r - s p q s r - q p ) ( u x y z ) . \begin{pmatrix}u^{\prime}\\ x^{\prime}\\ y^{\prime}\\ z^{\prime}\end{pmatrix}=\begin{pmatrix}p&-q&-r&-s\\ q&\;\,\,p&\;\,\,s&-r\\ r&-s&\;\,\,p&\;\,\,q\\ s&\;\,\,r&-q&\;\,\,p\end{pmatrix}\cdot\begin{pmatrix}u\\ x\\ y\\ z\end{pmatrix}.
  26. u + x i + y j + z k = ( a + b i + c j + d k ) ( u + x i + y j + z k ) ( p + q i + r j + s k ) , u^{\prime}+x^{\prime}i+y^{\prime}j+z^{\prime}k=(a+bi+cj+dk)(u+xi+yj+zk)(p+qi+% rj+sk),\,
  27. P = Q L P Q R . P^{\prime}=Q\text{L}PQ\text{R}.\,
  28. P = ( Q L P ) Q R = Q L ( P Q R ) , P^{\prime}=(Q\text{L}P)Q\text{R}=Q\text{L}(PQ\text{R}),\,
  29. ± \pm
  30. ( 1 0 0 0 0 a 11 a 12 a 13 0 a 21 a 22 a 23 0 a 31 a 32 a 33 ) . \begin{pmatrix}1&\,\,0&\,\,0&\,\,0\\ 0&a_{11}&a_{12}&a_{13}\\ 0&a_{21}&a_{22}&a_{23}\\ 0&a_{31}&a_{32}&a_{33}\end{pmatrix}.
  31. p = a , q = - b , r = - c , s = - d p=a,q=-b,r=-c,s=-d
  32. ( a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ) = ( a 2 + b 2 - c 2 - d 2 2 ( b c - a d ) 2 ( b d + a c ) 2 ( b c + a d ) a 2 - b 2 + c 2 - d 2 2 ( c d - a b ) 2 ( b d - a c ) 2 ( c d + a b ) a 2 - b 2 - c 2 + d 2 ) , \begin{pmatrix}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{pmatrix}=\begin{pmatrix}a^{2}+b^{2}-c^{2}-d^{2}&2(bc-% ad)&2(bd+ac)\\ 2(bc+ad)&a^{2}-b^{2}+c^{2}-d^{2}&2(cd-ab)\\ 2(bd-ac)&2(cd+ab)&a^{2}-b^{2}-c^{2}+d^{2}\end{pmatrix},
  33. P = Q P Q - 1 P^{\prime}=QPQ^{-1}
  34. x i + y j + z k = ( a + b i + c j + d k ) ( x i + y j + z k ) ( a - b i - c j - d k ) x^{\prime}i+y^{\prime}j+z^{\prime}k=(a+bi+cj+dk)(xi+yj+zk)(a-bi-cj-dk)
  35. x = sin θ cos ϕ x=\sin\theta\cos\phi
  36. y = sin θ sin ϕ y=\sin\theta\sin\phi
  37. z = cos θ z=\cos\theta
  38. x 2 + y 2 + z 2 = 1 x^{2}+y^{2}+z^{2}=1
  39. { θ 0 , ϕ 0 } \{\theta_{0},\phi_{0}\}
  40. ϕ \phi
  41. { θ 0 , ϕ 0 + ϕ } ) \{\theta_{0},\phi_{0}+\phi\})
  42. { ξ 1 , η , ξ 2 } \{\xi_{1},\eta,\xi_{2}\}
  43. u = cos ( ξ 1 ) sin ( η ) u=\cos(\xi_{1})\sin(\eta)
  44. z = sin ( ξ 1 ) sin ( η ) z=\sin(\xi_{1})\sin(\eta)
  45. x = cos ( ξ 2 ) cos ( η ) x=\cos(\xi_{2})\cos(\eta)
  46. y = sin ( ξ 2 ) cos ( η ) y=\sin(\xi_{2})\cos(\eta)
  47. u 2 + x 2 + y 2 + z 2 = 1 u^{2}+x^{2}+y^{2}+z^{2}=1
  48. ξ 1 \xi_{1}
  49. ξ 2 \xi_{2}
  50. { ξ 10 , η 0 , ξ 20 } \{\xi_{10},\eta_{0},\xi_{20}\}
  51. ξ 1 \xi_{1}
  52. ξ 2 \xi_{2}
  53. { ξ 10 + ξ 1 , η 0 , ξ 20 + ξ 2 } \{\xi_{10}+\xi_{1},\eta_{0},\xi_{20}+\xi_{2}\}
  54. { θ , ϕ } \{\theta,\phi\}
  55. θ \theta
  56. { θ 0 , ϕ 0 } \{\theta_{0},\phi_{0}\}
  57. { θ 0 , ϕ 0 + ϕ } \{\theta_{0},\phi_{0}+\phi\}
  58. ϕ \phi
  59. ϕ = ω t \phi=\omega t
  60. ω \omega
  61. { ξ 1 , η , ξ 2 } \{\xi_{1},\eta,\xi_{2}\}
  62. η \eta
  63. ξ 1 \xi_{1}
  64. ξ 2 \xi_{2}
  65. η = π / 4 \eta=\pi/4
  66. { ξ 10 , η 0 , ξ 20 } \{\xi_{10},\eta_{0},\xi_{20}\}
  67. η 0 \eta_{0}
  68. { ξ 10 + ω 1 t , η 0 , ξ 20 + ω 2 t } \{\xi_{10}+\omega_{1}t,\eta_{0},\xi_{20}+\omega_{2}t\}
  69. { 0 , π / 4 , 0 } \{0,\pi/4,0\}
  70. ω 1 = 1 \omega_{1}=1
  71. ω 2 = 5 \omega_{2}=5
  72. ω 1 = 5 \omega_{1}=5
  73. ω 2 = 1 \omega_{2}=1

Roughness_length.html

  1. z 0 z_{0}
  2. z 0 z_{0}

Routh–Hurwitz_theorem.html

  1. P 0 ( y ) P_{0}(y)
  2. P 1 ( y ) P_{1}(y)
  3. f ( i y ) = P 0 ( y ) + i P 1 ( y ) f(iy)=P_{0}(y)+iP_{1}(y)
  4. Δ arg f ( i y ) \Delta\arg f(iy)
  5. P 0 ( y ) P_{0}(y)
  6. P 1 ( y ) P_{1}(y)
  7. I - + r I_{-\infty}^{+\infty}r
  8. p - q = 1 π Δ arg f ( i y ) = { + I - + P 0 ( y ) P 1 ( y ) for odd degree - I - + P 1 ( y ) P 0 ( y ) for even degree } = w ( + ) - w ( - ) . p-q=\frac{1}{\pi}\Delta\arg f(iy)=\left.\begin{cases}+I_{-\infty}^{+\infty}% \frac{P_{0}(y)}{P_{1}(y)}&\,\text{for odd degree}\\ -I_{-\infty}^{+\infty}\frac{P_{1}(y)}{P_{0}(y)}&\,\text{for even degree}\end{% cases}\right\}=w(+\infty)-w(-\infty).

Routing_and_wavelength_assignment.html

  1. C 0 ( ρ , q ) = i = 1 N s d m i C_{0}(\rho,q)=\sum_{i=1}^{N_{sd}}m_{i}
  2. m i 0 , i n t e g e r , i = 1 , 2 , , N s d m_{i}\geq 0,integer,i=1,2,...,N_{sd}
  3. c i j 0 , 1 , i = 1 , 2 , , P , j = 1 , 2 , , W c_{ij}\in{0,1},i=1,2,...,P,j=1,2,...,W
  4. C T B l W × L C^{T}B\leq l_{W\times L}
  5. m 1 W C T A m\leq 1_{W}C^{T}A
  6. m i q i ρ , i = 1 , 2 , , N s d m_{i}\leq q_{i}\rho,i=1,2,...,N_{sd}
  7. N s d N_{sd}
  8. m i m_{i}
  9. L L
  10. W W
  11. P P
  12. A : P × N s d A:P\times N_{sd}
  13. B : P × L B:P\times L
  14. C : P × W C:P\times W
  15. n n
  16. O ( m + n log n ) O(m+n\log n)
  17. m m
  18. n n
  19. p p
  20. p p
  21. p > 1 p>1
  22. p p
  23. O ( p n ( m + n log n ) ) O(pn(m+n\log n))
  24. m m
  25. n n
  26. p p
  27. c o s t ( l ) = β u s a g e ( l ) cost(l)=\beta^{usage(l)}
  28. β \beta
  29. u s a g e ( l ) usage(l)
  30. l l
  31. β \beta
  32. β \beta
  33. β \beta
  34. p p
  35. p p
  36. p p
  37. N B E R N_{BER}
  38. N w a v e N_{wave}
  39. i t h i_{th}
  40. D i = j = 1 N i 10 log [ Q i , j ( s ) / Q i , j ( d ) ] N i D_{i}=\frac{\sum_{j=1}^{N_{i}}10\log[Q_{i,j}^{(s)}/Q_{i,j}^{(d)}]}{N_{i}}
  41. N i N_{i}
  42. i t h i_{th}
  43. Q i , j ( s ) Q_{i,j}^{(s)}
  44. Q i , j ( d ) Q_{i,j}^{(d)}
  45. j t h j_{th}
  46. i t h i_{th}
  47. O ( w ) O(w)
  48. w w
  49. w w

Roy's_identity.html

  1. V ( P , Y ) V(P,Y)
  2. i i
  3. x i m = - V p i V Y x_{i}^{m}=-\frac{\frac{\partial V}{\partial p_{i}}}{\frac{\partial V}{\partial Y}}
  4. P P
  5. Y Y
  6. i i
  7. Y Y
  8. V ( P , Y ) V(P,Y)
  9. u u
  10. V ( P , e ( P , u ) ) = u V(P,e(P,u))=u
  11. p p
  12. p i p_{i}
  13. V [ P , e ( P , u ) ] Y e ( P , u ) p i + V [ P , e ( P , u ) ] p i = 0 \frac{\partial V[P,e(P,u)]}{\partial Y}\frac{\partial e(P,u)}{\partial p_{i}}+% \frac{\partial V[P,e(P,u)]}{\partial p_{i}}=0
  14. - V [ P , e ( P , u ) ] p i V [ P , e ( P , u ) ] Y = e ( P , u ) p i = h i ( P , u ) = x i ( P , e ( P , u ) ) -\frac{\frac{\partial V[P,e(P,u)]}{\partial p_{i}}}{\frac{\partial V[P,e(P,u)]% }{\partial Y}}=\frac{\partial e(P,u)}{\partial p_{i}}=h_{i}(P,u)=x_{i}(P,e(P,u))
  15. V ( p 1 , p 2 , Y ) V(p_{1},p_{2},Y)
  16. = U ( x 1 , x 2 ) + λ ( Y - p 1 x 1 - p 2 x 2 ) \mathcal{L}=U(x_{1},x_{2})+\lambda(Y-p_{1}x_{1}-p_{2}x_{2})
  17. V ( p 1 , p 2 , Y ) V(p_{1},p_{2},Y)
  18. V p 1 = - λ x 1 m \frac{\partial V}{\partial p_{1}}=-\lambda x_{1}^{m}
  19. V Y = λ \frac{\partial V}{\partial Y}=\lambda
  20. x 1 m x_{1}^{m}
  21. - V p 1 V Y = - - λ x 1 m λ = x 1 m -\frac{\frac{\partial V}{\partial p_{1}}}{\frac{\partial V}{\partial Y}}=-% \frac{-\lambda x_{1}^{m}}{\lambda}=x_{1}^{m}

Rubik's_Cube_group.html

  1. F F
  2. F 2 F^{2}
  3. F F^{\prime}
  4. B B
  5. B 2 B^{2}
  6. B B^{\prime}
  7. U U
  8. U 2 U^{2}
  9. U U^{\prime}
  10. D D
  11. D 2 D^{2}
  12. D D^{\prime}
  13. L L
  14. L 2 L^{2}
  15. L L^{\prime}
  16. R R
  17. R 2 R^{2}
  18. R R^{\prime}
  19. E E
  20. L L L L LLLL
  21. E E
  22. R R R RRR
  23. R R^{\prime}
  24. { F , B , U , D , L , R } \{F,B,U,D,L,R\}
  25. | G | = 43 , 252 , 003 , 274 , 489 , 856 , 000 = 2 27 3 14 5 3 7 2 11 |G|=43{,}252{,}003{,}274{,}489{,}856{,}000\,\!=2^{27}3^{14}5^{3}7^{2}11
  26. ( R U 2 D - 1 B D - 1 ) (RU^{2}D^{-1}BD^{-1})
  27. F R FR
  28. R F RF
  29. B R D 2 R B U 2 B R D 2 R B U 2 , BR^{\prime}D^{2}RB^{\prime}U^{2}BR^{\prime}D^{2}RB^{\prime}U^{2},\,\!
  30. R U D B 2 U 2 B U B U B 2 D R U , RUDB^{2}U^{2}B^{\prime}UBUB^{2}D^{\prime}R^{\prime}U^{\prime},\,\!
  31. C p = [ U 2 , D 2 , F , B , L 2 , R 2 , R 2 U F B R 2 F B U R 2 ] . C_{p}=[U^{2},D^{2},F,B,L^{2},R^{2},R^{2}U^{\prime}FB^{\prime}R^{2}F^{\prime}BU% ^{\prime}R^{2}].\,\!
  32. G = C o C p . G=C_{o}\rtimes C_{p}.\,
  33. 3 7 × 2 11 , \mathbb{Z}_{3}^{7}\times\mathbb{Z}_{2}^{11},
  34. 3 \mathbb{Z}_{3}
  35. 2 \mathbb{Z}_{2}
  36. C p = ( A 8 × A 12 ) 2 . C_{p}=(A_{8}\times A_{12})\,\rtimes\mathbb{Z}_{2}.
  37. ( 3 7 × 2 11 ) ( ( A 8 × A 12 ) 2 ) . (\mathbb{Z}_{3}^{7}\times\mathbb{Z}_{2}^{11})\rtimes\,((A_{8}\times A_{12})% \rtimes\mathbb{Z}_{2}).
  38. [ ( 3 7 S 8 ) × ( 2 11 S 12 ) ] 1 2 [(\mathbb{Z}_{3}^{7}\rtimes\mathrm{S}_{8})\times(\mathbb{Z}_{2}^{11}\rtimes% \mathrm{S}_{12})]^{\frac{1}{2}}
  39. [ 4 6 × ( 3 7 S 8 ) × ( 2 11 S 12 ) ] 1 2 . [\mathbb{Z}_{4}^{6}\times(\mathbb{Z}_{3}^{7}\rtimes\mathrm{S}_{8})\times(% \mathbb{Z}_{2}^{11}\rtimes\mathrm{S}_{12})]^{\frac{1}{2}}.
  40. 4 6 × 3 S 8 × 2 S 12 . \mathbb{Z}_{4}^{6}\times\mathbb{Z}_{3}\wr\mathrm{S}_{8}\times\mathbb{Z}_{2}\wr% \mathrm{S}_{12}.
  41. A 8 A_{8}
  42. A 12 A_{12}
  43. 3 \mathbb{Z}_{3}
  44. 2 \mathbb{Z}_{2}

Rule_of_78s.html

  1. u = f * k k + 1 n ( n + 1 ) u=f*k\frac{k+1}{n(n+1)}
  2. u u
  3. f f
  4. k k
  5. n n

Rule_of_twelfths.html

  1. = ( 1 12 + 2 12 + 3 12 ) × 12 m = ( 6 12 ) × 12 m = ( 1 2 ) × 12 m = 6 m =\left({1\over 12}+{2\over 12}+{3\over 12}\right)\times 12\ \mathrm{m}=\left({% 6\over 12}\right)\times 12\ \mathrm{m}=\left({1\over 2}\right)\times 12\ % \mathrm{m}=6\ \mathrm{m}

Run-time_algorithm_specialisation.html

  1. 𝑎𝑙𝑔 ( A , B ) \mathit{alg}(A,B)
  2. A A
  3. B B
  4. 𝑎𝑙𝑔 \mathit{alg}
  5. A A
  6. 𝑎𝑙𝑔 A \mathit{alg}_{A}
  7. 𝑎𝑙𝑔 A ( B ) \mathit{alg}_{A}(B)
  8. 𝑎𝑙𝑔 ( A , B ) \mathit{alg}(A,B)
  9. A A
  10. 𝑎𝑙𝑔 A ( B ) \mathit{alg}_{A}(B)
  11. 𝑎𝑙𝑔 ( A , B ) \mathit{alg}(A,B)
  12. A A
  13. A A
  14. A A
  15. 𝑎𝑙𝑔 \mathit{alg}
  16. 𝑎𝑙𝑔 \mathit{alg}
  17. 𝑎𝑙𝑔 \mathit{alg}
  18. 𝑎𝑙𝑔 A \mathit{alg}_{A}
  19. 𝑎𝑙𝑔 \mathit{alg}
  20. 𝑎𝑙𝑔 \mathit{alg}
  21. A A
  22. B B
  23. 𝑎𝑙𝑔 A ( B ) \mathit{alg}_{A}(B)
  24. B B
  25. 𝑎𝑙𝑔 A \mathit{alg}_{A}
  26. A A
  27. i i
  28. i i
  29. A A
  30. 𝑎𝑙𝑔 ( A , B ) \mathit{alg}(A,B)
  31. B B
  32. 𝑎𝑙𝑔 ( A 1 , B 1 ) \mathit{alg}(A_{1},B_{1})
  33. 𝑎𝑙𝑔 ( A 2 , B 2 ) \mathit{alg}(A_{2},B_{2})
  34. 𝑎𝑙𝑔 ( A 1 , B 3 ) \mathit{alg}(A_{1},B_{3})
  35. A A^{\prime}
  36. A A
  37. A A
  38. 𝑎𝑙𝑔 \mathit{alg}^{\prime}
  39. 𝑎𝑙𝑔 ( A , B ) \mathit{alg}(A,B)
  40. 𝑎𝑙𝑔 ( A , B ) \mathit{alg}^{\prime}(A^{\prime},B)

Runcinated_5-cell.html

  1. ± ( 5 2 , 1 6 , 1 3 , ± 1 ) \pm\left(\sqrt{\frac{5}{2}},\ \frac{1}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\ \pm 1\right)
  2. ± ( 5 2 , 1 6 , - 2 3 , 0 ) \pm\left(\sqrt{\frac{5}{2}},\ \frac{1}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0\right)
  3. ± ( 5 2 , - 3 2 , 0 , 0 ) \pm\left(\sqrt{\frac{5}{2}},\ -\sqrt{\frac{3}{2}},\ 0,\ 0\right)
  4. ± ( 0 , 2 2 3 , 1 3 , ± 1 ) \pm\left(0,\ 2\sqrt{\frac{2}{3}},\ \frac{1}{\sqrt{3}},\ \pm 1\right)
  5. ± ( 0 , 2 2 3 , - 2 3 , 0 ) \pm\left(0,\ 2\sqrt{\frac{2}{3}},\ \frac{-2}{\sqrt{3}},\ 0\right)
  6. ( 0 , 0 , ± 3 , ± 1 ) \left(0,\ 0,\ \pm\sqrt{3},\ \pm 1\right)
  7. ( 0 , 0 , 0 , ± 2 ) \left(0,\ 0,\ 0,\ \pm 2\right)
  8. ( 7 10 , 3 2 , ± 3 , ± 1 ) \left(\frac{7}{\sqrt{10}},\ \sqrt{\frac{3}{2}},\ \pm\sqrt{3},\ \pm 1\right)
  9. ( 7 10 , 3 2 , 0 , ± 2 ) \left(\frac{7}{\sqrt{10}},\ \sqrt{\frac{3}{2}},\ 0,\ \pm 2\right)
  10. ( 7 10 , - 1 6 , 2 3 , ± 2 ) \left(\frac{7}{\sqrt{10}},\ \frac{-1}{\sqrt{6}},\ \frac{2}{\sqrt{3}},\ \pm 2\right)
  11. ( 7 10 , - 1 6 , - 4 3 , 0 ) \left(\frac{7}{\sqrt{10}},\ \frac{-1}{\sqrt{6}},\ \frac{-4}{\sqrt{3}},\ 0\right)
  12. ( 7 10 , - 5 6 , 1 3 , ± 1 ) \left(\frac{7}{\sqrt{10}},\ \frac{-5}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\ \pm 1\right)
  13. ( 7 10 , - 5 6 , - 2 3 , 0 ) \left(\frac{7}{\sqrt{10}},\ \frac{-5}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0\right)
  14. ( 2 5 , ± 6 , ± 3 , ± 1 ) \left(\sqrt{\frac{2}{5}},\ \pm\sqrt{6},\ \pm\sqrt{3},\ \pm 1\right)
  15. ( 2 5 , ± 6 , 0 , ± 2 ) \left(\sqrt{\frac{2}{5}},\ \pm\sqrt{6},\ 0,\ \pm 2\right)
  16. ( 2 5 , 2 3 , 5 3 , ± 1 ) \left(\sqrt{\frac{2}{5}},\ \sqrt{\frac{2}{3}},\ \frac{5}{\sqrt{3}},\ \pm 1\right)
  17. ( 2 5 , 2 3 , - 1 3 , ± 3 ) \left(\sqrt{\frac{2}{5}},\ \sqrt{\frac{2}{3}},\ \frac{-1}{\sqrt{3}},\ \pm 3\right)
  18. ( 2 5 , 2 3 , - 4 3 , ± 2 ) \left(\sqrt{\frac{2}{5}},\ \sqrt{\frac{2}{3}},\ \frac{-4}{\sqrt{3}},\ \pm 2\right)
  19. ( 2 5 , - 2 3 , 4 3 , ± 2 ) \left(\sqrt{\frac{2}{5}},\ -\sqrt{\frac{2}{3}},\ \frac{4}{\sqrt{3}},\ \pm 2\right)
  20. ( 2 5 , - 2 3 , 1 3 , ± 3 ) \left(\sqrt{\frac{2}{5}},\ -\sqrt{\frac{2}{3}},\ \frac{1}{\sqrt{3}},\ \pm 3\right)
  21. ( 2 5 , - 2 3 , - 5 3 , ± 1 ) \left(\sqrt{\frac{2}{5}},\ -\sqrt{\frac{2}{3}},\ \frac{-5}{\sqrt{3}},\ \pm 1\right)
  22. ( - 3 10 , 5 6 , 2 3 , ± 2 ) \left(\frac{-3}{\sqrt{10}},\ \frac{5}{\sqrt{6}},\ \frac{2}{\sqrt{3}},\ \pm 2\right)
  23. ( - 3 10 , 5 6 , - 4 3 , 0 ) \left(\frac{-3}{\sqrt{10}},\ \frac{5}{\sqrt{6}},\ \frac{-4}{\sqrt{3}},\ 0\right)
  24. ( - 3 10 , 1 6 , 4 3 , ± 2 ) \left(\frac{-3}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{4}{\sqrt{3}},\ \pm 2\right)
  25. ( - 3 10 , 1 6 , 1 3 , ± 3 ) \left(\frac{-3}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\ \pm 3\right)
  26. ( - 3 10 , 1 6 , - 5 3 , ± 1 ) \left(\frac{-3}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{-5}{\sqrt{3}},\ \pm 1\right)
  27. ( - 3 10 , - 7 6 , 2 3 , 0 ) \left(\frac{-3}{\sqrt{10}},\ \frac{-7}{\sqrt{6}},\ \frac{2}{\sqrt{3}},\ 0\right)
  28. ( - 3 10 , - 7 6 , - 1 3 , ± 1 ) \left(\frac{-3}{\sqrt{10}},\ \frac{-7}{\sqrt{6}},\ \frac{-1}{\sqrt{3}},\ \pm 1\right)
  29. ( - 4 2 5 , 2 2 3 , 1 3 , ± 1 ) \left(-4\sqrt{\frac{2}{5}},\ 2\sqrt{\frac{2}{3}},\ \frac{1}{\sqrt{3}},\ \pm 1\right)
  30. ( - 4 2 5 , 2 2 3 , - 2 3 , 0 ) \left(-4\sqrt{\frac{2}{5}},\ 2\sqrt{\frac{2}{3}},\ \frac{-2}{\sqrt{3}},\ 0\right)
  31. ( - 4 2 5 , 0 , ± 3 , ± 1 ) \left(-4\sqrt{\frac{2}{5}},\ 0,\ \pm\sqrt{3},\ \pm 1\right)
  32. ( - 4 2 5 , 0 , 0 , ± 2 ) \left(-4\sqrt{\frac{2}{5}},\ 0,\ 0,\ \pm 2\right)
  33. ( - 4 2 5 , - 2 2 3 , 2 3 , 0 ) \left(-4\sqrt{\frac{2}{5}},\ -2\sqrt{\frac{2}{3}},\ \frac{2}{\sqrt{3}},\ 0\right)
  34. ( - 4 2 5 , - 2 2 3 , - 1 3 , 1 ) \left(-4\sqrt{\frac{2}{5}},\ -2\sqrt{\frac{2}{3}},\ \frac{-1}{\sqrt{3}},\ 1\right)
  35. ( ± 10 , ± 6 , ± 3 , ± 1 ) \left(\pm\sqrt{10},\ \pm\sqrt{6},\ \pm\sqrt{3},\ \pm 1\right)
  36. ( ± 10 , ± 6 , 0 , ± 2 ) \left(\pm\sqrt{10},\ \pm\sqrt{6},\ 0,\ \pm 2\right)
  37. ± ( ± 10 , 2 3 , 5 3 , ± 1 ) \pm\left(\pm\sqrt{10},\ \sqrt{\frac{2}{3}},\ \frac{5}{\sqrt{3}},\ \pm 1\right)
  38. ± ( ± 10 , 2 3 , - 1 3 , ± 3 ) \pm\left(\pm\sqrt{10},\ \sqrt{\frac{2}{3}},\ \frac{-1}{\sqrt{3}},\ \pm 3\right)
  39. ± ( ± 10 , 2 3 , - 4 3 , ± 2 ) \pm\left(\pm\sqrt{10},\ \sqrt{\frac{2}{3}},\ \frac{-4}{\sqrt{3}},\ \pm 2\right)
  40. ( ± 5 2 , 3 3 2 , ± 3 , ± 1 ) \left(\pm\sqrt{\frac{5}{2}},\ 3\sqrt{\frac{3}{2}},\ \pm\sqrt{3},\ \pm 1\right)
  41. ( ± 5 2 , 3 3 2 , 0 , ± 2 ) \left(\pm\sqrt{\frac{5}{2}},\ 3\sqrt{\frac{3}{2}},\ 0,\ \pm 2\right)
  42. ± ( 5 2 , 1 6 , 7 3 , ± 1 ) \pm\left(\sqrt{\frac{5}{2}},\ \frac{1}{\sqrt{6}},\ \frac{7}{\sqrt{3}},\ \pm 1\right)
  43. ± ( 5 2 , 1 6 , - 2 3 , ± 4 ) \pm\left(\sqrt{\frac{5}{2}},\ \frac{1}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ \pm 4\right)
  44. ± ( 5 2 , 1 6 , - 5 3 , ± 3 ) \pm\left(\sqrt{\frac{5}{2}},\ \frac{1}{\sqrt{6}},\ \frac{-5}{\sqrt{3}},\ \pm 3\right)
  45. ± ( 5 2 , - 3 2 , ± 2 3 , ± 2 ) \pm\left(\sqrt{\frac{5}{2}},\ -\sqrt{\frac{3}{2}},\ \pm 2\sqrt{3},\ \pm 2\right)
  46. ± ( 5 2 , - 3 2 , 0 , ± 4 ) \pm\left(\sqrt{\frac{5}{2}},\ -\sqrt{\frac{3}{2}},\ 0,\ \pm 4\right)
  47. ± ( 5 2 , - 7 6 , 5 3 , ± 1 ) \pm\left(\sqrt{\frac{5}{2}},\ \frac{-7}{\sqrt{6}},\ \frac{5}{\sqrt{3}},\ \pm 1\right)
  48. ± ( 5 2 , - 7 6 , - 1 3 , ± 3 ) \pm\left(\sqrt{\frac{5}{2}},\ \frac{-7}{\sqrt{6}},\ \frac{-1}{\sqrt{3}},\ \pm 3\right)
  49. ± ( 5 2 , - 7 6 , - 4 3 , ± 2 ) \pm\left(\sqrt{\frac{5}{2}},\ \frac{-7}{\sqrt{6}},\ \frac{-4}{\sqrt{3}},\ \pm 2\right)
  50. ± ( 0 , 4 2 3 , 5 3 , ± 1 ) \pm\left(0,\ 4\sqrt{\frac{2}{3}},\ \frac{5}{\sqrt{3}},\ \pm 1\right)
  51. ± ( 0 , 4 2 3 , - 1 3 , ± 3 ) \pm\left(0,\ 4\sqrt{\frac{2}{3}},\ \frac{-1}{\sqrt{3}},\ \pm 3\right)
  52. ± ( 0 , 4 2 3 , - 4 3 , ± 2 ) \pm\left(0,\ 4\sqrt{\frac{2}{3}},\ \frac{-4}{\sqrt{3}},\ \pm 2\right)
  53. ± ( 0 , 2 2 3 , 7 3 , ± 1 ) \pm\left(0,\ 2\sqrt{\frac{2}{3}},\ \frac{7}{\sqrt{3}},\ \pm 1\right)
  54. ± ( 0 , 2 2 3 , - 2 3 , ± 4 ) \pm\left(0,\ 2\sqrt{\frac{2}{3}},\ \frac{-2}{\sqrt{3}},\ \pm 4\right)
  55. ± ( 0 , 2 2 3 , - 5 3 , ± 3 ) \pm\left(0,\ 2\sqrt{\frac{2}{3}},\ \frac{-5}{\sqrt{3}},\ \pm 3\right)

Runcinated_tesseracts.html

  1. ( ± 1 , ± 1 , ± 1 , ± ( 1 + 2 ) ) \left(\pm 1,\ \pm 1,\ \pm 1,\ \pm(1+\sqrt{2})\right)
  2. ( ± 1 , ± ( 1 + 2 ) , ± ( 1 + 2 ) , ± ( 1 + 2 2 ) ) \left(\pm 1,\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+2\sqrt{2})\right)
  3. ( ± 1 , ± 1 , ± ( 1 + 2 ) , ± ( 1 + 2 2 ) ) \left(\pm 1,\ \pm 1,\ \pm(1+\sqrt{2}),\ \pm(1+2\sqrt{2})\right)
  4. ( 1 , 1 + 2 , 1 + 2 2 , 1 + 3 2 ) \left(1,\ 1+\sqrt{2},\ 1+2\sqrt{2},\ 1+3\sqrt{2}\right)

Runtime_verification.html

  1. p a r a m e t e r s : φ \forall parameters:\varphi
  2. φ \varphi
  3. φ \varphi
  4. Iterator i i . next ( ) ( i . hasNext ( ) = = t r u e ) \forall~{}\,\text{Iterator}~{}i\quad i.\,\text{next}()~{}\rightarrow~{}\odot(i% .\,\text{hasNext}()==true)
  5. ϵ \epsilon
  6. t 1 t_{1}
  7. t 2 t_{2}
  8. t 3 t_{3}
  9. l 1 l_{1}
  10. l 2 l_{2}
  11. l 1 l_{1}
  12. l 2 l_{2}
  13. l 1 l_{1}
  14. l 2 l_{2}
  15. l 1 l_{1}
  16. l 2 l_{2}

S_plane.html

  1. e - s t e^{-st}
  2. 0
  3. \infty
  4. s = σ + j ω s=\sigma+j\omega
  5. 0 f ( t ) e - s t d t | s \int_{0}^{\infty}f(t)e^{-st}\,dt\;|\;s\;\in\mathbb{C}

Safety_stock.html

  1. s s = z α × E ( L ) σ D 2 + ( E ( D ) ) 2 σ L 2 ss=z_{\alpha}\times\sqrt{E(L)\sigma^{2}_{D}+(E(D))^{2}\sigma^{2}_{L}}
  2. R O P = E ( L ) . E ( D ) + s s ROP=E(L).E(D)+ss
  3. α \alpha
  4. z α z_{\alpha}
  5. α \alpha
  6. E ( L ) E(L)
  7. σ L \sigma_{L}
  8. E ( D ) E(D)
  9. L L
  10. σ D \sigma_{D}
  11. E ( L ) E ( D ) E(L)E(D)
  12. s s ss
  13. E ( L ) \sqrt{E(L)}
  14. σ L = 0 \sigma_{L}=0
  15. R O P = D L + z α σ D L ROP=DL+z_{\alpha}\sigma_{D}\sqrt{L}

Salt_metathesis_reaction.html

  1. \color O r a n g e A g + 1 \color B l u e N O 3 - 1 + \color G r e e n H + 1 \color M a g e n t a C l - 1 \color G r e e n H + 1 \color B l u e N O 3 - 1 + \color O r a n g e A g + 1 \color M a g e n t a C l - 1 \overset{+1}{{\color{Orange}Ag}}\overset{-1}{{\color{Blue}NO_{3}}}+\overset{+1% }{{\color{Green}H}}\overset{-1}{{\color{Magenta}Cl}}\rightarrow\overset{+1}{{% \color{Green}H}}\overset{-1}{{\color{Blue}NO_{3}}}+\overset{+1}{{\color{Orange% }Ag}}\overset{-1}{{\color{Magenta}Cl}}\downarrow

Sample_exclusion_dimension.html

  1. Y X Y\subseteq X

Sampling_fraction.html

  1. f = n N , f=\frac{n}{N},

Samuelson_condition.html

  1. i = 1 n MRS i = MRT \sum_{i=1}^{n}\,\text{MRS}_{i}=\,\text{MRT}
  2. i = 1 n MB i = MC \sum_{i=1}^{n}\,\text{MB}_{i}=\,\text{MC}
  3. MB i \,\text{MB}_{i}
  4. MB i \,\text{MB}_{i}

Saturable_absorption.html

  1. A A
  2. I I
  3. ( 1 ) A = α 1 + I / I 0 (1)~{}~{}~{}~{}A=\frac{\alpha}{1+I/I_{0}}
  4. α \alpha
  5. I 0 I_{0}
  6. N N
  7. σ \sigma
  8. τ \tau
  9. ( 2 ) d I d z = - A I (2)~{}~{}~{}~{}\frac{\mathrm{d}I}{\mathrm{d}z}=-AI
  10. z z
  11. ( 3 ) d I d z = - α I 1 + I / I 0 (3)~{}~{}~{}~{}\frac{\mathrm{d}I}{\mathrm{d}z}=-\frac{\alpha~{}I}{1+I/I_{0}}
  12. u = I / I 0 u=I/I_{0}
  13. t = α z t=\alpha z
  14. ( 4 ) d u d t = - u 1 + u (4)~{}~{}~{}~{}\frac{\mathrm{d}u}{\mathrm{d}t}=\frac{-u}{1+u}
  15. ω \omega
  16. ( 5 ) u = ω ( - t ) (5)~{}~{}~{}~{}u=\omega(-t)
  17. u = V ( - e t ) u=V\big(-\mathrm{e}^{t}\big)
  18. ( 6 ) - e t V ( - e t ) = - V ( - e t ) 1 + V ( - e t ) (6)~{}~{}~{}~{}-\mathrm{e}^{t}V^{\prime}\big(-\mathrm{e}^{t}\big)=-\frac{V\big% (-\mathrm{e}^{t}\big)}{1+V\big(-\mathrm{e}^{t}\big)}
  19. p = - e t p=-\mathrm{e}^{t}
  20. ( 7 ) V ( p ) = V ( p ) p ( 1 + V ( p ) ) (7)~{}~{}~{}~{}V^{\prime}(p)=\frac{V(p)}{p\cdot(1+V(p))}
  21. ( 8 ) V ( p ) = W ( p - p 0 ) (8)~{}~{}~{}~{}V(p)=W(p-p_{0})
  22. p 0 p_{0}
  23. V ( p 0 ) = 0 V(p_{0})=0
  24. ( 9 ) F = 0 t I ( t ) d t (9)~{}~{}~{}~{}F=\int_{0}^{t}I(t)\mathrm{d}t
  25. t t
  26. t < 0 t<0
  27. ( 10 ) A = α 1 + F / F 0 (10)~{}~{}~{}~{}A=\frac{\alpha}{1+F/F_{0}}
  28. F 0 F_{0}

Saturated_calomel_electrode.html

  1. Cl - ( 4 M ) | Hg 2 Cl 2 ( s ) | Hg ( l ) | Pt \,\text{Cl}^{-}\big(4M\big)\big|\,\text{Hg}_{2}\,\text{Cl}_{2}\big(\,\text{s}% \big)\big|\,\text{Hg}\big(\,\text{l}\big)\big|\,\text{Pt}
  2. Hg 2 2 + + 2 e - 2 Hg(l) \,\text{Hg}_{2}^{2+}+2\,\text{e}^{-}\rightleftarrows 2\,\text{Hg(l)}
  3. E = E Hg 2 2 + / Hg 0 - R T 2 F ln 1 a Hg 2 2 + E=E^{0}_{\,\text{Hg}_{2}^{2+}/\,\text{Hg}}-\frac{RT}{2F}\ln\frac{1}{a_{\,\text% {Hg}_{2}^{2+}}}
  4. Hg 2 2 + + 2 Cl - Hg 2 Cl 2 (s) , K sp = a Hg 2 2 + a Cl - 2 \,\text{Hg}_{2}^{2+}+2\,\text{Cl}^{-}\rightleftarrows\,\text{Hg}_{2}\,\text{Cl% }_{2}\,\text{(s)},\qquad K_{\,\text{sp}}=a_{\,\text{Hg}_{2}^{2+}}a_{\,\text{Cl% }^{-}}^{2}
  5. E = E Hg 2 2 + / Hg 0 + R T 2 F ln K sp - R T 2 F ln a Cl - 2 E=E^{0}_{\,\text{Hg}_{2}^{2+}/\,\text{Hg}}+\frac{RT}{2F}\ln K_{\,\text{sp}}-% \frac{RT}{2F}\ln a^{2}_{\,\text{Cl}^{-}}

Saturated_measure.html

  1. E E
  2. A A
  3. E A E\cap A
  4. σ \sigma

Saturation_(graph_theory).html

  1. G ( V , E ) G(V,E)
  2. M M
  3. G G
  4. v V ( G ) v\in V(G)
  5. M M
  6. M M
  7. v v
  8. v V ( G ) v\in V(G)
  9. M M
  10. M M
  11. v v

Saturation_(magnetic).html

  1. μ = B / H \mu=B/H
  2. μ r = μ / μ 0 \mu_{r}=\mu/\mu_{0}
  3. μ 0 \mu_{0}

Sauter_mean_diameter.html

  1. d s = A p π d_{s}=\sqrt{\frac{A_{p}}{\pi}}
  2. d v = ( 6 V p π ) 1 / 3 , d_{v}=\left(\frac{6V_{p}}{\pi}\right)^{1/3},
  3. S D = D [ 3 , 2 ] = d 32 = d v 3 d s 2 . SD=D[3,2]=d_{32}=\frac{d_{v}^{3}}{d_{s}^{2}}.
  4. V p A p = 4 3 π ( d v / 2 ) 3 4 π ( d s / 2 ) 2 = ( d v / 2 ) 3 3 ( d s / 2 ) 2 = d 32 6 \frac{V_{p}}{A_{p}}=\frac{\frac{4}{3}\pi(d_{v}/2)^{3}}{4\pi(d_{s}/2)^{2}}=% \frac{(d_{v}/2)^{3}}{3(d_{s}/2)^{2}}=\frac{d_{32}}{6}
  5. d 32 = 6 V p A p . d_{32}=6\frac{V_{p}}{A_{p}}.
  6. D s = 1 i f i d i D_{s}=\frac{1}{\sum_{i}\frac{f_{i}}{d_{i}}}
  7. f i f_{i}
  8. d i d_{i}

Scalar_field_solution.html

  1. g a b g_{ab}
  2. R a b c d R_{abcd}
  3. G a b G^{ab}
  4. ψ \psi
  5. g a b ψ ; a b = 0 g^{ab}\psi_{;ab}=0
  6. G a b = 8 π ( ψ ; a ψ ; b - 1 2 ψ ; m ψ ; m g a b ) G^{ab}=8\pi\left(\psi^{;a}\psi^{;b}-\frac{1}{2}\psi_{;m}\psi^{;m}g^{ab}\right)
  7. L = - g m n ψ ; m ψ ; n L=-g^{mn}\,\psi_{;m}\,\psi_{;n}
  8. δ L δ ψ = 0 \frac{\delta L}{\delta\psi}=0
  9. δ L δ g a b = 0 \frac{\delta L}{\delta g^{ab}}=0
  10. e 0 , e 1 , e 2 , e 3 \vec{e}_{0},\;\vec{e}_{1},\;\vec{e}_{2},\;\vec{e}_{3}
  11. G a ^ b ^ = 8 π σ [ - 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] G^{\hat{a}\hat{b}}=8\pi\sigma\,\left[\begin{matrix}-1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{matrix}\right]
  12. σ \sigma
  13. χ ( λ ) = ( λ + 8 π σ ) 3 ( λ - 8 π σ ) \chi(\lambda)=(\lambda+8\pi\sigma)^{3}\,(\lambda-8\pi\sigma)
  14. a 2 = 0 , a 1 3 + 4 a 3 = 0 , a 1 4 + 16 a 4 = 0 a_{2}=0,\;\;a_{1}^{3}+4a_{3}=0,\;\;a_{1}^{4}+16a_{4}=0
  15. t 2 = t 1 2 , t 3 = t 1 3 / 4 , t 4 = t 1 4 / 4 t_{2}=t_{1}^{2},\;t_{3}=t_{1}^{3}/4,\;t_{4}=t_{1}^{4}/4
  16. G a a = - R {G^{a}}_{a}=-R
  17. G a b G b a = R 2 {G^{a}}_{b}\,{G^{b}}_{a}=R^{2}
  18. G a b G b c G c a = R 3 / 4 {G^{a}}_{b}\,{G^{b}}_{c}\,{G^{c}}_{a}=R^{3}/4
  19. G a b G b c G c d G d a = R 4 / 4 {G^{a}}_{b}\,{G^{b}}_{c}\,{G^{c}}_{d}\,{G^{d}}_{a}=R^{4}/4

Scalar_theories_of_gravitation.html

  1. Φ \Phi
  2. Δ Φ = 4 π G ρ \Delta\Phi=4\pi G\rho
  3. ρ \rho
  4. F = m 1 m 2 G / r 2 F=m_{1}m_{2}G/r^{2}
  5. = t 2 - 2 \square=\partial_{t}^{2}-\nabla^{2}
  6. Φ = 4 π G ρ \square\Phi=4\pi G\rho
  7. Φ Φ = - 4 π G T \Phi\square\Phi=-4\pi GT
  8. T T
  9. g μ ν = A η μ ν g_{\mu\nu}=A\eta_{\mu\nu}
  10. A A
  11. T g μ ν = 1 4 π G [ μ ϕ ν ϕ - 1 2 η μ ν λ ϕ λ ϕ ] T^{\mu\nu}_{g}=\frac{1}{4\pi G}\left[\partial^{\mu}\phi\,\partial^{\nu}\phi\,-% \frac{1}{2}\eta^{\mu\nu}\partial_{\lambda}\phi\,\partial^{\lambda}\phi\right]
  12. T m μ ν = ρ ϕ u μ u ν T^{\mu\nu}_{m}=\rho\phi u^{\mu}u^{\nu}
  13. u μ u^{\mu}
  14. A μ A^{\mu}

Scatchard_equation.html

  1. L L
  2. P P
  3. r [ L ] = n K a - r K a \frac{r}{[L]}=nK_{a}-rK_{a}
  4. r = [ L ] b o u n d [ P ] r=\frac{[L]_{bound}}{[P]}
  5. K a = [ L P ] [ L ] [ P ] K_{a}=\frac{[LP]}{[L][P]}

Scattering_channel.html

  1. t ± t\to\pm\infty

Scattering_theory.html

  1. I I
  2. d I d x = - Q I \frac{dI}{dx}=-QI\,\!
  3. I = I o e - Q Δ x = I o e - Δ x λ = I o e - σ ( η Δ x ) = I o e - ρ Δ x τ , I=I_{o}e^{-Q\Delta x}=I_{o}e^{-\frac{\Delta x}{\lambda}}=I_{o}e^{-\sigma(\eta% \Delta x)}=I_{o}e^{-\frac{\rho\Delta x}{\tau}},

Schmidt_number.html

  1. Sc = ν D = μ ρ D = viscous diffusion rate molecular (mass) diffusion rate \mathrm{Sc}=\frac{\nu}{D}=\frac{\mu}{\rho D}=\frac{\mbox{viscous diffusion % rate}~{}}{\mbox{molecular (mass) diffusion rate}~{}}
  2. ν \nu
  3. μ {\mu}
  4. ρ {\rho}\,
  5. D D
  6. μ {\mu}
  7. ρ \rho
  8. Sc t = ν t K \mathrm{Sc}_{\mathrm{t}}=\frac{\nu_{\mathrm{t}}}{K}
  9. ν t \nu_{\mathrm{t}}
  10. K K
  11. Sc = | Q | p ¯ V s w \mathrm{Sc}=\frac{\sum{\left|{Q}\right|}}{\bar{p}V_{sw}}
  12. Sc \mathrm{Sc}
  13. Q Q
  14. p ¯ \bar{p}
  15. V s w V_{sw}

Schnyder's_theorem.html

  1. P ( G ) P(G)
  2. G G
  3. V V
  4. E E
  5. V E V∪E
  6. x x
  7. y y
  8. x x
  9. y y
  10. G G
  11. P ( G ) P(G)
  12. 1 + d 1+d
  13. d d
  14. Θ ( log log n ) \Theta(\log\log n)

Schottky_effect.html

  1. J ( F , T , W ) = A G T 2 e - ( W - Δ W ) k T J(F,T,W)=A_{\mathrm{G}}T^{2}e^{-(W-\Delta W)\over kT}
  2. Δ W = e 3 F 4 π ϵ 0 , \Delta W=\sqrt{e^{3}F\over 4\pi\epsilon_{0}},

Schrödinger_functional.html

  1. ϕ \phi
  2. H H
  3. 𝒮 [ ϕ 2 , t 2 ; ϕ 1 , t 1 ] = ϕ 2 | e - i H ( t 2 - t 1 ) / | ϕ 1 . \mathcal{S}[\phi_{2},t_{2};\phi_{1},t_{1}]=\langle\,\phi_{2}\,|e^{-iH(t_{2}-t_% {1})/\hbar}|\,\phi_{1}\,\rangle.
  4. Ψ [ ϕ 2 , t 2 ] = 𝒟 ϕ 1 𝒮 [ ϕ 2 , t 2 ; ϕ 1 , t 1 ] Ψ [ ϕ 1 , t 1 ] \Psi[\phi_{2},t_{2}]=\int\!\mathcal{D}\phi_{1}\,\,\mathcal{S}[\phi_{2},t_{2};% \phi_{1},t_{1}]\Psi[\phi_{1},t_{1}]

Schur's_inequality.html

  1. x t ( x - y ) ( x - z ) + y t ( y - z ) ( y - x ) + z t ( z - x ) ( z - y ) 0 x^{t}(x-y)(x-z)+y^{t}(y-z)(y-x)+z^{t}(z-x)(z-y)\geq 0
  2. t = 1 t=1
  3. x 3 + y 3 + z 3 + 3 x y z x y ( x + y ) + x z ( x + z ) + y z ( y + z ) x^{3}+y^{3}+z^{3}+3xyz\geq xy(x+y)+xz(x+z)+yz(y+z)
  4. x , y , z x,y,z
  5. x y z x\geq y\geq z
  6. ( x - y ) [ x t ( x - z ) - y t ( y - z ) ] + z t ( x - z ) ( y - z ) 0 (x-y)[x^{t}(x-z)-y^{t}(y-z)]+z^{t}(x-z)(y-z)\geq 0\,
  7. a ( x - y ) ( x - z ) + b ( y - z ) ( y - x ) + c ( z - x ) ( z - y ) 0. a(x-y)(x-z)+b(y-z)(y-x)+c(z-x)(z-y)\geq 0.
  8. a , b , c , x , y , z a,b,c,x,y,z\in\mathbb{R}
  9. a b c a\geq b\geq c
  10. x y z x\geq y\geq z
  11. z y x z\geq y\geq x
  12. k + k\in\mathbb{Z}^{+}
  13. f : 0 + f:\mathbb{R}\rightarrow\mathbb{R}_{0}^{+}
  14. f ( x ) ( a - b ) k ( a - c ) k + f ( y ) ( b - a ) k ( b - c ) k + f ( z ) ( c - a ) k ( c - b ) k 0. {f(x)(a-b)^{k}(a-c)^{k}+f(y)(b-a)^{k}(b-c)^{k}+f(z)(c-a)^{k}(c-b)^{k}\geq 0}.\,

Schur_polynomial.html

  1. a ( λ 1 + n - 1 , λ 2 + n - 2 , , λ n ) ( x 1 , x 2 , , x n ) = det [ x 1 λ 1 + n - 1 x 2 λ 1 + n - 1 x n λ 1 + n - 1 x 1 λ 2 + n - 2 x 2 λ 2 + n - 2 x n λ 2 + n - 2 x 1 λ n x 2 λ n x n λ n ] a_{(\lambda_{1}+n-1,\lambda_{2}+n-2,\dots,\lambda_{n})}(x_{1},x_{2},\dots,x_{n% })=\det\left[\begin{matrix}x_{1}^{\lambda_{1}+n-1}&x_{2}^{\lambda_{1}+n-1}&% \dots&x_{n}^{\lambda_{1}+n-1}\\ x_{1}^{\lambda_{2}+n-2}&x_{2}^{\lambda_{2}+n-2}&\dots&x_{n}^{\lambda_{2}+n-2}% \\ \vdots&\vdots&\ddots&\vdots\\ x_{1}^{\lambda_{n}}&x_{2}^{\lambda_{n}}&\dots&x_{n}^{\lambda_{n}}\end{matrix}\right]
  2. a ( n - 1 , n - 2 , , 0 ) ( x 1 , x 2 , , x n ) = det [ x 1 n - 1 x 2 n - 1 x n n - 1 x 1 n - 2 x 2 n - 2 x n n - 2 1 1 1 ] = 1 j < k n ( x j - x k ) . a_{(n-1,n-2,\dots,0)}(x_{1},x_{2},\dots,x_{n})=\det\left[\begin{matrix}x_{1}^{% n-1}&x_{2}^{n-1}&\dots&x_{n}^{n-1}\\ x_{1}^{n-2}&x_{2}^{n-2}&\dots&x_{n}^{n-2}\\ \vdots&\vdots&\ddots&\vdots\\ 1&1&\dots&1\end{matrix}\right]=\prod_{1\leq j<k\leq n}(x_{j}-x_{k}).
  3. s λ ( x 1 , x 2 , , x n ) = a ( λ 1 + n - 1 , λ 2 + n - 2 , , λ n + 0 ) ( x 1 , x 2 , , x n ) a ( n - 1 , n - 2 , , 0 ) ( x 1 , x 2 , , x n ) . s_{\lambda}(x_{1},x_{2},\dots,x_{n})=\frac{a_{(\lambda_{1}+n-1,\lambda_{2}+n-2% ,\dots,\lambda_{n}+0)}(x_{1},x_{2},\dots,x_{n})}{a_{(n-1,n-2,\dots,0)}(x_{1},x% _{2},\dots,x_{n})}.
  4. d d
  5. n n
  6. d d
  7. n n
  8. s λ ( x 1 , x 2 , , x n ) = T x T = T x 1 t 1 x n t n s_{\lambda}(x_{1},x_{2},\ldots,x_{n})=\sum_{T}x^{T}=\sum_{T}x_{1}^{t_{1}}% \cdots x_{n}^{t_{n}}
  9. T T
  10. λ λ
  11. T T
  12. i i
  13. T T
  14. s λ = μ K λ μ m μ . s_{\lambda}=\sum_{\mu}K_{\lambda\mu}m_{\mu}.
  15. s λ = det i j h λ i + j - i , 1 i , j n = | h λ 1 h λ 1 + 1 h λ 1 + n - 1 h λ 2 - 1 h λ 2 h λ 2 + n - 2 h λ n - n + 1 h λ n - n + 2 h λ n | , s_{\lambda}=\det_{ij}h_{\lambda_{i}+j-i},1\leq i,j\leq n=\left|\begin{matrix}h% _{\lambda_{1}}&h_{\lambda_{1}+1}&\dots&h_{\lambda_{1}+n-1}\\ h_{\lambda_{2}-1}&h_{\lambda_{2}}&\dots&h_{\lambda_{2}+n-2}\\ \vdots&\vdots&\ddots&\vdots\\ h_{\lambda_{n}-n+1}&h_{\lambda_{n}-n+2}&\dots&h_{\lambda_{n}}\end{matrix}% \right|,
  16. s λ = det i j e λ i + j - i , 1 i , j l = | e λ 1 e λ 1 + 1 e λ 1 + l - 1 e λ 2 - 1 e λ 2 e λ 2 + l - 2 e λ l - l + 1 e λ l - l + 2 e λ l | , s_{\lambda}=\det_{ij}e_{\lambda^{\prime}_{i}+j-i},1\leq i,j\leq l=\left|\begin% {matrix}e_{\lambda^{\prime}_{1}}&e_{\lambda^{\prime}_{1}+1}&\dots&e_{\lambda^{% \prime}_{1}+l-1}\\ e_{\lambda^{\prime}_{2}-1}&e_{\lambda^{\prime}_{2}}&\dots&e_{\lambda^{\prime}_% {2}+l-2}\\ \vdots&\vdots&\ddots&\vdots\\ e_{\lambda^{\prime}_{l}-l+1}&e_{\lambda^{\prime}_{l}-l+2}&\dots&e_{\lambda^{% \prime}_{l}}\end{matrix}\right|,
  17. λ λ
  18. ( a 1 , a r | b 1 , b r ) (a_{1},...a_{r}|b_{1},...b_{r})
  19. i i ii
  20. s ( a 1 , a r | b 1 , b r ) = det ( s ( a i | b j ) ) s_{(a_{1},...a_{r}|b_{1},...b_{r})}=\det(s_{(a_{i}|b_{j})})
  21. p r s λ = μ ( - 1 ) h t ( μ / λ ) + 1 s μ p_{r}\cdot s_{\lambda}=\sum_{\mu}(-1)^{ht(\mu/\lambda)+1}s_{\mu}
  22. λ , μ , ν \lambda,\mu,\nu
  23. λ \lambda
  24. μ \mu
  25. ν \nu
  26. c λ , μ ν c_{\lambda,\mu}^{\nu}
  27. s λ s μ = ν c λ , μ ν s ν . s_{\lambda}s_{\mu}=\sum_{\nu}c_{\lambda,\mu}^{\nu}s_{\nu}.
  28. c λ , μ ν c_{\lambda,\mu}^{\nu}
  29. ν / λ \nu/\lambda
  30. μ \mu
  31. h r s λ h_{r}s_{\lambda}
  32. ( 1 , 1 , , 1 ) (1,1,...,1)
  33. λ λ
  34. 1 , 2 , , n 1,2,...,n
  35. s λ ( 1 , 1 , , 1 ) = 1 i < j n λ i - λ j + j - i j - i . s_{\lambda}(1,1,\dots,1)=\prod_{1\leq i<j\leq n}\frac{\lambda_{i}-\lambda_{j}+% j-i}{j-i}.
  36. λ λ
  37. n n
  38. d d
  39. s ( 2 , 1 , 1 ) ( x 1 , x 2 , x 3 ) = 1 Δ det [ x 1 4 x 2 4 x 3 4 x 1 2 x 2 2 x 3 2 x 1 x 2 x 3 ] = x 1 x 2 x 3 ( x 1 + x 2 + x 3 ) s_{(2,1,1)}(x_{1},x_{2},x_{3})=\frac{1}{\Delta}\;\det\left[\begin{matrix}x_{1}% ^{4}&x_{2}^{4}&x_{3}^{4}\\ x_{1}^{2}&x_{2}^{2}&x_{3}^{2}\\ x_{1}&x_{2}&x_{3}\end{matrix}\right]=x_{1}\,x_{2}\,x_{3}\,(x_{1}+x_{2}+x_{3})
  40. s ( 2 , 2 , 0 ) ( x 1 , x 2 , x 3 ) = 1 Δ det [ x 1 4 x 2 4 x 3 4 x 1 3 x 2 3 x 3 3 1 1 1 ] = x 1 2 x 2 2 + x 1 2 x 3 2 + x 2 2 x 3 2 + x 1 2 x 2 x 3 + x 1 x 2 2 x 3 + x 1 x 2 x 3 2 s_{(2,2,0)}(x_{1},x_{2},x_{3})=\frac{1}{\Delta}\;\det\left[\begin{matrix}x_{1}% ^{4}&x_{2}^{4}&x_{3}^{4}\\ x_{1}^{3}&x_{2}^{3}&x_{3}^{3}\\ 1&1&1\end{matrix}\right]=x_{1}^{2}\,x_{2}^{2}+x_{1}^{2}\,x_{3}^{2}+x_{2}^{2}\,% x_{3}^{2}+x_{1}^{2}\,x_{2}\,x_{3}+x_{1}\,x_{2}^{2}\,x_{3}+x_{1}\,x_{2}\,x_{3}^% {2}
  41. s ( 2 , 1 , 1 ) = e 1 e 3 s_{(2,1,1)}=e_{1}\,e_{3}
  42. s ( 2 , 2 , 0 ) = e 2 2 - e 1 e 3 s_{(2,2,0)}=e_{2}^{2}-e_{1}\,e_{3}
  43. s ( 3 , 1 , 0 ) = e 1 2 e 2 - e 2 2 - e 1 e 3 s_{(3,1,0)}=e_{1}^{2}\,e_{2}-e_{2}^{2}-e_{1}\,e_{3}
  44. s ( 4 , 0 , 0 ) = e 1 4 - 3 e 1 2 e 2 + 2 e 1 e 3 + e 2 2 . s_{(4,0,0)}=e_{1}^{4}-3\,e_{1}^{2}\,e_{2}+2\,e_{1}\,e_{3}+e_{2}^{2}.
  45. ϕ ( x 1 , x 2 , x 3 ) = x 1 4 + x 2 4 + x 3 4 \phi(x_{1},x_{2},x_{3})=x_{1}^{4}+x_{2}^{4}+x_{3}^{4}
  46. ϕ = s ( 2 , 1 , 1 ) - s ( 3 , 1 , 0 ) + s ( 4 , 0 , 0 ) . \phi=s_{(2,1,1)}-s_{(3,1,0)}+s_{(4,0,0)}.\,\!
  47. p k = i x i k p_{k}=\sum_{i}x_{i}^{k}
  48. s λ = ν χ ν λ z ν p ν = ρ = ( 1 r 1 , 2 r 2 , 3 r 3 , ) χ ρ λ k p k r k r k ! k r k , s_{\lambda}=\sum_{\nu}\frac{\chi^{\lambda}_{\nu}}{z_{\nu}}p_{\nu}=\sum_{\rho=(% 1^{r_{1}},2^{r_{2}},3^{r_{3}},\dots)}\chi^{\lambda}_{\rho}\prod_{k}\frac{p^{r_% {k}}_{k}}{r_{k}!k^{r_{k}}},
  49. s λ / μ , s ν = s λ , s μ s ν . \langle s_{\lambda/\mu},s_{\nu}\rangle=\langle s_{\lambda},s_{\mu}s_{\nu}\rangle.
  50. s λ / μ = ( h λ i - μ j - i + j ) , 1 i , j l ( λ ) s_{\lambda/\mu}=(h_{\lambda_{i}-\mu_{j}-i+j}),1\leq i,j\leq l(\lambda)
  51. s λ / μ = ( e λ i - μ j - i + j ) , 1 i , j l ( λ ) s_{\lambda^{\prime}/\mu^{\prime}}=(e_{\lambda_{i}-\mu_{j}-i+j}),1\leq i,j\leq l% (\lambda)
  52. λ / μ \lambda/\mu
  53. λ λ
  54. s λ ( x | | a ) = T α λ ( x T ( α ) - a T ( α ) - c ( α ) ) s_{\lambda}(x||a)=\sum_{T}\prod_{\alpha\in\lambda}(x_{T(\alpha)}-a_{T(\alpha)-% c(\alpha)})
  55. T T
  56. λ λ
  57. 1 , , n 1,…,n
  58. T ( α ) T(α)
  59. α α
  60. T T
  61. c ( α ) c(α)
  62. s λ ( x | a ) = T α λ ( x T ( α ) - a T ( α ) + c ( α ) ) s_{\lambda}(x|a)=\sum_{T}\prod_{\alpha\in\lambda}(x_{T(\alpha)}-a_{T(\alpha)+c% (\alpha)})
  63. s λ ( x | a ) = det [ ( x j | a ) λ i + n - i ] 1 i , j n i < j ( x i - x j ) s_{\lambda}(x|a)=\frac{\det[(x_{j}|a)^{\lambda_{i}+n-i}]_{1\leq i,j\leq n}}{% \prod_{i<j}(x_{i}-x_{j})}

Schwartz–Zippel_lemma.html

  1. ( x 1 + 3 x 2 - x 3 ) ( 3 x 1 + x 4 - 1 ) ( x 7 - x 2 ) 0 ? (x_{1}+3x_{2}-x_{3})(3x_{1}+x_{4}-1)\cdots(x_{7}-x_{2})\equiv 0\ ?
  2. p ( x 1 , x 2 , , x n ) p(x_{1},x_{2},\ldots,x_{n})\,
  3. P F [ x 1 , x 2 , , x n ] P\in F[x_{1},x_{2},\ldots,x_{n}]
  4. Pr [ P ( r 1 , r 2 , , r n ) = 0 ] d | S | . \Pr[P(r_{1},r_{2},\ldots,r_{n})=0]\leq\frac{d}{|S|}.\,
  5. P ( x 1 , , x n ) = i = 0 d x 1 i P i ( x 2 , , x n ) . P(x_{1},\dots,x_{n})=\sum_{i=0}^{d}x_{1}^{i}P_{i}(x_{2},\dots,x_{n}).
  6. P P
  7. i i
  8. P i P_{i}
  9. i i
  10. deg P i d - i \deg P_{i}\leq d-i
  11. x 1 i P i x_{1}^{i}P_{i}
  12. r 2 , , r n r_{2},\dots,r_{n}
  13. S S
  14. Pr [ P i ( r 2 , , r n ) = 0 ] d - i | S | . \Pr[P_{i}(r_{2},\ldots,r_{n})=0]\leq\frac{d-i}{|S|}.
  15. P i ( r 2 , , r n ) 0 P_{i}(r_{2},\ldots,r_{n})\neq 0
  16. P ( x 1 , r 2 , , r n ) P(x_{1},r_{2},\ldots,r_{n})
  17. i i
  18. Pr [ P ( r 1 , r 2 , , r n ) = 0 | P i ( r 2 , , r n ) 0 ] i | S | . \Pr[P(r_{1},r_{2},\ldots,r_{n})=0|P_{i}(r_{2},\ldots,r_{n})\neq 0]\leq\frac{i}% {|S|}.
  19. P ( r 1 , r 2 , , r n ) = 0 P(r_{1},r_{2},\ldots,r_{n})=0
  20. A A
  21. P i ( r 2 , , r n ) = 0 P_{i}(r_{2},\ldots,r_{n})=0
  22. B B
  23. B B
  24. B c B^{c}
  25. Pr [ A ] \Pr[A]
  26. = Pr [ A B ] + Pr [ A B c ] =\Pr[A\cap B]+\Pr[A\cap B^{c}]
  27. = Pr [ B ] Pr [ A | B ] + Pr [ B c ] Pr [ A | B c ] =\Pr[B]\Pr[A|B]+\Pr[B^{c}]\Pr[A|B^{c}]
  28. Pr [ B ] + Pr [ A | B c ] \leq\Pr[B]+\Pr[A|B^{c}]
  29. d - i | S | + i | S | = d | S | . \leq\frac{d-i}{|S|}+\frac{i}{|S|}=\frac{d}{|S|}.
  30. p 1 ( x ) p_{1}(x)
  31. p 2 ( x ) p_{2}(x)
  32. p 1 ( x ) p 2 ( x ) p_{1}(x)\equiv p_{2}(x)
  33. [ p 1 ( x ) - p 2 ( x ) ] 0. [p_{1}(x)-p_{2}(x)]\equiv 0.
  34. p ( x ) 0 , p(x)\equiv 0,
  35. p ( x ) = p 1 ( x ) - p 2 ( x ) , p(x)=p_{1}(x)\;-\;p_{2}(x),
  36. p A ( x , y ) p_{A}(x,y)
  37. p B ( x , y ) p_{B}(x,y)
  38. n + n\in\mathbb{Z^{+}}
  39. n n
  40. n n
  41. ( 1 + z ) n = 1 + z n ( mod n ) . (1+z)^{n}=1+z^{n}(\mbox{mod}~{}\;n).
  42. 𝒫 n ( z ) = ( 1 + z ) n - 1 - z n . \mathcal{P}_{n}(z)=(1+z)^{n}-1-z^{n}.\,
  43. 𝒫 n ( z ) = 0 ( mod n ) \mathcal{P}_{n}(z)=0\;(\mbox{mod}~{}\;n)
  44. n n
  45. n n
  46. 𝒫 n \mathcal{P}_{n}
  47. 10 350 2 1024 10^{350}\approx 2^{1024}
  48. G = ( V , E ) G=(V,E)
  49. n \mathrm{n}
  50. n \mathrm{n}
  51. G \mathrm{G}
  52. 0 \mathrm{0}
  53. D \mathrm{D}
  54. E \mathrm{E}
  55. V \mathrm{V}
  56. D \mathrm{D}
  57. V \mathrm{V}
  58. D \mathrm{D}
  59. A \mathrm{A}
  60. A = [ a 11 a 12 a 1 n a 21 a 22 a 2 n a n 1 a n 2 a 𝑛𝑛 ] A=\begin{bmatrix}a_{11}&a_{12}&\cdots&a_{1\mathit{n}}\\ a_{21}&a_{22}&\cdots&a_{2\mathit{n}}\\ \vdots&\vdots&\ddots&\vdots\\ a_{\mathit{n}1}&a_{\mathit{n}2}&\ldots&a_{\mathit{nn}}\end{bmatrix}
  61. a i j = { x i j if ( i , j ) E and i < j - x j i if ( i , j ) E and i > j 0 otherwise . a_{ij}=\begin{cases}x_{ij}\;\;\mbox{if}~{}\;(i,j)\in E\mbox{ and }~{}i<j\\ -x_{ji}\;\;\mbox{if}~{}\;(i,j)\in E\mbox{ and }~{}i>j\\ 0\;\;\;\;\mbox{otherwise}~{}.\end{cases}
  62. G \mathrm{G}
  63. n = m + m n=m+m
  64. A = ( 0 X - X t 0 ) A=\begin{pmatrix}0&X\\ -X^{t}&0\end{pmatrix}

Schwarz_triangle.html

  1. 1 p + 1 q + 1 r \displaystyle\frac{1}{p}+\frac{1}{q}+\frac{1}{r}

Schwinger_parametrization.html

  1. 1 A n = 1 ( n - 1 ) ! 0 d u u n - 1 e - u A , \frac{1}{A^{n}}=\frac{1}{(n-1)!}\int^{\infty}_{0}du\,u^{n-1}e^{-uA},
  2. d p A ( p ) n = 1 Γ ( n ) d p 0 d u u n - 1 e - u A ( p ) = 1 Γ ( n ) 0 d u u n - 1 d p e - u A ( p ) , \int\frac{dp}{A(p)^{n}}=\frac{1}{\Gamma(n)}\int dp\int^{\infty}_{0}du\,u^{n-1}% e^{-uA(p)}=\frac{1}{\Gamma(n)}\int^{\infty}_{0}du\,u^{n-1}\int dp\,e^{-uA(p)},
  3. 1 A = - i 0 d u e i u A , \frac{1}{A}=-i\int^{\infty}_{0}du\,e^{iuA},

Search_coil.html

  1. Φ \Phi
  2. e e
  3. e = - N d Φ d t e=-N\frac{\mathrm{d}\Phi}{\mathrm{d}t}
  4. e = - N × S d B d t e=-N\times S\frac{\mathrm{d}B}{\mathrm{d}t}
  5. Φ = B × S \Phi=B\times S
  6. e e
  7. μ a p p \mu_{app}
  8. μ a p p = μ r 1 + N z × ( μ r - 1 ) \mu_{app}=\frac{\mathrm{\mu}_{r}}{\mathrm{1}+N_{z}\times(\mu_{r}-1)}
  9. μ r \mu_{r}
  10. N z N_{z}
  11. e = - N S μ a p p d B d t e=-NS\mu_{app}\frac{\mathrm{d}B}{\mathrm{d}t}
  12. 𝐁 \ \mathbf{B}

Seasonal_energy_efficiency_ratio.html

  1. C O P C a r n o t = T C T H - T C COP_{Carnot}=\frac{T_{C}}{T_{H}-T_{C}}
  2. T C T_{C}
  3. T H T_{H}
  4. E E R C a r n o t = 3.412 T C T H - T C EER_{Carnot}=3.412\frac{T_{C}}{T_{H}-T_{C}}

Second-generation_wavelet_transform.html

  1. f f
  2. γ 1 \gamma_{1}
  3. λ 1 \lambda_{1}
  4. γ 2 \gamma_{2}
  5. γ 1 \gamma_{1}
  6. γ 2 = γ 1 - P ( λ 1 ) \gamma_{2}=\gamma_{1}-P(\lambda_{1})\,
  7. λ 2 = λ 1 + U ( γ 2 ) \lambda_{2}=\lambda_{1}+U(\gamma_{2})\,
  8. P P
  9. U U

Second-order_fluid.html

  1. τ = - p 1 + C 1 A + C 2 A 2 + C 3 A u + C 4 A l , \tau=-p1+C_{1}A+C_{2}A^{2}+C_{3}A_{u}+C_{4}A_{l},
  2. A A
  3. A u A_{u}
  4. A A
  5. A l A_{l}
  6. A A
  7. C i C_{i}
  8. A A
  9. τ = - p 1 + C 1 A + C 2 A 2 + C 5 B , \tau=-p1+C_{1}A+C_{2}A^{2}+C_{5}B,
  10. C 5 C_{5}

Sector_mass_spectrometer.html

  1. 𝐅 = q ( 𝐄 + 𝐯 × 𝐁 ) , \mathbf{F}=q(\mathbf{E}+\mathbf{v}\times\mathbf{B}),
  2. F = q E F=qE\,
  3. F = q v B F=qvB\,
  4. ( π 2 ) \left(\frac{\pi}{\sqrt{2}}\right)
  5. v = E / B v=E/B\,
  6. π / 4 2 \pi/4\sqrt{2}

Seemingly_unrelated_regressions.html

  1. B B
  2. y i r = x i r 𝖳 β i + ε i r , i = 1 , , m . y_{ir}=x_{ir}^{\mathsf{T}}\;\!\beta_{i}+\varepsilon_{ir},\quad i=1,\ldots,m.
  3. x i r x_{ir}
  4. y i = X i β i + ε i , i = 1 , , m , y_{i}=X_{i}\beta_{i}+\varepsilon_{i},\quad i=1,\ldots,m,
  5. ( y 1 y 2 y m ) = ( X 1 0 0 0 X 2 0 0 0 X m ) ( β 1 β 2 β m ) + ( ε 1 ε 2 ε m ) = X β + ε . \begin{pmatrix}y_{1}\\ y_{2}\\ \vdots\\ y_{m}\end{pmatrix}=\begin{pmatrix}X_{1}&0&\ldots&0\\ 0&X_{2}&\ldots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\ldots&X_{m}\end{pmatrix}\begin{pmatrix}\beta_{1}\\ \beta_{2}\\ \vdots\\ \beta_{m}\end{pmatrix}+\begin{pmatrix}\varepsilon_{1}\\ \varepsilon_{2}\\ \vdots\\ \varepsilon_{m}\end{pmatrix}=X\beta+\varepsilon\,.
  6. Ω E [ ε ε 𝖳 | X ] = Σ I R , \Omega\equiv\operatorname{E}[\,\varepsilon\varepsilon^{\mathsf{T}}\,|X\,]=% \Sigma\otimes I_{R},
  7. Σ \Sigma
  8. σ ^ i j = 1 R ε ^ i 𝖳 ε ^ j . \hat{\sigma}_{ij}=\frac{1}{R}\,\hat{\varepsilon}_{i}^{\mathsf{T}}\hat{% \varepsilon}_{j}.
  9. Ω ^ = Σ ^ I R \scriptstyle\hat{\Omega}\;=\;\hat{\Sigma}\,\otimes\,I_{R}
  10. β ^ = ( X 𝖳 ( Σ ^ - 1 I R ) X ) - 1 X 𝖳 ( Σ ^ - 1 I R ) y . \hat{\beta}=\Big(X^{\mathsf{T}}(\hat{\Sigma}^{-1}\otimes I_{R})X\Big)^{\!-1}X^% {\mathsf{T}}(\hat{\Sigma}^{-1}\otimes I_{R})\,y.
  11. R ( β ^ - β ) 𝑑 𝒩 ( 0 , ( 1 R X 𝖳 ( Σ - 1 I R ) X ) - 1 ) . \sqrt{R}(\hat{\beta}-\beta)\ \xrightarrow{d}\ \mathcal{N}\Big(\,0,\;\Big(% \tfrac{1}{R}X^{\mathsf{T}}(\Sigma^{-1}\otimes I_{R})X\Big)^{\!-1}\,\Big).
  12. Σ ^ \scriptstyle\hat{\Sigma}
  13. β ^ \scriptstyle\hat{\beta}

Seiberg_duality.html

  1. N f > N c + 1 N_{f}>N_{c}+1
  2. 1 3 N f < N c < 2 3 N f {1\over 3}N_{f}<N_{c}<{2\over 3}N_{f}
  3. D = 3 2 R D=\frac{3}{2}R
  4. S U ( N c ) SU(N_{c})
  5. S U ( N f - N c ) SU(N_{f}-N_{c})
  6. S U ( N f ) L × S U ( N f ) R × U ( 1 ) B × U ( 1 ) R SU(N_{f})_{L}\times SU(N_{f})_{R}\times U(1)_{B}\times U(1)_{R}
  7. S U ( N f ) L × S U ( N f ) R × U ( 1 ) B × U ( 1 ) R SU(N_{f})_{L}\times SU(N_{f})_{R}\times U(1)_{B}\times U(1)_{R}
  8. Q ( N f , 1 ) 1 / N c , ( N f - N c ) / N f Q\,(N_{f},1)_{1/N_{c},(N_{f}-N_{c})/N_{f}}
  9. Q ~ ( 1 , N f ) - 1 / ( N f - N c ) , N c / N f \tilde{Q}\,(1,N_{f})_{-1/(N_{f}-N_{c}),N_{c}/N_{f}}
  10. Q c ( 1 , N f ¯ ) - 1 / N c , ( N f - N c ) / N f Q^{c}\,(1,\overline{N_{f}})_{-1/N_{c},(N_{f}-N_{c})/N_{f}}
  11. Q c ~ ( N f ¯ , 1 ) 1 / ( N f - N c ) , N c / N f \tilde{Q^{c}}\,(\overline{N_{f}},1)_{1/(N_{f}-N_{c}),N_{c}/N_{f}}
  12. M ( N f , N f ¯ ) 0 , 2 ( N f - N c ) / N f M\,(N_{f},\overline{N_{f}})_{0,2(N_{f}-N_{c})/N_{f}}
  13. W = α M Q c ~ Q ~ W=\alpha M\tilde{Q^{c}}\tilde{Q}
  14. M Q c Q M\equiv Q^{c}Q
  15. S U ( N f ) L 3 SU(N_{f})_{L}^{3}
  16. N c d ( 3 ) ( N f ) N_{c}d^{(3)}(N_{f})
  17. N c d ( 3 ) ( N f ) N_{c}d^{(3)}(N_{f})
  18. S U ( N f ) L 2 U ( 1 ) B SU(N_{f})_{L}^{2}U(1)_{B}
  19. d ( 2 ) ( N f ) d^{(2)}(N_{f})
  20. d ( 2 ) ( N f ) d^{(2)}(N_{f})
  21. S U ( N f ) L 2 U ( 1 ) R SU(N_{f})_{L}^{2}U(1)_{R}
  22. - N c 2 N f d ( 2 ) ( N f ) -\frac{N_{c}^{2}}{N_{f}}d^{(2)}(N_{f})
  23. - N c 2 N f d ( 2 ) ( N f ) \frac{-N_{c}^{2}}{N_{f}}d^{(2)}(N_{f})
  24. U ( 1 ) R U(1)_{R}
  25. - N c 2 - 1 -N_{c}^{2}-1
  26. - N c 2 - 1 -N_{c}^{2}-1
  27. U ( 1 ) R 3 U(1)_{R}^{3}
  28. - 2 N c 4 N f 2 + N c 2 - 1 -2\frac{N_{c}^{4}}{N_{f}^{2}}+N_{c}^{2}-1
  29. - 2 N c 4 N f 2 + N c 2 - 1 -2\frac{N_{c}^{4}}{N_{f}^{2}}+N_{c}^{2}-1
  30. U ( 1 ) B 2 U ( 1 ) R U(1)_{B}^{2}U(1)_{R}
  31. - 2 -2
  32. - 2 -2
  33. χ \chi

Selberg_class.html

  1. F ( s ) = n = 1 a n n s F(s)=\sum_{n=1}^{\infty}\frac{a_{n}}{n^{s}}
  2. b p n = O ( p n θ ) . b_{p^{n}}=O(p^{n\theta}).\,
  3. F p ( s ) = n = 0 a p n p n s for Re ( s ) > 0. F_{p}(s)=\sum_{n=0}^{\infty}\frac{a_{p^{n}}}{p^{ns}}\,\text{ for Re}(s)>0.
  4. L ( s , Δ ) = n = 1 a n n s L(s,\Delta)=\sum_{n=1}^{\infty}\frac{a_{n}}{n^{s}}
  5. a n = τ ( n ) / n 11 / 2 a_{n}=\tau(n)/n^{11/2}
  6. N F ( T ) = d F T log ( T + C ) 2 π + O ( log T ) . N_{F}(T)=d_{F}\frac{T\log(T+C)}{2\pi}+O(\log T).
  7. d F = 2 i = 1 k ω i . d_{F}=2\sum_{i=1}^{k}\omega_{i}.
  8. d F G = d F + d G . d_{FG}=d_{F}+d_{G}.
  9. p x | a p | 2 p = n F log log x + O ( 1 ) \sum_{p\leq x}\frac{|a_{p}|^{2}}{p}=n_{F}\log\log x+O(1)
  10. p x a p a p p = O ( 1 ) . \sum_{p\leq x}\frac{a_{p}a_{p}^{\prime}}{p}=O(1).
  11. F = i = 1 m F i , F=\prod_{i=1}^{m}F_{i},
  12. F χ ( s ) = n = 1 χ ( n ) a n n s F^{\chi}(s)=\sum_{n=1}^{\infty}\frac{\chi(n)a_{n}}{n^{s}}
  13. F F
  14. \mathbb{Q}
  15. ζ F ( s ) \zeta_{F}(s)
  16. ζ ( s ) \zeta(s)
  17. ζ F ( s ) / ζ ( s ) \zeta_{F}(s)/\zeta(s)
  18. K K
  19. F F
  20. ζ K ( s ) / ζ F ( s ) \zeta_{K}(s)/\zeta_{F}(s)

Selection_(relational_algebra).html

  1. σ a θ b ( R ) \sigma_{a\theta b}(R)
  2. σ a θ v ( R ) \sigma_{a\theta v}(R)
  3. a a
  4. b b
  5. θ \theta
  6. { < , , = , , , > } \{\;<,\leq,=,\neq,\geq,\;>\}
  7. v v
  8. R R
  9. σ a θ b ( R ) \sigma_{a\theta b}(R)
  10. R R
  11. θ \theta
  12. a a
  13. b b
  14. σ a θ v ( R ) \sigma_{a\theta v}(R)
  15. R R
  16. θ \theta
  17. a a
  18. v v
  19. P e r s o n Person
  20. σ A g e 34 ( P e r s o n ) \sigma_{Age\geq 34}(Person)
  21. σ A g e = W e i g h t ( P e r s o n ) \sigma_{Age=Weight}(Person)
  22. P e r s o n Person
  23. σ A g e 34 ( P e r s o n ) \sigma_{Age\geq 34}(Person)
  24. σ A g e = W e i g h t ( P e r s o n ) \sigma_{Age=Weight}(Person)
  25. σ a θ b ( R ) = { t : t R , t ( a ) θ t ( b ) } \sigma_{a\theta b}(R)=\{\ t:t\in R,\ t(a)\ \theta\ t(b)\ \}
  26. σ a θ v ( R ) = { t : t R , t ( a ) θ v } \sigma_{a\theta v}(R)=\{\ t:t\in R,\ t(a)\ \theta\ v\ \}

Self-phase_modulation.html

  1. I ( t ) = I 0 exp ( - t 2 τ 2 ) I(t)=I_{0}\exp\left(-\frac{t^{2}}{\tau^{2}}\right)
  2. n ( I ) = n 0 + n 2 I n(I)=n_{0}+n_{2}\cdot I
  3. d n ( I ) d t = n 2 d I d t = n 2 I 0 - 2 t τ 2 exp ( - t 2 τ 2 ) . \frac{dn(I)}{dt}=n_{2}\frac{dI}{dt}=n_{2}\cdot I_{0}\cdot\frac{-2t}{\tau^{2}}% \cdot\exp\left(\frac{-t^{2}}{\tau^{2}}\right).
  4. ϕ ( t ) = ω 0 t - k z = ω 0 t - 2 π λ 0 n ( I ) L \phi(t)=\omega_{0}t-kz=\omega_{0}t-\frac{2\pi}{\lambda_{0}}\cdot n(I)L
  5. ω 0 \omega_{0}
  6. λ 0 \lambda_{0}
  7. L L
  8. ω ( t ) = d ϕ ( t ) d t = ω 0 - 2 π L λ 0 d n ( I ) d t , \omega(t)=\frac{d\phi(t)}{dt}=\omega_{0}-\frac{2\pi L}{\lambda_{0}}\frac{dn(I)% }{dt},
  9. ω ( t ) = ω 0 + 4 π L n 2 I 0 λ 0 τ 2 t exp ( - t 2 τ 2 ) . \omega(t)=\omega_{0}+\frac{4\pi Ln_{2}I_{0}}{\lambda_{0}\tau^{2}}\cdot t\cdot% \exp\left(\frac{-t^{2}}{\tau^{2}}\right).
  10. ω ( t ) = ω 0 + α t \omega(t)=\omega_{0}+\alpha\cdot t
  11. α = d ω d t | 0 = 4 π L n 2 I 0 λ 0 τ 2 . \alpha=\left.\frac{d\omega}{dt}\right|_{0}=\frac{4\pi Ln_{2}I_{0}}{\lambda_{0}% \tau^{2}}.

Semi-differentiability.html

  1. + f ( a ) := lim < m t p l > x a + x I f ( x ) - f ( a ) x - a \partial_{+}f(a):=\lim_{<}mtpl>{{\scriptstyle x\to a+\atop\scriptstyle x\in I}% }\frac{f(x)-f(a)}{x-a}
  2. - f ( a ) := lim < m t p l > x a - x I f ( x ) - f ( a ) x - a \partial_{-}f(a):=\lim_{<}mtpl>{{\scriptstyle x\to a-\atop\scriptstyle x\in I}% }\frac{f(x)-f(a)}{x-a}
  3. c = inf { x ( a , b ] | f ( x ) - f ( a ) | > ε ( x - a ) } . c=\inf\{\,x\in(a,b]\mid|f(x)-f(a)|>\varepsilon(x-a)\,\}.
  4. | f ( x ) - f ( a ) | | f ( x ) - f ( c ) | + | f ( c ) - f ( a ) | ε ( x - a ) |f(x)-f(a)|\leq|f(x)-f(c)|+|f(c)-f(a)|\leq\varepsilon(x-a)
  5. u f ( a ) = lim h 0 + f ( a + h u ) - f ( a ) h \partial_{u}f(a)=\lim_{h\to 0^{+}}\frac{f(a+h\,u)-f(a)}{h}

Semi-supervised_learning.html

  1. l l
  2. x 1 , , x l X x_{1},\dots,x_{l}\in X
  3. y 1 , , y l Y y_{1},\dots,y_{l}\in Y
  4. u u
  5. x l + 1 , , x l + u X x_{l+1},\dots,x_{l+u}\in X
  6. x l + 1 , , x l + u x_{l+1},\dots,x_{l+u}
  7. X X
  8. Y Y
  9. p ( x | y ) p(x|y)
  10. p ( y | x ) p(y|x)
  11. x x
  12. y y
  13. p ( x | y ) p ( y ) p(x|y)p(y)
  14. p ( x ) p(x)
  15. p ( x | y , θ ) p(x|y,\theta)
  16. θ \theta
  17. p ( x , y | θ ) = p ( y | θ ) p ( x | y , θ ) p(x,y|\theta)=p(y|\theta)p(x|y,\theta)
  18. θ \theta
  19. f θ ( x ) = argmax 𝑦 p ( y | x , θ ) f_{\theta}(x)=\underset{y}{\operatorname{argmax}}\ p(y|x,\theta)
  20. λ \lambda
  21. argmax Θ ( log p ( { x i , y i } i = 1 l | θ ) + λ log p ( { x i } i = l + 1 l + u | θ ) ) \underset{\Theta}{\operatorname{argmax}}\left(\log p(\{x_{i},y_{i}\}_{i=1}^{l}% |\theta)+\lambda\log p(\{x_{i}\}_{i=l+1}^{l+u}|\theta)\right)
  22. ( 1 - y f ( x ) ) + (1-yf(x))_{+}
  23. ( 1 - | f ( x ) | ) + (1-|f(x)|)_{+}
  24. y = sign f ( x ) y=\operatorname{sign}{f(x)}
  25. f * ( x ) = h * ( x ) + b f^{*}(x)=h^{*}(x)+b
  26. \mathcal{H}
  27. f * = argmin 𝑓 ( i = 1 l ( 1 - y i f ( x i ) ) + + λ 1 || h || 2 + λ 2 i = l + 1 l + u ( 1 - | f ( x i ) | ) + ) f^{*}=\underset{f}{\operatorname{argmin}}\left(\displaystyle\sum_{i=1}^{l}(1-y% _{i}f(x_{i}))_{+}+\lambda_{1}||h||_{\mathcal{H}}^{2}+\lambda_{2}\sum_{i=l+1}^{% l+u}(1-|f(x_{i})|)_{+}\right)
  28. ( 1 - | f ( x ) | ) + (1-|f(x)|)_{+}
  29. k k
  30. ϵ \epsilon
  31. W i j W_{ij}
  32. x i x_{i}
  33. x j x_{j}
  34. e - || x i - x j || 2 ϵ e^{\frac{-||x_{i}-x_{j}||^{2}}{\epsilon}}
  35. argmin f ( 1 l i = 1 l V ( f ( x i ) , y i ) + λ A || f || 2 + λ I || f ( x ) || 2 d p ( x ) ) \underset{f\in\mathcal{H}}{\operatorname{argmin}}\left(\frac{1}{l}% \displaystyle\sum_{i=1}^{l}V(f(x_{i}),y_{i})+\lambda_{A}||f||^{2}_{\mathcal{H}% }+\lambda_{I}\int_{\mathcal{M}}||\nabla_{\mathcal{M}}f(x)||^{2}dp(x)\right)
  36. \mathcal{H}
  37. \mathcal{M}
  38. λ A \lambda_{A}
  39. λ I \lambda_{I}
  40. L = D - W L=D-W
  41. D i i = j = 1 l + u W i j D_{ii}=\sum_{j=1}^{l+u}W_{ij}
  42. 𝐟 \mathbf{f}
  43. [ f ( x 1 ) f ( x l + u ) ] [f(x_{1})\dots f(x_{l+u})]
  44. 𝐟 T L 𝐟 = i , j = 1 l + u W i j ( f i - f j ) 2 || f ( x ) || 2 d p ( x ) \mathbf{f}^{T}L\mathbf{f}=\displaystyle\sum_{i,j=1}^{l+u}W_{ij}(f_{i}-f_{j})^{% 2}\approx\int_{\mathcal{M}}||\nabla_{\mathcal{M}}f(x)||^{2}dp(x)
  45. x 1 , , x l + u x_{1},\dots,x_{l+u}

Semi-Thue_system.html

  1. R R
  2. s t s\rightarrow t
  3. u s v u t v usv\rightarrow utv
  4. s s
  5. t t
  6. u u
  7. v v
  8. ( Σ , R ) (\Sigma,R)
  9. Σ \Sigma
  10. Σ * \Sigma^{*}
  11. Σ \Sigma
  12. R R
  13. Σ \Sigma
  14. R Σ * × Σ * . R\subseteq\Sigma^{*}\times\Sigma^{*}.
  15. ( u , v ) R (u,v)\in R
  16. u v u\rightarrow v
  17. R R
  18. R R
  19. Σ * \Sigma^{*}
  20. R R
  21. R \rightarrow_{R}
  22. R R
  23. Σ * \Sigma^{*}
  24. s s
  25. t t
  26. Σ * \Sigma^{*}
  27. s R t s\rightarrow_{R}t
  28. x x
  29. y y
  30. u u
  31. v v
  32. Σ * \Sigma^{*}
  33. s = x u y s=xuy
  34. t = x v y t=xvy
  35. u v u\rightarrow v
  36. R \rightarrow_{R}
  37. Σ * \Sigma^{*}
  38. ( Σ * , R ) (\Sigma^{*},\rightarrow_{R})
  39. R R
  40. R \rightarrow_{R}
  41. R \rightarrow_{R}
  42. R \Rightarrow_{R}
  43. R R
  44. \rightarrow
  45. R R
  46. R R
  47. s 0 Σ * s_{0}\in\Sigma^{*}
  48. s 0 R s 1 R s 2 R s_{0}\ \rightarrow_{R}\ s_{1}\ \rightarrow_{R}\ s_{2}\ \rightarrow_{R}\ \ldots
  49. R \rightarrow_{R}
  50. * R \stackrel{*}{\rightarrow}_{R}
  51. Σ * \Sigma^{*}
  52. R R
  53. Σ * \Sigma^{*}
  54. \cdot
  55. * R \stackrel{*}{\rightarrow}_{R}
  56. x * R y x\stackrel{*}{\rightarrow}_{R}y
  57. u x v * R u y v uxv\stackrel{*}{\rightarrow}_{R}uyv
  58. x x
  59. y y
  60. u u
  61. v v
  62. Σ * \Sigma^{*}
  63. * R \stackrel{*}{\rightarrow}_{R}
  64. ( Σ * , , * R ) (\Sigma^{*},\cdot,\stackrel{*}{\rightarrow}_{R})
  65. R \rightarrow_{R}
  66. * R \stackrel{*}{\leftrightarrow}_{R}
  67. * R \stackrel{*}{\leftrightarrow}_{R}
  68. R R
  69. R R
  70. * R \stackrel{*}{\rightarrow}_{R}
  71. * R \stackrel{*}{\leftrightarrow}_{R}
  72. * R \stackrel{*}{\leftrightarrow}_{R}
  73. R = Σ * / * R \mathcal{M}_{R}=\Sigma^{*}/\stackrel{*}{\leftrightarrow}_{R}
  74. Σ * \Sigma^{*}
  75. \mathcal{M}
  76. R \mathcal{M}_{R}
  77. ( Σ , R ) (\Sigma,R)
  78. \mathcal{M}
  79. ( Σ , R ) (\Sigma,R)
  80. Σ \Sigma
  81. \mathcal{M}
  82. R R
  83. \mathcal{M}
  84. \mathcal{M}
  85. Σ \Sigma
  86. Σ \Sigma
  87. R R
  88. T := ( Σ , R ) T:=(\Sigma,R)
  89. u , v Σ * u,v\in\Sigma^{*}
  90. u u
  91. v v
  92. R R
  93. f 2 ( f 1 ( x ) ) g ( x ) f_{2}(f_{1}(x))\rightarrow g(x)
  94. f 1 f 2 g f_{1}f_{2}\rightarrow g
  95. ( Σ , A , R ) (\Sigma,A,R)
  96. A Σ * A\subseteq\Sigma^{*}

Semicubical_parabola.html

  1. x = t 2 x=t^{2}\,
  2. y = a t 3 . y=at^{3}.\,
  3. y 2 - a 2 x 3 = 0 , y^{2}-a^{2}x^{3}=0,
  4. y y
  5. y = ± a x 3 2 . y=\pm ax^{3\over 2}.
  6. u = a t u=at
  7. X = u 2 X=u^{2}
  8. Y = u 3 . Y=u^{3}.
  9. a a
  10. a = 1 a=1
  11. a a
  12. x = 3 4 ( 2 y ) 2 3 + 1 2 . x={3\over 4}(2y)^{2\over 3}+{1\over 2}.
  13. x = 3 ( t 2 - 3 ) = 3 t 2 - 9 x=3(t^{2}-3)=3t^{2}-9\,
  14. y = t ( t 2 - 3 ) = t 3 - 3 t . y=t(t^{2}-3)=t^{3}-3t.\,

Sentence_(logic).html

  1. y x ( x 2 = y ) \forall y\exists x(x^{2}=y)
  2. x ( x 2 = y ) \exists x(x^{2}=y)

Sentience_quotient.html

  1. S Q = log 10 ( I M ) SQ=\log_{10}\left(\frac{I}{M}\right)
  2. I I
  3. M M
  4. I M \frac{I}{M}

Separable_state.html

  1. H 1 H_{1}
  2. H 2 H_{2}
  3. { | a i } i = 1 n \{|{a_{i}}\rangle\}_{i=1}^{n}
  4. { | b j } j = 1 m \{|{b_{j}}\rangle\}_{j=1}^{m}
  5. H 1 H 2 H_{1}\otimes H_{2}
  6. { | a i | b j } \{|{a_{i}}\rangle\otimes|{b_{j}}\rangle\}
  7. { | a i b j } \{|a_{i}b_{j}\rangle\}
  8. | ψ = i , j c i , j ( | a i | b j ) = i , j c i , j | a i b j |\psi\rangle=\sum_{i,j}c_{i,j}(|a_{i}\rangle\otimes|b_{j}\rangle)=\sum_{i,j}c_% {i,j}|a_{i}b_{j}\rangle
  9. | ψ H 1 H 2 |\psi\rangle\in H_{1}\otimes H_{2}
  10. | ψ = | ψ 1 | ψ 2 |\psi\rangle=|\psi_{1}\rangle\otimes|\psi_{2}\rangle
  11. | ψ i |\psi_{i}\rangle
  12. ρ \rho
  13. H 1 H 2 H_{1}\otimes H_{2}
  14. p k 0 p_{k}\geq 0
  15. { ρ 1 k } \{\rho_{1}^{k}\}
  16. { ρ 2 k } \{\rho_{2}^{k}\}
  17. ρ = k p k ρ 1 k ρ 2 k \rho=\sum_{k}p_{k}\rho_{1}^{k}\otimes\rho_{2}^{k}
  18. k p k = 1. \;\sum_{k}p_{k}=1.
  19. ρ \rho
  20. { ρ 1 k } \{\rho_{1}^{k}\}
  21. { ρ 2 k } \{\rho_{2}^{k}\}
  22. { ρ 1 k } \{\rho_{1}^{k}\}
  23. { ρ 2 k } \{\rho_{2}^{k}\}
  24. k p k = 1. \;\sum_{k}p_{k}=1.
  25. p k p_{k}
  26. H = H 1 H n H=H_{1}\otimes\cdots\otimes H_{n}
  27. | ψ H |\psi\rangle\in H
  28. | ψ = | ψ 1 | ψ n . |\psi\rangle=|\psi_{1}\rangle\otimes\cdots\otimes|\psi_{n}\rangle.
  29. ρ = k p k ρ 1 k ρ n k . \rho=\sum_{k}p_{k}\rho_{1}^{k}\otimes\cdots\otimes\rho_{n}^{k}.
  30. 1 1 1\oplus 1
  31. 1 n 1\oplus n
  32. 2 2 2\oplus 2

Separatrix_(dynamical_systems).html

  1. d 2 θ d t 2 + g l sin θ = 0. {d^{2}\theta\over dt^{2}}+{g\over l}\sin\theta=0.
  2. l l
  3. g g
  4. θ \theta
  5. H = θ ˙ 2 2 - g l cos θ . H=\frac{\dot{\theta}^{2}}{2}-\frac{g}{l}\cos\theta.
  6. θ \theta
  7. θ ˙ \dot{\theta}
  8. H < - g l H<-\frac{g}{l}
  9. θ ˙ \dot{\theta}
  10. - g l < H < g l -\frac{g}{l}<H<\frac{g}{l}
  11. g l < H \frac{g}{l}<H
  12. H = g l H=\frac{g}{l}

Sequence_transformation.html

  1. S = { s n } n 𝒩 , S=\{s_{n}\}_{n\in\mathcal{N}},\,
  2. 𝐓 ( S ) = S = { s n } n 𝒩 , \mathbf{T}(S)=S^{\prime}=\{s^{\prime}_{n}\}_{n\in\mathcal{N}},\,
  3. s n = T ( s n , s n + 1 , , s n + k ) s_{n}^{\prime}=T(s_{n},s_{n+1},\dots,s_{n+k})
  4. k k
  5. n n
  6. s n s_{n}
  7. s n s^{\prime}_{n}
  8. lim n s n - s n - = 0 \lim_{n\to\infty}\frac{s^{\prime}_{n}-\ell}{s_{n}-\ell}=0
  9. \ell
  10. S S
  11. \ell
  12. T T
  13. s n = m = 0 k c m s n + m s^{\prime}_{n}=\sum_{m=0}^{k}c_{m}s_{n+m}
  14. c 0 , , c k c_{0},\dots,c_{k}
  15. 𝐓 \mathbf{T}
  16. s n = s n + k s^{\prime}_{n}=s_{n+k}

Serpentine_curve.html

  1. x 2 y + a 2 y - a b x = 0 , a b > 0. x^{2}y+a^{2}y-abx=0,\quad ab>0.
  2. x = a cot ( t ) x=a\cot(t)
  3. y = b sin ( t ) cos ( t ) , y=b\sin(t)\cos(t),
  4. y = a b x x 2 + a 2 . y=\frac{abx}{x^{2}+a^{2}}.

Serre's_modularity_conjecture.html

  1. G G_{\mathbb{Q}}
  2. \mathbb{Q}
  3. ρ \rho
  4. G G_{\mathbb{Q}}
  5. F = 𝔽 r F=\mathbb{F}_{\ell^{r}}
  6. ρ : G GL 2 ( F ) . \rho:G_{\mathbb{Q}}\rightarrow\mathrm{GL}_{2}(F).
  7. ρ \rho
  8. f = q + a 2 q 2 + a 3 q 3 + f=q+a_{2}q^{2}+a_{3}q^{3}+\cdots
  9. N = N ( ρ ) N=N(\rho)
  10. k = k ( ρ ) k=k(\rho)
  11. χ : / N F * \chi:\mathbb{Z}/N\mathbb{Z}\rightarrow F^{*}
  12. f f
  13. ρ f : G GL 2 ( 𝒪 ) , \rho_{f}:G_{\mathbb{Q}}\rightarrow\mathrm{GL}_{2}(\mathcal{O}),
  14. 𝒪 \mathcal{O}
  15. \mathbb{Q}_{\ell}
  16. p p
  17. N N\ell
  18. Trace ( ρ f ( Frob p ) ) = a p \operatorname{Trace}(\rho_{f}(\operatorname{Frob}_{p}))=a_{p}
  19. det ( ρ f ( Frob p ) ) = p k - 1 χ ( p ) . \det(\rho_{f}(\operatorname{Frob}_{p}))=p^{k-1}\chi(p).
  20. 𝒪 \mathcal{O}
  21. \ell
  22. ρ f ¯ \overline{\rho_{f}}
  23. G G_{\mathbb{Q}}
  24. ρ \rho
  25. f f
  26. ρ f ¯ ρ \overline{\rho_{f}}\cong\rho
  27. f f

Serre's_property_FA.html

  1. a , b : a A = b B = ( a b ) C = 1 \left\langle{a,b:a^{A}=b^{B}=(ab)^{C}=1}\right\rangle

Serre_spectral_sequence.html

  1. E 2 p , q = H p ( B , H q ( F ) ) H p + q ( X ) . E_{2}^{p,q}=H^{p}(B,H^{q}(F))\Rightarrow H^{p+q}(X).
  2. A = p , q H q ( X p ) A=\bigoplus_{p,q}H^{q}(X_{p})
  3. E 1 p , q = C = p , q H q ( X p , X p - 1 ) , E_{1}^{p,q}=C=\bigoplus_{p,q}H^{q}(X_{p},X_{p-1}),
  4. E r p , q × E r s , t E r p + s , q + t , E_{r}^{p,q}\times E_{r}^{s,t}\to E_{r}^{p+s,q+t},
  5. E p , q 2 = H p ( B , H q ( F ) ) H p + q ( X ) . E^{2}_{p,q}=H_{p}(B,H_{q}(F))\Rightarrow H_{p+q}(X).
  6. Ω 𝐒 n + 1 𝐏𝐒 n + 1 𝐒 n + 1 . \Omega\mathbf{S}^{n+1}\to\mathbf{PS}^{n+1}\to\mathbf{S}^{n+1}.
  7. E p , q 2 = H p ( 𝐒 n + 1 ; H q ( Ω 𝐒 n + 1 ) ) . E^{2}_{p,q}=H_{p}(\mathbf{S}^{n+1};H_{q}(\Omega\mathbf{S}^{n+1})).
  8. 𝐒 1 𝐒 2 n + 1 𝐂𝐏 n \mathbf{S}^{1}\hookrightarrow\mathbf{S}^{2n+1}\to\mathbf{CP}^{n}
  9. X 𝐒 3 K ( 𝐙 , 3 ) . X\rightarrow\mathbf{S}^{3}\rightarrow K(\mathbf{Z},3).

Service_level.html

  1. α p \alpha_{p}
  2. α p = P r o b { P e r i o d d e m a n d I n v e n t o r y o n h a n d a t t h e b e g i n n i n g o f a p e r i o d } \alpha_{p}=Prob\{Period~{}demand\leq\;{Inventory~{}on~{}hand~{}at~{}the~{}% beginning~{}of~{}a~{}period}\}
  3. α p \alpha_{p}
  4. α c \alpha_{c}
  5. α c = P r o b { D e m a n d d u r i n g r e p l e n i s h m e n t l e a d t i m e I n v e n t o r y o n h a n d a t t h e b e g i n n i n g o f t h e l e a d t i m e } \alpha_{c}=Prob\{Demand~{}during~{}replenishment~{}lead~{}time\leq Inventory~{% }on~{}hand~{}at~{}the~{}beginning~{}of~{}the~{}lead~{}time\}
  6. β = 1 - E x p e c t e d b a c k o r d e r s p e r t i m e p e r i o d E x p e c t e d p e r i o d d e m a n d \beta=1-\frac{Expected~{}backorders~{}per~{}time~{}period}{Expected~{}period~{% }demand}
  7. α \alpha
  8. β \beta
  9. α β \alpha\leq\beta
  10. γ = 1 - E x p e c t e d b a c k o r d e r l e v e l p e r t i m e p e r i o d E x p e c t e d p e r i o d d e m a n d \gamma=1-\frac{Expected~{}backorder~{}level~{}per~{}time~{}period}{Expected~{}% period~{}demand}
  11. μ \mu

Sessile_drop_technique.html

  1. θ C \theta_{C}
  2. γ S G \gamma_{SG}
  3. γ L G \gamma_{LG}
  4. γ S L \gamma_{SL}
  5. σ \sigma
  6. σ \sigma
  7. σ \sigma
  8. σ \sigma
  9. σ \sigma
  10. σ \sigma
  11. σ \sigma
  12. σ \sigma
  13. σ \sigma
  14. σ \sigma
  15. σ \sigma
  16. σ \sigma
  17. cos θ = ( F - F b ) / I σ \cos{\theta}=(F-Fb)/I\sigma
  18. σ L ( cos θ + 1 ) 2 σ L D \sigma_{L}\ (\cos\theta+1)\over 2\sqrt{\sigma_{L}^{D}}
  19. σ S P σ L P σ L D \sqrt{\sigma_{S}^{P}}\sqrt{\sigma_{L}^{P}}\over\sqrt{\sigma_{L}^{D}}
  20. σ S D \sqrt{\sigma_{S}^{D}}
  21. σ L ( cos θ + 1 ) 2 σ L D \sigma_{L}\ (\cos\theta+1)\over 2\sqrt{\sigma_{L}^{D}}
  22. σ S P \sqrt{\sigma_{S}^{P}}
  23. σ L P σ L D \sqrt{\sigma_{L}^{P}}\over\sqrt{\sigma_{L}^{D}}
  24. σ S D \sqrt{\sigma_{S}^{D}}
  25. σ S P = 0 {\sigma_{S}^{P}}=0
  26. σ S D = 18.0 mN / m {\sigma_{S}^{D}}=18.0~{}\mathrm{mN/m}
  27. σ L D \sigma_{L}^{D}
  28. ( σ L ( cos θ + 1 ) ) 2 72 (\sigma_{L}\ (\cos\theta+1))^{2}\over 72
  29. σ L ( cos θ + 1 ) 2 \sigma_{L}\ (\cos\theta+1)\over 2
  30. σ S P σ L P \sqrt{\sigma_{S}^{P}}\sqrt{\sigma_{L}^{P}}
  31. σ S D σ L D \sqrt{\sigma_{S}^{D}}\sqrt{\sigma_{L}^{D}}
  32. σ L D \sqrt{\sigma_{L}^{D}}
  33. σ L P \sigma_{L}^{P}
  34. σ L \sigma_{L}
  35. σ L D \sigma_{L}^{D}
  36. σ + \sigma^{+}
  37. σ - \sigma^{-}
  38. σ L ( cos θ + 1 ) \sigma_{L}(\cos\theta+1)
  39. 2 [ σ L D σ S D + σ L - σ S + + σ L + σ S - ] 2[\sqrt{\sigma_{L}^{D}\sigma_{S}^{D}}+\sqrt{\sigma_{L}^{-}\sigma_{S}^{+}}+% \sqrt{\sigma_{L}^{+}\sigma_{S}^{-}}]
  40. σ L = σ L D \sigma_{L}=\sigma_{L}^{D}
  41. σ L = σ L D + σ L ± \sigma_{L}=\sigma_{L}^{D}+\sigma_{L}^{\pm}
  42. σ L = σ L D + σ L \sigma_{L}=\sigma_{L}^{D}+\sigma_{L}^{\mp}
  43. σ L = σ L D + σ L + + σ L - \sigma_{L}=\sigma_{L}^{D}+\sigma_{L}^{+}+\sigma_{L}^{-}

Set_notation.html

  1. S S
  2. A A
  3. B B
  4. C C
  5. \mathbf{∅}
  6. \emptyset
  7. \varnothing
  8. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  9. 𝐙 \mathbf{Z}
  10. 𝐍 \mathbf{N}
  11. 𝐐 \mathbf{Q}
  12. 𝐑 \mathbf{R}
  13. 𝐂 \mathbf{C}
  14. \mathbb{C}
  15. \mathbb{N}
  16. 22 22
  17. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  18. π π , 6 , 1 / 2 ππ,6,1/2
  19. 1 / 2 , π , 6 1/2,π,6
  20. 1 , 2 , 2 , 3 , 3 , 33 = 1 , 2 , 33 1,2,2,3,3,33=1,2,33
  21. 1 , 2 , 3 , , 100 1,2,3,...,100
  22. n n
  23. n n
  24. 𝐍 \mathbf{N}
  25. 1 , 2 , 3 , . 1,2,3,....
  26. 0 , 1 , 3 , , k ( k - 1 ) / 2 , 0,1,3,...,k(k-1)/2,...
  27. x x : P ( x ) xx:P(x)
  28. x x
  29. P ( x ) P(x)
  30. x x
  31. r r
  32. ( a , b ) (a,b)
  33. ( 1 , 1 ) (−1,1)
  34. x x
  35. ( 2 , 5 (2,5
  36. 0 , ) ) 0,∞))
  37. b b
  38. a a
  39. b = 2 a b=2a
  40. 22 a : a 𝐙 22a:a∈\mathbf{Z}
  41. b b 𝐙 : a 𝐙 : b = 2 a bb∈\mathbf{Z}:∃a∈\mathbf{Z}:b=2a
  42. 2 𝐙 2\mathbf{Z}
  43. 2 a + 1 2a+1
  44. a a
  45. 2 𝐙 + 1 2\mathbf{Z}+1
  46. 1 , , n 1,...,n
  47. n n
  48. n n nn
  49. n n nn
  50. 𝐧 \mathbf{n}
  51. a ¯ n \bar{a}_{n}
  52. a a
  53. n n
  54. 0 ¯ 2 \bar{0}_{2}
  55. S S
  56. S S
  57. A A
  58. B B
  59. ( ( a , b ) : a A , b B ((a,b):a∈A,b∈B
  60. A A
  61. B B
  62. A × B A×B
  63. A × B A×B
  64. A A
  65. B B
  66. A A
  67. B B
  68. A A
  69. B B
  70. A A
  71. B B
  72. S S
  73. S S
  74. S S
  75. ( X k ) {\textstyle\left({{X}\atop{k}}\right)}
  76. k k
  77. X X
  78. X ! X!
  79. X X
  80. S < s u b > X S<sub>X

Seven-dimensional_cross_product.html

  1. 𝐞 < s u b > 1 \mathbf{e}<sub>1
  2. 𝐞 1 × 𝐞 2 = 𝐞 3 = - 𝐞 2 × 𝐞 1 \mathbf{e}_{1}\times\mathbf{e}_{2}=\mathbf{e}_{3}=-\mathbf{e}_{2}\times\mathbf% {e}_{1}
  3. ( 𝐱 × 𝐲 ) 1 = x 2 y 3 - x 3 y 2 + x 4 y 5 - x 5 y 4 + x 7 y 6 - x 6 y 7 . \left(\mathbf{x\times y}\right)_{1}=x_{2}y_{3}-x_{3}y_{2}+x_{4}y_{5}-x_{5}y_{4% }+x_{7}y_{6}-x_{6}y_{7}.
  4. 𝐞 i × 𝐞 j = ε i j k 𝐞 k , \mathbf{e}_{i}\mathbf{\times}\mathbf{e}_{j}=\varepsilon_{ijk}\mathbf{e}_{k},
  5. ε i j k \varepsilon_{ijk}
  6. 𝐱 ( 𝐱 × 𝐲 ) = ( 𝐱 × 𝐲 ) 𝐲 = 0 , \mathbf{x}\cdot(\mathbf{x}\times\mathbf{y})=(\mathbf{x}\times\mathbf{y})\cdot% \mathbf{y}=0,
  7. | 𝐱 × 𝐲 | 2 = | 𝐱 | 2 | 𝐲 | 2 - ( 𝐱 𝐲 ) 2 |\mathbf{x}\times\mathbf{y}|^{2}=|\mathbf{x}|^{2}|\mathbf{y}|^{2}-(\mathbf{x}% \cdot\mathbf{y})^{2}
  8. | 𝐱 × 𝐲 | = | 𝐱 | | 𝐲 | sin θ , |\mathbf{x}\times\mathbf{y}|=|\mathbf{x}||\mathbf{y}|\sin\theta,
  9. | 𝐱 × 𝐲 | = | 𝐱 | | 𝐲 | if ( 𝐱 𝐲 ) = 0. |\mathbf{x}\times\mathbf{y}|=|\mathbf{x}||\mathbf{y}|~{}\mbox{if}~{}\ \left(% \mathbf{x}\cdot\mathbf{y}\right)=0.
  10. 𝐱 × 𝐲 = - 𝐲 × 𝐱 \mathbf{x}\times\mathbf{y}=-\mathbf{y}\times\mathbf{x}
  11. 𝐱 ( 𝐲 × 𝐳 ) = 𝐲 ( 𝐳 × 𝐱 ) = 𝐳 ( 𝐱 × 𝐲 ) \mathbf{x}\cdot(\mathbf{y}\times\mathbf{z})=\mathbf{y}\cdot(\mathbf{z}\times% \mathbf{x})=\mathbf{z}\cdot(\mathbf{x}\times\mathbf{y})
  12. ( 𝐱 × 𝐲 ) × ( 𝐱 × 𝐳 ) = ( ( 𝐱 × 𝐲 ) × 𝐳 ) × 𝐱 + ( ( 𝐲 × 𝐳 ) × 𝐱 ) × 𝐱 + ( ( 𝐳 × 𝐱 ) × 𝐱 ) × 𝐲 (\mathbf{x}\times\mathbf{y})\times(\mathbf{x}\times\mathbf{z})=((\mathbf{x}% \times\mathbf{y})\times\mathbf{z})\times\mathbf{x}+((\mathbf{y}\times\mathbf{z% })\times\mathbf{x})\times\mathbf{x}+((\mathbf{z}\times\mathbf{x})\times\mathbf% {x})\times\mathbf{y}
  13. 𝐱 × ( 𝐱 × 𝐲 ) = - | 𝐱 | 2 𝐲 + ( 𝐱 𝐲 ) 𝐱 . \mathbf{x}\times(\mathbf{x}\times\mathbf{y})=-|\mathbf{x}|^{2}\mathbf{y}+(% \mathbf{x}\cdot\mathbf{y})\mathbf{x}.
  14. 𝐱 × ( 𝐲 × 𝐳 ) = ( 𝐱 𝐳 ) 𝐲 - ( 𝐱 𝐲 ) 𝐳 \mathbf{x}\times(\mathbf{y}\times\mathbf{z})=(\mathbf{x}\cdot\mathbf{z})% \mathbf{y}-(\mathbf{x}\cdot\mathbf{y})\mathbf{z}
  15. 𝐱 × ( 𝐲 × 𝐳 ) + 𝐲 × ( 𝐳 × 𝐱 ) + 𝐳 × ( 𝐱 × 𝐲 ) = 0 \mathbf{x}\times(\mathbf{y}\times\mathbf{z})+\mathbf{y}\times(\mathbf{z}\times% \mathbf{x})+\mathbf{z}\times(\mathbf{x}\times\mathbf{y})=0
  16. 𝐞 1 × 𝐞 2 = 𝐞 4 , 𝐞 2 × 𝐞 4 = 𝐞 1 , 𝐞 4 × 𝐞 1 = 𝐞 2 , \mathbf{e}_{1}\times\mathbf{e}_{2}=\mathbf{e}_{4},\quad\mathbf{e}_{2}\times% \mathbf{e}_{4}=\mathbf{e}_{1},\quad\mathbf{e}_{4}\times\mathbf{e}_{1}=\mathbf{% e}_{2},
  17. 𝐞 2 × 𝐞 3 = 𝐞 5 , 𝐞 3 × 𝐞 5 = 𝐞 2 , 𝐞 5 × 𝐞 2 = 𝐞 3 , \mathbf{e}_{2}\times\mathbf{e}_{3}=\mathbf{e}_{5},\quad\mathbf{e}_{3}\times% \mathbf{e}_{5}=\mathbf{e}_{2},\quad\mathbf{e}_{5}\times\mathbf{e}_{2}=\mathbf{% e}_{3},
  18. 𝐞 3 × 𝐞 4 = 𝐞 6 , 𝐞 4 × 𝐞 6 = 𝐞 3 , 𝐞 6 × 𝐞 3 = 𝐞 4 , \mathbf{e}_{3}\times\mathbf{e}_{4}=\mathbf{e}_{6},\quad\mathbf{e}_{4}\times% \mathbf{e}_{6}=\mathbf{e}_{3},\quad\mathbf{e}_{6}\times\mathbf{e}_{3}=\mathbf{% e}_{4},
  19. 𝐞 4 × 𝐞 5 = 𝐞 7 , 𝐞 5 × 𝐞 7 = 𝐞 4 , 𝐞 7 × 𝐞 4 = 𝐞 5 , \mathbf{e}_{4}\times\mathbf{e}_{5}=\mathbf{e}_{7},\quad\mathbf{e}_{5}\times% \mathbf{e}_{7}=\mathbf{e}_{4},\quad\mathbf{e}_{7}\times\mathbf{e}_{4}=\mathbf{% e}_{5},
  20. 𝐞 5 × 𝐞 6 = 𝐞 1 , 𝐞 6 × 𝐞 1 = 𝐞 5 , 𝐞 1 × 𝐞 5 = 𝐞 6 , \mathbf{e}_{5}\times\mathbf{e}_{6}=\mathbf{e}_{1},\quad\mathbf{e}_{6}\times% \mathbf{e}_{1}=\mathbf{e}_{5},\quad\mathbf{e}_{1}\times\mathbf{e}_{5}=\mathbf{% e}_{6},
  21. 𝐞 6 × 𝐞 7 = 𝐞 2 , 𝐞 7 × 𝐞 2 = 𝐞 6 , 𝐞 2 × 𝐞 6 = 𝐞 7 , \mathbf{e}_{6}\times\mathbf{e}_{7}=\mathbf{e}_{2},\quad\mathbf{e}_{7}\times% \mathbf{e}_{2}=\mathbf{e}_{6},\quad\mathbf{e}_{2}\times\mathbf{e}_{6}=\mathbf{% e}_{7},
  22. 𝐞 7 × 𝐞 1 = 𝐞 3 , 𝐞 1 × 𝐞 3 = 𝐞 7 , 𝐞 3 × 𝐞 7 = 𝐞 1 . \mathbf{e}_{7}\times\mathbf{e}_{1}=\mathbf{e}_{3},\quad\mathbf{e}_{1}\times% \mathbf{e}_{3}=\mathbf{e}_{7},\quad\mathbf{e}_{3}\times\mathbf{e}_{7}=\mathbf{% e}_{1}.
  23. 𝐞 i × 𝐞 i + 1 = 𝐞 i + 3 \mathbf{e}_{i}\times\mathbf{e}_{i+1}=\mathbf{e}_{i+3}
  24. 𝐞 i × ( 𝐞 i × 𝐞 i + 1 ) = - 𝐞 i + 1 = 𝐞 i × 𝐞 i + 3 , \mathbf{e}_{i}\times\left(\mathbf{e}_{i}\times\mathbf{e}_{i+1}\right)=-\mathbf% {e}_{i+1}=\mathbf{e}_{i}\times\mathbf{e}_{i+3}\ ,
  25. 𝐱 × 𝐲 = ( x 2 y 4 - x 4 y 2 + x 3 y 7 - x 7 y 3 + x 5 y 6 - x 6 y 5 ) 𝐞 1 + ( x 3 y 5 - x 5 y 3 + x 4 y 1 - x 1 y 4 + x 6 y 7 - x 7 y 6 ) 𝐞 2 + ( x 4 y 6 - x 6 y 4 + x 5 y 2 - x 2 y 5 + x 7 y 1 - x 1 y 7 ) 𝐞 3 + ( x 5 y 7 - x 7 y 5 + x 6 y 3 - x 3 y 6 + x 1 y 2 - x 2 y 1 ) 𝐞 4 + ( x 6 y 1 - x 1 y 6 + x 7 y 4 - x 4 y 7 + x 2 y 3 - x 3 y 2 ) 𝐞 5 + ( x 7 y 2 - x 2 y 7 + x 1 y 5 - x 5 y 1 + x 3 y 4 - x 4 y 3 ) 𝐞 6 + ( x 1 y 3 - x 3 y 1 + x 2 y 6 - x 6 y 2 + x 4 y 5 - x 5 y 4 ) 𝐞 7 . \begin{aligned}\displaystyle\mathbf{x}\times\mathbf{y}=(x_{2}y_{4}-x_{4}y_{2}+% x_{3}y_{7}-x_{7}y_{3}+x_{5}y_{6}-x_{6}y_{5})&\displaystyle\mathbf{e}_{1}\\ \displaystyle{}+(x_{3}y_{5}-x_{5}y_{3}+x_{4}y_{1}-x_{1}y_{4}+x_{6}y_{7}-x_{7}y% _{6})&\displaystyle\mathbf{e}_{2}\\ \displaystyle{}+(x_{4}y_{6}-x_{6}y_{4}+x_{5}y_{2}-x_{2}y_{5}+x_{7}y_{1}-x_{1}y% _{7})&\displaystyle\mathbf{e}_{3}\\ \displaystyle{}+(x_{5}y_{7}-x_{7}y_{5}+x_{6}y_{3}-x_{3}y_{6}+x_{1}y_{2}-x_{2}y% _{1})&\displaystyle\mathbf{e}_{4}\\ \displaystyle{}+(x_{6}y_{1}-x_{1}y_{6}+x_{7}y_{4}-x_{4}y_{7}+x_{2}y_{3}-x_{3}y% _{2})&\displaystyle\mathbf{e}_{5}\\ \displaystyle{}+(x_{7}y_{2}-x_{2}y_{7}+x_{1}y_{5}-x_{5}y_{1}+x_{3}y_{4}-x_{4}y% _{3})&\displaystyle\mathbf{e}_{6}\\ \displaystyle{}+(x_{1}y_{3}-x_{3}y_{1}+x_{2}y_{6}-x_{6}y_{2}+x_{4}y_{5}-x_{5}y% _{4})&\displaystyle\mathbf{e}_{7}\end{aligned}.
  26. T 𝐱 = [ 0 - x 4 - x 7 x 2 - x 6 x 5 x 3 x 4 0 - x 5 - x 1 x 3 - x 7 x 6 x 7 x 5 0 - x 6 - x 2 x 4 - x 1 - x 2 x 1 x 6 0 - x 7 - x 3 x 5 x 6 - x 3 x 2 x 7 0 - x 1 - x 4 - x 5 x 7 - x 4 x 3 x 1 0 - x 2 - x 3 - x 6 x 1 - x 5 x 4 x 2 0 ] . T_{\mathbf{x}}=\begin{bmatrix}0&-x_{4}&-x_{7}&x_{2}&-x_{6}&x_{5}&x_{3}\\ x_{4}&0&-x_{5}&-x_{1}&x_{3}&-x_{7}&x_{6}\\ x_{7}&x_{5}&0&-x_{6}&-x_{2}&x_{4}&-x_{1}\\ -x_{2}&x_{1}&x_{6}&0&-x_{7}&-x_{3}&x_{5}\\ x_{6}&-x_{3}&x_{2}&x_{7}&0&-x_{1}&-x_{4}\\ -x_{5}&x_{7}&-x_{4}&x_{3}&x_{1}&0&-x_{2}\\ -x_{3}&-x_{6}&x_{1}&-x_{5}&x_{4}&x_{2}&0\end{bmatrix}.
  27. 𝐱 × 𝐲 = T 𝐱 ( 𝐲 ) . \mathbf{x}\times\mathbf{y}=T_{\mathbf{x}}(\mathbf{y}).
  28. 𝐞 6 × ( 𝐞 6 × e 1 ) = - 𝐞 1 = 𝐞 6 × 𝐞 5 , \mathbf{e}_{6}\times\left(\mathbf{e}_{6}\times e_{1}\right)=-\mathbf{e}_{1}=% \mathbf{e}_{6}\times\mathbf{e}_{5},
  29. 𝐞 5 × 𝐞 6 = 𝐞 1 , \mathbf{e}_{5}\times\mathbf{e}_{6}=\mathbf{e}_{1},
  30. 𝐁 = 𝐱 𝐲 = 1 2 ( 𝐱𝐲 - 𝐲𝐱 ) . \mathbf{B}=\mathbf{x}\wedge\mathbf{y}=\frac{1}{2}(\mathbf{xy}-\mathbf{yx}).
  31. 𝐯 = 𝐞 124 + 𝐞 235 + 𝐞 346 + 𝐞 457 + 𝐞 561 + 𝐞 672 + 𝐞 713 . \mathbf{v}=\mathbf{e}_{124}+\mathbf{e}_{235}+\mathbf{e}_{346}+\mathbf{e}_{457}% +\mathbf{e}_{561}+\mathbf{e}_{672}+\mathbf{e}_{713}.
  32. 𝐱 × 𝐲 = - ( 𝐱 𝐲 ) 𝐯 \mathbf{x}\times\mathbf{y}=-(\mathbf{x}\wedge\mathbf{y})~{}\lrcorner~{}\mathbf% {v}
  33. \lrcorner
  34. 𝐱 × 𝐲 = Im ( 𝐱𝐲 ) = 1 2 ( 𝐱𝐲 - 𝐲𝐱 ) . \mathbf{x}\times\mathbf{y}=\mathrm{Im}(\mathbf{xy})=\frac{1}{2}(\mathbf{xy}-% \mathbf{yx}).
  35. ( a , 𝐱 ) ( b , 𝐲 ) = ( a b - 𝐱 𝐲 , a 𝐲 + b 𝐱 + 𝐱 × 𝐲 ) . (a,\mathbf{x})(b,\mathbf{y})=(ab-\mathbf{x}\cdot\mathbf{y},a\mathbf{y}+b% \mathbf{x}+\mathbf{x}\times\mathbf{y}).
  36. 𝐱 × ( 𝐲 × 𝐳 ) + 𝐲 × ( 𝐳 × 𝐱 ) + 𝐳 × ( 𝐱 × 𝐲 ) = - 3 2 [ 𝐱 , 𝐲 , 𝐳 ] \mathbf{x}\times(\mathbf{y}\times\mathbf{z})+\mathbf{y}\times(\mathbf{z}\times% \mathbf{x})+\mathbf{z}\times(\mathbf{x}\times\mathbf{y})=-\frac{3}{2}[\mathbf{% x},\mathbf{y},\mathbf{z}]
  37. ( 𝐚 1 × × 𝐚 k ) 𝐚 j = 0 \left(\mathbf{a}_{1}\times\ \cdots\ \times\mathbf{a}_{k}\right)\cdot\mathbf{a}% _{j}=0
  38. | 𝐚 1 × × 𝐚 k | 2 = det ( 𝐚 i 𝐚 j ) = | 𝐚 1 𝐚 1 𝐚 1 𝐚 2 𝐚 1 𝐚 k 𝐚 2 𝐚 1 𝐚 2 𝐚 2 𝐚 2 𝐚 k 𝐚 k 𝐚 1 𝐚 k 𝐚 2 𝐚 k 𝐚 k | |\mathbf{a}_{1}\times\ \cdots\ \times\mathbf{a}_{k}|^{2}=\det(\mathbf{a}_{i}% \cdot\mathbf{a}_{j})=\begin{vmatrix}\mathbf{a}_{1}\cdot\mathbf{a}_{1}&\mathbf{% a}_{1}\cdot\mathbf{a}_{2}&\dots&\mathbf{a}_{1}\cdot\mathbf{a}_{k}\\ \mathbf{a}_{2}\cdot\mathbf{a}_{1}&\mathbf{a}_{2}\cdot\mathbf{a}_{2}&\dots&% \mathbf{a}_{2}\cdot\mathbf{a}_{k}\\ \dots&\dots&\dots&\dots\\ \mathbf{a}_{k}\cdot\mathbf{a}_{1}&\mathbf{a}_{k}\cdot\mathbf{a}_{2}&\dots&% \mathbf{a}_{k}\cdot\mathbf{a}_{k}\\ \end{vmatrix}
  39. 𝐚 × 𝐛 × 𝐜 = ( 𝐚 𝐛 𝐜 ) ( 𝐰 - 𝐯𝐞 8 ) \mathbf{a}\times\mathbf{b}\times\mathbf{c}=(\mathbf{a}\wedge\mathbf{b}\wedge% \mathbf{c})~{}\lrcorner~{}(\mathbf{w}-\mathbf{ve}_{8})
  40. \lrcorner

Shadow_mapping.html

  1. [ 0.5 0 0 0.5 0 0.5 0 0.5 0 0 0.5 0.5 0 0 0 1 ] \begin{bmatrix}0.5&0&0&0.5\\ 0&0.5&0&0.5\\ 0&0&0.5&0.5\\ 0&0&0&1\end{bmatrix}

Shapiro–Wilk_test.html

  1. W = ( i = 1 n a i x ( i ) ) 2 i = 1 n ( x i - x ¯ ) 2 W={\left(\sum_{i=1}^{n}a_{i}x_{(i)}\right)^{2}\over\sum_{i=1}^{n}(x_{i}-% \overline{x})^{2}}
  2. x ( i ) x_{(i)}
  3. x ¯ = ( x 1 + + x n ) / n \overline{x}=\left(x_{1}+\cdots+x_{n}\right)/n
  4. a i a_{i}
  5. ( a 1 , , a n ) = m 𝖳 V - 1 ( m 𝖳 V - 1 V - 1 m ) 1 / 2 (a_{1},\dots,a_{n})={m^{\mathsf{T}}V^{-1}\over(m^{\mathsf{T}}V^{-1}V^{-1}m)^{1% /2}}
  6. m = ( m 1 , , m n ) 𝖳 m=(m_{1},\dots,m_{n})^{\mathsf{T}}\,
  7. m 1 , , m n m_{1},\ldots,m_{n}
  8. V V
  9. W W

Shear_rate.html

  1. γ ˙ = v h , \dot{\gamma}=\frac{v}{h},
  2. γ ˙ \dot{\gamma}
  3. v v
  4. h h
  5. γ ˙ i j = v i x j + v j x i . \dot{\gamma}_{ij}=\frac{\partial v_{i}}{\partial x_{j}}+\frac{\partial v_{j}}{% \partial x_{i}}.
  6. γ ˙ = 8 v d , \dot{\gamma}=\frac{8v}{d},
  7. γ ˙ \dot{\gamma}
  8. v v
  9. d d
  10. v = Q A , v=\frac{Q}{A},
  11. A = π r 2 , A=\pi r^{2},
  12. v = Q π r 2 . v=\frac{Q}{\pi r^{2}}.
  13. γ ˙ = 8 v d = 8 ( Q π r 2 ) 2 r , \dot{\gamma}=\frac{8v}{d}=\frac{8\left(\frac{Q}{\pi r^{2}}\right)}{2r},
  14. γ ˙ = 4 Q π r 3 . \dot{\gamma}=\frac{4Q}{\pi r^{3}}.
  15. τ w \tau_{w}
  16. τ w = γ ˙ x μ \tau_{w}=\dot{\gamma}_{x}\mu
  17. μ \mu

Shear_strength.html

  1. τ \tau
  2. τ = σ 1 - σ 3 2 , \tau=\frac{\sigma_{1}-\sigma_{3}}{2},
  3. σ 1 \sigma_{1}
  4. σ 3 \sigma_{3}
  5. τ \tau
  6. τ = F A = F π r b o l t 2 = 4 F π d b o l t 2 \tau=\frac{F}{A}=\frac{F}{\pi r_{bolt}^{2}}=\frac{4F}{\pi d_{bolt}^{2}}

Shearing_interferometer.html

  1. d f = λ 2 n θ d_{f}=\frac{\lambda}{2n\theta}
  2. λ \lambda
  3. θ \theta
  4. R R
  5. R = s d f λ sin γ R=\frac{s\cdot d_{f}}{\lambda\sin\gamma}
  6. s s
  7. d f d_{f}
  8. λ \lambda
  9. γ \gamma

Shephard's_lemma.html

  1. i i
  2. p i p_{i}
  3. h i ( 𝐩 , u ) = e ( 𝐩 , u ) p i h_{i}(\mathbf{p},u)=\frac{\partial e(\mathbf{p},u)}{\partial p_{i}}
  4. i i
  5. u u
  6. x i ( 𝐰 , y ) = c ( 𝐰 , y ) w i x_{i}(\mathbf{w},y)=\frac{\partial c(\mathbf{w},y)}{\partial w_{i}}
  7. i i
  8. y y
  9. e ( p 1 , p 2 , u ) e(p_{1},p_{2},u)
  10. = p 1 x 1 + p 2 x 2 + λ ( u - U ( x 1 , x 2 ) ) \mathcal{L}=p_{1}x_{1}+p_{2}x_{2}+\lambda(u-U(x_{1},x_{2}))
  11. e ( p 1 , p 2 , u ) e(p_{1},p_{2},u)
  12. p 1 p_{1}
  13. e p 1 = p 1 = x 1 h \frac{\partial e}{\partial p_{1}}=\frac{\partial\mathcal{L}}{\partial p_{1}}=x% _{1}^{h}
  14. x 1 h x_{1}^{h}

Sherman–Morrison_formula.html

  1. A A
  2. u v T uv^{T}
  3. u u
  4. v v
  5. A A
  6. u u
  7. v v
  8. 1 + v T A - 1 u 0 1+v^{T}A^{-1}u\neq 0
  9. ( A + u v T ) - 1 = A - 1 - A - 1 u v T A - 1 1 + v T A - 1 u . (A+uv^{T})^{-1}=A^{-1}-{A^{-1}uv^{T}A^{-1}\over 1+v^{T}A^{-1}u}.
  10. u v T uv^{T}
  11. u u
  12. v v
  13. A A
  14. A A
  15. u v T uv^{T}
  16. A + u v T A+uv^{T}
  17. A - 1 A^{-1}
  18. u u
  19. v v
  20. A A
  21. A - 1 A^{-1}
  22. n n
  23. n n
  24. u u
  25. v v
  26. n n
  27. 3 n 2 3n^{2}
  28. u u
  29. 2 n 2 2n^{2}
  30. v v
  31. u u
  32. v v
  33. n 2 n^{2}
  34. Y Y
  35. X X
  36. A + u v T A+uv^{T}
  37. X Y = Y X = I XY=YX=I
  38. Y Y
  39. X Y = I XY=I
  40. X Y = ( A + u v T ) ( A - 1 - A - 1 u v T A - 1 1 + v T A - 1 u ) XY=(A+uv^{T})\left(A^{-1}-{A^{-1}uv^{T}A^{-1}\over 1+v^{T}A^{-1}u}\right)
  41. = A A - 1 + u v T A - 1 - A A - 1 u v T A - 1 + u v T A - 1 u v T A - 1 1 + v T A - 1 u =AA^{-1}+uv^{T}A^{-1}-{AA^{-1}uv^{T}A^{-1}+uv^{T}A^{-1}uv^{T}A^{-1}\over 1+v^{% T}A^{-1}u}
  42. = I + u v T A - 1 - u v T A - 1 + u v T A - 1 u v T A - 1 1 + v T A - 1 u =I+uv^{T}A^{-1}-{uv^{T}A^{-1}+uv^{T}A^{-1}uv^{T}A^{-1}\over 1+v^{T}A^{-1}u}
  43. = I + u v T A - 1 - u ( 1 + v T A - 1 u ) v T A - 1 1 + v T A - 1 u =I+uv^{T}A^{-1}-{u(1+v^{T}A^{-1}u)v^{T}A^{-1}\over 1+v^{T}A^{-1}u}
  44. v T A - 1 u v^{T}A^{-1}u
  45. ( 1 + v T A - 1 u ) (1+v^{T}A^{-1}u)
  46. X Y = I + u v T A - 1 - u v T A - 1 = I . XY=I+uv^{T}A^{-1}-uv^{T}A^{-1}=I.\,
  47. Y X = ( A - 1 - A - 1 u v T A - 1 1 + v T A - 1 u ) ( A + u v T ) = I . YX=\left(A^{-1}-{A^{-1}uv^{T}A^{-1}\over 1+v^{T}A^{-1}u}\right)(A+uv^{T})=I.
  48. ( I + w v T ) - 1 = I - w v T 1 + v T w (I+wv^{T})^{-1}=I-\frac{wv^{T}}{1+v^{T}w}
  49. u = A w u=Aw
  50. A + u v T = A ( I + w v T ) A+uv^{T}=A\left(I+wv^{T}\right)
  51. ( A + u v T ) - 1 = ( I + w v T ) - 1 A - 1 = ( I - w v T 1 + v T w ) A - 1 (A+uv^{T})^{-1}=(I+wv^{T})^{-1}{A^{-1}}=\left(I-\frac{wv^{T}}{1+v^{T}w}\right)% A^{-1}
  52. w = A - 1 u w={{A}^{-1}}u
  53. ( A + u v T ) - 1 = ( I - A - 1 u v T 1 + v T A - 1 u ) A - 1 = A - 1 - A - 1 u v T A - 1 1 + v T A - 1 u (A+uv^{T})^{-1}=\left(I-\frac{A^{-1}uv^{T}}{1+v^{T}A^{-1}u}\right)A^{-1}={A^{-% 1}}-\frac{A^{-1}uv^{T}A^{-1}}{1+v^{T}A^{-1}u}

Shortest_common_supersequence_problem.html

  1. [ 1.. m ] = a b c b d a b [1..m]=abcbdab
  2. [ 1.. n ] = b d c a b a [1..n]=bdcaba
  3. [ 1.. r ] = b c b a [1..r]=bcba
  4. [ 1.. t ] = a b d c a b d a b [1..t]=abdcabdab
  5. r + t = m + n r+t=m+n

Shriek.html

  1. f ! f_{!}
  2. f ! f^{!}

Sigma_model.html

  1. ( ϕ 1 , ϕ 2 , , ϕ n ) = i = 1 n j = 1 n g i j d ϕ i * d ϕ j \mathcal{L}(\phi_{1},\phi_{2},\ldots,\phi_{n})=\sum_{i=1}^{n}\sum_{j=1}^{n}g_{% ij}\;\mathrm{d}\phi_{i}\wedge{*\mathrm{d}\phi_{j}}
  2. g < s u b > i j g<sub>ij

Signed_measure.html

  1. μ : Σ { , - } \mu:\Sigma\to\mathbb{R}\cup\{\infty,-\infty\}
  2. μ ( ) = 0 \mu(\emptyset)=0
  3. μ \mu
  4. μ ( n = 1 A n ) = n = 1 μ ( A n ) \mu\left(\bigcup_{n=1}^{\infty}A_{n}\right)=\sum_{n=1}^{\infty}\mu(A_{n})
  5. X | f ( x ) | d ν ( x ) < . \int_{X}\!|f(x)|\,d\nu(x)<\infty.
  6. μ ( A ) = A f ( x ) d ν ( x ) \mu(A)=\int_{A}\!f(x)\,d\nu(x)
  7. X f - ( x ) d ν ( x ) < , \int_{X}\!f^{-}(x)\,d\nu(x)<\infty,
  8. μ + ( E ) = μ ( P E ) \mu^{+}(E)=\mu(P\cap E)
  9. μ - ( E ) = - μ ( N E ) \mu^{-}(E)=-\mu(N\cap E)

Signed_zero.html

  1. 0 + 0^{+}
  2. 0 - 0^{-}
  3. - 0 | x | = - 0 \frac{-0}{\left|x\right|}=-0\,\!
  4. x x
  5. ( - 0 ) ( - 0 ) = + 0 (-0)\cdot(-0)=+0\,\!
  6. x + ( ± 0 ) = x x+(\pm 0)=x\,\!
  7. x x
  8. ( - 0 ) + ( - 0 ) = ( - 0 ) - ( + 0 ) = - 0 (-0)+(-0)=(-0)-(+0)=-0\,\!
  9. ( + 0 ) + ( + 0 ) = ( + 0 ) - ( - 0 ) = + 0 (+0)+(+0)=(+0)-(-0)=+0\,\!
  10. x - x = x + ( - x ) = + 0 x-x=x+(-x)=+0\,\!
  11. x x
  12. - 0 = - 0 \sqrt{-0}=-0\,\!
  13. - 0 - = + 0 \frac{-0}{-\infty}=+0\,\!
  14. | x | - 0 = - \frac{\left|x\right|}{-0}=-\infty\,\!
  15. x x
  16. ± 0 × ± = NaN {\pm 0}\times{\pm\infty}=\mbox{NaN}~{}\,\!
  17. ± 0 ± 0 = NaN \frac{\pm 0}{\pm 0}=\mbox{NaN}~{}\,\!

Simple_linear_regression.html

  1. n n
  2. y y
  3. x x
  4. ( x ¯ , y ¯ ) (\overline{x},\overline{y})
  5. n n
  6. y i = α + β x i + ε i . y_{i}=\alpha+\beta x_{i}+\varepsilon_{i}.
  7. y = α + β x , y=\alpha+\beta x,
  8. α α
  9. y y
  10. β β
  11. Find min α , β Q ( α , β ) , for Q ( α , β ) = i = 1 n ε i 2 = i = 1 n ( y i - α - β x i ) 2 \,\text{Find }\min_{\alpha,\,\beta}Q(\alpha,\beta),\qquad\,\text{for }Q(\alpha% ,\beta)=\sum_{i=1}^{n}\varepsilon_{i}^{\,2}=\sum_{i=1}^{n}(y_{i}-\alpha-\beta x% _{i})^{2}
  12. α α
  13. β β
  14. α α
  15. β β
  16. Q Q
  17. β ^ \displaystyle\hat{\beta}
  18. x x
  19. y y
  20. x x
  21. y y
  22. x y ¯ = 1 n i = 1 n x i y i . \overline{xy}=\tfrac{1}{n}\sum_{i=1}^{n}x_{i}y_{i}.
  23. α ^ \hat{\alpha}
  24. β ^ \hat{\beta}
  25. f = α ^ + β ^ x , f=\hat{\alpha}+\hat{\beta}x,
  26. f - y ¯ s y = r x y x - x ¯ s x \frac{f-\bar{y}}{s_{y}}=r_{xy}\frac{x-\bar{x}}{s_{x}}
  27. r x y = x y ¯ - x ¯ y ¯ ( x 2 ¯ - x ¯ 2 ) ( y 2 ¯ - y ¯ 2 ) r_{xy}=\frac{\overline{xy}-\bar{x}\bar{y}}{\sqrt{(\overline{x^{2}}-\bar{x}^{2}% )(\overline{y^{2}}-\bar{y}^{2})}}
  28. r x y 2 r_{xy}^{2}
  29. y = β x y=βx
  30. β β
  31. β ^ = i = 1 n x i y i i = 1 n x i 2 = x y ¯ x 2 ¯ \hat{\beta}=\frac{\sum_{i=1}^{n}{x_{i}y_{i}}}{\sum_{i=1}^{n}{x_{i}^{2}}}=\frac% {\overline{xy}}{\overline{x^{2}}}
  32. r x y = x y ¯ ( x 2 ¯ ) ( y 2 ¯ ) r_{xy}=\frac{\overline{xy}}{\sqrt{(\overline{x^{2}})(\overline{y^{2}})}}
  33. ( x , y ) ( x - h , y - k ) (x,y)→(x-h,y-k)
  34. ( h , k ) (h,k)
  35. β ^ \displaystyle\hat{\beta}
  36. ( x ¯ , y ¯ ) (\overline{x},\overline{y})
  37. i = 1 n ε ^ i = 0. \textstyle\sum_{i=1}^{n}\hat{\varepsilon}_{i}=0.
  38. x x
  39. i = 1 n x i ε ^ i = 0. \textstyle\sum_{i=1}^{n}x_{i}\hat{\varepsilon}_{i}=0.
  40. α ^ \hat{\alpha}
  41. β ^ \hat{\beta}
  42. x x
  43. y y
  44. α + β x α+βx
  45. ε ε
  46. x x
  47. α ^ \hat{\alpha}
  48. β ^ \hat{\beta}
  49. α α
  50. β β
  51. α α
  52. β β
  53. α ^ \hat{\alpha}
  54. β ^ \hat{\beta}
  55. n n
  56. β β
  57. σ 2 / ( x i - x ¯ ) 2 , \sigma^{2}/\sum(x_{i}-\bar{x})^{2},
  58. Q Q
  59. n 2 n−2
  60. β ^ . \hat{\beta}.
  61. t t
  62. t = β ^ - β s β ^ t n - 2 , t=\frac{\hat{\beta}-\beta}{s_{\hat{\beta}}}\ \sim\ t_{n-2},
  63. s β ^ = 1 n - 2 i = 1 n ε ^ i 2 i = 1 n ( x i - x ¯ ) 2 s_{\hat{\beta}}=\sqrt{\frac{\tfrac{1}{n-2}\sum_{i=1}^{n}\hat{\varepsilon}_{i}^% {\,2}}{\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}}}
  64. β ^ . \hat{\beta}.
  65. t t
  66. t t
  67. n 2 n−2
  68. β β
  69. β [ β ^ - s β ^ t n - 2 * , β ^ + s β ^ t n - 2 * ] , \beta\in\left[\hat{\beta}-s_{\hat{\beta}}t^{*}_{n-2},\ \hat{\beta}+s_{\hat{% \beta}}t^{*}_{n-2}\right],
  70. ( 1 γ ) (1−γ)
  71. t n - 2 * t^{*}_{n-2}
  72. ( 1 γ 2 ) (1−\frac{γ}{2})
  73. γ = 0.05 γ=0.05
  74. α α
  75. α [ α ^ - s α ^ t n - 2 * , α ^ + s α ^ t n - 2 * ] , \alpha\in\left[\hat{\alpha}-s_{\hat{\alpha}}t^{*}_{n-2},\ \hat{\alpha}+s_{\hat% {\alpha}}t^{*}_{n-2}\right],
  76. s α ^ = s β ^ 1 n i = 1 n x i 2 = 1 n ( n - 2 ) ( j = 1 n ε ^ j 2 ) i = 1 n x i 2 i = 1 n ( x i - x ¯ ) 2 s_{\hat{\alpha}}=s_{\hat{\beta}}\sqrt{\tfrac{1}{n}\textstyle\sum_{i=1}^{n}x_{i% }^{2}}=\sqrt{\tfrac{1}{n(n-2)}\left(\textstyle\sum_{j=1}^{n}\hat{\varepsilon}_% {j}^{\,2}\right)\frac{\sum_{i=1}^{n}x_{i}^{2}}{\sum_{i=1}^{n}(x_{i}-\bar{x})^{% 2}}}
  77. α α
  78. β β
  79. α ^ = 0.859 , β ^ = - 1.817. \hat{\alpha}=0.859,\qquad\hat{\beta}=-1.817.
  80. α [ 0.76 , 0.96 ] , β [ - 2.06 , - 1.58 ] . \alpha\in\left[0.76,0.96\right],\qquad\beta\in\left[-2.06,-1.58\right].
  81. y ^ | x = ξ [ α ^ + β ^ ξ ± t n - 2 * ( 1 n - 2 ε ^ i 2 ) ( 1 n + ( ξ - x ¯ ) 2 ( x i - x ¯ ) 2 ) ] . \hat{y}|_{x=\xi}\in\left[\hat{\alpha}+\hat{\beta}\xi\pm t^{*}_{n-2}\sqrt{\left% (\frac{1}{n-2}\sum\hat{\varepsilon}_{i}^{\,2}\right)\cdot\left(\frac{1}{n}+% \frac{(\xi-\bar{x})^{2}}{\sum(x_{i}-\bar{x})^{2}}\right)}\right].
  82. 1 n 2 1\frac{n}{−2}
  83. 1 n \frac{1}{n}
  84. n n
  85. S x = x i = 24.76 , S y = y i = 931.17 \displaystyle S_{x}=\sum x_{i}=24.76,\quad S_{y}=\sum y_{i}=931.17
  86. β ^ \displaystyle\hat{\beta}
  87. α α
  88. β β
  89. α [ α ^ t 13 * s α ] = [ - 45.4 , - 32.7 ] \displaystyle\alpha\in[\,\hat{\alpha}\mp t^{*}_{13}s_{\alpha}\,]=[\,{-45.4},\ % {-32.7}\,]
  90. r ^ = n S x y - S x S y ( n S x x - S x 2 ) ( n S y y - S y 2 ) = 0.9945 \hat{r}=\frac{nS_{xy}-S_{x}S_{y}}{\sqrt{(nS_{xx}-S_{x}^{2})(nS_{yy}-S_{y}^{2})% }}=0.9945
  91. β ^ = 61.6746 , α ^ = - 39.7468. \hat{\beta}=61.6746,\qquad\hat{\alpha}=-39.7468.
  92. α ^ , β ^ \hat{\alpha},\hat{\beta}
  93. min α ^ , β ^ SSE ( α ^ , β ^ ) \underset{\hat{\alpha},\hat{\beta}}{\mathrm{min}}\,\mathrm{SSE}\left(\hat{% \alpha},\hat{\beta}\right)
  94. SSE ( α ^ , β ^ ) = i = 1 n ( y i - α ^ - β ^ x i ) 2 \mathrm{SSE}\left(\hat{\alpha},\hat{\beta}\right)=\sum_{i=1}^{n}\left(y_{i}-% \hat{\alpha}-\hat{\beta}x_{i}\right)^{2}
  95. α ^ \hat{\alpha}
  96. β ^ \hat{\beta}
  97. SSE ( α ^ , β ^ ) α ^ = - 2 i = 1 n ( y i - α ^ - β ^ x i ) = 0 \displaystyle\frac{\partial\,\mathrm{SSE}\left(\hat{\alpha},\hat{\beta}\right)% }{\partial\hat{\alpha}}=-2\sum_{i=1}^{n}\left(y_{i}-\hat{\alpha}-\hat{\beta}x_% {i}\right)=0
  98. i = 1 n ( y i - α ^ - β ^ x i ) = 0 \displaystyle\sum_{i=1}^{n}\left(y_{i}-\hat{\alpha}-\hat{\beta}x_{i}\right)=0
  99. i = 1 n y i = i = 1 n α ^ + β ^ i = 1 n x i \displaystyle\sum_{i=1}^{n}y_{i}=\sum_{i=1}^{n}\hat{\alpha}+\hat{\beta}\sum_{i% =1}^{n}x_{i}
  100. 1 n \frac{1}{n}
  101. 1 n i = 1 n y i = α ^ 1 n i = 1 n 1 + β ^ 1 n i = 1 n x i . \displaystyle\frac{1}{n}\sum_{i=1}^{n}y_{i}=\hat{\alpha}\frac{1}{n}\sum_{i=1}^% {n}1+\hat{\beta}\frac{1}{n}\sum_{i=1}^{n}x_{i}.
  102. y ¯ = α ^ + β ^ x ¯ \displaystyle\bar{y}=\hat{\alpha}+\hat{\beta}\bar{x}
  103. β ^ \hat{\beta}
  104. α ^ \hat{\alpha}
  105. min α ^ , β ^ i = 1 n ( y i - ( y ¯ - β ^ x ¯ ) - β ^ x i ) 2 \displaystyle\underset{\hat{\alpha},\hat{\beta}}{\mathrm{min}}\sum_{i=1}^{n}% \left(y_{i}-\left(\bar{y}-\hat{\beta}\bar{x}\right)-\hat{\beta}x_{i}\right)^{2}
  106. min α ^ , β ^ i = 1 n [ ( y i - y ¯ ) - β ^ ( x i - x ¯ ) ] 2 \displaystyle\underset{\hat{\alpha},\hat{\beta}}{\mathrm{min}}\sum_{i=1}^{n}% \left[\left(y_{i}-\bar{y}\right)-\hat{\beta}\left(x_{i}-\bar{x}\right)\right]^% {2}
  107. β ^ \hat{\beta}
  108. SSE ( α ^ , β ^ ) β ^ = - 2 i = 1 n [ ( y i - y ¯ ) - β ^ ( x i - x ¯ ) ] ( x i - x ¯ ) = 0 \displaystyle\frac{\partial\,\mathrm{SSE}\left(\hat{\alpha},\hat{\beta}\right)% }{\partial\hat{\beta}}=-2\sum_{i=1}^{n}\left[\left(y_{i}-\bar{y}\right)-\hat{% \beta}\left(x_{i}-\bar{x}\right)\right]\left(x_{i}-\bar{x}\right)=0
  109. i = 1 n ( y i - y ¯ ) ( x i - x ¯ ) - β ^ i = 1 n ( x i - x ¯ ) 2 = 0 \displaystyle\sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)\left(x_{i}-\bar{x}\right% )-\hat{\beta}\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2}=0
  110. β ^ = i = 1 n ( y i - y ¯ ) ( x i - x ¯ ) i = 1 n ( x i - x ¯ ) 2 = C o v ( x , y ) V a r ( x ) \displaystyle\hat{\beta}=\frac{}{}\frac{\sum_{i=1}^{n}\left(y_{i}-\bar{y}% \right)\left(x_{i}-\bar{x}\right)}{\sum_{i=1}^{n}\left(x_{i}-\bar{x}\right)^{2% }}=\frac{Cov\left(x,y\right)}{Var\left(x\right)}
  111. β ^ \hat{\beta}
  112. α ^ \hat{\alpha}
  113. α ^ = y ¯ - β ^ x ¯ \displaystyle\hat{\alpha}=\bar{y}-\hat{\beta}\bar{x}

Simple_set.html

  1. I I\subseteq\mathbb{N}
  2. I I
  3. e e
  4. W e infinite W e I W_{e}\,\text{ infinite}\implies W_{e}\not\subseteq I
  5. I I
  6. S S\subseteq\mathbb{N}
  7. I I\subseteq\mathbb{N}
  8. I I
  9. f f
  10. e e
  11. W e I # ( W e ) < f ( e ) W_{e}\subseteq I\implies\#(W_{e})<f(e)
  12. S S\subseteq\mathbb{N}
  13. I I\subseteq\mathbb{N}
  14. I I
  15. p I p_{I}
  16. p I p_{I}
  17. I I
  18. S S\subseteq\mathbb{N}

Simple_shear.html

  1. V x = f ( x , y ) \ V_{x}=f(x,y)
  2. V y = V z = 0 \ V_{y}=V_{z}=0
  3. V x y = γ ˙ \frac{\partial V_{x}}{\partial y}=\dot{\gamma}
  4. γ ˙ \dot{\gamma}
  5. V x x = V x z = 0 \frac{\partial V_{x}}{\partial x}=\frac{\partial V_{x}}{\partial z}=0
  6. Γ \Gamma
  7. Γ = [ 0 γ ˙ 0 0 0 0 0 0 0 ] \Gamma=\begin{bmatrix}0&{\dot{\gamma}}&0\\ 0&0&0\\ 0&0&0\end{bmatrix}
  8. γ ˙ \dot{\gamma}
  9. γ ˙ 2 \dot{\gamma}\over 2
  10. γ ˙ 2 \dot{\gamma}\over 2
  11. Γ = [ 0 γ ˙ 0 0 0 0 0 0 0 ] simple shear = [ 0 γ ˙ 2 0 γ ˙ 2 0 0 0 0 0 ] pure shear + [ 0 γ ˙ 2 0 - γ ˙ 2 0 0 0 0 0 ] solid rotation 10.12 11 𝐞 1 10.13 12 𝐞 1 - 𝐞 2 10.14 13 symbolF = [ 1 γ 0 0 1 0 0 0 1 ] . 10.15 14 symbolF = symbol 1 + γ 𝐞 1 𝐞 2 . \Gamma=\begin{matrix}\underbrace{}\begin{bmatrix}0&{\dot{\gamma}}&0\\ 0&0&0\\ 0&0&0\end{bmatrix}\\ \mbox{simple shear}=\begin{matrix}\underbrace{}\begin{bmatrix}0&{\dot{\gamma}% \over 2}&0\\ {\dot{\gamma}\over 2}&0&0\\ 0&0&0\end{bmatrix}\\ \mbox{pure shear}+\begin{matrix}\underbrace{}\begin{bmatrix}0&{\dot{\gamma}% \over 2}&0\\ {-{\dot{\gamma}\over 2}}&0&0\\ 0&0&0\end{bmatrix}\\ \mbox{solid rotation}$$\par \par \@@section{subsection}{S10.SS12}{10.12}{10.12% }{{\@tag[][]{10.12}11}}{{\@tag[][]{10.12}11}}\par $\mathbf{e}_{1}$\par \par % \@@section{subsection}{S10.SS13}{10.13}{10.13}{{\@tag[][]{10.13}12}}{{\@tag[][% ]{10.13}12}}\par $\mathbf{e}_{1}-\mathbf{e}_{2}$\par \par \@@section{% subsection}{S10.SS14}{10.14}{10.14}{{\@tag[][]{10.14}13}}{{\@tag[][]{10.14}13}% }\par $symbol{F} = \begin{bmatrix}1&\gamma&0\\ 0&1&0\\ 0&0&1\end{bmatrix}.$\par \par \@@section{subsection}{S10.SS15}{10.15}{10.15}{{% \@tag[][]{10.15}14}}{{\@tag[][]{10.15}14}}\par $symbol{F} = symbol{\mathit{1}}% + \gamma\mathbf{e}_{1}\otimes\mathbf{e}_{2}.$\end{document}\end{matrix}\end{% matrix}\end{matrix}
  12. 𝐞 1 \mathbf{e}_{1}
  13. 𝐞 1 - 𝐞 2 \mathbf{e}_{1}-\mathbf{e}_{2}
  14. s y m b o l F = [ 1 γ 0 0 1 0 0 0 1 ] . symbol{F}=\begin{bmatrix}1&\gamma&0\\ 0&1&0\\ 0&0&1\end{bmatrix}.
  15. s y m b o l F = s y m b o l 1 + γ 𝐞 1 𝐞 2 . symbol{F}=symbol{\mathit{1}}+\gamma\mathbf{e}_{1}\otimes\mathbf{e}_{2}.

Simplicial_manifold.html

  1. G n G^{n}

Sine_and_cosine_transforms.html

  1. f ( t ) f(t)
  2. f ^ s {\hat{f}}^{s}
  3. s ( f ) {\mathcal{F}}_{s}(f)
  4. - f ( t ) sin ( 2 π ν t ) d t . \int_{-\infty}^{\infty}f(t)\sin(2\pi\nu t)\,dt.
  5. t t
  6. ν ν
  7. ν ν
  8. f ^ s ( ν ) = - f ^ s ( - ν ) . {\hat{f}}^{s}(\nu)=-{\hat{f}}^{s}(-\nu).
  9. 1 2 . \tfrac{1}{\sqrt{2}}.
  10. f ( t ) f(t)
  11. f ^ c {\hat{f}}^{c}
  12. c ( f ) {\mathcal{F}}_{c}(f)
  13. - f ( t ) cos ( 2 π ν t ) d t . \int_{-\infty}^{\infty}f(t)\cos(2\pi\nu t)\,dt.
  14. ν ν
  15. f ^ c ( ν ) = f ^ c ( - ν ) . {\hat{f}}^{c}(\nu)={\hat{f}}^{c}(-\nu).
  16. t t
  17. 2 0 f ( t ) cos ( 2 π ν t ) d t . 2\int_{0}^{\infty}f(t)\cos(2\pi\nu t)\,dt.
  18. f f
  19. 2 0 f ( t ) sin ( 2 π ν t ) d t . 2\int_{0}^{\infty}f(t)\sin(2\pi\nu t)\,dt.
  20. f f
  21. f f
  22. f ( t ) = 0 f ^ c cos ( 2 π ν t ) d ν + 0 f ^ s sin ( 2 π ν t ) d ν , f(t)=\int_{0}^{\infty}{\hat{f}}^{c}\cos(2\pi\nu t)d\nu+\int_{0}^{\infty}{\hat{% f}}^{s}\sin(2\pi\nu t)d\nu,
  23. π 2 ( f ( x + 0 ) + f ( x - 0 ) ) = 0 - f ( t ) cos ( ω ( t - x ) ) d t d ω , \tfrac{\pi}{2}\left(f(x+0)+f(x-0)\right)=\int_{0}^{\infty}\int_{-\infty}^{% \infty}f(t)\cos(\omega(t-x))dtd\omega,
  24. f ( x + 0 ) f(x+0)
  25. f f
  26. x x
  27. f ( x 0 ) f(x−0)
  28. f f
  29. x x
  30. f f
  31. f f
  32. f ^ ( ν ) = - f ( t ) e - 2 π i ν t d t = - f ( t ) ( cos ( 2 π ν t ) - i sin ( 2 π ν t ) ) d t Euler’s Formula = ( - f ( t ) cos ( 2 π ν t ) d t ) - i ( - f ( t ) sin ( 2 π ν t ) d t ) = f ^ c ( ν ) - i f ^ s ( ν ) \begin{aligned}\displaystyle\hat{f}(\nu)&\displaystyle=\int_{-\infty}^{\infty}% f(t)e^{-2\pi i\nu t}\,dt\\ &\displaystyle=\int_{-\infty}^{\infty}f(t)(\cos(2\pi\nu t)-i\,\sin(2\pi\nu t))% \,dt&&\displaystyle\,\text{Euler's Formula}\\ &\displaystyle=\left(\int_{-\infty}^{\infty}f(t)\cos(2\pi\nu t)\,dt\right)-i% \left(\int_{-\infty}^{\infty}f(t)\sin(2\pi\nu t)\,dt\right)\\ &\displaystyle={\hat{f}}^{c}(\nu)-i{\hat{f}}^{s}(\nu)\end{aligned}

Singly_and_doubly_even.html

  1. q = 2 ν a b q=2^{\nu}\frac{a}{b}
  2. | n | 2 = 2 - ν 2 ( n ) , |n|_{2}=2^{-\nu_{2}(n)},
  3. 2 = a / b , \sqrt{2}=a/b,
  4. 1 2 = ν 2 ( a ) - ν 2 ( b ) , \frac{1}{2}=\nu_{2}(a)-\nu_{2}(b),
  5. | 2 b 2 - a 2 | 1 |2b^{2}-a^{2}|\geq 1
  6. 1 3 b 2 \frac{1}{3b^{2}}
  7. | 2 - a / b | |\sqrt{2}-a/b|
  8. tanh 1 2 = e - 1 e + 1 = 0 + 1 2 + 1 6 + 1 10 + 1 14 + 1 \tanh\frac{1}{2}=\frac{e-1}{e+1}=0+\cfrac{1}{2+\cfrac{1}{6+\cfrac{1}{10+\cfrac% {1}{14+\cfrac{1}{\ddots}}}}}
  9. e e

Sipser–Lautemann_theorem.html

  1. R = { 1 , 0 } | r | R=\{1,0\}^{|r|}
  2. | A ( x ) | | R | > 1 - 1 2 | x | \frac{|A(x)|}{|R|}>1-\frac{1}{2^{|x|}}
  3. t 1 , t 2 , , t | r | \exists t_{1},t_{2},\ldots,t_{|r|}
  4. t i { 1 , 0 } | r | t_{i}\in\{1,0\}^{|r|}
  5. i A ( x ) t i = R . \bigcup_{i}A(x)\oplus t_{i}=R.
  6. S = i A ( x ) t i S=\bigcup_{i}A(x)\oplus t_{i}
  7. Pr [ r S ] = Pr [ r A ( x ) t 1 ] Pr [ r A ( x ) t 2 ] Pr [ r A ( x ) t | r | ] 1 2 | x | | r | . \Pr[r\notin S]=\Pr[r\notin A(x)\oplus t_{1}]\cdot\Pr[r\notin A(x)\oplus t_{2}]% \cdots\Pr[r\notin A(x)\oplus t_{|r|}]\leq{\frac{1}{2^{|x|\cdot|r|}}}.
  8. Pr [ i ( r i S ) ] i 1 2 | x | | r | = 1 2 | x | < 1. \Pr\Bigl[\bigvee_{i}(r_{i}\notin S)\Bigr]\leq\sum_{i}\frac{1}{2^{|x|\cdot|r|}}% =\frac{1}{2^{|x|}}<1.
  9. Pr [ S = R ] 1 - 1 2 | x | . \Pr[S=R]\geq 1-\frac{1}{2^{|x|}}.
  10. t 1 , t 2 , , t | r | t_{1},t_{2},\ldots,t_{|r|}
  11. i A ( x ) t i = R . \bigcup_{i}A(x)\oplus t_{i}=R.
  12. R = i A ( x ) t i R=\bigcup_{i}A(x)\oplus t_{i}
  13. | A ( x ) | | R | 1 2 | k | ¬ t 1 , t 2 , , t r \frac{|A(x)|}{|R|}\leq\frac{1}{2^{|k|}}\implies\neg\exists t_{1},t_{2},\dots,t% _{r}
  14. x L t 1 , t 2 , , t | r | r R 1 i | r | ( M ( x , r t i ) accepts ) . x\in L\iff\exists t_{1},t_{2},\dots,t_{|r|}\,\forall r\in R\bigvee_{1\leq i% \leq|r|}(M(x,r\oplus t_{i})\,\text{ accepts}).
  15. Pr r ( A ( x , r ) = right answer ) 1 - 1 3 m , {\rm Pr}_{r}(A(x,r)=\mbox{right answer}~{})\geq 1-\frac{1}{3m},
  16. | r | = m = | x | O ( 1 ) |r|=m=|x|^{O(1)}
  17. | x | O ( 1 ) |x|^{O(1)}
  18. Pr r ( A ( x , r ) = wrong answer ) 1 / 3 \Pr_{r}(A^{\prime}(x,r)=\mbox{wrong answer}~{})\leq 1/3
  19. Pr r ( A ( x , r ) = wrong answer ) 2 - c k ( n ) . {\rm Pr}_{r}(A(x,r)=\mbox{wrong answer}~{})\leq 2^{-ck(n)}.
  20. k ( n ) = Θ ( log m ( n ) ) k(n)=\Theta(\log m^{\prime}(n))
  21. 1 2 c k ( n ) 1 3 k ( n ) m ( n ) . \frac{1}{2^{ck(n)}}\leq\frac{1}{3k(n)m^{\prime}(n)}.
  22. x L y 1 , , y m { 0 , 1 } m z { 0 , 1 } m i = 1 m A ( x , y i z ) = 1. x\in L\iff\exists y_{1},\dots,y_{m}\in\{0,1\}^{m}\,\forall z\in\{0,1\}^{m}% \bigvee_{i=1}^{m}A(x,y_{i}\oplus z)=1.
  23. x L x\in L
  24. Pr y 1 , , y m ( z A ( x , y 1 z ) = = A ( x , y m z ) = 0 ) z { 0 , 1 } m Pr y 1 , , y m ( A ( x , y 1 z ) = = A ( x , y m z ) = 0 ) 2 m 1 ( 3 m ) m < 1. \begin{aligned}\displaystyle{\rm Pr}_{y_{1},\dots,y_{m}}(\exists z&% \displaystyle A(x,y_{1}\oplus z)=\dots=A(x,y_{m}\oplus z)=0)\\ &\displaystyle\leq\sum_{z\in\{0,1\}^{m}}{\rm Pr}_{y_{1},\dots,y_{m}}(A(x,y_{1}% \oplus z)=\dots=A(x,y_{m}\oplus z)=0)\\ &\displaystyle\leq 2^{m}\frac{1}{(3m)^{m}}\\ &\displaystyle<1.\end{aligned}
  25. Pr y 1 , , y m ( z i A ( x , y i z ) ) = 1 - Pr y 1 , , y m ( z A ( x , y 1 z ) = = A ( x , y m z ) = 0 ) . {\rm Pr}_{y_{1},\dots,y_{m}}\Bigl(\forall z\bigvee_{i}A(x,y_{i}\oplus z)\Bigr)% =1-{\rm Pr}_{y_{1},...,y_{m}}(\exists zA(x,y_{1}\oplus z)=\dots=A(x,y_{m}% \oplus z)=0).
  26. ( y 1 , , y m ) (y_{1},\dots,y_{m})
  27. x L x\notin L
  28. Pr z ( i A ( x , y i z ) ) i Pr z ( A ( x , y i z ) = 1 ) m 1 3 m = 1 3 . {\rm Pr}_{z}\Bigl(\bigvee_{i}A(x,y_{i}\oplus z)\Bigr)\leq\sum_{i}{\rm Pr}_{z}(% A(x,y_{i}\oplus z)=1)\leq m\frac{1}{3m}=\frac{1}{3}.
  29. Pr z ( A ( x , y 1 z ) = = A ( x , y m z ) = 0 ) = 1 - Pr z ( i A ( x , y i z ) ) 2 3 > 0. {\rm Pr}_{z}(A(x,y_{1}\oplus z)=\dots=A(x,y_{m}\oplus z)=0)=1-{\rm Pr}_{z}% \Bigl(\bigvee_{i}A(x,y_{i}\oplus z)\Bigr)\geq\frac{2}{3}>0.
  30. i A ( x , y i z ) = 0 \bigvee_{i}A(x,y_{i}\oplus z)=0
  31. y 1 , , y m { 0 , 1 } m . y_{1},...,y_{m}\in\{0,1\}^{m}.
  32. 𝖡𝖯𝖯 𝖬𝖠 𝖲 2 P Σ 2 Π 2 \mathsf{BPP}\subseteq\mathsf{MA}\subseteq\mathsf{S}_{2}^{P}\subseteq\Sigma_{2}% \cap\Pi_{2}

Situation_calculus.html

  1. ( x , y ) (x,y)
  2. P o s s Poss
  3. m o v e ( x , y ) move(x,y)
  4. ( x , y ) (x,y)
  5. p i c k u p ( o ) pickup(o)
  6. o o
  7. S 0 S_{0}
  8. d o do
  9. r e s u l t result
  10. d o ( a , s ) do(a,s)
  11. d o ( a , s ) do(a^{\prime},s^{\prime})
  12. a = a a=a^{\prime}
  13. s = s s=s^{\prime}
  14. ( 2 , 3 ) (2,3)
  15. m o v e ( 2 , 3 ) move(2,3)
  16. d o ( m o v e ( 2 , 3 ) , S 0 ) do(move(2,3),S_{0})
  17. d o ( p i c k u p ( B a l l ) , d o ( m o v e ( 2 , 3 ) , S 0 ) ) do(pickup(Ball),do(move(2,3),S_{0}))
  18. d o ( m o v e ( 2 , 3 ) , S 0 ) do(move(2,3),S_{0})
  19. d o ( p i c k u p ( B a l l ) , d o ( m o v e ( 2 , 3 ) , S 0 ) ) do(pickup(Ball),do(move(2,3),S_{0}))
  20. i s _ c a r r y i n g ( o , s ) is\_carrying(o,s)
  21. i s _ c a r r y i n g ( B a l l , S 0 ) is\_carrying(Ball,S_{0})
  22. i s _ c a r r y i n g ( B a l l , d o ( p i c k u p ( B a l l ) , S 0 ) ) is\_carrying(Ball,do(pickup(Ball),S_{0}))
  23. l o c a t i o n ( s ) location(s)
  24. ( x , y ) (x,y)
  25. P o s s ( a , s ) Poss(a,s)
  26. a a
  27. s s
  28. P o s s Poss
  29. P o s s ( d r o p ( o ) , s ) i s _ c a r r y i n g ( o , s ) Poss(drop(o),s)\leftrightarrow is\_carrying(o,s)
  30. h e a v y heavy
  31. P o s s ( p i c k u p ( o ) , s ) ( z ¬ i s _ c a r r y i n g ( z , s ) ) ¬ h e a v y ( o ) Poss(pickup(o),s)\leftrightarrow(\forall z\neg is\_carrying(z,s))\wedge\neg heavy% (o)
  32. P o s s ( p i c k u p ( o ) , s ) i s _ c a r r y i n g ( o , d o ( p i c k u p ( o ) , s ) ) Poss(pickup(o),s)\rightarrow is\_carrying(o,do(pickup(o),s))
  33. f r a g i l e fragile
  34. b r o k e n broken
  35. P o s s ( d r o p ( o ) , s ) f r a g i l e ( o ) b r o k e n ( o , d o ( d r o p ( o ) , s ) ) Poss(drop(o),s)\wedge fragile(o)\rightarrow broken(o,do(drop(o),s))
  36. P o s s ( p i c k u p ( o ) , s ) l o c a t i o n ( s ) = ( x , y ) l o c a t i o n ( d o ( p i c k u p ( o ) , s ) ) = ( x , y ) Poss(pickup(o),s)\wedge location(s)=(x,y)\rightarrow location(do(pickup(o),s))% =(x,y)
  37. F ( x , s ) F(\overrightarrow{x},s)
  38. P o s s ( a , s ) γ F + ( x , a , s ) F ( x , d o ( a , s ) ) Poss(a,s)\wedge\gamma_{F}^{+}(\overrightarrow{x},a,s)\rightarrow F(% \overrightarrow{x},do(a,s))
  39. P o s s ( a , s ) γ F - ( x , a , s ) ¬ F ( x , d o ( a , s ) ) Poss(a,s)\wedge\gamma_{F}^{-}(\overrightarrow{x},a,s)\rightarrow\neg F(% \overrightarrow{x},do(a,s))
  40. γ F + \gamma_{F}^{+}
  41. a a
  42. s s
  43. F F
  44. d o ( a , s ) do(a,s)
  45. γ F - \gamma_{F}^{-}
  46. a a
  47. s s
  48. F F
  49. F F
  50. P o s s ( a , s ) [ F ( x , d o ( a , s ) ) γ F + ( x , a , s ) ( F ( x , s ) ¬ γ F - ( x , a , s ) ) ] Poss(a,s)\rightarrow\left[F(\overrightarrow{x},do(a,s))\leftrightarrow\gamma_{% F}^{+}(\overrightarrow{x},a,s)\vee\left(F(\overrightarrow{x},s)\wedge\neg% \gamma_{F}^{-}(\overrightarrow{x},a,s)\right)\right]
  51. a a
  52. s s
  53. F F
  54. d o ( a , s ) do(a,s)
  55. a a
  56. s s
  57. s s
  58. a a
  59. s s
  60. b r o k e n broken
  61. P o s s ( a , s ) [ b r o k e n ( o , d o ( a , s ) ) a = d r o p ( o ) f r a g i l e ( o ) b r o k e n ( o , s ) a r e p a i r ( o ) ] Poss(a,s)\rightarrow\left[broken(o,do(a,s))\leftrightarrow a=drop(o)\wedge fragile% (o)\vee broken(o,s)\wedge a\neq repair(o)\right]
  62. S 0 S_{0}
  63. ( 0 , 0 ) (0,0)
  64. z ¬ i s _ c a r r y i n g ( z , S 0 ) \forall z\,\neg is\_carrying(z,S_{0})
  65. l o c a t i o n ( S 0 ) = ( 0 , 0 ) location(S_{0})=(0,0)\,
  66. o ¬ b r o k e n ( o , S 0 ) \forall o\,\neg broken(o,S_{0})
  67. d o ( a , s ) = d o ( a , s ) a = a and s = s do(a,s)=do(a^{\prime},s^{\prime})\iff a=a^{\prime}\and s=s^{\prime}
  68. d o ( a , s ) do(a,s)
  69. a a
  70. s s
  71. d o ( a , s ) do(a,s)
  72. S 0 S_{0}
  73. x x
  74. s s
  75. r a i n i n g ( x , s ) raining(x,s)
  76. x x
  77. s s
  78. l o c a t i o n ( x , s ) location(x,s)
  79. l o c a t i o n location
  80. l o c a t i o n ( x , s ) = l o c a t i o n ( x , s ) location(x,s)=location(x,s^{\prime})
  81. x x
  82. s s
  83. s s^{\prime}
  84. r e s u l t result
  85. a a
  86. s s
  87. r e s u l t ( a , s ) result(a,s)
  88. s s
  89. r e s u l t ( a , s ) result(a,s)
  90. ¬ l o c k e d ( d o o r , s ) o p e n ( d o o r , r e s u l t ( o p e n s , s ) ) \neg locked(door,s)\rightarrow open(door,result(opens,s))
  91. l o c k e d locked
  92. o p e n open
  93. o p e n s opens
  94. ¬ l o c k e d ( d o o r , r e s u l t ( o p e n s , s ) ) \neg locked(door,result(opens,s))
  95. ¬ l o c k e d ( d o o r , s ) \neg locked(door,s)
  96. ¬ l o c k e d ( d o o r , s ) ¬ l o c k e d ( d o o r , r e s u l t ( o p e n s , s ) ) \neg locked(door,s)\rightarrow\neg locked(door,result(opens,s))
  97. S 0 S_{0}
  98. s s
  99. ¬ l o c k e d ( d o o r , s ) \neg locked(door,s)
  100. o p e n ( d o o r , r e s u l t ( o p e n s , s ) ) open(door,result(opens,s))
  101. l o c a t i o n ( x , s ) location(x,s)
  102. x x
  103. s s
  104. H o l d s ( f , d o ( a , s ) ) P o s s ( a , s ) I n i t i a t e s ( a , f , s ) Holds(f,do(a,s))\leftarrow Poss(a,s)\wedge Initiates(a,f,s)
  105. H o l d s ( f , d o ( a , s ) ) P o s s ( a , s ) H o l d s ( f , s ) ¬ T e r m i n a t e s ( a , f , s ) Holds(f,do(a,s))\leftarrow Poss(a,s)\wedge Holds(f,s)\wedge\neg Terminates(a,f% ,s)
  106. H o l d s Holds
  107. f f
  108. P o s s Poss
  109. I n i t i a t e s Initiates
  110. T e r m i n a t e s Terminates
  111. P o s s Poss
  112. γ F + ( x , a , s ) \gamma_{F}^{+}(\overrightarrow{x},a,s)
  113. γ F - ( x , a , s ) \gamma_{F}^{-}(\overrightarrow{x},a,s)
  114. \leftarrow
  115. \leftrightarrow

Skewb_Diamond.html

  1. 4 ! × 6 ! × 2 5 4 = 138 , 240. \frac{4!\times 6!\times 2^{5}}{4}=138,240.

Skid_mark.html

  1. V = 2 μ g d s k i d V=\sqrt{2\mu gd_{skid}}
  2. μ = 0.7 \mu=0.7
  3. d s k i d d_{skid}
  4. V 177.8 d s k i d V\approx\sqrt{177.8d_{skid}}
  5. V 20.9 d s k i d V\approx\sqrt{20.9d_{skid}}

Smeed's_law.html

  1. D = .0003 ( n p 2 ) 1 3 D=.0003(np^{2})^{1\over 3}
  2. D p = .0003 × n p 3 {D\over p}=.0003\times{\sqrt[3]{n\over p}}

Smn_theorem.html

  1. φ \varphi
  2. φ s ( p , x ) ( y ) \varphi_{s(p,x)}(y)
  3. f ( x , y ) f(x,y)
  4. φ s ( p , x ) λ y . φ p ( x , y ) . \varphi_{s(p,x)}\simeq\lambda y.\varphi_{p}(x,y).\,
  5. s n m s^{m}_{n}
  6. φ s n m ( p , x 1 , , x m ) λ y 1 , , y n . φ p ( x 1 , , x m , y 1 , , y n ) . \varphi_{s^{m}_{n}(p,x_{1},\dots,x_{m})}\simeq\lambda y_{1},\dots,y_{n}.% \varphi_{p}(x_{1},\dots,x_{m},y_{1},\dots,y_{n}).\,
  7. s 1 1 s^{1}_{1}

Smoluchowski_coagulation_equation.html

  1. n ( x , t ) t = 1 2 0 x K ( x - y , y ) n ( x - y , t ) n ( y , t ) d y - 0 K ( x , y ) n ( x , t ) n ( y , t ) d y . \frac{\partial n(x,t)}{\partial t}=\frac{1}{2}\int^{x}_{0}K(x-y,y)n(x-y,t)n(y,% t)\,dy-\int^{\infty}_{0}K(x,y)n(x,t)n(y,t)\,dy.
  2. n ( x i , t ) t = 1 2 j = 1 i - 1 K ( x i - x j , x j ) n ( x i - x j , t ) n ( x j , t ) - j = 1 K ( x i , x j ) n ( x i , t ) n ( x j , t ) . \frac{\partial n(x_{i},t)}{\partial t}=\frac{1}{2}\sum^{i-1}_{j=1}K(x_{i}-x_{j% },x_{j})n(x_{i}-x_{j},t)n(x_{j},t)-\sum^{\infty}_{j=1}K(x_{i},x_{j})n(x_{i},t)% n(x_{j},t).
  3. x 1 x_{1}
  4. x 2 x_{2}
  5. K = 1 , K = x 1 + x 2 , K = x 1 x 2 , K=1,\quad K=x_{1}+x_{2},\quad K=x_{1}x_{2},
  6. K = π k B T 2 ( 1 m ( x 1 ) + 1 m ( x 2 ) ) 1 / 2 ( d ( x 1 ) + d ( x 2 ) ) 2 . K=\sqrt{\frac{\pi k_{B}T}{2}}\left(\frac{1}{m(x_{1})}+\frac{1}{m(x_{2})}\right% )^{1/2}\left(d(x_{1})+d(x_{2})\right)^{2}.
  7. K = 2 3 k B T η ( x 1 1 / y 1 + x 2 1 / y 2 ) ( x 1 - 1 / y 1 + x 2 - 1 / y 2 ) , K=\frac{2}{3}\frac{k_{B}T}{\eta}\left(x_{1}^{1/y_{1}}+x_{2}^{1/y_{2}}\right)% \left(x_{1}^{-1/y_{1}}+x_{2}^{-1/y_{2}}\right),
  8. K = 2 3 k B T η ( x 1 x 2 ) γ W ( x 1 1 / y 1 + x 2 1 / y 2 ) ( x 1 - 1 / y 1 + x 2 - 1 / y 2 ) , K=\frac{2}{3}\frac{k_{B}T}{\eta}\frac{(x_{1}x_{2})^{\gamma}}{W}\left(x_{1}^{1/% y_{1}}+x_{2}^{1/y_{2}}\right)\left(x_{1}^{-1/y_{1}}+x_{2}^{-1/y_{2}}\right),
  9. y 1 , y 2 y_{1},y_{2}
  10. k B k_{B}
  11. T T
  12. W W
  13. η \eta
  14. γ \gamma

SNARE_(protein).html

  1. α \alpha
  2. α \alpha
  3. α \alpha
  4. α \alpha
  5. α \alpha
  6. α \alpha
  7. α \alpha
  8. α \alpha
  9. α \alpha

Solar_balloon.html

  1. π r 2 \mathrm{\pi}r^{2}
  2. Area = π × ( 5 m ) 2 78.54 m 2 \mathrm{Area}=\pi\times(5m)^{2}\approx 78{.}54m^{2}

Solenoid_(mathematics).html

  1. Λ = i 0 T i \Lambda=\bigcap_{i\geq 0}T_{i}
  2. f ( t , z ) = ( 2 t , 1 4 z + 1 2 e i t ) . f(t,z)=\left(2t,\tfrac{1}{4}z+\tfrac{1}{2}e^{it}\right).

Solenoid_valve.html

  1. F s = P A = P π d 2 / 4 F_{s}=PA=P\pi d^{2}/4

Solid_solution.html

  1. α \alpha
  2. β \beta
  3. A A
  4. B B
  5. α \alpha
  6. B B
  7. A A
  8. β \beta
  9. A A
  10. B B
  11. α \alpha
  12. β \beta
  13. α \alpha
  14. β \beta
  15. A A
  16. B B
  17. B B
  18. A A

Solvable_Lie_algebra.html

  1. 𝐠 \mathbf{g}
  2. 𝐠 \mathbf{g}
  3. [ 𝔤 , 𝔤 ] [\mathfrak{g},\mathfrak{g}]
  4. 𝐠 \mathbf{g}
  5. 𝔤 [ 𝔤 , 𝔤 ] [ [ 𝔤 , 𝔤 ] , [ 𝔤 , 𝔤 ] ] [ [ [ 𝔤 , 𝔤 ] , [ 𝔤 , 𝔤 ] ] , [ [ 𝔤 , 𝔤 ] , [ 𝔤 , 𝔤 ] ] ] \mathfrak{g}\geq[\mathfrak{g},\mathfrak{g}]\geq[[\mathfrak{g},\mathfrak{g}],[% \mathfrak{g},\mathfrak{g}]]\geq[[[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},% \mathfrak{g}]],[[\mathfrak{g},\mathfrak{g}],[\mathfrak{g},\mathfrak{g}]]]\geq...
  6. 𝐠 \mathbf{g}
  7. 0
  8. 𝐠 \mathbf{g}
  9. a d ( 𝐠 ) ad(\mathbf{g})
  10. 𝐠 \mathbf{g}
  11. 𝐠 \mathbf{g}
  12. 𝔤 = 𝔞 0 𝔞 1 𝔞 r = 0 , i [ 𝔞 i , 𝔞 i ] 𝔞 i + 1 . \mathfrak{g}=\mathfrak{a}_{0}\supset\mathfrak{a}_{1}\supset...\mathfrak{a}_{r}% =0,\quad\forall i[\mathfrak{a}_{i},\mathfrak{a}_{i}]\subset\mathfrak{a}_{i+1}.
  13. g 𝐠 , 𝐠 g\mathbf{g},\mathbf{g}
  14. 𝐠 \mathbf{g}
  15. n n
  16. 𝐠 \mathbf{g}
  17. 𝔤 = 𝔞 0 𝔞 1 𝔞 n = 0 , i dim 𝔞 i / 𝔞 i + 1 = 1 , \mathfrak{g}=\mathfrak{a}_{0}\supset\mathfrak{a}_{1}\supset...\mathfrak{a}_{n}% =0,\quad\forall i\operatorname{dim}\mathfrak{a}_{i}/\mathfrak{a}_{i+1}=1,
  18. 𝐠 \mathbf{g}
  19. 𝔤 = 𝔤 0 𝔤 1 𝔤 r = 0 , \mathfrak{g}=\mathfrak{g}_{0}\supset\mathfrak{g}_{1}\supset...\mathfrak{g}_{r}% =0,
  20. 𝐠 \mathbf{g}
  21. B B
  22. B ( X , Y ) = 0 B(X,Y)=0
  23. X X
  24. 𝐠 \mathbf{g}
  25. Y Y
  26. g 𝐠 , 𝐠 g\mathbf{g},\mathbf{g}
  27. V V
  28. 𝐊 \mathbf{K}
  29. 𝐠 \mathbf{g}
  30. 𝐤 \mathbf{k}
  31. 𝐊 \mathbf{K}
  32. π π
  33. 𝐠 \mathbf{g}
  34. V V
  35. v V v∈V
  36. π ( X ) π(X)
  37. X 𝐠 X∈\mathbf{g}
  38. π ( X ) π(X)
  39. 𝐊 \mathbf{K}
  40. X 𝐠 X∈\mathbf{g}
  41. 𝐚 \mathbf{a}
  42. 𝐠 \mathbf{g}
  43. 𝐠 / 𝐚 \mathbf{g}/\mathbf{a}
  44. 𝐠 \mathbf{g}
  45. 𝐠 \mathbf{g}
  46. 𝐫 𝐠 \mathbf{r}⊂\mathbf{g}
  47. 𝐠 \mathbf{g}
  48. 𝐠 \mathbf{g}
  49. r a d 𝐠 rad\mathbf{g}
  50. 𝐚 , 𝐛 𝐠 \mathbf{a},\mathbf{b}⊂\mathbf{g}
  51. 𝐚 + 𝐛 \mathbf{a}+\mathbf{b}
  52. 𝐠 \mathbf{g}
  53. 𝐧 \mathbf{n}
  54. X 𝐠 X∈\mathbf{g}
  55. D D
  56. 𝐠 \mathbf{g}
  57. D ( 𝐠 ) 𝐧 D(\mathbf{g})⊂\mathbf{n}
  58. 𝐠 \mathbf{g}
  59. 𝐠 \mathbf{g}
  60. 0
  61. 𝐠 \mathbf{g}
  62. 𝐠 \mathbf{g}
  63. 𝐤 \mathbf{k}
  64. X X
  65. 𝐠 \mathbf{g}
  66. 𝐠 \mathbf{g}
  67. X = ( 0 θ x - θ 0 y 0 0 0 ) , θ , x , y . X=\left(\begin{matrix}0&\theta&x\\ -\theta&0&y\\ 0&0&0\end{matrix}\right),\quad\theta,x,y\in\mathbb{R}.
  68. 𝐠 \mathbf{g}

Solvatochromism.html

  1. π \pi

Somos'_quadratic_recurrence_constant.html

  1. σ = 1 2 3 = 1 1 / 2 2 1 / 4 3 1 / 8 . \sigma=\sqrt{1\sqrt{2\sqrt{3\cdots}}}=1^{1/2}\;2^{1/4}\;3^{1/8}\cdots.\,
  2. σ = σ 2 / σ = ( 2 1 ) 1 / 2 ( 3 2 ) 1 / 4 ( 4 3 ) 1 / 8 ( 5 4 ) 1 / 16 . \sigma=\sigma^{2}/\sigma=\left(\frac{2}{1}\right)^{1/2}\left(\frac{3}{2}\right% )^{1/4}\left(\frac{4}{3}\right)^{1/8}\left(\frac{5}{4}\right)^{1/16}\cdots.
  3. g 0 = 1 ; g n = n g n - 1 2 , n > 1 , g_{0}=1\,;\,g_{n}=ng_{n-1}^{2},\qquad n>1,\,
  4. g n σ 2 n n + 2 + O ( 1 n ) . g_{n}\sim\frac{\sigma^{2^{n}}}{n+2+O(\frac{1}{n})}.
  5. ln σ = - 1 2 Φ s ( 1 2 , 0 , 1 ) \ln\sigma=\frac{-1}{2}\frac{\partial\Phi}{\partial s}\left(\frac{1}{2},0,1\right)
  6. Φ \Phi
  7. ln σ = n = 1 k = 0 n ( - 1 ) n - k ( n k ) ln ( k + 1 ) . \ln\sigma=\sum_{n=1}^{\infty}\sum_{k=0}^{n}(-1)^{n-k}{n\choose k}\ln(k+1).
  8. σ = 1.661687949633594121296 \sigma=1.661687949633594121296\dots\;

Soundfield_microphone.html

  1. 2 W + X \sqrt{2}W+X

Space–time_block_code.html

  1. time-slots transmit antennas [ s 11 s 12 s 1 n T s 21 s 22 s 2 n T s T 1 s T 2 s T n T ] \,\text{time-slots}\begin{matrix}\,\text{transmit antennas}\\ \left\downarrow\overrightarrow{\begin{bmatrix}s_{11}&s_{12}&\cdots&s_{1n_{T}}% \\ s_{21}&s_{22}&\cdots&s_{2n_{T}}\\ \vdots&\vdots&&\vdots\\ s_{T1}&s_{T2}&\cdots&s_{Tn_{T}}\end{bmatrix}}\right.\end{matrix}
  2. s i j s_{ij}
  3. i i
  4. j j
  5. T T
  6. n T n_{T}
  7. n R n_{R}
  8. T T
  9. k k
  10. r = k T . r=\frac{k}{T}.
  11. 𝐜 = c 1 1 c 1 2 c 1 n T c 2 1 c 2 2 c 2 n T c T 1 c T 2 c T n T \mathbf{c}=c_{1}^{1}c_{1}^{2}\cdots c_{1}^{n_{T}}c_{2}^{1}c_{2}^{2}\cdots c_{2% }^{n_{T}}\cdots c_{T}^{1}c_{T}^{2}\cdots c_{T}^{n_{T}}
  12. 𝐞 = e 1 1 e 1 2 e 1 n T e 2 1 e 2 2 e 2 n T e T 1 e T 2 e T n T . \mathbf{e}=e_{1}^{1}e_{1}^{2}\cdots e_{1}^{n_{T}}e_{2}^{1}e_{2}^{2}\cdots e_{2% }^{n_{T}}\cdots e_{T}^{1}e_{T}^{2}\cdots e_{T}^{n_{T}}.
  13. 𝐁 ( 𝐜 , 𝐞 ) = [ e 1 1 - c 1 1 e 2 1 - c 2 1 e T 1 - c T 1 e 1 2 - c 1 2 e 2 2 - c 2 2 e T 2 - c T 2 e 1 n T - c 1 n T e 2 n T - c 2 n T e T n T - c T n T ] \mathbf{B}(\mathbf{c},\mathbf{e})=\begin{bmatrix}e_{1}^{1}-c_{1}^{1}&e_{2}^{1}% -c_{2}^{1}&\cdots&e_{T}^{1}-c_{T}^{1}\\ e_{1}^{2}-c_{1}^{2}&e_{2}^{2}-c_{2}^{2}&\cdots&e_{T}^{2}-c_{T}^{2}\\ \vdots&\vdots&\ddots&\vdots\\ e_{1}^{n_{T}}-c_{1}^{n_{T}}&e_{2}^{n_{T}}-c_{2}^{n_{T}}&\cdots&e_{T}^{n_{T}}-c% _{T}^{n_{T}}\end{bmatrix}
  14. 𝐜 \mathbf{c}
  15. 𝐞 \mathbf{e}
  16. n T n R n_{T}n_{R}
  17. 𝐁 ( 𝐜 , 𝐞 ) \mathbf{B}(\mathbf{c},\mathbf{e})
  18. b b
  19. b n R bn_{R}
  20. C 2 = [ c 1 c 2 - c 2 * c 1 * ] , C_{2}=\begin{bmatrix}c_{1}&c_{2}\\ -c_{2}^{*}&c_{1}^{*}\end{bmatrix},
  21. 2 n R 2n_{R}
  22. n R n_{R}
  23. C 3 , 1 / 2 = [ c 1 c 2 c 3 - c 2 c 1 - c 4 - c 3 c 4 c 1 - c 4 - c 3 c 2 c 1 * c 2 * c 3 * - c 2 * c 1 * - c 4 * - c 3 * c 4 * c 1 * - c 4 * - c 3 * c 2 * ] and C 3 , 3 / 4 = [ c 1 c 2 c 3 2 - c 2 * c 1 * c 3 2 c 3 * 2 c 3 * 2 ( - c 1 - c 1 * + c 2 - c 2 * ) 2 c 3 * 2 - c 3 * 2 ( c 2 + c 2 * + c 1 - c 1 * ) 2 . ] C_{3,1/2}=\begin{bmatrix}c_{1}&c_{2}&c_{3}\\ -c_{2}&c_{1}&-c_{4}\\ -c_{3}&c_{4}&c_{1}\\ -c_{4}&-c_{3}&c_{2}\\ c_{1}^{*}&c_{2}^{*}&c_{3}^{*}\\ -c_{2}^{*}&c_{1}^{*}&-c_{4}^{*}\\ -c_{3}^{*}&c_{4}^{*}&c_{1}^{*}\\ -c_{4}^{*}&-c_{3}^{*}&c_{2}^{*}\end{bmatrix}\quad\,\text{and}\quad C_{3,3/4}=% \begin{bmatrix}c_{1}&c_{2}&\frac{c_{3}}{\sqrt{2}}\\ -c_{2}^{*}&c_{1}^{*}&\frac{c_{3}}{\sqrt{2}}\\ \frac{c_{3}^{*}}{\sqrt{2}}&\frac{c_{3}^{*}}{\sqrt{2}}&\frac{\left(-c_{1}-c_{1}% ^{*}+c_{2}-c_{2}*\right)}{2}\\ \frac{c_{3}^{*}}{\sqrt{2}}&-\frac{c_{3}^{*}}{\sqrt{2}}&\frac{\left(c_{2}+c_{2}% ^{*}+c_{1}-c_{1}^{*}\right)}{2}.\end{bmatrix}
  24. C 3 , 3 / 4 C_{3,3/4}
  25. C 4 , 1 / 2 = [ c 1 c 2 c 3 c 4 - c 2 c 1 - c 4 c 3 - c 3 c 4 c 1 - c 2 - c 4 - c 3 c 2 c 1 c 1 * c 2 * c 3 * c 4 * - c 2 * c 1 * - c 4 * c 3 * - c 3 * c 4 * c 1 * - c 2 * - c 4 * - c 3 * c 2 * c 1 * ] and C 4 , 3 / 4 = [ c 1 c 2 c 3 2 c 3 2 - c 2 * c 1 * c 3 2 - c 3 2 c 3 * 2 c 3 * 2 ( - c 1 - c 1 * + c 2 - c 2 * ) 2 ( - c 2 - c 2 * + c 1 - c 1 * ) 2 c 3 * 2 - c 3 * 2 ( c 2 + c 2 * + c 1 - c 1 * ) 2 - ( c 1 + c 1 * + c 2 - c 2 * ) 2 ] . C_{4,1/2}=\begin{bmatrix}c_{1}&c_{2}&c_{3}&c_{4}\\ -c_{2}&c_{1}&-c_{4}&c_{3}\\ -c_{3}&c_{4}&c_{1}&-c_{2}\\ -c_{4}&-c_{3}&c_{2}&c_{1}\\ c_{1}^{*}&c_{2}^{*}&c_{3}^{*}&c_{4}^{*}\\ -c_{2}^{*}&c_{1}^{*}&-c_{4}^{*}&c_{3}^{*}\\ -c_{3}^{*}&c_{4}^{*}&c_{1}^{*}&-c_{2}^{*}\\ -c_{4}^{*}&-c_{3}^{*}&c_{2}^{*}&c_{1}^{*}\end{bmatrix}\quad\,\text{and}\quad{}% C_{4,3/4}=\begin{bmatrix}c_{1}&c_{2}&\frac{c_{3}}{\sqrt{2}}&\frac{c_{3}}{\sqrt% {2}}\\ -c_{2}^{*}&c_{1}^{*}&\frac{c_{3}}{\sqrt{2}}&-\frac{c_{3}}{\sqrt{2}}\\ \frac{c_{3}^{*}}{\sqrt{2}}&\frac{c_{3}^{*}}{\sqrt{2}}&\frac{\left(-c_{1}-c_{1}% ^{*}+c_{2}-c_{2}^{*}\right)}{2}&\frac{\left(-c_{2}-c_{2}^{*}+c_{1}-c_{1}^{*}% \right)}{2}\\ \frac{c_{3}^{*}}{\sqrt{2}}&-\frac{c_{3}^{*}}{\sqrt{2}}&\frac{\left(c_{2}+c_{2}% ^{*}+c_{1}-c_{1}^{*}\right)}{2}&-\frac{\left(c_{1}+c_{1}^{*}+c_{2}-c_{2}^{*}% \right)}{2}\end{bmatrix}.
  26. C 4 , 3 / 4 C_{4,3/4}
  27. C 3 , 3 / 4 C_{3,3/4}
  28. C 4 , 3 / 4 C_{4,3/4}
  29. C 4 , 3 / 4 = [ c 1 c 2 c 3 0 - c 2 * c 1 * 0 c 3 - c 3 * 0 c 1 * - c 2 0 - c 3 * c 2 * c 1 ] , C_{4,3/4}=\begin{bmatrix}c_{1}&c_{2}&c_{3}&0\\ -c_{2}^{*}&c_{1}^{*}&0&c_{3}\\ -c_{3}^{*}&0&c_{1}^{*}&-c_{2}\\ 0&-c_{3}^{*}&c_{2}^{*}&c_{1}\end{bmatrix},
  30. t t
  31. r t j r_{t}^{j}
  32. j j
  33. r t j = i = 1 n T α i j s t i + n t j , r_{t}^{j}=\sum_{i=1}^{n_{T}}\alpha_{ij}s_{t}^{i}+n_{t}^{j},
  34. α i j \alpha_{ij}
  35. i i
  36. j j
  37. s t i s_{t}^{i}
  38. i i
  39. n t j n_{t}^{j}
  40. R i = t = 1 n T j = 1 n R r t j α ϵ t ( i ) j δ t ( i ) R_{i}=\sum_{t=1}^{n_{T}}\sum_{j=1}^{n_{R}}r_{t}^{j}\alpha_{\epsilon_{t}(i)j}% \delta_{t}(i)
  41. δ k ( i ) \delta_{k}(i)
  42. s i s_{i}
  43. k k
  44. ϵ k ( p ) = q \epsilon_{k}(p)=q
  45. s p s_{p}
  46. ( k , q ) (k,q)
  47. i = 1 , 2 , , n T i=1,2,\ldots,n_{T}
  48. s i s_{i}
  49. s i = arg min s 𝒜 ( | R i - s | 2 + ( - 1 + k , l | α k l | 2 ) | s | 2 ) , s_{i}=\arg{}\min_{s\in\mathcal{A}}\left(\left|R_{i}-s\right|^{2}+\left(-1+\sum% _{k,l}\left|\alpha_{kl}\right|^{2}\right)\left|s\right|^{2}\right),
  50. 𝒜 \mathcal{A}
  51. T T
  52. n T n_{T}
  53. r max = n 0 + 1 2 n 0 , r_{\max}=\frac{n_{0}+1}{2n_{0}},
  54. n T = 2 n 0 n_{T}=2n_{0}
  55. n T = 2 n 0 - 1 n_{T}=2n_{0}-1
  56. 0 , c i , - c i , c i * , 0,c_{i},-c_{i},c_{i}^{*},
  57. - c i * -c_{i}^{*}
  58. c i c_{i}
  59. C 4 , 1 = [ c 1 c 2 c 3 c 4 - c 2 * c 1 * - c 4 * c 3 * - c 3 * - c 4 * c 1 * c 2 * c 4 - c 3 - c 2 c 1 ] . C_{4,1}=\begin{bmatrix}c_{1}&c_{2}&c_{3}&c_{4}\\ -c_{2}^{*}&c_{1}^{*}&-c_{4}^{*}&c_{3}^{*}\\ -c_{3}^{*}&-c_{4}^{*}&c_{1}^{*}&c_{2}^{*}\\ c_{4}&-c_{3}&-c_{2}&c_{1}\end{bmatrix}.

Spatial_ecology.html

  1. r e = 1 2 ϱ r_{e}=\frac{1}{2\sqrt{\varrho}}
  2. Z = r o - r e S E Z=\frac{r_{o}-r_{e}}{SE}
  3. S E = 0.0863 A N SE=\sqrt{\frac{0.0863A}{N}}
  4. α = π d ω \alpha=\pi d\omega
  5. χ 2 n 2 = 2 n α \chi^{2}_{2n}=2n\alpha

Spatial_frequency.html

  1. ξ \xi
  2. ν \nu
  3. λ \lambda
  4. ξ = 1 λ . \xi=\frac{1}{\lambda}.
  5. k k
  6. k = 2 π ξ = 2 π λ . k=2\pi\xi=\frac{2\pi}{\lambda}.
  7. I max - I min I max + I min . \frac{I_{\mathrm{max}}-I_{\mathrm{min}}}{I_{\mathrm{max}}+I_{\mathrm{min}}}.

Speaker_wire.html

  1. X c = 1 2 π f C X_{c}=\frac{1}{2\pi fC}
  2. f f
  3. C C
  4. X i = 2 π f L X_{i}=2\pi fL
  5. f f
  6. L L

Special_right_triangles.html

  1. 0 2 = 0 \tfrac{\sqrt{0}}{2}=0
  2. 4 2 = 1 \tfrac{\sqrt{4}}{2}=1
  3. 0
  4. π 6 \tfrac{\pi}{6}
  5. 1 2 = 1 2 \tfrac{\sqrt{1}}{2}=\tfrac{1}{2}
  6. 3 2 \tfrac{\sqrt{3}}{2}
  7. 1 3 \tfrac{1}{\sqrt{3}}
  8. π 4 \tfrac{\pi}{4}
  9. 2 2 = 1 2 \tfrac{\sqrt{2}}{2}=\tfrac{1}{\sqrt{2}}
  10. 2 2 = 1 2 \tfrac{\sqrt{2}}{2}=\tfrac{1}{\sqrt{2}}
  11. 1 1
  12. π 3 \tfrac{\pi}{3}
  13. 3 2 \tfrac{\sqrt{3}}{2}
  14. 1 2 = 1 2 \tfrac{\sqrt{1}}{2}=\tfrac{1}{2}
  15. 3 \sqrt{3}
  16. π 2 \tfrac{\pi}{2}
  17. 4 2 = 1 \tfrac{\sqrt{4}}{2}=1
  18. 0 2 = 0 \tfrac{\sqrt{0}}{2}=0
  19. 2 / 2. \sqrt{2}/2.
  20. 2 / 4. \sqrt{2}/4.
  21. m 2 - n 2 : 2 m n : m 2 + n 2 m^{2}-n^{2}:2mn:m^{2}+n^{2}\,
  22. ( x - 1 2 ) 2 + ( x + 1 2 ) 2 = y 2 (\tfrac{x-1}{2})^{2}+(\tfrac{x+1}{2})^{2}=y^{2}
  23. x 2 - 2 y 2 = - 1 x^{2}-2y^{2}=-1
  24. 1 : φ : φ . 1:\sqrt{\varphi}:\varphi.\,
  25. a = 2 sin π 10 = - 1 + 5 2 = 1 φ a=2\sin\tfrac{\pi}{10}=\tfrac{-1+\sqrt{5}}{2}=\tfrac{1}{\varphi}
  26. φ \varphi
  27. b = 2 sin π 6 = 1 b=2\sin\tfrac{\pi}{6}=1
  28. c = 2 sin π 5 = 5 - 5 2 c=2\sin\tfrac{\pi}{5}=\sqrt{\tfrac{5-\sqrt{5}}{2}}
  29. a 2 + b 2 = c 2 a^{2}+b^{2}=c^{2}