wpmath0000013_11

Quasireversibility.html

  1. π \pi
  2. π ( 𝐱 ) q ( 𝐱 , 𝐱 ) = π ( 𝐱 ) q ( 𝐱 , 𝐱 ) \pi(\mathbf{x})q^{\prime}(\mathbf{x},\mathbf{x^{\prime}})=\pi(\mathbf{x^{% \prime}})q(\mathbf{x^{\prime}},\mathbf{x})
  3. α \alpha
  4. α = 𝐱 M 𝐱 q ( 𝐱 , 𝐱 ) = 𝐱 M 𝐱 q ( 𝐱 , 𝐱 ) \alpha=\sum_{\mathbf{x^{\prime}}\in M_{\mathbf{x}}}q(\mathbf{x},\mathbf{x^{% \prime}})=\sum_{\mathbf{x^{\prime}}\in M_{\mathbf{x}}}q^{\prime}(\mathbf{x},% \mathbf{x^{\prime}})
  5. 𝐱 M 𝐱 \scriptstyle{\mathbf{x^{\prime}}\in M_{\mathbf{x}}}

Quaternionic_analysis.html

  1. f ( q ) = u q u - 1 f(q)=uqu^{-1}
  2. f ( q ) = q - 1 f(q)=q^{-1}
  3. f ( q ) = a q + b , a , b . f(q)=aq+b,\quad a,b\in\mathbb{H}.
  4. q u q v , q\mapsto uqv,
  5. u u
  6. v v
  7. f ( q ) = - 1 2 ( q + i q i + j q j + k q k ) f(q)=-\tfrac{1}{2}(q+iqi+jqj+kqk)
  8. f ( 1 ) = - 1 2 ( 1 - 1 - 1 - 1 ) = 1 , f ( i ) = - 1 2 ( i - i + i + i ) = - i , f ( j ) = - j , f ( k ) = - k f(1)=-\tfrac{1}{2}(1-1-1-1)=1,f(i)=-\tfrac{1}{2}(i-i+i+i)=-i,f(j)=-j,f(k)=-k
  9. f ( q ) = f ( w + x i + y j + z k ) = w f ( 1 ) + x f ( i ) + y f ( j ) + z f ( k ) = w - x i - y j - z k = q * . f(q)\ =f(w+xi+yj+zk)\ =wf(1)+xf(i)+yf(j)+zf(k)\ =w-xi-yj-zk\ =q^{*}.
  10. f ( z ) = u ( x , y ) + i v ( x , y ) f(z)=u(x,y)+iv(x,y)
  11. z = x + i y z=x+iy
  12. f ( q ) = u ( x , y ) + r v ( x , y ) f(q)=u(x,y)+r\ v(x,y)
  13. q = x + y r , r 2 = - 1 , r q=x+yr,\quad r^{2}=-1,\quad r\in\mathbb{H}
  14. u ( x , y ) = u ( x , - y ) , v ( x , y ) = - v ( x , - y ) u(x,y)=u(x,-y),\quad v(x,y)=-v(x,-y)\quad
  15. f ( x - y r * ) = u ( x , - y ) + r * v ( x , - y ) = u ( x , y ) + r v ( x , y ) = f ( q ) . f(x-yr^{*})=u(x,-y)+r^{*}v(x,-y)=u(x,y)+r\ v(x,y)=f(q).
  16. U ( q , 1 ) ( u 0 0 u ) = U ( q u , u ) U ( u - 1 q u , 1 ) , U(q,1)\begin{pmatrix}u&0\\ 0&u\end{pmatrix}=U(qu,u)\thicksim U(u^{-1}qu,1),
  17. u = exp ( θ r ) = cos θ + r sin θ u=\exp(\theta r)=\cos\theta+r\sin\theta
  18. q q + p q\mapsto q+p
  19. U ( q , 1 ) ( 1 0 p 1 ) = U ( q + p , 1 ) . U(q,1)\begin{pmatrix}1&0\\ p&1\end{pmatrix}=U(q+p,1).
  20. U ( q , 1 ) ( u 0 u x r u ) = U ( q u + u x r , u ) U ( u - 1 q u + x r , 1 ) . U(q,1)\begin{pmatrix}u&0\\ uxr&u\end{pmatrix}=U(qu+uxr,u)\thicksim U(u^{-1}qu+xr,1).
  21. ( 1 0 - s 1 ) ( u 0 0 u ) ( 1 0 s 1 ) = ( u 0 z u ) , \begin{pmatrix}1&0\\ -s&1\end{pmatrix}\begin{pmatrix}u&0\\ 0&u\end{pmatrix}\begin{pmatrix}1&0\\ s&1\end{pmatrix}=\begin{pmatrix}u&0\\ z&u\end{pmatrix},
  22. z = u s - s u = sin θ ( r s - s r ) = 2 t sin θ . z=us-su=\sin\theta(rs-sr)=2t\sin\theta.
  23. u - 1 z = u - 1 ( 2 t sin θ ) = 2 sin θ ( t cos θ - s sin θ ) . u^{-1}z=u^{-1}(2t\sin\theta)=2\sin\theta(t\cos\theta-s\sin\theta).
  24. { w t + x s : x > 0 } . \{wt+xs:x>0\}.
  25. { u - 1 z : 0 < θ < π } \{u^{-1}z:0<\theta<\pi\}
  26. p = a u - 1 z , a > 0. p=au^{-1}z,\ \ a>0.
  27. ( u 0 a z u ) \begin{pmatrix}u&0\\ az&u\end{pmatrix}
  28. f ( q ) = q 2 f(q)=q^{2}
  29. f ( x ) ( h ) = lim t 0 R ( t - 1 ( f ( x + t h ) - f ( x ) ) ) \partial f(x)(h)=\lim_{t\to 0\in R}(t^{-1}(f(x+th)-f(x)))
  30. f ( x ) ( h ) = s ( s ) 0 f ( x ) x h ( s ) 1 f ( x ) x . \partial f(x)(h)=\sum_{s}\frac{{}_{(s)0}\partial f(x)}{\partial x}h\frac{{}_{(% s)1}\partial f(x)}{\partial x}.\,\!
  31. ( s ) p f ( x ) x , p = 0 , 1 \frac{{}_{(s)p}\partial f(x)}{\partial x},\ p=0,1\,\!
  32. f ( x ) ( h ) = a h b \partial f(x)(h)=ahb\,\!
  33. ( 1 ) 0 a x b x = a \frac{{}_{(1)0}\partial axb}{\partial x}=a\,\!
  34. ( 1 ) 1 a x b x = b \frac{{}_{(1)1}\partial axb}{\partial x}=b\,\!
  35. f ( x ) ( h ) = x h + h x \partial f(x)(h)=xh+hx
  36. ( 1 ) 0 x 2 x = x \frac{{}_{(1)0}\partial x^{2}}{\partial x}=x\,\!
  37. ( 1 ) 1 x 2 x = 1 \frac{{}_{(1)1}\partial x^{2}}{\partial x}=1\,\!
  38. ( 2 ) 0 x 2 x = 1 \frac{{}_{(2)0}\partial x^{2}}{\partial x}=1\,\!
  39. ( 2 ) 1 x 2 x = x \frac{{}_{(2)1}\partial x^{2}}{\partial x}=x\,\!
  40. f ( x ) ( h ) = - x - 1 h x - 1 \partial f(x)(h)=-x^{-1}hx^{-1}
  41. ( 1 ) 0 x - 1 x = - x - 1 \frac{{}_{(1)0}\partial x^{-1}}{\partial x}=-x^{-1}\,\!
  42. ( 1 ) 1 x - 1 x = x - 1 \frac{{}_{(1)1}\partial x^{-1}}{\partial x}=x^{-1}\,\!

Quaternionic_matrix.html

  1. ( A + B ) i j = A i j + B i j . (A+B)_{ij}=A_{ij}+B_{ij}.\,
  2. ( A B ) i j = s A i s B s j . (AB)_{ij}=\sum_{s}A_{is}B_{sj}.\,
  3. U = ( u 11 u 12 u 21 u 22 ) , V = ( v 11 v 12 v 21 v 22 ) , U=\begin{pmatrix}u_{11}&u_{12}\\ u_{21}&u_{22}\\ \end{pmatrix},\quad V=\begin{pmatrix}v_{11}&v_{12}\\ v_{21}&v_{22}\\ \end{pmatrix},
  4. U V = ( u 11 v 11 + u 12 v 21 u 11 v 12 + u 12 v 22 u 21 v 11 + u 22 v 21 u 21 v 12 + u 22 v 22 ) . UV=\begin{pmatrix}u_{11}v_{11}+u_{12}v_{21}&u_{11}v_{12}+u_{12}v_{22}\\ u_{21}v_{11}+u_{22}v_{21}&u_{21}v_{12}+u_{22}v_{22}\\ \end{pmatrix}.
  5. trace ( A B ) trace ( B A ) . \operatorname{trace}(AB)\neq\operatorname{trace}(BA).
  6. ( c A ) i j = c A i j , ( A c ) i j = A i j c . (cA)_{ij}=cA_{ij},\qquad(Ac)_{ij}=A_{ij}c.\,
  7. [ a + b i c + d i - c + d i a - b i ] . \begin{bmatrix}~{}~{}a+bi&c+di\\ -c+di&a-bi\end{bmatrix}.

Quintic_threefold.html

  1. V 5 + W 5 + X 5 + Y 5 + Z 5 = 0 V^{5}+W^{5}+X^{5}+Y^{5}+Z^{5}=0

Quorum_(distributed_computing).html

  1. \leq
  2. \geq
  3. \geq

R-algebroid.html

  1. R 𝖦 R\mathsf{G}
  2. 𝖦 \mathsf{G}
  3. R 𝖦 R\mathsf{G}
  4. 𝖦 \mathsf{G}
  5. R 𝖦 ( b , c ) R\mathsf{G}(b,c)
  6. 𝖦 ( b , c ) \mathsf{G}(b,c)
  7. 𝖦 \mathsf{G}
  8. 𝖦 \mathsf{G}
  9. 𝖦 \mathsf{G}
  10. R ¯ 𝖦 := R 𝖦 ( b , c ) {\bar{R}}\mathsf{G}:=R\mathsf{G}(b,c)
  11. 𝖦 ( b , c ) R \mathsf{G}(b,c){\longrightarrow}R
  12. ( f * g ) ( z ) = { ( f x ) ( g y ) z = x y } \displaystyle(f*g)(z)=\sum\{(fx)(gy)\mid z=x\circ y\}
  13. R R\cong\mathbb{C}

R-hadron.html

  1. p p pp
  2. p p pp
  3. p p pp
  4. p p pp

R-Phase.html

  1. d σ d T \frac{d\sigma}{dT}

Radial_function.html

  1. Φ ( x , y ) = φ ( r ) , r = x 2 + y 2 \Phi(x,y)=\varphi(r),\quad r=\sqrt{x^{2}+y^{2}}
  2. f ρ = f f\circ\rho=f\,
  3. S [ ϕ ] = S [ φ ρ ] S[\phi]=S[\varphi\circ\rho]
  4. ϕ ( x ) = 1 ω n - 1 S n - 1 f ( r x ) d x \phi(x)=\frac{1}{\omega_{n-1}}\int_{S^{n-1}}f(rx^{\prime})\,dx^{\prime}

Radial_trajectory.html

  1. ϵ = v 2 2 - μ x \epsilon=\frac{v^{2}}{2}-\frac{\mu}{x}
  2. μ = G ( m 1 + m 2 ) \mu={G}(m_{1}+m_{2})
  3. w = 1 x - v 2 2 μ = - ϵ μ w=\frac{1}{x}-\frac{v^{2}}{2\mu}=\frac{-\epsilon}{\mu}
  4. - v 2 2 μ \textstyle\frac{-v_{\infty}^{2}}{2\mu}
  5. v \textstyle v_{\infty}
  6. w = 1 x 0 - v 0 2 2 μ w=\frac{1}{x_{0}}-\frac{v_{0}^{2}}{2\mu}
  7. x 0 \textstyle x_{0}
  8. v 0 \textstyle v_{0}
  9. t ( x ) = 2 x 3 9 μ t(x)=\sqrt{\frac{2x^{3}}{9\mu}}
  10. t ( x , w ) = arcsin ( w x ) - w x ( 1 - w x ) 2 μ w 3 / 2 t(x,w)=\frac{\arcsin(\sqrt{w\,x})-\sqrt{w\,x\ (1-w\,x)}}{\sqrt{2\mu}\,w^{3/2}}
  11. t ( x , w ) = ( | w | x ) 2 + | w | x - ln ( | w | x + 1 + | w | x ) 2 μ | w | 3 / 2 t(x,w)=\frac{\sqrt{(|w|x)^{2}+|w|x}-\ln(\sqrt{|w|x}+\sqrt{1+|w|x})}{\sqrt{2\mu% }\,|w|^{3/2}}
  12. t ( x , w ) = lim u w arcsin ( u x ) - u x ( 1 - u x ) 2 μ u 3 / 2 t(x,w)=\lim_{u\to w}\frac{\arcsin(\sqrt{u\,x})-\sqrt{u\,x\ (1-u\,x)}}{\sqrt{2% \mu}\,u^{3/2}}
  13. t ( x , w ) = 1 2 μ ( 2 3 x 3 / 2 + 1 5 w x 5 / 2 + 3 28 w 2 x 7 / 2 + 5 72 w 3 x 9 / 2 + 35 704 w 4 x 11 / 2 ) | - 1 < w x < 1 t(x,w)=\frac{1}{\sqrt{2\mu}}\left.(\frac{2}{3}x^{3/2}+\frac{1}{5}wx^{5/2}+% \frac{3}{28}w^{2}x^{7/2}+\frac{5}{72}w^{3}x^{9/2}+\frac{35}{704}w^{4}x^{11/2}% \cdots)\right|_{-1<w\cdot x<1}
  14. w = 1 x 0 - v 0 2 2 μ w=\frac{1}{x_{0}}-\frac{v_{0}^{2}}{2\mu}
  15. x 0 \textstyle x_{0}
  16. v 0 \textstyle v_{0}
  17. x ( t ) = ( 9 2 μ t 2 ) 1 3 x(t)=\left(\frac{9}{2}\mu t^{2}\right)^{\frac{1}{3}}
  18. w = 1 x 0 - v 0 2 2 μ and p = ( 9 2 μ t 2 ) 1 3 w=\frac{1}{x_{0}}-\frac{v_{0}^{2}}{2\mu}\quad\,\text{and}\quad p=\left(\frac{9% }{2}\mu t^{2}\right)^{\frac{1}{3}}
  19. x 0 x_{0}
  20. v 0 v_{0}
  21. μ = G ( m 1 + m 2 ) \mu={G}(m_{1}+m_{2})
  22. x ( t ) = n = 1 ( lim r 0 ( w n - 1 p n n ! d n - 1 d r n - 1 ( r n ( 3 2 ( arcsin ( r ) - r - r 2 ) ) - 2 3 n ) ) ) x(t)=\sum_{n=1}^{\infty}\left(\lim_{r\to 0}\left({\frac{w^{n-1}p^{n}}{n!}}% \frac{\mathrm{d}^{\,n-1}}{\mathrm{d}r^{\,n-1}}\left(r^{n}\left(\frac{3}{2}(% \arcsin(\sqrt{r})-\sqrt{r-r^{2}})\right)^{-\frac{2}{3}n}\right)\right)\right)
  23. x ( t ) = p - 1 5 w p 2 - 3 175 w 2 p 3 - 23 7875 w 3 p 4 - 1894 3931875 w 4 p 5 - 3293 21896875 w 5 p 6 - 2418092 62077640625 w 6 p 7 x(t)=p-\frac{1}{5}wp^{2}-\frac{3}{175}w^{2}p^{3}-\frac{23}{7875}w^{3}p^{4}-% \frac{1894}{3931875}w^{4}p^{5}-\frac{3293}{21896875}w^{5}p^{6}-\frac{2418092}{% 62077640625}w^{6}p^{7}\cdots

Radially_unbounded_function.html

  1. f : n f:\mathbb{R}^{n}\rightarrow\mathbb{R}
  2. x f ( x ) . \|x\|\to\infty\Rightarrow f(x)\to\infty.\,
  3. n \mathbb{R}^{n}
  4. x \|x\|\to\infty\,
  5. f 1 ( x ) = ( x 1 - x 2 ) 2 \ f_{1}(x)=(x_{1}-x_{2})^{2}\,
  6. f 2 ( x ) = ( x 1 2 + x 2 2 ) / ( 1 + x 1 2 + x 2 2 ) + ( x 1 - x 2 ) 2 \ f_{2}(x)=(x_{1}^{2}+x_{2}^{2})/(1+x_{1}^{2}+x_{2}^{2})+(x_{1}-x_{2})^{2}\,
  7. x 1 = x 2 x_{1}=x_{2}

Radiative_equilibrium.html

  1. 𝐅 ν \mathbf{F}_{\nu}
  2. h ν = - 𝐅 ν h_{\nu}=-\nabla\cdot\mathbf{F}_{\nu}
  3. 𝐅 ν = 0 \nabla\cdot\mathbf{F}_{\nu}=0
  4. h = 0 h ν d ν = 0 h=\int_{0}^{\infty}h_{\nu}d\nu=0

Radioactive_Instability_in_the_Nucleus_–_Formula.html

  1. p = 1 / 2 n p=1/2^{n}
  2. p p
  3. n n
  4. p = 1 / 2 n p=1/2^{n}
  5. p = 1 / 2 45 p=1/2^{45}

Radioanalytical_chemistry.html

  1. X Z A Y Z - 2 A - 4 + α 4 2 {}_{Z}^{A}\!X\to{}_{Z-2}^{A-4}\!Y+{}_{4}^{2}\alpha
  2. X Z A Y Z + 1 A + ν ¯ + β - {}_{Z}^{A}\!X\to{}_{Z+1}^{A}\!Y+\bar{\nu}+\beta^{-}
  3. X Z A Y Z - 1 A + ν + β + {}_{Z}^{A}\!X\to{}_{Z-1}^{A}\!Y+\nu+\beta^{+}
  4. X * A Y A + γ {}^{A}\!X^{*}\to{}^{A}\!Y+\gamma

Rainscreen.html

  1. w e t s a m p l e w e i g h t - d r y s a m p l e w e i g h t d r y s a m p l e w e i g h t * 100 = W M E {\frac{wet\;sample\;weight-dry\;sample\;weight}{dry\;sample\;weight}}\;*100\;=WME\;

Randić's_molecular_connectivity_index.html

  1. 1 / ( d i d j ) 1 / 2 1/(d_{i}d_{j})^{1/2}
  2. d i d_{i}
  3. d j d_{j}

Random_binary_tree.html

  1. x x
  2. n n
  3. x x
  4. 2 l o g n + O ( 1 ) 2logn+O(1)
  5. l o g log
  6. O O
  7. x x
  8. y y
  9. y y
  10. x x
  11. y y
  12. x x
  13. y y
  14. x x , y xx,y
  15. x x
  16. 1 / 2 1/2
  17. x x
  18. 1 / 3 1/3
  19. x x
  20. n n
  21. 1 β log n 4.311 log n , \frac{1}{\beta}\log n\approx 4.311\log n,
  22. β β
  23. 2 β e 1 - β = 1. \displaystyle 2\beta e^{1-\beta}=1.
  24. 1 / 3 1/3
  25. 1 / 2 1/2
  26. n 2 n≥2
  27. ( n + 1 ) / 3 (n+1)/3
  28. 1 1
  29. n n
  30. n n
  31. n = 1 , 2 , 3 , n= 1,2,3,...
  32. 1 , 2 , 5 , 14 , 42 , 132 , 429 , 1430 , 4862 , 16796 , 1,2,5,14,42,132,429,1430,4862,16796,…
  33. n n
  34. i i
  35. i 1 i−1
  36. n n
  37. x x
  38. ( 0 , 1 ) (0,1)
  39. x n xn
  40. x x
  41. x x
  42. x x

Random_laser.html

  1. k l < 1 kl<1
  2. k l > 1 kl>1

Random_walk_model_of_consumption.html

  1. E 1 u ( c 2 ) = ( 1 + δ 1 + r ) u ( c 1 ) E_{1}u^{\prime}(c_{2})=\left(\frac{1+\delta}{1+r}\right)u^{\prime}(c_{1})
  2. δ \delta
  3. r r
  4. E 1 E_{1}
  5. δ = r \delta=r
  6. E 1 c 2 = c 1 E_{1}c_{2}=c_{1}
  7. c 2 = c 1 + ϵ 2 c_{2}=c_{1}+\epsilon_{2}
  8. ϵ 2 \epsilon_{2}

Random_walker_algorithm.html

  1. L L
  2. v i v_{i}
  3. e i j e_{ij}
  4. v i v_{i}
  5. v j v_{j}
  6. g i g_{i}
  7. v i v_{i}
  8. w i j = exp ( - β ( g i - g j ) 2 ) . w_{ij}=\exp{\left(-\beta(g_{i}-g_{j})^{2}\right)}.
  9. Q ( x ) = x T L x = e i j w i j ( x i - x j ) 2 Q(x)=x^{T}Lx=\sum_{e_{ij}}w_{ij}\left(x_{i}-x_{j}\right)^{2}
  10. x i x_{i}
  11. x i = 1 x_{i}=1
  12. v i F v_{i}\in F
  13. x i = 0 x_{i}=0
  14. v i B v_{i}\in B
  15. F F
  16. B B
  17. S S
  18. S = F B S=F\cup B
  19. S ¯ \overline{S}
  20. S S ¯ = V S\cup\overline{S}=V
  21. V V
  22. L S ¯ , S ¯ x S ¯ = - L S ¯ , S x S , L_{\overline{S},\overline{S}}x_{\overline{S}}=-L_{\overline{S},S}x_{S},
  23. L L
  24. Q ( x ) = x T L x + γ ( ( 1 - x ) T F ( 1 - x ) + x T B x ) = e i j w i j ( x i - x j ) 2 + γ ( v i f i ( 1 - x i ) 2 + v i b i x i 2 ) , Q(x)=x^{T}Lx+\gamma\left((1-x)^{T}F(1-x)+x^{T}Bx\right)=\sum_{e_{ij}}w_{ij}% \left(x_{i}-x_{j}\right)^{2}+\gamma\left(\sum_{v_{i}}f_{i}(1-x_{i})^{2}+\sum_{% v_{i}}b_{i}x_{i}^{2}\right),
  25. F F
  26. B B
  27. ( L S ¯ , S ¯ + γ F S ¯ , S ¯ + γ B S ¯ , S ¯ ) x S ¯ = - L S ¯ , S x S - γ F S ¯ , S ¯ . \left(L_{\overline{S},\overline{S}}+\gamma F_{\overline{S},\overline{S}}+% \gamma B_{\overline{S},\overline{S}}\right)x_{\overline{S}}=-L_{\overline{S},S% }x_{S}-\gamma F_{\overline{S},\overline{S}}.
  28. S S
  29. S ¯ = V \overline{S}=V
  30. f i f_{i}
  31. v i v_{i}
  32. f i f_{i}
  33. v i v_{i}
  34. b i b_{i}
  35. v i v_{i}
  36. r i j r_{ij}
  37. e i j e_{ij}
  38. r i j = 1 w i j r_{ij}=\frac{1}{w_{ij}}
  39. v i B v_{i}\in B
  40. v i F v_{i}\in F
  41. v i F v_{i}\in F
  42. x i x_{i}
  43. v i v_{i}
  44. v i v_{i}

Randomized_algorithms_as_zero-sum_games.html

  1. min R max D T ( A , D ) = max D min A T ( A , D ) \min_{R}\max_{D}T(A,D)=\max_{D}\min_{A}T(A,D)\,
  2. T ( A , D ) = E x D [ T ( A , X ) ] . T(A,D)=\,\underset{x\sim D}{\operatorname{E}}[T(A,X)].\,

Randomized_Hough_transform.html

  1. a ( x - p ) 2 + 2 b ( x - p ) ( y - q ) + c ( y - q ) 2 = 1 a(x-p)^{2}+2b(x-p)(y-q)+c(y-q)^{2}=1
  2. a c - b 2 > 0 ac-b^{2}>0
  3. X 1 X 2 X_{1}X_{2}
  4. X 2 X 3 X_{2}X_{3}
  5. T 12 M 12 T_{12}M_{12}
  6. T 23 M 23 T_{23}M_{23}
  7. x = x - x 0 x^{\prime}=x-x_{0}
  8. y = y - y 0 y^{\prime}=y-y_{0}
  9. a x 2 + 2 b x y + c y 2 = 1 ax^{\prime 2}+2bx^{\prime}y^{\prime}+cy^{\prime 2}=1

Randomized_rounding.html

  1. x x
  2. x x
  3. x x^{\prime}
  4. x x
  5. x x^{\prime}
  6. x x
  7. x x^{\prime}
  8. x x
  9. x x
  10. n n
  11. d d
  12. n / ( d + 1 ) n/(d+1)
  13. n / ( d + 1 ) n/(d+1)
  14. c , 𝒮 \langle c,\mathcal{S}\rangle
  15. 𝒰 \mathcal{U}
  16. x * x^{*}
  17. x x
  18. s 𝒮 s\in\mathcal{S}
  19. x s x_{s}
  20. e 𝒰 e\in\mathcal{U}
  21. x x^{\prime}
  22. e e
  23. e e
  24. s e x s 1. \sum_{s\ni e}x_{s}\geq 1.
  25. x * x^{*}
  26. s 𝒰 c ( S ) x s * \sum_{s\in\mathcal{U}}c(S)x^{*}_{s}
  27. 𝒞 \mathcal{C}
  28. 𝒮 \mathcal{S}
  29. x x
  30. x s = 1 x_{s}=1
  31. s 𝒞 s\in\mathcal{C}
  32. x s = 0 x_{s}=0
  33. 𝒞 \mathcal{C}
  34. x x
  35. s 𝒞 c ( s ) = s 𝒮 c ( s ) x s . \sum_{s\in\mathcal{C}}c(s)=\sum_{s\in\mathcal{S}}c(s)x_{s}.
  36. x * x^{*}
  37. x * x^{*}
  38. x * x^{*}
  39. x x^{\prime}
  40. x x^{\prime}
  41. x * x^{*}
  42. x * x^{*}
  43. x x^{\prime}
  44. s 𝒮 s\in\mathcal{S}
  45. x s = 1 x^{\prime}_{s}=1
  46. min ( 1 , x s * ) \min(1,x^{*}_{s})
  47. x s = 0 x^{\prime}_{s}=0
  48. s c ( s ) x s * \sum_{s}c(s)x^{*}_{s}
  49. x s * x^{*}_{s}
  50. e e
  51. s e 1 - x s * s e exp ( - x s * ) = exp ( - s e x s * ) exp ( - 1 ) . \prod_{s\ni e}1-x^{*}_{s}\approx\prod_{s\ni e}\exp(-x^{*}_{s})=\exp\Big(-\sum_% {s\ni e}x^{*}_{s}\Big)\approx\exp(-1).
  52. x x^{\prime}
  53. λ > 1 \lambda>1
  54. λ 1 \lambda\geq 1
  55. s 𝒮 s\in\mathcal{S}
  56. x s = 1 x^{\prime}_{s}=1
  57. min ( λ x s * , 1 ) \min(\lambda x^{*}_{s},1)
  58. x s = 0 x^{\prime}_{s}=0
  59. λ \lambda
  60. λ \lambda
  61. λ \lambda
  62. λ = ln ( 2 | 𝒰 | ) \lambda=\ln(2|\mathcal{U}|)
  63. x x^{\prime}
  64. 2 ln ( 2 | 𝒰 | ) c x * 2\ln(2|\mathcal{U}|)c\cdot x^{*}
  65. O ( log | 𝒰 | ) O(\log|\mathcal{U}|)
  66. O ( log | 𝒰 | ) O(\log|\mathcal{U}|)
  67. ln ( | 𝒰 | ) + O ( log log | 𝒰 | ) \ln(|\mathcal{U}|)+O(\log\log|\mathcal{U}|)
  68. x x^{\prime}
  69. c x c\cdot x^{\prime}
  70. x x^{\prime}
  71. 2 λ c x * 2\lambda c\cdot x^{*}
  72. e e
  73. x x^{\prime}
  74. e e
  75. x s x^{\prime}_{s}
  76. λ x s * \lambda x_{s}^{*}
  77. c x c\cdot x^{\prime}
  78. s c ( s ) λ x s * = λ c x * \sum_{s}c(s)\lambda x_{s}^{*}=\lambda c\cdot x^{*}
  79. 1 / 2 1/2
  80. e e
  81. s e x s * 1 \sum_{s\ni e}x^{*}_{s}\geq 1
  82. e e
  83. e e
  84. s e ( 1 - min ( λ x s * , 1 ) ) \displaystyle\prod_{s\ni e}\big(1-\min(\lambda x^{*}_{s},1)\big)
  85. 1 + z e z 1+z\leq e^{z}
  86. z 0 z\neq 0
  87. | 𝒰 | |\mathcal{U}|
  88. 1 / ( 2 𝒰 ) 1/(2\mathcal{U})
  89. 1 + | 𝒰 | 1+|\mathcal{U}|
  90. 1 / 2 + | 𝒰 | / ( 2 𝒰 ) = 1 1/2+|\mathcal{U}|/(2\mathcal{U})=1
  91. x x^{\prime}
  92. 2 λ c x * 2\lambda c\cdot x^{*}
  93. O ( log ( | 𝒰 | ) c x * O(\log(|\mathcal{U}|)c\cdot x^{*}
  94. λ \lambda
  95. s 𝒮 s\in\mathcal{S}
  96. x s { 0 , 1 } x^{\prime}_{s}\in\{0,1\}
  97. x * x^{*}
  98. x s x^{\prime}_{s}
  99. F = c x 2 λ c x * + | 𝒰 ( m ) | F=\frac{c\cdot x^{\prime}}{2\lambda c\cdot x^{*}}+|\mathcal{U}^{(m)}|
  100. 𝒰 ( m ) = { e : s e ( 1 - x s ) = 1 } \mathcal{U}^{(m)}=\Big\{e:\prod_{s\ni e}(1-x^{\prime}_{s})=1\Big\}
  101. F F
  102. F F
  103. F F
  104. x x^{\prime}
  105. F F
  106. x x^{\prime}
  107. F F
  108. x x^{\prime}
  109. F 1 F\geq 1
  110. E [ F ] E[F]
  111. E [ F ] < 1 E[F]<1
  112. F 1 F\geq 1
  113. F F
  114. F F
  115. F F
  116. t t
  117. S ( t ) S^{(t)}
  118. t t
  119. 𝒮 \mathcal{S}
  120. x ( t ) x^{(t)}
  121. x x^{\prime}
  122. x s ( t ) x^{(t)}_{s}
  123. s S ( t ) s\in S^{(t)}
  124. s S ( t ) s\not\in S^{(t)}
  125. p s = min ( λ x s * , 1 ) p_{s}=\min(\lambda x^{*}_{s},1)
  126. x s x^{\prime}_{s}
  127. 𝒰 ( t ) \mathcal{U}^{(t)}
  128. F F
  129. x ( t ) x^{(t)}
  130. E [ F | x ( t ) ] = s S ( t ) c ( s ) x s + s S ( t ) c ( s ) p s 2 λ c x * + e 𝒰 ( t ) s S ( t ) , s e ( 1 - p s ) . E[F|x^{(t)}]~{}=~{}\frac{\sum_{s\in S^{(t)}}c(s)x^{\prime}_{s}+\sum_{s\not\in S% ^{(t)}}c(s)p_{s}}{2\lambda c\cdot x^{*}}~{}+~{}\sum_{e\in\mathcal{U}^{(t)}}% \prod_{s\not\in S^{(t)},s\ni e}(1-p_{s}).
  131. E [ F | x ( t ) ] E[F|x^{(t)}]
  132. t t
  133. F F
  134. F F
  135. x s x^{\prime}_{s}
  136. E [ F | x ( m ) ] E [ F | x ( m - 1 ) ] E [ F | x ( 1 ) ] E [ F | x ( 0 ) ] < 1 E[F|x^{(m)}]\leq E[F|x^{(m-1)}]\leq\cdots\leq E[F|x^{(1)}]\leq E[F|x^{(0)}]<1
  137. m = | 𝒮 | m=|\mathcal{S}|
  138. t t
  139. x s x^{\prime}_{s^{\prime}}
  140. E [ F | x ( t ) ] E [ F | S ( t - 1 ) ] E[F|x^{(t)}]\leq E[F|S^{(t-1)}]
  141. x s x^{\prime}_{s^{\prime}}
  142. E [ F | x ( t ) ] E[F|x^{(t)}]
  143. t t
  144. E [ F | x ( t - 1 ) ] E[F|x^{(t-1)}]
  145. E [ F | x ( t ) ] E[F|x^{(t)}]
  146. x s x^{\prime}_{s^{\prime}}
  147. t t
  148. E ( t - 1 ) E^{(t-1)}
  149. E [ F | x ( t - 1 ) ] E[F|x^{\prime(t-1)}]
  150. E 0 ( t ) E^{(t)}_{0}
  151. E 1 ( t ) E^{(t)}_{1}
  152. E [ F | x ( t ) ] E[F|x^{(t)}]
  153. x s x^{\prime}_{s^{\prime}}
  154. E ( t - 1 ) = Pr [ x s = 0 ] E 0 ( t ) + Pr [ x s = 1 ] E 1 ( t ) . E^{(t-1)}~{}=~{}\Pr[x^{\prime}_{s^{\prime}}=0]E^{(t)}_{0}+\Pr[x^{\prime}_{s^{% \prime}}=1]E^{(t)}_{1}.
  155. E ( t - 1 ) min ( E 0 ( t ) , E 1 ( t ) ) . E^{(t-1)}~{}\geq~{}\min(E^{(t)}_{0},E^{(t)}_{1}).
  156. x s x^{\prime}_{s^{\prime}}
  157. E [ F | x ( t ) ] E[F|x^{(t)}]
  158. E [ F | x ( t ) ] E [ F | x ( t - 1 ) ] E[F|x^{(t)}]\leq E[F|x^{(t-1)}]
  159. x s x^{\prime}_{s^{\prime}}
  160. E [ F | x ( t ) ] E[F|x^{(t)}]
  161. x s x^{\prime}_{s^{\prime}}
  162. x s x^{\prime}_{s^{\prime}}
  163. c s 2 λ c x * - e s 𝒰 t - 1 s S ( t ) , s e ( 1 - p s ) . \frac{c_{s^{\prime}}}{2\lambda c\cdot x^{*}}~{}-~{}\sum_{e\in s^{\prime}\cap% \mathcal{U}_{t-1}}\prod_{s\not\in S^{(t)},s\ni e}(1-p_{s}).
  164. x s x^{\prime}_{s^{\prime}}
  165. 𝒮 \mathcal{S}
  166. 𝒰 \mathcal{U}
  167. c c
  168. x x^{\prime}
  169. x * x^{*}
  170. λ ln ( 2 | 𝒰 | ) \lambda\leftarrow\ln(2|\mathcal{U}|)
  171. p s min ( λ x s * , 1 ) p_{s}\leftarrow\min(\lambda x^{*}_{s},1)
  172. s 𝒮 s\in\mathcal{S}
  173. s 𝒮 s^{\prime}\in\mathcal{S}
  174. 𝒮 𝒮 - { s } \mathcal{S}\leftarrow\mathcal{S}-\{s^{\prime}\}
  175. 𝒮 \mathcal{S}
  176. c s 2 λ c x * > e s 𝒰 s 𝒮 , s e ( 1 - p s ) \frac{c_{s^{\prime}}}{2\lambda c\cdot x^{*}}>\sum_{e\in s^{\prime}\cap\mathcal% {U}}\prod_{s\in\mathcal{S},s\ni e}(1-p_{s})
  177. x s 0 x^{\prime}_{s}\leftarrow 0
  178. x s 1 x^{\prime}_{s}\leftarrow 1
  179. 𝒰 𝒰 - s \mathcal{U}\leftarrow\mathcal{U}-s^{\prime}
  180. 𝒰 \mathcal{U}
  181. x x^{\prime}
  182. x x^{\prime}
  183. 2 ln ( 2 | 𝒰 | ) 2\ln(2|\mathcal{U}|)
  184. F F
  185. E [ F | x ( t ) ] E[F\,|\,x^{(t)}]
  186. F F
  187. x x^{\prime}
  188. 2 ln ( 2 | 𝒰 | ) 2\ln(2|\mathcal{U}|)
  189. F F

Randomized_weighted_majority_algorithm.html

  1. n n
  2. m m
  3. 2.4 ( log 2 n + m ) 2.4(\log_{2}n+m)
  4. 2.4 ( log 2 n + m ) 2.4(\log_{2}n+m)
  5. N = 100 N=100
  6. n = 10 n=10
  7. m = 20 m=20
  8. 2.4 ( log 2 10 + 20 ) 56 2.4(\log_{2}10+20)\approx 56
  9. m m
  10. w i w_{i}
  11. i i
  12. W = i w i W=\sum_{i}w_{i}
  13. i i
  14. w i W \frac{w_{i}}{W}
  15. m m
  16. log 2 n \log_{2}n
  17. β < 1 \beta<1
  18. 1 2 \frac{1}{2}
  19. t \ t
  20. F t \ F_{t}
  21. F t \ F_{t}
  22. t \ t
  23. M \ M
  24. E [ M ] = t F t E[M]=\ \sum_{t}F_{t}
  25. t \ t
  26. W W
  27. W ( 1 - ( 1 - β ) F t ) \ W(1-(1-\beta)F_{t})
  28. F t \ F_{t}
  29. β \ \beta
  30. W f i n a l = n * ( 1 - ( 1 - β ) F 1 ) * ( 1 - ( 1 - β ) F 2 ) \ W_{final}=n*(1-(1-\beta)F_{1})*(1-(1-\beta)F_{2})...
  31. m \ m
  32. W β m \ W\geq\beta^{m}
  33. m l n β l n ( n ) + t l n ( 1 - ( 1 - β ) F t ) \ mln\beta\leq ln(n)+\sum_{t}ln(1-(1-\beta)F_{t})
  34. l n ( 1 - x ) = - x - x 2 2 - x 3 3 - \ ln(1-x)=-x-\frac{x^{2}}{2}-\frac{x^{3}}{3}-...
  35. l n ( 1 - ( 1 - β ) F t ) < - ( 1 - β ) F t \ ln(1-(1-\beta)F_{t})<-(1-\beta)F_{t}
  36. m l n β l n ( n ) - ( 1 - β ) * t F t \ mln\beta\leq ln(n)-(1-\beta)*\sum_{t}F_{t}
  37. E [ M ] = t F t \ E[M]=\ \sum_{t}F_{t}
  38. E [ M ] m l n ( 1 / β ) + l n ( n ) 1 - β \ E[M]\leq\frac{mln(1/\beta)+ln(n)}{1-\beta}
  39. β = 1 2 \ \beta=\frac{1}{2}
  40. 1.39 m + 2 l n ( n ) . \ 1.39m+2ln(n).
  41. β = 3 4 \ \beta=\frac{3}{4}
  42. 1.15 m + 4 l n ( n ) \ 1.15m+4ln(n)
  43. ( 1 + ϵ ) * m + ϵ - 1 * l n ( n ) \ (1+\epsilon)*m+\epsilon^{-1}*ln(n)
  44. β l o s s \beta^{loss}
  45. 1 2 \ \frac{1}{2}

Range_concatenation_grammars.html

  1. G = ( N , T , V , S , P ) G=(N,~{}T,~{}V,~{}S,~{}P)
  2. N N
  3. T T
  4. V V
  5. d i m : N { 0 } dim:N\rightarrow\mathbb{N}\setminus\{0\}
  6. S N S\in N
  7. d i m ( S ) = 1 dim(S)=1
  8. P P
  9. ψ 0 ψ 1 ψ m \psi_{0}\rightarrow\psi_{1}\ldots\psi_{m}
  10. ψ i \psi_{i}
  11. A i ( α 1 , , α d i m ( A i ) ) A_{i}(\alpha_{1},\ldots,\alpha_{dim(A_{i})})
  12. A i N A_{i}\in N
  13. α i ( T V ) \alpha_{i}\in(T\cup V)^{\star}
  14. A i ( α 1 , , α d i m ( A i ) ) ¯ \overline{A_{i}(\alpha_{1},\ldots,\alpha_{dim(A_{i})})}
  15. w T w\in T^{\star}
  16. l , r w \langle l,r\rangle_{w}
  17. 0 l r n 0\leq l\leq r\leq n
  18. n n
  19. w w
  20. l 1 , r 1 w \langle l_{1},r_{1}\rangle_{w}
  21. l 2 , r 2 w \langle l_{2},r_{2}\rangle_{w}
  22. r 1 = l 2 r_{1}=l_{2}
  23. l 1 , r 1 w l 2 , r 2 w = l 1 , r 2 w \langle l_{1},r_{1}\rangle_{w}\cdot\langle l_{2},r_{2}\rangle_{w}=\langle l_{1% },r_{2}\rangle_{w}
  24. w = w 1 w 2 w n w=w_{1}w_{2}\ldots w_{n}
  25. w i T w_{i}\in T
  26. l , r w = w 1 w l - 1 w l w r - 1 w r w n \langle l,r\rangle_{w}=w_{1}\ldots w_{l-1}\bullet w_{l}\ldots w_{r-1}\bullet w% _{r}\ldots w_{n}
  27. A ( x 1 , , x n ) α A(x_{1},...,x_{n})\to\alpha
  28. α \alpha
  29. x i x_{i}
  30. x y xy
  31. a b ab
  32. x = ϵ , y = a b ; x = a , y = b ; x = a b , y = ϵ x=\epsilon,\ y=ab;\ x=a,\ y=b;\ x=ab,\ y=\epsilon
  33. A ( x 1 , , x n ) ¯ \overline{A(x_{1},...,x_{n})}
  34. A ( α 1 , , α n ) A(\alpha_{1},...,\alpha_{n})
  35. α i \alpha_{i}
  36. A ( x 1 , , x n ) β A(x_{1},...,x_{n})\to\beta
  37. β \beta
  38. x i x_{i}
  39. A ( x , a y b ) B ( a x b , y ) A(x,ayb)\to B(axb,y)
  40. x x
  41. y y
  42. a a
  43. b b
  44. A ( a , a b b ) A(a,abb)
  45. B ( a a b , b ) B(aab,b)
  46. A ( a , a b b ) A(a,abb)
  47. A ( x , a y b ) A(x,ayb)
  48. x = a , y = b x=a,\ y=b
  49. A ( x , a y b ) A ( x , x ) A ( y , y ) A(x,ayb)\to A(x,x)\ A(y,y)
  50. A ( a , a b b ) A(a,abb)
  51. A ( a , a ) A ( b , b ) A(a,a)\ A(b,b)
  52. α \alpha
  53. S ( α ) S(\alpha)
  54. S ( α ) S(\alpha)
  55. { w w w : w { a , b } * } \{www:w\in\{a,b\}^{*}\}
  56. S ( x y z ) A ( x , y , z ) S(xyz)\to A(x,y,z)
  57. A ( a x , a y , a z ) A ( x , y , z ) A(ax,ay,az)\to A(x,y,z)
  58. A ( b x , b y , b z ) A ( x , y , z ) A(bx,by,bz)\to A(x,y,z)
  59. A ( ϵ , ϵ , ϵ ) ϵ A(\epsilon,\epsilon,\epsilon)\to\epsilon
  60. S ( a b b a b b a b b ) A ( a b b , a b b , a b b ) A ( b b , b b , b b ) A ( b , b , b ) A ( ϵ , ϵ , ϵ ) ϵ S(abbabbabb)\Rightarrow A(abb,abb,abb)\Rightarrow A(bb,bb,bb)\Rightarrow A(b,b% ,b)\Rightarrow A(\epsilon,\epsilon,\epsilon)\Rightarrow\epsilon
  61. S ( a b b a b b a b b ) A ( a b b a b b a b b , a b b a b b a b b , a b b a b b a b b ) A ( a b b a b b a b b , a b b a b b a b b , a b b a b b a b b ) S(\bullet{}abbabbabb\bullet{})\Rightarrow A(\bullet{}abb\bullet{}abbabb,abb% \bullet{}abb\bullet{}abb,abbabb\bullet{}abb\bullet{})\Rightarrow A(a\bullet{}% bb\bullet{}abbabb,abba\bullet{}bb\bullet{}abb,abbabba\bullet{}bb\bullet{})
  62. A ( a b b a b b a b b , a b b a b b a b b , a b b a b b a b b ) A ( ϵ , ϵ , ϵ ) ϵ \Rightarrow A(ab\bullet{}b\bullet{}abbabb,abbab\bullet{}b\bullet{}abb,abbabbab% \bullet{}b\bullet{})\Rightarrow A(\epsilon,\epsilon,\epsilon)\Rightarrow\epsilon

Rank_factorization.html

  1. m × n m\times n
  2. A A
  3. r r
  4. A A
  5. A = C F A=CF
  6. C C
  7. m × r m\times r
  8. F F
  9. r × n r\times n
  10. A A
  11. m × n m\times n
  12. r r
  13. r r
  14. A A
  15. A A
  16. r r
  17. c 1 , c 2 , , c r c_{1},c_{2},\ldots,c_{r}
  18. A A
  19. m × r m\times r
  20. C = [ c 1 : c 2 : : c r ] C=[c_{1}:c_{2}:\ldots:c_{r}]
  21. A A
  22. C C
  23. A = [ a 1 : a 2 : : a n ] A=[a_{1}:a_{2}:\ldots:a_{n}]
  24. m × n m\times n
  25. a j a_{j}
  26. j j
  27. a j = f 1 j c 1 + f 2 j c 2 + + f r j c r , a_{j}=f_{1j}c_{1}+f_{2j}c_{2}+\cdots+f_{rj}c_{r},
  28. f i j f_{ij}
  29. a j a_{j}
  30. c 1 , c 2 , , c r c_{1},c_{2},\ldots,c_{r}
  31. A = C F A=CF
  32. f i j f_{ij}
  33. ( i , j ) (i,j)
  34. F F
  35. A A
  36. A T A\text{T}
  37. A A
  38. A T A\text{T}
  39. A A
  40. A = C F A=CF
  41. A T = F T C T A\text{T}=F\text{T}C\text{T}
  42. A T A\text{T}
  43. F T F\text{T}
  44. A T A\text{T}
  45. F T F\text{T}
  46. A T A\text{T}
  47. F T F\text{T}
  48. F T F\text{T}
  49. n n
  50. r r
  51. r r
  52. F T F\text{T}
  53. A T A\text{T}
  54. r r
  55. A A
  56. A T ) A\text{T})
  57. A A
  58. A T A\text{T}
  59. ( A T ) T (A\text{T})\text{T}
  60. A A
  61. A A
  62. ( A T ) T ) (A\text{T})\text{T})
  63. A T A\text{T}
  64. A ) A)
  65. A T A\text{T}
  66. A T ) A\text{T})
  67. A A
  68. A A
  69. A T A\text{T}
  70. A A
  71. A T A\text{T}
  72. B B
  73. A A
  74. C C
  75. A A
  76. F F
  77. B B
  78. A = [ 1 3 1 4 2 7 3 9 1 5 3 1 1 2 0 8 ] [ 1 0 - 2 0 0 1 1 0 0 0 0 1 0 0 0 0 ] = B . A=\begin{bmatrix}1&3&1&4\\ 2&7&3&9\\ 1&5&3&1\\ 1&2&0&8\end{bmatrix}\sim\begin{bmatrix}1&0&-2&0\\ 0&1&1&0\\ 0&0&0&1\\ 0&0&0&0\end{bmatrix}=B\,\text{.}
  79. B B
  80. C C
  81. A A
  82. F F
  83. C = [ 1 3 4 2 7 9 1 5 1 1 2 8 ] , F = [ 1 0 - 2 0 0 1 1 0 0 0 0 1 ] . C=\begin{bmatrix}1&3&4\\ 2&7&9\\ 1&5&1\\ 1&2&8\end{bmatrix}\,\text{,}\qquad F=\begin{bmatrix}1&0&-2&0\\ 0&1&1&0\\ 0&0&0&1\end{bmatrix}\,\text{.}
  84. A = [ 1 3 1 4 2 7 3 9 1 5 3 1 1 2 0 8 ] = [ 1 3 4 2 7 9 1 5 1 1 2 8 ] [ 1 0 - 2 0 0 1 1 0 0 0 0 1 ] = C F . A=\begin{bmatrix}1&3&1&4\\ 2&7&3&9\\ 1&5&3&1\\ 1&2&0&8\end{bmatrix}=\begin{bmatrix}1&3&4\\ 2&7&9\\ 1&5&1\\ 1&2&8\end{bmatrix}\begin{bmatrix}1&0&-2&0\\ 0&1&1&0\\ 0&0&0&1\end{bmatrix}=CF\,\text{.}
  85. P P
  86. n × n n\times n
  87. A P = ( C , D ) AP=(C,D)
  88. C C
  89. r r
  90. A A
  91. D D
  92. C C
  93. G G
  94. D = C G D=CG
  95. G G
  96. A P = ( C , C G ) = C ( I r , G ) AP=(C,CG)=C(I_{r},G)
  97. I r I_{r}
  98. r × r r\times r
  99. ( I r , G ) = F P (I_{r},G)=FP
  100. A P AP
  101. E E
  102. E A P = B P = E C ( I r , G ) EAP=BP=EC(I_{r},G)
  103. E C = ( I r 0 ) EC=\begin{pmatrix}I_{r}\\ 0\end{pmatrix}
  104. B P = ( I r G 0 0 ) BP=\begin{pmatrix}I_{r}&G\\ 0&0\end{pmatrix}
  105. ( I r , G ) = F P (I_{r},G)=FP
  106. r r
  107. A A
  108. A P = C F P AP=CFP
  109. P P
  110. A = C F A=CF

Ranklet.html

  1. W s = i = 1 N π i V i where π i = rank of element i and V i = { 0 for π i C 1 for π i T W_{s}=\sum^{N}_{i=1}\pi_{i}V_{i}\,\text{ where }\pi_{i}=\,\text{rank of % element }i\,\text{ and }V_{i}=\begin{cases}0&\,\text{ for }\pi_{i}\in C\\ 1&\,\text{ for }\pi_{i}\in T\end{cases}
  2. M W = W s - m ( m + 1 ) 2 MW=W_{s}-\frac{m(m+1)}{2}
  3. R = M W m n / 2 - 1 R=\frac{MW}{mn/2}-1
  4. T = { 5 , 9 , 1 , 10 , 15 } T=\{5,9,1,10,15\}
  5. C = { 20 , 4 , 7 , 13 , 19 , 11 } C=\{20,4,7,13,19,11\}
  6. W s = { 1 + 3 + 5 + 6 + 9 } = 24 W_{s}=\Big\{1+3+5+6+9\Big\}=24
  7. M W = 24 - [ 5 × ( 5 + 1 ) / 2 ] = 9 MW=24-[5\times(5+1)/2]=9
  8. R = [ 9 / [ 5 × 6 / 2 ] ] - 1 = - 0.4 R=[9/[5\times 6/2]]-1=-0.4

Rare_disasters.html

  1. P t {P_{t}}
  2. R = A t + 1 P t 1 R=\frac{A_{t+1}}{P_{t1}}
  3. w ~ t + 1 \tilde{w}_{t+1}
  4. v ~ \tilde{v}
  5. log A t + 1 = log A t + g ¯ + u ~ t + 1 + v ~ t + 1 \log A_{t+1}=\log A_{t}+\bar{g}+\tilde{u}_{t+1}+\tilde{v}_{t+1}
  6. v ~ t + 1 \tilde{v}_{t+1}
  7. u ~ t + 1 \tilde{u}_{t+1}
  8. v ~ t + 1 \tilde{v}_{t+1}
  9. { 1 ( e - p ) no disaster 1 - b ( 1 - e - p ) disaster \begin{cases}1&(e^{-p})\mbox{ no disaster}\\ 1-b&(1-e^{-p})\mbox{ disaster}\\ \end{cases}

Rastrigin_function.html

  1. f ( 𝐱 ) = A n + i = 1 n [ x i 2 - A cos ( 2 π x i ) ] f(\mathbf{x})=An+\sum_{i=1}^{n}\left[x_{i}^{2}-A\cos(2\pi x_{i})\right]
  2. A = 10 A=10
  3. x i [ - 5.12 , 5.12 ] x_{i}\in[-5.12,5.12]
  4. 𝐱 = 𝟎 \mathbf{x}=\mathbf{0}
  5. f ( 𝐱 ) = 0 f(\mathbf{x})=0

Rat-race_coupler.html

  1. 2 \sqrt{2}
  2. S = - i 2 ( 0 1 0 - 1 1 0 1 0 0 1 0 1 - 1 0 1 0 ) S=\frac{-i}{\sqrt{2}}\begin{pmatrix}0&1&0&-1\\ 1&0&1&0\\ 0&1&0&1\\ -1&0&1&0\end{pmatrix}

Rate_of_return_on_a_portfolio.html

  1. r = A 1 r 1 + A 2 r 2 + + A n r n r=A_{1}r_{1}+A_{2}r_{2}+\cdots+A_{n}r_{n}
  2. A 1 r 1 A_{1}r_{1}
  3. A 2 r 2 A n r n A_{2}r_{2}\cdots A_{n}r_{n}
  4. r r
  5. A i A_{i}
  6. r i r_{i}

Rate_ratio.html

  1. Rate Ratio = Incidence Rate 1 Incidence Rate 2 \,\text{Rate Ratio}=\frac{\,\text{Incidence Rate 1}}{\,\text{Incidence Rate 2}}
  2. Incidence Rate = events person time \,\text{Incidence Rate}=\frac{\,\text{events}}{\,\text{person time}}

Rating_curve.html

  1. Q = C r ( G - a ) β Q=C_{r}(G-a)^{\beta}
  2. C r C_{r}
  3. β \beta
  4. a a
  5. a a

Rational_difference_equation.html

  1. x n + 1 = α + i = 0 k β i x n - i A + i = 0 k B i x n - i , x_{n+1}=\frac{\alpha+\sum_{i=0}^{k}\beta_{i}x_{n-i}}{A+\sum_{i=0}^{k}B_{i}x_{n% -i}}~{},
  2. x 0 , x - 1 , , x - k x_{0},x_{-1},\dots,x_{-k}
  3. n n
  4. w t + 1 = a w t + b c w t + d . w_{t+1}=\frac{aw_{t}+b}{cw_{t}+d}.
  5. a , b , c , d a,b,c,d
  6. w 0 w_{0}
  7. w t w_{t}
  8. x t x_{t}
  9. x t x_{t}
  10. x t x_{t}
  11. a d - b c 0 ad-bc\neq 0
  12. y t + 1 = α - β y t y_{t+1}=\alpha-\frac{\beta}{y_{t}}
  13. α = ( a + d ) / c \alpha=(a+d)/c
  14. β = ( a d - b c ) / c 2 \beta=(ad-bc)/c^{2}
  15. w t = y t - d / c w_{t}=y_{t}-d/c
  16. y t = x t + 1 / x t y_{t}=x_{t+1}/x_{t}
  17. x t + 2 - α x t + 1 + β x t = 0. x_{t+2}-\alpha x_{t+1}+\beta x_{t}=0.\,
  18. x t x_{t}
  19. ( d - a ) 2 + 4 b c (d-a)^{2}+4bc
  20. x t = 1 / ( η + w t ) x_{t}=1/(\eta+w_{t})
  21. w t = ( 1 - η x t ) / x t w_{t}=(1-\eta x_{t})/x_{t}
  22. η \eta
  23. η = ( d - a + r ) / 2 c \eta=(d-a+r)/2c
  24. r = ( d - a ) 2 + 4 b c r=\sqrt{(d-a)^{2}+4bc}
  25. x t x_{t}
  26. x t + 1 = ( d - η c η c + a ) x t + c η c + a . x_{t+1}=\left(\frac{d-\eta c}{\eta c+a}\right)x_{t}+\frac{c}{\eta c+a}.
  27. w t + 1 = a w t + b c w t + d w_{t+1}=\frac{aw_{t}+b}{cw_{t}+d}
  28. X t + 1 = - ( E + B X t ) ( C + A X t ) - 1 , X_{t+1}=-(E+BX_{t})(C+AX_{t})^{-1},
  29. X t = N t D t - 1 X_{t}=N_{t}D_{t}^{-1}
  30. ( N t D t ) = ( - B - E A C ) t ( X 0 I ) . \begin{pmatrix}N_{t}\\ D_{t}\end{pmatrix}=\begin{pmatrix}-B&-E\\ A&C\end{pmatrix}^{t}\begin{pmatrix}X_{0}\\ I\end{pmatrix}.
  31. H t - 1 = K + A H t A - A H t C ( C H t C ) - 1 C H t A , H_{t-1}=K+A^{\prime}H_{t}A-A^{\prime}H_{t}C(C^{\prime}H_{t}C)^{-1}C^{\prime}H_% {t}A,\,

Rational_quadratic_covariance_function.html

  1. C ( d ) = ( 1 + d 2 2 α k 2 ) - α C(d)=\Bigg(1+\frac{d^{2}}{2\alpha k^{2}}\Bigg)^{-\alpha}

Rayleigh_distance.html

  1. Z = 2 D 2 λ Z=\frac{2D^{2}}{\lambda}

Reach_(mathematics).html

  1. reach ( X ) := sup { r : x n X with dist ( x , X ) < r exists a unique closest point y X such that dist ( x , y ) = dist ( x , X ) } . \,\text{reach}(X):=\sup\{r\in\mathbb{R}:\forall x\in\mathbb{R}^{n}\setminus X% \,\text{ with }{\rm dist}(x,X)<r\,\text{ exists a unique closest point }y\in X% \,\text{ such that }{\rm dist}(x,y)={\rm dist}(x,X)\}.

Real-valued_function.html

  1. X X
  2. ( X , ) {\mathcal{F}}(X,{\mathbb{R}})
  3. X X
  4. 𝐑 \mathbf{R}
  5. 𝐑 \mathbf{R}
  6. ( X , ) {\mathcal{F}}(X,{\mathbb{R}})
  7. f + g : x f ( x ) + g ( x ) \ f+g:x\mapsto f(x)+g(x)
  8. 0 : x 0 \ \mathbf{0}:x\mapsto 0
  9. c f : x c f ( x ) , c \ cf:x\mapsto cf(x),\quad c\in{\mathbb{R}}
  10. f g : x f ( x ) g ( x ) \ fg:x\mapsto f(x)g(x)
  11. 𝐑 \mathbf{R}
  12. ( X , ) {\mathcal{F}}(X,{\mathbb{R}})
  13. f g x : f ( x ) g ( x ) \ f\leq g\quad\iff\quad\forall x:f(x)\leq g(x)
  14. ( X , ) {\mathcal{F}}(X,{\mathbb{R}})
  15. X X
  16. f f
  17. B B
  18. f f
  19. X X
  20. X X
  21. Ω Ω
  22. X X
  23. x X x∈X
  24. f ( x ) f(x)
  25. p p
  26. \sdot : L 1 / α × L 1 / β L 1 / ( α + β ) , 0 α , β 1 , α + β 1. \sdot:L^{1/\alpha}\times L^{1/\beta}\to L^{1/(\alpha+\beta)},\quad 0\leq\alpha% ,\beta\leq 1,\quad\alpha+\beta\leq 1.
  27. [ 0 , + ] [0, +∞]

Real_form_(Lie_theory).html

  1. 𝔤 𝔤 0 . \mathfrak{g}\simeq\mathfrak{g}_{0}\otimes_{\mathbb{R}}\mathbb{C}.
  2. X X t . X\to{X}^{t}.
  3. 𝔤 0 = 𝔨 0 𝔭 0 . \mathfrak{g}_{0}=\mathfrak{k}_{0}\oplus\mathfrak{p}_{0}.
  4. 𝔲 0 = 𝔨 0 i 𝔭 0 \mathfrak{u}_{0}=\mathfrak{k}_{0}\oplus i\mathfrak{p}_{0}

Recombination_(cosmology).html

  1. x e = n e n p + n H . x\text{e}=\frac{n\text{e}}{n\text{p}+n\text{H}}.
  2. p + e - H + γ p+e^{-}\longleftrightarrow H+\gamma
  3. n p n e n H = ( m e k B T 2 π 2 ) 3 / 2 exp ( - E I k B T ) , \frac{n\text{p}n\text{e}}{n\text{H}}=\left(\frac{m\text{e}k\text{B}T}{2\pi% \hbar^{2}}\right)^{3/2}\exp\left(-\frac{E\text{I}}{k\text{B}T}\right),
  4. x 2 e 1 - x e = ( n H + n p ) - 1 ( m e k B T 2 π 2 ) 3 / 2 exp ( - E I k B T ) . \frac{{x^{2}\text{e}}}{1-x\text{e}}=(n\text{H}+n\text{p})^{-1}\left(\frac{m% \text{e}k\text{B}T}{2\pi\hbar^{2}}\right)^{3/2}\exp\left(-\frac{E\text{I}}{k% \text{B}T}\right).
  5. d x e d t = - C ( α B ( T ) n p x e - 4 ( 1 - x e ) β B ( T ) e - E 21 / T ) , \frac{dx\text{e}}{dt}=-C\left(\alpha\text{B}(T)n\text{p}x_{e}-4(1-x\text{e})% \beta\text{B}(T)e^{-E_{21}/T}\right),
  6. He + + + e - He + + γ \mathrm{He}^{++}+\mathrm{e}^{-}\longrightarrow\mathrm{He}^{+}+\gamma
  7. He + + e - He + γ \mathrm{He}^{+}+\mathrm{e}^{-}\longrightarrow\mathrm{He}+\gamma

Recovery_consistency_objective.html

  1. RCO = 1 - ( number of inconsistent entities ) n ( number of entities ) n \,\text{RCO}=1-\frac{(\,\text{number of inconsistent entities})_{n}}{(\,\text{% number of entities})_{n}}

Rectified_5-cubes.html

  1. 2 \sqrt{2}
  2. ( 0 , ± 1 , ± 1 , ± 1 , ± 1 ) (0,\ \pm 1,\ \pm 1,\ \pm 1,\ \pm 1)
  3. 2 \sqrt{2}
  4. ( 0 , 0 , ± 1 , ± 1 , ± 1 ) \left(0,\ 0,\ \pm 1,\ \pm 1,\ \pm 1\right)

Rectified_5-orthoplexes.html

  1. 2 \sqrt{2}

Rectified_6-cubes.html

  1. ( 0 , ± 1 , ± 1 , ± 1 , ± 1 , ± 1 ) (0,\ \pm 1,\ \pm 1,\ \pm 1,\ \pm 1,\ \pm 1)
  2. ( 0 , 0 , ± 1 , ± 1 , ± 1 , ± 1 ) (0,\ 0,\ \pm 1,\ \pm 1,\ \pm 1,\ \pm 1)

Rectified_6-orthoplexes.html

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

Rectified_7-orthoplexes.html

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

Rectified_8-orthoplexes.html

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

Recurrence_tracking_microscope.html

  1. H = P 2 2 m + m g z + V 0 e - k z ( 1 ) H={P^{2}\over 2m}+mgz+V_{0}e^{-kz}\quad(1)
  2. p p
  3. m m
  4. g g

Recurrent_tensor.html

  1. \nabla
  2. T = ω T . \nabla T=\omega\otimes T.\,
  3. A = 0 \nabla A=0
  4. \nabla
  5. ( M , g ) (M,g)
  6. L C g = 0 \nabla^{LC}g=0
  7. X = 0 \nabla X=0
  8. X X
  9. X = ω X \nabla X=\omega\otimes X
  10. ω \omega
  11. T ( X , Y ) = X Y - Y X - [ X , Y ] = 0 T^{\nabla}(X,Y)=\nabla_{X}Y-\nabla_{Y}X-[X,Y]=0
  12. \nabla^{\prime}
  13. g = φ g \nabla^{\prime}g=\varphi\otimes g
  14. φ \varphi
  15. \nabla^{\prime}
  16. ( M , g ) (M,g)
  17. \nabla
  18. g g
  19. g g
  20. g e λ g g\rightarrow e^{\lambda}g
  21. ϕ \phi
  22. φ φ - d λ \varphi\rightarrow\varphi-d\lambda
  23. F : [ g ] Λ 1 ( M ) F:[g]\rightarrow\Lambda^{1}(M)
  24. ( M , [ g ] ) (M,[g])
  25. F ( e λ g ) := φ - d λ F(e^{\lambda}g):=\varphi-d\lambda
  26. [ g ] [g]
  27. F F
  28. F ( e λ g ) = F ( g ) - d λ F(e^{\lambda}g)=F(g)-d\lambda
  29. \mathcal{R}
  30. = ω \nabla\mathcal{R}=\omega\otimes\mathcal{R}

Recursive_language.html

  1. L = { w { a , b , c } * w = a n b n c n for some n 1 } L=\{\,w\in\{a,b,c\}^{*}\mid w=a^{n}b^{n}c^{n}\mbox{ for some }~{}n\geq 1\,\}
  2. 2 2 c n 2^{2^{cn}}
  3. c n c^{n}
  4. L * L^{*}
  5. L P L\circ P
  6. L P L\cup P
  7. L P L\cap P
  8. L L
  9. L - P L-P

Redheffer_matrix.html

  1. ( 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 1 1 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 1 ) \left(\begin{smallmatrix}1&1&1&1&1&1&1&1&1&1&1&1\\ 1&1&0&1&0&1&0&1&0&1&0&1\\ 1&0&1&0&0&1&0&0&1&0&0&1\\ 1&0&0&1&0&0&0&1&0&0&0&1\\ 1&0&0&0&1&0&0&0&0&1&0&0\\ 1&0&0&0&0&1&0&0&0&0&0&1\\ 1&0&0&0&0&0&1&0&0&0&0&0\\ 1&0&0&0&0&0&0&1&0&0&0&0\\ 1&0&0&0&0&0&0&0&1&0&0&0\\ 1&0&0&0&0&0&0&0&0&1&0&0\\ 1&0&0&0&0&0&0&0&0&0&1&0\\ 1&0&0&0&0&0&0&0&0&0&0&1\end{smallmatrix}\right)

Redistribution_of_income_and_wealth.html

  1. W = min ( Y 1 , Y 2 , , Y n ) W=\min(Y_{1},Y_{2},\cdots,Y_{n})

ReDoS.html

  1. 2 m 2^{m}
  2. m m
  3. n n
  4. n n

Reduced_dynamics.html

  1. ρ S E ( 0 ) \rho_{SE}(0)\,
  2. U t U_{t}\,
  3. ρ S ( t ) = Tr E [ U t ρ S E ( 0 ) U t ] \rho_{S}(t)=\mathrm{Tr}_{E}[U_{t}\rho_{SE}(0)U_{t}^{\dagger}]
  4. ρ S ( 0 ) ρ S ( t ) \rho_{S}(0)\mapsto\rho_{S}(t)
  5. ρ S = i F i ρ S ( 0 ) F i \rho_{S}=\sum_{i}F_{i}\rho_{S}(0)F_{i}^{\dagger}
  6. F i F_{i}\,
  7. ρ S E ( 0 ) = ρ S ( 0 ) ρ E ( 0 ) \rho_{SE}(0)=\rho_{S}(0)\otimes\rho_{E}(0)

Reductio_ad_absurdum.html

  1. Q : ( P and P ) Q \forall Q:(P\and{\sim}P)\rightarrow Q\;

Reeb_sphere_theorem.html

  1. s s
  2. p p
  3. L p L\setminus p
  4. M n M^{n}
  5. n 2 n\geq 2
  6. M n M^{n}
  7. C 1 C^{1}
  8. F F
  9. F F
  10. M n M^{n}
  11. S n S^{n}
  12. c > s 0. c>s\geq 0.
  13. c s + 2 c\leq s+2
  14. c > s c>s
  15. c = s + 2 , c=s+2,\,
  16. c = s + 1. c=s+1.\,
  17. M n M^{n}
  18. F F
  19. c c
  20. s s
  21. c s + 2 c\leq s+2
  22. c = s + 2 c=s+2
  23. M M
  24. S n S^{n}
  25. S n - 1 S^{n-1}
  26. c = s + 1 c=s+1
  27. M n M^{n}
  28. F F
  29. M M
  30. s = c + 1 s=c+1
  31. n = 2 , 4 , 8 n=2,4,8
  32. 16 16
  33. M n M^{n}

Rees_algebra.html

  1. I I
  2. R R
  3. n = 0 I n t n = R [ I t ] R [ t ] . \oplus_{n=0}^{\infty}I^{n}t^{n}=R[It]\subset R[t].

Reference_Point_Indentation_(RPI).html

  1. H H
  2. P m a x P_{max}
  3. A r A_{r}
  4. H = P m a x A r H=\frac{P_{max}}{A_{r}}

Reflections_of_signals_on_conducting_lines.html

  1. V u ( t ) \scriptstyle Vu(t)
  2. V \scriptstyle V
  3. u ( t ) \scriptstyle u(t)
  4. t \scriptstyle t
  5. κ \scriptstyle\kappa
  6. v i \scriptstyle v_{\mathrm{i}}
  7. x \scriptstyle x
  8. v i = V u ( κ t - x ) v_{\mathrm{i}}=Vu(\kappa t-x)\,\!
  9. i i \scriptstyle i_{\mathrm{i}}
  10. Z 0 \scriptstyle Z_{0}
  11. i i = v i Z 0 = I u ( κ t - x ) i_{\mathrm{i}}=\frac{v_{\mathrm{i}}}{Z_{0}}=Iu(\kappa t-x)
  12. l \scriptstyle l
  13. t = l / κ \scriptstyle t=l/\kappa
  14. i r \scriptstyle i_{\mathrm{r}}
  15. i r = v r Z 0 i_{\mathrm{r}}=\frac{v_{\mathrm{r}}}{Z_{0}}
  16. v r \scriptstyle v_{\mathrm{r}}
  17. v i i i \scriptstyle v_{\mathrm{i}}i_{\mathrm{i}}
  18. v r = v i v_{\mathrm{r}}=v_{\mathrm{i}}\,\!
  19. t = l / κ \scriptstyle t=l/\kappa
  20. Z 0 \scriptstyle Z_{0}
  21. Z 0 \scriptstyle Z_{0}
  22. V \scriptstyle V
  23. Z 0 \scriptstyle Z_{0}
  24. 2 V \scriptstyle 2V
  25. Z 0 \scriptstyle Z_{0}
  26. V \scriptstyle V
  27. 2 V \scriptstyle 2V
  28. v r = - v i v_{\mathrm{r}}=-v_{\mathrm{i}}\,\!
  29. i r = - i i i_{\mathrm{r}}=-i_{\mathrm{i}}\,\!
  30. V 0 \scriptstyle V_{0}
  31. Z L \scriptstyle Z_{\mathrm{L}}
  32. V o = 2 V i Z L Z 0 + Z L V_{\mathrm{o}}=2V_{\mathrm{i}}\frac{Z_{\mathrm{L}}}{Z_{\mathrm{0}}+Z_{\mathrm{% L}}}
  33. V r \scriptstyle V_{\mathrm{r}}
  34. V i + V r = V o \scriptstyle V_{\mathrm{i}}+V_{\mathrm{r}}=V_{\mathrm{o}}
  35. V r = V o - V i = 2 V i Z L Z 0 + Z L - V i = V i Z L - Z 0 Z L + Z 0 V_{\mathrm{r}}=V_{\mathrm{o}}-V_{\mathrm{i}}=2V_{\mathrm{i}}\frac{Z_{\mathrm{L% }}}{Z_{\mathrm{0}}+Z_{\mathrm{L}}}-V_{\mathrm{i}}=V_{\mathrm{i}}\frac{Z_{% \mathrm{L}}-Z_{\mathrm{0}}}{Z_{\mathrm{L}}+Z_{\mathrm{0}}}
  36. Γ \scriptstyle\mathit{\Gamma}
  37. Γ := V r V i \mathit{\Gamma}:=\frac{V_{\mathrm{r}}}{V_{\mathrm{i}}}
  38. V r \scriptstyle V_{\mathrm{r}}
  39. Γ = V r V i = I r I i = Z L - Z 0 Z L + Z 0 \mathit{\Gamma}=\frac{V_{\mathrm{r}}}{V_{\mathrm{i}}}=\frac{I_{\mathrm{r}}}{I_% {\mathrm{i}}}=\frac{Z_{\mathrm{L}}-Z_{\mathrm{0}}}{Z_{\mathrm{L}}+Z_{\mathrm{0% }}}
  40. Γ \scriptstyle\mathit{\Gamma}
  41. | Γ | 1 |\mathit{\Gamma}|\leq 1
  42. ( Z L ) , ( Z 0 ) > 0 \Re(Z_{\mathrm{L}}),\Re(Z_{0})>0
  43. Γ \mathit{\Gamma}\,\!
  44. Γ = 1 \mathit{\Gamma}=1\,\!
  45. Γ = - 1 \mathit{\Gamma}=-1\,\!
  46. Z L = R L Z_{\mathrm{L}}=R_{\mathrm{L}}\,\!
  47. Z 0 = R 0 Z_{0}=R_{0}\,\!
  48. ( Γ ) < 1 , \Re(\mathit{\Gamma})<1\ ,
  49. ( Γ ) = 0 \Im(\mathit{\Gamma})=0
  50. Z 0 \scriptstyle Z_{0}
  51. Z L \scriptstyle Z_{\mathrm{L}}
  52. Γ \scriptstyle\mathit{\Gamma}
  53. Γ \scriptstyle\mathit{\Gamma}
  54. Z 0 \scriptstyle Z_{0}
  55. R 0 \scriptstyle R_{0}
  56. Z L \scriptstyle Z_{\mathrm{L}}
  57. j X L \scriptstyle jX_{\mathrm{L}}
  58. Γ = j X L - R 0 j X L + R 0 \mathit{\Gamma}=\frac{jX_{\mathrm{L}}-R_{\mathrm{0}}}{jX_{\mathrm{L}}+R_{% \mathrm{0}}}
  59. | j X L - R 0 | = | j X L + R 0 | |jX_{\mathrm{L}}-R_{\mathrm{0}}|=|jX_{\mathrm{L}}+R_{\mathrm{0}}|\,
  60. | Γ | = 1 |\mathit{\Gamma}|=1\,
  61. θ \scriptstyle\theta
  62. θ = { π - 2 arctan X L R 0 if X L > 0 - π - 2 arctan X L R 0 if X L < 0 \theta=\begin{cases}\pi-2\arctan\frac{X_{\mathrm{L}}}{R_{\mathrm{0}}}&\mbox{if% }~{}{X_{\mathrm{L}}}>0\\ -\pi-2\arctan\frac{X_{\mathrm{L}}}{R_{\mathrm{0}}}&\mbox{if }~{}{X_{\mathrm{L}% }}<0\\ \end{cases}
  63. Γ = Z 02 - Z 01 Z 02 + Z 01 \mathit{\Gamma}=\frac{Z_{02}-Z_{01}}{Z_{02}+Z_{01}}
  64. T \scriptstyle T
  65. V t \scriptstyle V_{\mathrm{t}}
  66. T = V t V i = 2 Z 02 Z 02 + Z 01 T=\frac{V_{\mathrm{t}}}{V_{\mathrm{i}}}=\frac{2Z_{02}}{Z_{02}+Z_{01}}
  67. Z L \scriptstyle Z_{\mathrm{L}}
  68. Γ = - Z 0 Z 0 + 2 Z L \mathit{\Gamma}=\frac{-Z_{0}}{Z_{0}+2Z_{\mathrm{L}}}
  69. T = 2 Z L Z 0 + 2 Z L T=\frac{2Z_{\mathrm{L}}}{Z_{0}+2Z_{\mathrm{L}}}
  70. γ \scriptstyle\gamma
  71. V \scriptstyle V
  72. x \scriptstyle x^{\prime}
  73. V i = V e - γ x V_{\mathrm{i}}=Ve^{-\gamma x^{\prime}}\,\!
  74. x = l - x \scriptstyle x=l-x^{\prime}
  75. V iL \scriptstyle V_{\mathrm{iL}}
  76. V i = V iL e γ x V_{\mathrm{i}}=V_{\mathrm{iL}}e^{\gamma x}\,\!
  77. x \scriptstyle x
  78. V r = Γ V iL e - γ x V_{\mathrm{r}}=\mathit{\Gamma}V_{\mathrm{iL}}e^{-\gamma x}\,\!
  79. V T = V i + V r = V iL ( e γ x + Γ e - γ x ) V_{\mathrm{T}}=V_{\mathrm{i}}+V_{\mathrm{r}}=V_{\mathrm{iL}}(e^{\gamma x}+% \mathit{\Gamma}e^{-\gamma x})\,\!
  80. V T = V iL [ ( 1 + Γ ) cosh ( γ x ) + ( 1 - Γ ) sinh ( γ x ) ] V_{\mathrm{T}}=V_{\mathrm{iL}}[(1+\mathit{\Gamma})\cosh(\gamma x)+(1-\mathit{% \Gamma})\sinh(\gamma x)]\,\!
  81. I T = I iL [ ( 1 - Γ ) cosh ( γ x ) + ( 1 + Γ ) sinh ( γ x ) ] I_{\mathrm{T}}=I_{\mathrm{iL}}[(1-\mathit{\Gamma})\cosh(\gamma x)+(1+\mathit{% \Gamma})\sinh(\gamma x)]\,\!
  82. | V T | x = 0 \frac{\partial|V_{\mathrm{T}}|}{\partial x}=0
  83. γ \scriptstyle\gamma
  84. i β \scriptstyle i\beta
  85. β \scriptstyle\beta
  86. V T = V iL [ ( 1 + Γ ) cos ( β x ) + i ( 1 - Γ ) sin ( β x ) ] V_{\mathrm{T}}=V_{\mathrm{iL}}[(1+\mathit{\Gamma})\cos(\beta x)+i(1-\mathit{% \Gamma})\sin(\beta x)]\,\!
  87. - 2 ( Γ ) = tan ( 2 β x ) -2\Im(\mathit{\Gamma})=\tan(2\beta x)
  88. β \scriptstyle\beta
  89. λ \scriptstyle\lambda
  90. x \scriptstyle x
  91. λ \scriptstyle\lambda
  92. - 2 ( Γ ) = tan ( 4 π λ x ) -2\Im(\mathit{\Gamma})=\tan\left(\frac{4\pi}{\lambda}x\right)
  93. Γ \scriptstyle\mathit{\Gamma}
  94. Z 0 \scriptstyle Z_{0}
  95. Z L \scriptstyle Z_{\mathrm{L}}
  96. tan ( 4 π λ x ) = 0 \tan\left(\frac{4\pi}{\lambda}x\right)=0
  97. x \scriptstyle x
  98. x = 0 , λ 4 , λ 2 , 3 λ 4 , x=0,~{}~{}\frac{\lambda}{4},~{}~{}\frac{\lambda}{2},~{}~{}\frac{3\lambda}{4},~% {}\dots
  99. R L < R 0 \scriptstyle R_{\mathrm{L}}<R_{0}
  100. R L > R 0 \scriptstyle R_{\mathrm{L}}>R_{0}
  101. Γ \scriptstyle\mathit{\Gamma}
  102. | V T | \scriptstyle|V_{\mathrm{T}}|
  103. VSWR = 1 + | Γ | 1 - | Γ | \mathrm{VSWR}=\frac{1+|\mathit{\Gamma}|}{1-|\mathit{\Gamma}|}
  104. Z 0 \scriptstyle Z_{0}
  105. Z in = V T I T = Z 0 ( 1 + Γ ) cosh ( γ x ) + ( 1 - Γ ) sinh ( γ x ) ( 1 - Γ ) cosh ( γ x ) + ( 1 + Γ ) sinh ( γ x ) Z_{\mathrm{in}}=\frac{V_{\mathrm{T}}}{I_{\mathrm{T}}}=Z_{0}\frac{(1+\mathit{% \Gamma})\cosh(\gamma x)+(1-\mathit{\Gamma})\sinh(\gamma x)}{(1-\mathit{\Gamma}% )\cosh(\gamma x)+(1+\mathit{\Gamma})\sinh(\gamma x)}
  106. x = l \scriptstyle x=l
  107. ( 1 + Γ ) cosh ( γ x ) \scriptstyle(1+\mathit{\Gamma})\cosh(\gamma x)
  108. Z in = Z 0 Z L + Z 0 tanh ( γ l ) Z 0 + Z L tanh ( γ l ) Z_{\mathrm{in}}=Z_{0}\frac{Z_{\mathrm{L}}+Z_{0}\tanh(\gamma l)}{Z_{0}+Z_{% \mathrm{L}}\tanh(\gamma l)}
  109. γ \scriptstyle\gamma
  110. j β \scriptstyle j\beta
  111. Z in = Z 0 Z L + j Z 0 tan ( β l ) Z 0 + j Z L tan ( β l ) Z_{\mathrm{in}}=Z_{0}\frac{Z_{\mathrm{L}}+jZ_{0}\tan(\beta l)}{Z_{0}+jZ_{% \mathrm{L}}\tan(\beta l)}
  112. X in = Z 0 tan ( β l ) X_{\mathrm{in}}=Z_{0}\tan(\beta l)\,\!
  113. λ / 4 \scriptstyle\lambda/4
  114. β l = π / 2 \scriptstyle\beta l=\pi/2
  115. Z in = Z 0 2 Z L Z_{\mathrm{in}}=\frac{{Z_{0}}^{2}}{Z_{\mathrm{L}}}

Reflexive_closure.html

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

Regressive_discrete_Fourier_series.html

  1. x n = x ( t n ) x_{n}=x(t_{n})
  2. x n = k = - q q X k e - i 2 π k t n T + ε n , t n arbitrary , n = 1 , , N . x_{n}=\sum_{k=-q}^{q}X_{k}e^{\frac{-i2\pi kt_{n}}{T}}+\varepsilon_{n},t_{n}\,% \text{ arbitrary },\quad n=1,\dots,N.\,
  3. t n = n Δ t t_{n}=n\,\Delta t
  4. W X = x + ε . WX=x+\varepsilon.\,
  5. X ^ = ( W H W ) - 1 W H x \hat{X}=(W^{H}W)^{-1}W^{H}x\,
  6. x ^ = W x \hat{x}=Wx\,
  7. x ^ \hat{x}
  8. d x d t ( t n ) = k = - q q - i 2 π k T X k e - i 2 π k t n T , n = 1 , , N . \frac{dx}{dt}(t_{n})=\sum_{k=-q}^{q}\frac{-i2\pi k}{T}X_{k}e^{\frac{-i2\pi kt_% {n}}{T}},\quad n=1,\dots,N.\,
  9. x m n = x ( ξ m , ν n ) , m = 1 , , M ; n = 1 , , N ; x_{mn}=x(\xi_{m},\nu_{n}),m=1,\dots,M;\ n=1,\dots,N;
  10. x m n = k = - p p l = - q q X k l e - i 2 π k ξ m L ξ e - i 2 π l ν n L ν + ε m n , m = 1 , , M ; n = 1 , , N . x_{mn}=\sum_{k=-p}^{p}\sum_{l=-q}^{q}X_{kl}e^{\frac{-i2\pi k\xi_{m}}{L_{\xi}}}% e^{\frac{-i2\pi l\nu_{n}}{L_{\nu}}}+\varepsilon_{mn},m=1,\dots,M;\ n=1,\dots,N.\,
  11. ξ m = m Δ ξ , ν n = n Δ ν \xi_{m}=m\Delta\xi,\nu_{n}=n\Delta\nu\,
  12. x m n = k = - p p l = - q q X k l e - i 2 π k ξ m L ξ e - i 2 π l ν n L ν + ϵ m n , m = 1 , , M ; n = 1 , , N . x_{mn}=\sum_{k=-p}^{p}\sum_{l=-q}^{q}X_{kl}e^{\frac{-i2\pi k\xi_{m}}{L_{\xi}}}% e^{\frac{-i2\pi l\nu_{n}}{L_{\nu}}}+\epsilon_{mn},\quad m=1,\dots,M;\ n=1,% \dots,N.\,
  13. X ^ = ( W L ξ H W L ξ ) - 1 W L ξ H x W L ν H ( W L ν W L ν H ) - 1 \hat{X}=(W^{H}_{L_{\xi}}W_{L_{\xi}})^{-1}W^{H}_{L_{\xi}}xW^{H}_{L_{\nu}}(W_{L_% {\nu}}W^{H}_{L_{\nu}})^{-1}\,
  14. x ^ = W L ξ X ^ W L ν \hat{x}=W_{L_{\xi}}\hat{X}W_{L_{\nu}}\,
  15. ξ and ν \xi\,\text{ and }\nu

Regular_extension.html

  1. L / k L/k
  2. k = k ^ k=\hat{k}
  3. k ^ \hat{k}
  4. L k k ¯ L\otimes_{k}\overline{k}
  5. k ¯ \overline{k}
  6. k k
  7. L , k ¯ L,\overline{k}
  8. L / k L/k
  9. L k L L\otimes_{k}L

Regular_ideal.html

  1. 𝔦 \mathfrak{i}
  2. e x - x 𝔦 ex-x\in\mathfrak{i}
  3. x A x\in A
  4. 𝔦 \mathfrak{i}
  5. 𝔦 \mathfrak{i}
  6. 𝔦 \mathfrak{i}
  7. 𝔦 \mathfrak{i}
  8. A / 𝔦 A/\mathfrak{i}
  9. e = 4 e=4
  10. M = { x R R x R is a von Neumann regular ideal } M=\{x\in R\mid RxR\,\text{ is a von Neumann regular ideal }\}

Regular_map_(graph_theory).html

  1. χ ( M ) = | V | - | E | + | F | \chi(M)=|V|-|E|+|F|
  2. 2 - 2 g 2-2g
  3. 2 - g 2-g
  4. Ω \Omega
  5. Γ \Gamma
  6. Γ \Gamma
  7. Γ \Gamma
  8. Ω \Omega
  9. Ω \Omega
  10. ϕ \phi

Regulatory_feedback_network.html

  1. x x
  2. y y
  3. y j y_{j}
  4. F F j FF_{j}
  5. x x
  6. F B FB
  7. F F j FF_{j}
  8. F B FB
  9. F B i FB_{i}
  10. x i x_{i}
  11. | F F j | |FF_{j}|
  12. y j y_{j}
  13. s i s_{i}
  14. x i x_{i}
  15. y j ( t + Δ t ) = y j ( t ) | F F j | k F F j s k y_{j}(t+\Delta t)=\frac{y_{j}(t)}{|FF_{j}|}\sum_{k\in FF_{j}}s_{k}
  16. s i s_{i}
  17. x i x_{i}
  18. s i = x i r F B i y r ( t ) s_{i}=\frac{x_{i}}{\sum_{r\in{FB_{i}}}y_{r}(t)}

Reinhardt_domain.html

  1. G G
  2. ( z 1 , , z n ) G (z_{1},\dots,z_{n})\in G
  3. ( e i θ 1 z 1 , , e i θ n z n ) G (e^{i\theta_{1}}z_{1},\dots,e^{i\theta_{n}}z_{n})\in G
  4. θ 1 , , θ n \theta_{1},\dots,\theta_{n}
  5. { ( z , w ) 𝐂 2 ; | z | < 1 , | w | < 1 } \{(z,w)\in\mathbf{C}^{2};~{}|z|<1,~{}|w|<1\}
  6. { ( z , w ) 𝐂 2 ; | z | 2 + | w | 2 < 1 } \{(z,w)\in\mathbf{C}^{2};~{}|z|^{2}+|w|^{2}<1\}
  7. { ( z , w ) 𝐂 2 ; | z | 2 + | w | 2 / p < 1 } ( p > 0 , 1 ) \{(z,w)\in\mathbf{C}^{2};~{}|z|^{2}+|w|^{2/p}<1\}(p>0,\neq 1)
  8. n n
  9. G 1 G_{1}
  10. G 2 G_{2}
  11. φ : 𝐂 n 𝐂 n \varphi:\mathbf{C}^{n}\longrightarrow\mathbf{C}^{n}
  12. z i r i z σ ( i ) ( r i > 0 ) z_{i}\mapsto r_{i}z_{\sigma(i)}(r_{i}>0)
  13. σ \sigma
  14. φ ( G 1 ) = G 2 \varphi(G_{1})=G_{2}

Relative_canonical_model.html

  1. X X
  2. X X
  3. f : Y X f:Y\to X
  4. f * ω Y n ; f_{*}\omega_{Y}^{\otimes n};
  5. ω X \omega_{X}
  6. f * ω Y n = I n ω X n f_{*}\omega_{Y}^{\otimes n}=I_{n}\omega_{X}^{\otimes n}
  7. I n I_{n}
  8. n f * ω Y n \oplus_{n}f_{*}\omega_{Y}^{\otimes n}
  9. P r o j n f * ω Y n X Proj\oplus_{n}f_{*}\omega_{Y}^{\otimes n}\to X
  10. Y Y
  11. X X
  12. r r
  13. Y Y
  14. X X
  15. Y Y
  16. X X

Relative_purchasing_power_parity.html

  1. 1 C = P ( 1 ) × S ( 1 ) Q ( 1 ) = P ( 2 ) × S ( 2 ) Q ( 2 ) \tfrac{1}{C}=\tfrac{P(1)\times S(1)}{Q(1)}=\tfrac{P(2)\times S(2)}{Q(2)}
  2. S ( 2 ) S ( 1 ) = ( Q ( 2 ) Q ( 1 ) ) ( P ( 2 ) P ( 1 ) ) \tfrac{S(2)}{S(1)}=\frac{\big(\tfrac{Q(2)}{Q(1)}\big)}{\big(\tfrac{P(2)}{P(1)}% \big)}
  3. 1.01 1.03 \tfrac{1.01}{1.03}

Relative_species_abundance.html

  1. S = α ln ( 1 + N α ) S=\alpha\ln{\left(1+{N\over{\alpha\,\!}}\right)}
  2. α \alpha\,\!
  3. S n = α x n n S_{n}={\alpha\,\!x^{n}\over n}
  4. α , α x 2 2 , α x 3 3 , , α x n n \alpha,{\alpha\,\!x^{2}\over{2}},{\alpha\,\!x^{3}\over{3}},\dots,{\alpha\,\!x^% {n}\over{n}}
  5. n = n 0 e - ( a R ) 2 n=n_{0}e^{-(aR)^{2}}\,
  6. N = n 0 π a N={n_{0}\sqrt{\pi}\over a}
  7. α \alpha\,\!

Relatively_hyperbolic_group.html

  1. Γ ^ ( G , H ) \hat{\Gamma}(G,H)
  2. Γ ^ ( G , H ) \hat{\Gamma}(G,H)
  3. Γ ^ ( 2 , ) \hat{\Gamma}(\mathbb{Z}^{2},\mathbb{Z})

Replacement_migration.html

  1. R ( t ) = - Δ P ( t , t + 1 ) A ( t ) R_{(t)}\ ^{\prime}=\frac{-\Delta P_{(t,t+1)}}{A_{(t)}}

Reservoir_sampling.html

  1. w i w_{i}
  2. W W
  3. P i = w i / W P_{i}=w_{i}/W
  4. i i
  5. j j
  6. w i w_{i}
  7. w j w_{j}
  8. i i
  9. p i p_{i}
  10. p j = p i max ( 1 , w j / w i ) p_{j}=p_{i}\max(1,w_{j}/w_{i})

Resolvent_cubic.html

  1. f ( x ) = x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 0 f(x)=x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}\,
  2. g ( x ) = x 3 + b 2 x 2 + b 1 x + b 0 g(x)=x^{3}+b_{2}x^{2}+b_{1}x+b_{0}\,
  3. b 2 = - a 2 b_{2}=-a_{2}\,
  4. b 1 = a 1 a 3 - 4 a 0 b_{1}=a_{1}a_{3}-4a_{0}\,
  5. b 0 = 4 a 0 a 2 - a 1 2 - a 0 a 3 2 . b_{0}=4a_{0}a_{2}-a_{1}^{2}-a_{0}a_{3}^{2}.\,
  6. α i \alpha_{i}
  7. β i \beta_{i}
  8. β 1 = α 1 α 2 + α 3 α 4 \beta_{1}=\alpha_{1}\alpha_{2}+\alpha_{3}\alpha_{4}
  9. β 2 = α 1 α 3 + α 2 α 4 \beta_{2}=\alpha_{1}\alpha_{3}+\alpha_{2}\alpha_{4}
  10. β 3 = α 1 α 4 + α 2 α 3 . \beta_{3}=\alpha_{1}\alpha_{4}+\alpha_{2}\alpha_{3}.

Resonant-cavity-enhanced_photo_detector.html

  1. r 1 e - j ϕ 1 r_{1}e^{-j\phi_{1}}
  2. r 2 e - j ϕ 2 r_{2}e^{-j\phi_{2}}
  3. η \eta
  4. η = ( 1 - R 1 ) ( 1 - e - α d ) [ ( e - α e x L 1 + r 2 2 e - α e x L 2 - α c L ) 1 - 2 r 1 r 2 e - α c L cos ( 2 β L + ϕ 1 + ϕ 2 ) + ( r 1 r 2 ) 2 e - α c L ] \eta=(1-R_{1})(1-e^{-\alpha d})[\frac{(e^{-\alpha_{ex}L_{1}}+r_{2}^{2}e^{-% \alpha_{ex}L_{2}-\alpha_{c}L})}{1-2r_{1}r_{2}e^{-\alpha_{c}L}\cos(2\beta L+% \phi_{1}+\phi_{2})+(r_{1}r_{2})^{2}e^{-\alpha_{c}L}}]
  5. η \eta
  6. η = ( 1 - R 1 ) ( 1 - e - α d ) [ ( 1 + R 2 e - α d ) 1 - 2 R 1 R 2 e - α c d cos ( 2 β L + ϕ 1 + ϕ 2 ) + ( R 1 R 2 ) e - α c d ] \eta=(1-R_{1})(1-e^{-\alpha d})[\frac{(1+R_{2}e^{-\alpha d})}{1-2\sqrt{R_{1}R_% {2}}e^{-\alpha_{c}d}\cos(2\beta L+\phi_{1}+\phi_{2})+(R_{1}R_{2})e^{-\alpha_{c% }d}}]
  7. η = ( 1 - R 1 ) ( 1 - e - α d ) [ ( 1 + R 2 e - α d ) ( 1 - R 1 R 2 e - α c d ) 2 ] \eta=(1-R_{1})(1-e^{-\alpha d})[\frac{(1+R_{2}e^{-\alpha d})}{(1-\sqrt{R_{1}R_% {2}}e^{-\alpha_{c}d})^{2}}]
  8. η = ( 1 - R ) α L \eta=(1-R)\alpha L
  9. f t r a n s i t = 0.45 v h L f_{transit}=0.45\frac{v_{h}}{L}
  10. η = ( 1 - R ) α L \eta=(1-R)\alpha L
  11. v e v_{e}
  12. v h v_{h}
  13. F S R = λ 2 2 n e f f ( L + L e f f , 1 + L e f f , 2 ) FSR=\frac{\lambda^{2}}{2n_{eff}(L+L_{eff,1}+L_{eff,2})}
  14. F i n e s s e = π ( R 1 R 2 ) 1 / 4 e - α d 2 ( 1 - R 1 R 2 e - α c d ) 2 Finesse=\frac{\pi(R_{1}R_{2})^{1/4}e^{-\frac{\alpha d}{2}}}{(1-\sqrt{R_{1}R_{2% }}e^{-\alpha_{c}d})^{2}}

Resonant_inductive_coupling.html

  1. Q = 1 R L C Q=\frac{1}{R}\sqrt{\frac{L}{C}}\,
  2. U = k Q 1 Q 2 U=k\sqrt{Q_{1}Q_{2}}
  3. η o p t = U 2 ( 1 + 1 + U 2 ) 2 \eta_{opt}=\frac{U^{2}}{(1+\sqrt{1+U^{2}})^{2}}
  4. A = k L 2 L 1 A=k\sqrt{\frac{L_{2}}{L_{1}}}\,

Response_reactions.html

  1. ( S C + 1 ) S\choose C+1

Restricted_isometry_property.html

  1. δ s ( 0 , 1 ) \delta_{s}\in(0,1)
  2. ( 1 - δ s ) y 2 2 A s y 2 2 ( 1 + δ s ) y 2 2 . (1-\delta_{s})\|y\|_{2}^{2}\leq\|A_{s}y\|_{2}^{2}\leq(1+\delta_{s})\|y\|_{2}^{% 2}.\,
  3. δ s \delta_{s}
  4. δ \delta
  5. A n * m A\in\mathbb{R}^{n*m}
  6. δ K = inf [ δ : ( 1 - δ ) y 2 2 A s y 2 2 ( 1 + δ ) y 2 2 ] , | s | K , y R | s | \delta_{K}=\inf\left[\delta:(1-\delta)\|y\|_{2}^{2}\leq\|A_{s}y\|_{2}^{2}\leq(% 1+\delta)\|y\|_{2}^{2}\right],\ \forall|s|\leq K,\forall y\in R^{|s|}
  7. δ K \delta_{K}
  8. δ k \delta_{k}
  9. 1 - δ K λ m i n ( A τ * A τ ) λ m a x ( A τ * A τ ) 1 + δ k 1-\delta_{K}\leq\lambda_{min}(A_{\tau}^{*}A_{\tau})\leq\lambda_{max}(A_{\tau}^% {*}A_{\tau})\leq 1+\delta_{k}

Reversible_hydrogen_electrode.html

  1. E 0 = 0.000 - 0.059 * p H E_{0}=0.000-0.059*pH
  2. 2 H 3 O + + 2 e - H 2 + 2 H 2 O 2\;\mathrm{H_{3}O^{+}}\;+\;2\;\mathrm{e^{-}}\quad\overrightarrow{\leftarrow}% \quad\mathrm{H_{2}}\;+\;2\;\mathrm{H_{2}O}
  3. E E
  4. p [ H 2 ] p\mathrm{[H_{2}]}
  5. a [ H 3 O + ] a\mathrm{[H_{3}O^{+}]}
  6. E = E 00 + R T F ( ln ( a [ H 3 O + ] ) - 1 2 ln ( p [ H 2 ] ) ) E=E_{00}+\frac{R\;T}{F}\left(\ln\left(a[\mathrm{H_{3}O^{+}}]\right)-\frac{1}{2% }\ln\left(\;p[\mathrm{H_{2}}]\right)\right)
  7. E 00 E_{00}

Reynolds_number.html

  1. Re = inertial forces viscous forces = ρ 𝐯 L μ = 𝐯 L ν \mathrm{Re}=\dfrac{\mbox{inertial forces}~{}}{\mbox{viscous forces}~{}}={{\rho% {\mathbf{v}}L}\over{\mu}}={{{\mathbf{v}}L}\over{\nu}}
  2. 𝐯 {\mathbf{v}}
  3. L {L}
  4. μ {\mu}
  5. ν {\nu}
  6. ν = μ / ρ {\nu}=\mu/{\rho}
  7. ρ {\rho}\,
  8. ρ L v ρ L v \rho L\mathrm{v}\over\rho L\mathrm{v}
  9. ρ v 2 L 2 μ v L \rho\mathrm{v}^{2}L^{2}\over\mu\mathrm{v}L
  10. Re = ρ 𝐯 D H μ = 𝐯 D H ν = 𝐐 D H ν A \mathrm{Re}={{\rho{\mathbf{v}}D_{H}}\over{\mu}}={{{\mathbf{v}}D_{H}}\over{\nu}% }={{{\mathbf{Q}}D_{H}}\over{\nu}A}
  11. D H {D_{H}}
  12. L {L}
  13. 𝐐 {\mathbf{Q}}
  14. A {A}
  15. 𝐯 {\mathbf{v}}
  16. μ {\mu}
  17. ν {\nu}
  18. ν = μ / ρ ) \nu=\mu/{\rho})
  19. ρ {\rho}\,
  20. D H D_{H}
  21. D H = 4 A P , D_{H}=\frac{4A}{P},
  22. D D
  23. D H = D . D_{H}=D.
  24. D H , annulus = D o - D i D_{H,\mathrm{annulus}}=D_{o}-D_{i}
  25. D o D_{o}
  26. D i D_{i}
  27. Re = ρ 𝐯 s D μ . \mathrm{Re}={{\rho{\mathbf{v}_{s}}D}\over{\mu}}.
  28. Re = ρ 𝐯 s D μ ϵ . \mathrm{Re}={{\rho{\mathbf{v}_{s}}D}\over{\mu\epsilon}}.
  29. Re = ρ 𝐯 s D μ ( 1 - ϵ ) . \mathrm{Re}={{\rho{\mathbf{v}_{s}}D}\over{\mu(1-\epsilon)}}.
  30. D D
  31. V V
  32. N D ND
  33. N N
  34. Re = ρ N D 2 μ . \mathrm{Re}={{\rho ND^{2}}\over{\mu}}.
  35. Re = ρ V D μ . \mathrm{Re}={{\rho VD}\over{\mu}}.
  36. Re x 5 × 10 5 \mathrm{Re}_{x}\approx 5\times 10^{5}
  37. x x
  38. D D
  39. Re D < 2100 \mathrm{Re}_{D}<2100
  40. Re D > 4000 \mathrm{Re}_{D}>4000
  41. f f
  42. Re {\mathrm{Re}}
  43. ϵ / D \epsilon/D
  44. Re m = Re \mathrm{Re}_{m}=\mathrm{Re}\;
  45. Eu m = Eu i.e. p m ρ m v m 2 = p ρ v 2 , \mathrm{Eu}_{m}=\mathrm{Eu}\;\quad\quad\mbox{i.e.}~{}\quad{p_{m}\over\rho_{m}{% v_{m}}^{2}}={p\over\rho v^{2}}\;,
  46. ρ D 𝐯 D t = - p + μ 2 𝐯 + ρ 𝐟 . \rho\frac{D\mathbf{v}}{Dt}=-\nabla p+\mu\nabla^{2}\mathbf{v}+\rho\mathbf{f}.
  47. L ρ V 2 L\over\rho V^{2}
  48. V V\,
  49. v v\,
  50. 𝐯 ¯ \mathbf{\bar{v}}
  51. L L\,
  52. ρ \rho\,
  53. 𝐯 = 𝐯 V , p = p 1 ρ V 2 , 𝐟 = 𝐟 L V 2 , t = L V t , = L \mathbf{v^{\prime}}=\frac{\mathbf{v}}{V},\ p^{\prime}=p\frac{1}{\rho V^{2}},\ % \mathbf{f^{\prime}}=\mathbf{f}\frac{L}{V^{2}},\ \frac{\partial}{\partial t^{% \prime}}=\frac{L}{V}\frac{\partial}{\partial t},\ \nabla^{\prime}=L\nabla
  54. D 𝐯 D t = - p + μ ρ L V 2 𝐯 + 𝐟 \frac{D\mathbf{v^{\prime}}}{Dt^{\prime}}=-\nabla^{\prime}p^{\prime}+\frac{\mu}% {\rho LV}\nabla^{\prime 2}\mathbf{v^{\prime}}+\mathbf{f^{\prime}}
  55. μ ρ L V = 1 Re . \frac{\mu}{\rho LV}=\frac{1}{\mathrm{Re}}.
  56. D 𝐯 D t = - p + 1 Re 2 𝐯 + 𝐟 . \frac{D\mathbf{v}}{Dt}=-\nabla p+\frac{1}{\mathrm{Re}}\nabla^{2}\mathbf{v}+% \mathbf{f}.
  57. Re \mathrm{Re}\to\infty

Riemann–Siegel_formula.html

  1. ζ ( s ) = n = 1 N 1 n s + γ ( 1 - s ) n = 1 M 1 n 1 - s + R ( s ) \zeta(s)=\sum_{n=1}^{N}\frac{1}{n^{s}}+\gamma(1-s)\sum_{n=1}^{M}\frac{1}{n^{1-% s}}+R(s)
  2. γ ( s ) = π 1 2 - s Γ ( s 2 ) Γ ( 1 2 ( 1 - s ) ) \gamma(s)=\pi^{\tfrac{1}{2}-s}\frac{\Gamma\left(\tfrac{s}{2}\right)}{\Gamma% \left(\tfrac{1}{2}(1-s)\right)}
  3. ζ ( s ) = γ ( 1 - s ) ζ ( 1 s ) ζ(s)=γ(1-s)ζ(1−s)
  4. R ( s ) = - Γ ( 1 - s ) 2 π i ( - x ) s - 1 e - N x e x - 1 d x R(s)=\frac{-\Gamma(1-s)}{2\pi i}\int\frac{(-x)^{s-1}e^{-Nx}}{e^{x}-1}dx
  5. 2 π M 2πM
  6. 0 1 e - i π u 2 + 2 π i p u e π i u - e - π i u d u = e i π p 2 - e i π p e i π p - e - i π p \int_{0\searrow 1}\frac{e^{-i\pi u^{2}+2\pi ipu}}{e^{\pi iu}-e^{-\pi iu}}du=% \frac{e^{i\pi p^{2}}-e^{i\pi p}}{e^{i\pi p}-e^{-i\pi p}}
  7. π - s 2 Γ ( s 2 ) ζ ( s ) = π - s 2 Γ ( s 2 ) 0 1 x - s e π i x 2 e π i x - e - π i x d x + π - 1 - s 2 Γ ( 1 - s 2 ) 0 1 x s - 1 e - π i x 2 e π i x - e - π i x d x \pi^{-\tfrac{s}{2}}\Gamma\left(\tfrac{s}{2}\right)\zeta(s)=\pi^{-\tfrac{s}{2}}% \Gamma\left(\tfrac{s}{2}\right)\int_{0\swarrow 1}\frac{x^{-s}e^{\pi ix^{2}}}{e% ^{\pi ix}-e^{-\pi ix}}\,dx+\pi^{-\frac{1-s}{2}}\Gamma\left(\tfrac{1-s}{2}% \right)\int_{0\searrow 1}\frac{x^{s-1}e^{-\pi ix^{2}}}{e^{\pi ix}-e^{-\pi ix}}% \,dx

Right_conoid.html

  1. x = v cos u , y = v sin u , z = h ( u ) x=v\cos u,y=v\sin u,z=h(u)\,
  2. x = v cos u , y = v sin u , z = 2 sin u x=v\cos u,y=v\sin u,z=2\sin u\,
  3. x = v cos u , y = v sin u , z = c u . x=v\cos u,y=v\sin u,z=cu.\,
  4. x = v u , y = v , z = u 2 . x=vu,y=v,z=u^{2}.\,
  5. x = v cos u , y = v sin u , z = c a 2 - b 2 cos 2 u . x=v\cos u,y=v\sin u,z=c\sqrt{a^{2}-b^{2}\cos^{2}u}.\,
  6. x = v cos u , y = v sin u , z = c sin n u . x=v\cos u,y=v\sin u,z=c\sin nu.\,
  7. x = v , y = u , z = u v x=v,y=u,z=uv\,

Rigid_category.html

  1. η X : 𝟏 X Y \eta_{X}:\mathbf{1}\to X\otimes Y
  2. ϵ X : Y X 𝟏 \epsilon_{X}:Y\otimes X\to\mathbf{1}
  3. X η X id X ( X Y ) X α X , Y , X - 1 X ( Y X ) id X ϵ X X X~{}\xrightarrow{\eta_{X}\otimes\mathrm{id}_{X}}~{}(X\otimes Y)\otimes X~{}% \xrightarrow{\alpha^{-1}_{X,Y,X}}~{}X\otimes(Y\otimes X)~{}\xrightarrow{% \mathrm{id}_{X}\otimes\epsilon_{X}}~{}X
  4. Y id X η X Y ( X Y ) α X , Y , X ( Y X ) Y ϵ X id X Y Y~{}\xrightarrow{\mathrm{id}_{X}\otimes\eta_{X}}~{}Y\otimes(X\otimes Y)~{}% \xrightarrow{~{}\alpha_{X,Y,X}~{}}~{}(Y\otimes X)\otimes Y~{}\xrightarrow{% \epsilon_{X}\otimes\mathrm{id}_{X}}~{}Y
  5. ϕ X , Y : { Hom ( 𝟏 , X * Y ) Hom ( X , Y ) f ( ϵ X i d Y ) ( i d X f ) \phi_{X,Y}:\left\{\begin{array}[]{rcl}\mathrm{Hom}(\mathbf{1},X^{*}\otimes Y)&% \longrightarrow&\mathrm{Hom}(X,Y)\\ f&\longmapsto&(\epsilon_{X}\otimes id_{Y})\circ(id_{X}\otimes f)\end{array}\right.
  6. ψ X , Y : { Hom ( X , Y ) Hom ( 𝟏 , X * Y ) g ( i d X * g ) η X \psi_{X,Y}:\left\{\begin{array}[]{rcl}\mathrm{Hom}(X,Y)&\longrightarrow&% \mathrm{Hom}(\mathbf{1},X^{*}\otimes Y)\\ g&\longmapsto&(id_{X^{*}}\otimes g)\circ\eta_{X}\end{array}\right.
  7. f : X X f:X\to X
  8. tr f : 𝟏 ψ X , X ( f ) X * X γ X , X X X * ϵ X 𝟏 , \mathop{\mathrm{tr}}f:\mathbf{1}\xrightarrow{\psi_{X,X}(f)}X^{*}\otimes X% \xrightarrow{\gamma_{X,X}}X\otimes X^{*}\xrightarrow{\epsilon_{X}}\mathbf{1},
  9. dim X := tr id X \dim X:=\mathop{\mathrm{tr}}\ \mathrm{id}_{X}

Ringing_artifacts.html

  1. - a \int_{-\infty}^{a}
  2. a , \int_{a}^{\infty},
  3. ( sinc rect ) = rect * sinc . \mathcal{F}(\mathrm{sinc}\cdot\mathrm{rect})=\mathrm{rect}*\mathrm{sinc}.
  4. e - x 2 e^{-x^{2}}
  5. - x 2 . -x^{2}.
  6. - 6 n -6n
  7. rect ( t ) h ( t ) \mathrm{rect}(t)\cdot h(t)
  8. sinc ( t ) * h ^ ( t ) . \mathrm{sinc}(t)*\hat{h}(t).
  9. rect ( t ) sinc ( t ) \mathrm{rect}(t)\cdot\mathrm{sinc}(t)
  10. sinc ( t ) * rect ( t ) \mathrm{sinc}(t)*\mathrm{rect}(t)
  11. J 0 , J_{0},

Risk.html

  1. R = ( probability of the accident occurring ) × ( expected loss in case of the accident ) \,\text{R}=(\,\text{probability of the accident occurring})\times(\,\text{% expected loss in case of the accident})
  2. R = For all accidents ( probability of the accident occurring ) × ( expected loss in case of the accident ) \,\text{R}=\sum\text{For all accidents}(\,\text{probability of the accident % occurring})\times(\,\text{expected loss in case of the accident})

RLC_circuit.html

  1. f 0 f_{0}\,
  2. ω 0 \omega_{0}\,
  3. ω 0 = 2 π f 0 \omega_{0}=2\pi f_{0}\,
  4. ω 0 = 1 L C \omega_{0}=\frac{1}{\sqrt{LC}}
  5. ω 0 = 1 L C \scriptstyle\omega_{0}=\frac{1}{\sqrt{LC}}
  6. ζ = α ω 0 \zeta=\frac{\alpha}{\omega_{0}}
  7. Δ ω = ω 2 - ω 1 \Delta\omega=\omega_{2}-\omega_{1}\,
  8. Δ ω \scriptstyle\Delta\omega
  9. ω 1 \scriptstyle\omega_{1}
  10. ω 2 \scriptstyle\omega_{2}
  11. Δ ω = 2 α \Delta\omega=2\alpha\,
  12. F b = Δ ω ω 0 F_{\mathrm{b}}=\frac{\Delta\omega}{\omega_{0}}
  13. Q = 1 F b = ω 0 Δ ω Q={1\over F_{\mathrm{b}}}=\frac{\omega_{0}}{\Delta\omega}
  14. Q = 1 ω 0 R C = ω 0 L R = 1 R L C Q=\frac{1}{\omega_{0}RC}=\frac{\omega_{0}L}{R}=\frac{1}{R}\,\sqrt{\frac{L}{C}}
  15. v R + v L + v C = v ( t ) v_{R}+v_{L}+v_{C}=v(t)\,
  16. v R , v L , v C \textstyle v_{R},v_{L},v_{C}
  17. v ( t ) \textstyle v(t)
  18. R i ( t ) + L d i d t + 1 C - τ = t i ( τ ) d τ = v ( t ) Ri(t)+L{{di}\over{dt}}+{1\over C}\int_{-\infty}^{\tau=t}i(\tau)\,d\tau=v(t)
  19. d 2 i ( t ) d t 2 + R L d i ( t ) d t + 1 L C i ( t ) = 0 {{d^{2}i(t)}\over{dt^{2}}}+{R\over L}{{di(t)}\over{dt}}+{1\over{LC}}i(t)=0
  20. d 2 i ( t ) d t 2 + 2 α d i ( t ) d t + ω 0 2 i ( t ) = 0 {{d^{2}i(t)}\over{dt^{2}}}+2\alpha{{di(t)}\over{dt}}+{\omega_{0}}^{2}i(t)=0
  21. α \alpha\,
  22. ω 0 \omega_{0}\,
  23. α \alpha\,
  24. ω 0 \omega_{0}\,
  25. α = R 2 L \alpha={R\over 2L}
  26. ω 0 = 1 L C \omega_{0}={1\over\sqrt{LC}}
  27. ζ \zeta
  28. ζ = α ω 0 \zeta=\frac{\alpha}{\omega_{0}}
  29. ζ = R 2 C L \zeta={R\over 2}\sqrt{C\over L}
  30. ζ \zeta\,
  31. α \alpha\,
  32. ζ \scriptstyle\zeta\,
  33. ζ < 1 \scriptstyle\zeta<1\,
  34. ζ > 1 \scriptstyle\zeta>1\,
  35. ζ = 1 \scriptstyle\zeta=1\,
  36. s 2 + 2 α s + ω 0 2 = 0 s^{2}+2\alpha s+{\omega_{0}}^{2}=0
  37. s 1 = - α + α 2 - ω 0 2 s_{1}=-\alpha+\sqrt{\alpha^{2}-{\omega_{0}}^{2}}
  38. s 2 = - α - α 2 - ω 0 2 s_{2}=-\alpha-\sqrt{\alpha^{2}-{\omega_{0}}^{2}}
  39. i ( t ) = A 1 e s 1 t + A 2 e s 2 t i(t)=A_{1}e^{s_{1}t}+A_{2}e^{s_{2}t}
  40. ζ > 1 \scriptstyle\zeta>1\,
  41. i ( t ) = A 1 e - ω 0 ( ζ + ζ 2 - 1 ) t + A 2 e - ω 0 ( ζ - ζ 2 - 1 ) t i(t)=A_{1}e^{-\omega_{0}\left(\zeta+\sqrt{\zeta^{2}-1}\right)t}+A_{2}e^{-% \omega_{0}\left(\zeta-\sqrt{\zeta^{2}-1}\right)t}
  42. ζ < 1 \scriptstyle\zeta<1\,
  43. i ( t ) = B 1 e - α t cos ( ω d t ) + B 2 e - α t sin ( ω d t ) i(t)=B_{1}e^{-\alpha t}\cos(\omega_{d}t)+B_{2}e^{-\alpha t}\sin(\omega_{d}t)\,
  44. i ( t ) = B 3 e - α t sin ( ω d t + φ ) i(t)=B_{3}e^{-\alpha t}\sin(\omega_{d}t+\varphi)\,
  45. ω d \omega_{d}\,
  46. α \alpha\,
  47. α \alpha\,
  48. φ \varphi\,
  49. ω d \omega_{d}\,
  50. ω d = ω 0 2 - α 2 = ω 0 1 - ζ 2 \omega_{d}=\sqrt{{\omega_{0}}^{2}-\alpha^{2}}=\omega_{0}\sqrt{1-\zeta^{2}}
  51. ω 0 \omega_{0}\,
  52. ζ = 1 \scriptstyle\zeta=1\,
  53. i ( t ) = D 1 t e - α t + D 2 e - α t i(t)=D_{1}te^{-\alpha t}+D_{2}e^{-\alpha t}\,
  54. s = σ + i ω s=\sigma+i\omega\,
  55. V ( s ) = I ( s ) ( R + L s + 1 C s ) V(s)=I(s)\left(R+Ls+\frac{1}{Cs}\right)
  56. I ( s ) = 1 R + L s + 1 C s V ( s ) I(s)=\frac{1}{R+Ls+\frac{1}{Cs}}V(s)
  57. I ( s ) = s L ( s 2 + R L s + 1 L C ) V ( s ) I(s)=\frac{s}{L\left(s^{2}+{R\over L}s+\frac{1}{LC}\right)}V(s)
  58. Y ( s ) = I ( s ) V ( s ) = s L ( s 2 + R L s + 1 L C ) Y(s)={I(s)\over V(s)}=\frac{s}{L\left(s^{2}+{R\over L}s+\frac{1}{LC}\right)}
  59. Y ( s ) = I ( s ) V ( s ) = s L ( s 2 + 2 α s + ω 0 2 ) Y(s)={I(s)\over V(s)}=\frac{s}{L\left(s^{2}+2\alpha s+{\omega_{0}}^{2}\right)}
  60. Y ( s ) = 0 Y(s)=0
  61. s = 0 and | s | s=0\,\text{ and }|s|\rightarrow\infty
  62. Y ( s ) Y(s)\rightarrow\infty
  63. s = - α ± α 2 - ω 0 2 s=-\alpha\pm\sqrt{\alpha^{2}-{\omega_{0}}^{2}}
  64. s 1 s_{1}
  65. s 2 s_{2}
  66. I ( t ) = 1 L 0 t E ( t - τ ) e - α τ ( cos ω d τ - α ω d sin ω d τ ) d τ in the underdamped case ( ω 0 > α ) I(t)=\frac{1}{L}\int_{0}^{t}E(t-\tau)e^{-\alpha\tau}\left(\cos\omega_{d}\tau-{% \alpha\over\omega_{d}}\sin\omega_{d}\tau\right)\,d\tau\,\text{ in the % underdamped case }(\omega_{0}>\alpha)
  67. I ( t ) = 1 L 0 t E ( t - τ ) e - α τ ( 1 - α τ ) d τ in the critically damped case ( ω 0 = α ) I(t)=\frac{1}{L}\int_{0}^{t}E(t-\tau)e^{-\alpha\tau}(1-\alpha\tau)\,d\tau\,% \text{ in the critically damped case }(\omega_{0}=\alpha)
  68. I ( t ) = 1 L 0 t E ( t - τ ) e - α τ ( cosh ω r τ - α ω r sinh ω r τ ) d τ in the overdamped case ( ω 0 < α ) I(t)=\frac{1}{L}\int_{0}^{t}E(t-\tau)e^{-\alpha\tau}\left(\cosh\omega_{r}\tau-% {\alpha\over\omega_{r}}\sinh\omega_{r}\tau\right)\,d\tau\,\text{ in the % overdamped case }(\omega_{0}<\alpha)
  69. ω r = α 2 - ω 0 2 \omega_{r}=\sqrt{\alpha^{2}-{\omega_{0}}^{2}}
  70. s = j ω s=j\omega\,
  71. j j
  72. | Y ( s = j ω ) | = 1 R 2 + ( ω L - 1 ω C ) 2 . \displaystyle|Y(s=j\omega)|=\frac{1}{\sqrt{R^{2}+\left(\omega L-\frac{1}{% \omega C}\right)^{2}}}.
  73. | I ( j ω ) | = | Y ( j ω ) | | V ( j ω ) | . \displaystyle|I(j\omega)|=|Y(j\omega)||V(j\omega)|.\,
  74. | I ( j ω ) | |I(j\omega)|
  75. ω 0 = 1 L C . \omega_{0}=\frac{1}{\sqrt{LC}}.
  76. α = 1 2 R C \alpha={1\over 2RC}
  77. ζ = 1 2 R L C \zeta={1\over 2R}\,\sqrt{L\over C}
  78. F b = 1 R L C F_{\mathrm{b}}={1\over R}\,\sqrt{L\over C}
  79. Q = R C L Q=R\,\sqrt{C\over L}
  80. 1 Z = 1 Z L + 1 Z C + 1 Z R = 1 j ω L + j ω C + 1 R {1\over Z}={1\over Z_{L}}+{1\over Z_{C}}+{1\over Z_{R}}={1\over{j\omega L}}+{j% \omega C}+{1\over R}
  81. ω 0 = 1 L C \omega_{0}={1\over\sqrt{LC}}
  82. ω 0 = 1 L C - ( R L ) 2 \omega_{0}=\sqrt{\frac{1}{LC}-\left(\frac{R}{L}\right)^{2}}
  83. s 2 + 2 α s + ω 0 2 = 0 s^{2}+2\alpha s+{\omega^{\prime}_{0}}^{2}=0
  84. ω 0 = 1 L C \omega^{\prime}_{0}=\sqrt{\frac{1}{LC}}
  85. ω m \omega_{m}
  86. ω m = ω 0 - 1 Q L 2 + 1 + 2 Q L 2 \omega_{m}=\omega^{\prime}_{0}\sqrt{\frac{-1}{Q^{2}_{L}}+\sqrt{1+\frac{2}{Q^{2% }_{L}}}}
  87. Q L = ω 0 L R Q_{L}=\frac{\omega^{\prime}_{0}L}{R}
  88. ω m ω 0 1 - 1 2 Q L 4 \omega_{m}\approx\omega^{\prime}_{0}\sqrt{1-\frac{1}{2Q^{4}_{L}}}
  89. | Z | m a x = R Q L 2 1 2 Q L Q L 2 + 2 - 2 Q L 2 - 1 |Z|_{max}=RQ^{2}_{L}\sqrt{\frac{1}{2Q_{L}\sqrt{Q^{2}_{L}+2}-2Q^{2}_{L}-1}}
  90. Q L Q_{L}
  91. | Z | m a x R Q L 2 |Z|_{max}\approx{RQ^{2}_{L}}
  92. ω 0 = 1 L C - 1 ( R C ) 2 \omega_{0}=\sqrt{\frac{1}{LC}-\frac{1}{(RC)^{2}}}
  93. ω m \omega_{m}
  94. ω m = ω 0 - 1 Q C 2 + 1 + 2 Q C 2 \omega_{m}=\omega^{\prime}_{0}\sqrt{\frac{-1}{Q^{2}_{C}}+\sqrt{1+\frac{2}{Q^{2% }_{C}}}}
  95. Q C = ω 0 R C Q_{C}=\omega^{\prime}_{0}{R}{C}
  96. ω c = 1 L C \omega_{\mathrm{c}}=\frac{1}{\sqrt{LC}}
  97. ζ = 1 2 R L L C \zeta=\frac{1}{2R_{\mathrm{L}}}\sqrt{\frac{L}{C}}
  98. ω c = 1 L C \omega_{\mathrm{c}}=\frac{1}{\sqrt{LC}}
  99. ω c = 1 L C \omega_{\mathrm{c}}=\frac{1}{\sqrt{LC}}
  100. Δ ω = R L L \Delta\omega=\frac{R_{\mathrm{L}}}{L}
  101. Δ ω = 1 C R L \Delta\omega=\frac{1}{CR_{\mathrm{L}}}
  102. α ω 0 . \alpha\ll\omega_{0}.\,
  103. ζ 1. \zeta\ll 1.\,
  104. ω d ω 0 . \omega_{d}\approx\omega_{0}.\,
  105. I = V R I=\frac{V}{R}
  106. V L = V R ω 0 L V_{\mathrm{L}}=\frac{V}{R}\omega_{0}L
  107. V L V = Q \frac{V_{\mathrm{L}}}{V}=Q
  108. I ( t ) = I 0 ( e - α t - e - β t ) I(t)=I_{0}\left(e^{-\alpha t}-e^{-\beta t}\right)
  109. V 0 V_{0}
  110. L L
  111. C = 1 L α β C=\frac{1}{L\alpha\beta}
  112. R = L ( α + β ) R=L(\alpha+\beta)
  113. V 0 = - I 0 L α β ( 1 β - 1 α ) V_{0}=-I_{0}L\alpha\beta\left(\frac{1}{\beta}-\frac{1}{\alpha}\right)
  114. C = ( α + β ) R α β C=\frac{(\alpha+\beta)}{R\alpha\beta}
  115. L = R ( α + β ) L=\frac{R}{(\alpha+\beta)}
  116. V 0 = - I 0 R α β ( α + β ) ( 1 β - 1 α ) V_{0}=\frac{-I_{0}R\alpha\beta}{(\alpha+\beta)}\left(\frac{1}{\beta}-\frac{1}{% \alpha}\right)

Rng_(algebra).html

  1. r = Σ i r i s i r=\Sigma_{i}r_{i}s_{i}

Robertson_graph.html

  1. ( x - 4 ) ( x - 1 ) 2 ( x 2 - 3 ) 2 ( x 2 + x - 5 ) (x-4)(x-1)^{2}(x^{2}-3)^{2}(x^{2}+x-5)
  2. ( x 2 + x - 4 ) 2 ( x 2 + x - 3 ) 2 ( x 2 + x - 1 ) . (x^{2}+x-4)^{2}(x^{2}+x-3)^{2}(x^{2}+x-1).

Robinson–Foulds_metric.html

  1. α \alpha
  2. α - 1 \alpha^{-1}
  3. α \alpha
  4. α - 1 \alpha^{-1}
  5. α \alpha
  6. T 1 T_{1}
  7. T 2 T_{2}
  8. T 1 T 2 T_{1}\wedge T_{2}
  9. α - 1 \alpha^{-1}
  10. T 1 T 2 T_{1}\wedge T_{2}
  11. T 2 T_{2}
  12. T 1 T_{1}
  13. T 2 T_{2}
  14. T 2 T_{2}
  15. T 1 T_{1}
  16. T 1 T_{1}
  17. T 2 T_{2}

Robinson–Schensted–Knuth_correspondence.html

  1. A A
  2. ( P , Q ) (P,Q)
  3. A A
  4. P P
  5. A A
  6. Q Q
  7. A A
  8. ( P , Q ) (P,Q)
  9. A A
  10. P , Q P,Q
  11. σ = ( 1 2 n σ 1 σ 2 σ n ) \sigma=\begin{pmatrix}1&2&\ldots&n\\ \sigma_{1}&\sigma_{2}&\ldots&\sigma_{n}\end{pmatrix}
  12. P P
  13. Q Q
  14. P P
  15. P P
  16. Q Q
  17. Q Q
  18. A A
  19. w A = ( i 1 i 2 i m j 1 j 2 j m ) w_{A}=\begin{pmatrix}i_{1}&i_{2}&\ldots&i_{m}\\ j_{1}&j_{2}&\ldots&j_{m}\end{pmatrix}
  20. ( i , j ) (i,j)
  21. A A
  22. ( i j ) {\textstyle\left({{i}\atop{j}}\right)}
  23. i 1 i 2 i 3 i m i_{1}\leq i_{2}\leq i_{3}\cdots\leq i_{m}
  24. i r = i s i_{r}=i_{s}\,
  25. r s r\leq s
  26. j r j s j_{r}\leq j_{s}
  27. A = ( 1 0 2 0 2 0 1 1 0 ) A=\begin{pmatrix}1&0&2\\ 0&2&0\\ 1&1&0\end{pmatrix}
  28. w A = ( 1 1 1 2 2 3 3 1 3 3 2 2 1 2 ) w_{A}=\begin{pmatrix}1&1&1&2&2&3&3\\ 1&3&3&2&2&1&2\end{pmatrix}
  29. P P
  30. Q Q
  31. b b
  32. b b
  33. A A
  34. ( P , Q ) (P,Q)
  35. P P
  36. Q Q
  37. j j
  38. P P
  39. j j
  40. A A
  41. i i
  42. Q Q
  43. i i
  44. A A
  45. P P
  46. P = 1 1 2 2 2 3 3 , Q 0 = 1 2 3 7 4 5 6 , P\quad=\quad\begin{matrix}1&1&2&2\\ 2&3\\ 3\end{matrix},\qquad Q_{0}\quad=\quad\begin{matrix}1&2&3&7\\ 4&5\\ 6\end{matrix},
  47. P = 1 1 2 2 2 3 3 , Q = 1 1 1 3 2 2 3 . P\quad=\quad\begin{matrix}1&1&2&2\\ 2&3\\ 3\end{matrix},\qquad Q\quad=\quad\begin{matrix}1&1&1&3\\ 2&2\\ 3\end{matrix}.
  48. Q Q
  49. b b
  50. A = ( 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 ) A=\begin{pmatrix}0&0&0&0&0&0&0\\ 0&0&0&1&0&1&0\\ 0&0&0&1&0&0&0\\ 0&0&0&0&0&0&1\\ 0&0&0&0&1&0&0\\ 0&0&1&1&0&0&0\\ 0&0&0&0&0&0&0\\ 1&0&0&0&0&0&0\\ \end{pmatrix}
  51. w A = ( 2 2 3 4 5 6 6 8 4 6 4 7 5 3 4 1 ) . w_{A}=\begin{pmatrix}2&2&3&4&5&6&6&8\\ 4&6&4&7&5&3&4&1\end{pmatrix}.
  52. ( 2 4 ) {\textstyle\left({{2}\atop{4}}\right)}
  53. ( 2 6 ) {\textstyle\left({{2}\atop{6}}\right)}
  54. ( 3 4 ) {\textstyle\left({{3}\atop{4}}\right)}
  55. ( 4 7 ) {\textstyle\left({{4}\atop{7}}\right)}
  56. ( 5 5 ) {\textstyle\left({{5}\atop{5}}\right)}
  57. ( 6 3 ) {\textstyle\left({{6}\atop{3}}\right)}
  58. ( 6 4 ) {\textstyle\left({{6}\atop{4}}\right)}
  59. ( 8 1 ) {\textstyle\left({{8}\atop{1}}\right)}
  60. P P
  61. 4 \begin{matrix}4\end{matrix}
  62. 4 6 \begin{matrix}4&6\end{matrix}
  63. 4 4 6 \begin{matrix}4&4\\ 6\end{matrix}
  64. 4 4 7 6 \begin{matrix}4&4&7\\ 6\end{matrix}
  65. 4 4 5 6 7 \begin{matrix}4&4&5\\ 6&7\end{matrix}
  66. 3 4 5 4 7 6 \begin{matrix}3&4&5\\ 4&7\\ 6\end{matrix}
  67. 3 4 4 4 5 6 7 \begin{matrix}3&4&4\\ 4&5\\ 6&7\end{matrix}
  68. 1 4 4 3 5 4 7 6 \begin{matrix}1&4&4\\ 3&5\\ 4&7\\ 6\end{matrix}
  69. Q Q
  70. 2 \begin{matrix}2\end{matrix}
  71. 2 2 \begin{matrix}2&2\end{matrix}
  72. 2 2 3 \begin{matrix}2&2\\ 3\end{matrix}
  73. 2 2 4 3 \begin{matrix}2&2&4\\ 3\end{matrix}
  74. 2 2 4 3 5 \begin{matrix}2&2&4\\ 3&5\end{matrix}
  75. 2 2 4 3 5 6 \begin{matrix}2&2&4\\ 3&5\\ 6\end{matrix}
  76. 2 2 4 3 5 6 6 \begin{matrix}2&2&4\\ 3&5\\ 6&6\end{matrix}
  77. 2 2 4 3 5 6 6 8 \begin{matrix}2&2&4\\ 3&5\\ 6&6\\ 8\end{matrix}
  78. A A
  79. P , Q P,Q
  80. λ \lambda
  81. P , Q P,Q
  82. λ \lambda
  83. A A
  84. n 1 n\geq 1
  85. λ n ( t λ ) 2 = n ! \sum_{\lambda\vdash n}(t_{\lambda})^{2}=n!
  86. λ n \lambda\vdash n
  87. λ \lambda
  88. n n
  89. t λ t_{\lambda}
  90. λ \lambda
  91. A A
  92. A A
  93. ( P , Q ) (P,Q)
  94. A T A^{T}
  95. ( Q , P ) (Q,P)
  96. A T A^{T}
  97. A A
  98. σ \sigma
  99. ( λ , P , Q ) (\lambda,P,Q)
  100. σ - 1 \sigma^{-1}
  101. ( λ , Q , P ) (\lambda,Q,P)
  102. S n S_{n}
  103. S n S_{n}
  104. { 1 , 2 , 3 , , n } \{1,2,3,\ldots,n\}
  105. { 1 , 2 , 3 , , n } \{1,2,3,\ldots,n\}
  106. π \pi
  107. ( P , Q ) (P,Q)
  108. π = π - \pi=\pi^{-}
  109. ( Q , P ) (Q,P)
  110. P = Q P=Q
  111. π \pi
  112. ( P , P ) (P,P)
  113. π - \pi^{-}
  114. ( P , P ) (P,P)
  115. π = π - \pi=\pi^{-}
  116. π \pi
  117. P P
  118. { 1 , 2 , 3 , , n } \{1,2,3,\ldots,n\}
  119. a ( n ) = a ( n - 1 ) + ( n - 1 ) a ( n - 2 ) a(n)=a(n-1)+(n-1)a(n-2)\,
  120. a ( 1 ) = 1 , a ( 2 ) = 2 a(1)=1,a(2)=2
  121. { 1 , 2 , 3 , , n } \{1,2,3,\ldots,n\}
  122. I ( n ) = n ! k = 0 n / 2 1 2 k k ! ( n - 2 k ) ! I(n)=n!\sum_{k=0}^{\lfloor n/2\rfloor}\frac{1}{2^{k}k!(n-2k)!}
  123. A = A T A=A^{T}
  124. A A
  125. ( P , P ) (P,P)
  126. P P
  127. α \alpha
  128. α = ( α 1 , α 2 , ) \alpha=(\alpha_{1},\alpha_{2},\ldots)
  129. α i N \alpha_{i}\in N
  130. α i < \sum\alpha_{i}<\infty
  131. A P A\longmapsto P
  132. A A
  133. = α =\alpha
  134. α \alpha
  135. i , j ( 1 - x i y j ) - 1 = λ s λ ( x ) s λ ( y ) \prod_{i,j}(1-x_{i}y_{j})^{-1}=\sum_{\lambda}s_{\lambda}(x)s_{\lambda}(y)
  136. s λ s_{\lambda}
  137. μ , ν n \mu,\nu\vdash n
  138. λ n K λ μ K λ ν = N μ ν \sum_{\lambda\vdash n}K_{\lambda\mu}K_{\lambda\nu}=N_{\mu\nu}
  139. K λ μ K_{\lambda\mu}
  140. K λ ν K_{\lambda\nu}
  141. N μ ν N_{\mu\nu}
  142. A A
  143. A A
  144. = μ =\mu
  145. A A
  146. = ν =\nu

Robot_end_effector.html

  1. F = μ W n F=\mu Wn
  2. F \,F
  3. μ \,\mu
  4. n \,n
  5. W \,W
  6. F = μ W n g F=\mu Wng
  7. g \,g
  8. g \,g
  9. g \,g

Robust_measures_of_scale.html

  1. σ 1.4826 MAD . \sigma\approx 1.4826\ \operatorname{MAD}.\,
  2. S n \displaystyle S_{n}
  3. c n c_{n}
  4. n n
  5. n * i = 1 n ( x i - Q ) 2 ( 1 - u i 2 ) 4 I ( | u i | < 1 ) ( i ( 1 - u i 2 ) ( 1 - 5 u i 2 ) I ( | u i | < 1 ) ) 2 , \frac{n*\sum_{i=1}^{n}(x_{i}-Q)^{2}(1-u_{i}^{2})^{4}I(|u_{i}|<1)}{(\sum_{i}(1-% u_{i}^{2})(1-5u_{i}^{2})I(|u_{i}|<1))^{2}},
  6. u i = x i - Q 9 MAD . u_{i}=\frac{x_{i}-Q}{9\cdot{\rm MAD}}.

Rocchio_algorithm.html

  1. Q m = ( a Q o ) + ( b 1 | D r | D j D r D j ) - ( c 1 | D n r | D k D n r D k ) \overrightarrow{Q_{m}}=\bigl(a\cdot\overrightarrow{Q_{o}}\bigr)+\biggl(b\cdot{% \tfrac{1}{|D_{r}|}}\cdot\sum_{\overrightarrow{D_{j}}\in D_{r}}\overrightarrow{% D_{j}}\biggr)-\biggl(c\cdot{\tfrac{1}{|D_{nr}|}}\cdot\sum_{\overrightarrow{D_{% k}}\in D_{nr}}\overrightarrow{D_{k}}\biggr)
  2. Q m \overrightarrow{Q_{m}}
  3. Q o \overrightarrow{Q_{o}}
  4. D j \overrightarrow{D_{j}}
  5. D k \overrightarrow{D_{k}}
  6. a a
  7. b b
  8. c c
  9. D r D_{r}
  10. D n r D_{nr}
  11. D j \overrightarrow{Dj}
  12. D k \overrightarrow{Dk}
  13. 𝔻 \mathbb{D}
  14. L a v e L_{ave}
  15. \mathbb{C}
  16. V V
  17. L a L_{a}
  18. M a M_{a}
  19. Θ ( | | M a ) \Theta(|\mathbb{C}|M_{a})
  20. Θ ( | 𝔻 | L a v e + | | | V | ) \Theta(|\mathbb{D}|L_{ave}+|\mathbb{C}||V|)
  21. Θ ( L a + | | M a ) = Θ ( | | M a ) \Theta(L_{a}+|\mathbb{C}|M_{a})=\Theta(|\mathbb{C}|M_{a})

Rohn_Emergency_Scale.html

  1. E = E m e r g e n c y = f ( S , T , D ) E=Emergency=f(S,T,D)
  2. Scope = RawScope MaxScope \hbox{Scope}=\tfrac{\hbox{RawScope}}{\hbox{MaxScope}}
  3. RawScope = ( Victims Population + Monetary Losses GNP ) W \hbox{RawScope}=\left(\tfrac{\hbox{Victims}}{\hbox{Population}}+\tfrac{\hbox{% Monetary Losses}}{\hbox{GNP}}\right)^{W}
  4. W = ( ln ( Victims ) ln ( Monetary Losses ) ) β W=\left(\tfrac{\ln(\hbox{Victims})}{\ln(\hbox{Monetary Losses})}\right)^{\beta}
  5. MaxScope = ( 0.7 * Population Population + 0.5 * GNP GNP ) V \hbox{MaxScope}=\left(\tfrac{0.7*\hbox{Population}}{\hbox{Population}}+\tfrac{% 0.5*\hbox{GNP}}{\hbox{GNP}}\right)^{V}
  6. V = ln ( Victims ) ln ( Monetary Losses ) V=\tfrac{\ln(\hbox{Victims})}{\ln(\hbox{Monetary Losses})}
  7. Volume before the event Volume after the event \tfrac{\hbox{Volume before the event}}{\hbox{Volume after the event}}
  8. d ( Victims ) d ( Time ) \tfrac{d(\hbox{Victims})}{d(\hbox{Time})}
  9. d ( Losses ) d ( Time ) \tfrac{d(\hbox{Losses})}{d(\hbox{Time})}

Roll-off.html

  1. A = V o V i = 1 1 + i ω R C A=\frac{V_{o}}{V_{i}}=\frac{1}{1+i\omega RC}
  2. | A | 2 = 1 1 + ( ω ω c ) 2 = 1 1 + ω 2 |A|^{2}=\frac{1}{1+\left({\omega\over\omega_{c}}\right)^{2}}=\frac{1}{1+\omega% ^{2}}
  3. 10 log ( 1 1 + ω 2 ) 10\log\left({\frac{1}{1+\omega^{2}}}\right)
  4. L = 10 log ( 1 + ω 2 ) dB L=10\log\left({1+\omega^{2}}\right)\ \mathrm{dB}
  5. L 10 log ( ω 2 ) = 20 log ω dB L\approx 10\log\left(\omega^{2}\right)=20\log\omega\ \mathrm{dB}
  6. Δ L = 20 log ( ω 2 ω 1 ) dB / interval 2 , 1 \Delta L=20\log\left({\omega_{2}\over\omega_{1}}\right)\ \mathrm{dB/interval_{% 2,1}}
  7. Δ L = 20 log 10 = 20 dB / decade \Delta L=20\log 10=20\ \mathrm{dB/decade}
  8. Δ L = 20 log 2 20 × 0.3 = 6 dB / 8 v e \Delta L=20\log 2\approx 20\times 0.3=6\ \mathrm{dB/8ve}
  9. A T = A n A_{\mathrm{T}}=A^{n}
  10. Δ L T = n Δ L = 6 n dB / 8 v e \Delta L_{\mathrm{T}}=n\Delta L=6n\ \mathrm{dB/8ve}

Roller_screw.html

  1. s d s_{d}
  2. r d r_{d}
  3. n d n_{d}
  4. t t
  5. l l
  6. p p
  7. n d = s d t t - 2 n_{d}=\frac{s_{d}t}{t-2}
  8. \therefore
  9. ( n d : s d ) = 1 : t t - 2 (n_{d}:s_{d})=1:\frac{t}{t-2}
  10. r d = s d t - 2 r_{d}=\frac{s_{d}}{t-2}
  11. \therefore
  12. ( r d : s d ) = 1 : t - 2 (r_{d}:s_{d})=1:t-2
  13. r d = n d t r_{d}=\frac{n_{d}}{t}
  14. \therefore
  15. ( r d : n d ) = 1 : t (r_{d}:n_{d})=1:t
  16. r d = n d - s d 2 r_{d}=\frac{n_{d}-s_{d}}{2}
  17. p = l t p=\frac{l}{t}
  18. \therefore
  19. ( p : l ) = 1 : t (p:l)=1:t

Ronen's_golden_rule_for_cluster_radioactivity.html

  1. A z A_{z}
  2. A z = A z - 2 + 4 + B z / 2 A_{z}=A_{z-2}+4+B_{z/2}
  3. Z Z
  4. A 0 = 0 A_{0}=0
  5. B z / 2 = 0 B_{z/2}=0
  6. Z / 2 Z/2
  7. B z / 2 = 2 B_{z/2}=2
  8. Z / 2 Z/2

Root_system_of_a_semi-simple_Lie_algebra.html

  1. 𝔤 λ := { a 𝔤 : [ h , a ] = λ ( h ) a for all h 𝔥 } . \mathfrak{g}_{\lambda}:=\{a\in\mathfrak{g}:[h,a]=\lambda(h)a\,\text{ for all }% h\in\mathfrak{h}\}.
  2. 𝔤 = 𝔥 λ R 𝔤 λ . \mathfrak{g}=\mathfrak{h}\oplus\bigoplus_{\lambda\in R}\mathfrak{g}_{\lambda}.
  3. H λ , X λ , Y λ for λ Δ H_{\lambda},X_{\lambda},Y_{\lambda}\,\text{ for }\lambda\in\Delta
  4. [ H λ , H μ ] = 0 for all λ , μ Δ [H_{\lambda},H_{\mu}]=0\,\text{ for all }\lambda,\mu\in\Delta
  5. [ H λ , X μ ] = ( λ , μ ) X μ , [H_{\lambda},X_{\mu}]=(\lambda,\mu)X_{\mu},
  6. [ H λ , Y μ ] = - ( λ , μ ) Y μ , [H_{\lambda},Y_{\mu}]=-(\lambda,\mu)Y_{\mu},
  7. [ X μ , Y λ ] = δ μ λ H μ , [X_{\mu},Y_{\lambda}]=\delta_{\mu\lambda}H_{\mu},
  8. ad X λ - ( μ , λ ) + 1 X μ = 0 for λ μ , \mathrm{ad}_{X_{\lambda}}^{-(\mu,\lambda)+1}X_{\mu}=0\,\text{ for }\lambda\neq\mu,
  9. ad Y λ - ( μ , λ ) + 1 Y μ = 0 for λ μ . \mathrm{ad}_{Y_{\lambda}}^{-(\mu,\lambda)+1}Y_{\mu}=0\,\text{ for }\lambda\neq\mu.
  10. ( λ , μ ) (\lambda,\mu)

Rossmo's_formula.html

  1. S i , j S_{i,j}
  2. p i , j p_{i,j}
  3. ( X i , Y j ) (X_{i},Y_{j})
  4. p i , j = k n = 1 ( total crimes ) [ ϕ i j ( | X i - x n | + | Y j - y n | ) f 1 st term + ( 1 - ϕ i j ) ( B g - f ) ( 2 B - X i - x n - Y j - y n ) g 2 nd term ] , p_{i,j}=k\sum_{n=1}^{(\mathrm{total\;crimes})}\left[\underbrace{\frac{\phi_{ij% }}{(|X_{i}-x_{n}|+|Y_{j}-y_{n}|)^{f}}}_{1^{\mathrm{st}}\mathrm{\;term}}+% \underbrace{\frac{(1-\phi_{ij})(B^{g-f})}{(2B-\mid X_{i}-x_{n}\mid-\mid Y_{j}-% y_{n}\mid)^{g}}}_{2^{\mathrm{nd}}\mathrm{\;term}}\right],
  5. ϕ i j = { 1 , if ( X i - x n + Y j - y n ) > B 0 , else \phi_{ij}=\begin{cases}1,&\mathrm{\quad if\;}(\mid X_{i}-x_{n}\mid+\mid Y_{j}-% y_{n}\mid)>B\\ 0,&\mathrm{\quad else}\end{cases}
  6. ϕ i j \phi_{ij}
  7. ( X i , Y j ) (X_{i},Y_{j})
  8. ϕ i j \phi_{ij}
  9. ϕ i j = 0 \phi_{ij}=0
  10. ϕ i j = 1 \phi_{ij}=1
  11. p i , j p_{i,j}
  12. X i - x n + Y j - y n \mid X_{i}-x_{n}\mid+\mid Y_{j}-y_{n}\mid
  13. ( X i , Y j ) (X_{i},Y_{j})
  14. ( x n , y n ) (x_{n},y_{n})
  15. ϕ \phi
  16. B B
  17. k k
  18. f f
  19. g g

RSA::Intuitive.html

  1. a ( p - 1 ) 1 ( mod p ) a^{(p-1)}\equiv 1\;\;(\mathop{{\rm mod}}p)
  2. e d 1 ( mod p - 1 ) ed\equiv 1\;\;(\mathop{{\rm mod}}p-1)
  3. e d - 1 = k ( p - 1 ) ed-1=k(p-1)
  4. m e d m e d m ( e d - 1 ) m 1 m k ( p - 1 ) m 1 k m m ( mod p ) m^{e^{d}}\equiv m^{ed}\equiv m^{(ed-1)}\cdot m^{1}\equiv m^{k(p-1)}m\equiv 1^{% k}m\equiv m\;\;(\mathop{{\rm mod}}p)
  5. n = p q n=pq
  6. e d 1 ( mod ( p - 1 ) ( q - 1 ) ) ed\equiv 1\;\;(\mathop{{\rm mod}}(p-1)(q-1))
  7. e d - 1 = k ( p - 1 ) ( q - 1 ) ed-1=k(p-1)(q-1)
  8. m e d m e d m ( e d - 1 ) m m k ( p - 1 ) ( q - 1 ) m 1 k ( q - 1 ) m m ( mod p ) m^{e^{d}}\equiv m^{ed}\equiv m^{(ed-1)}m\equiv m^{k(p-1)(q-1)}m\equiv 1^{k(q-1% )}m\equiv m\;\;(\mathop{{\rm mod}}p)
  9. m e d m e d m ( e d - 1 ) m m k ( p - 1 ) ( q - 1 ) m 1 k ( p - 1 ) m m ( mod q ) m^{e^{d}}\equiv m^{ed}\equiv m^{(ed-1)}m\equiv m^{k(p-1)(q-1)}m\equiv 1^{k(p-1% )}m\equiv m\;\;(\mathop{{\rm mod}}q)
  10. a b ( mod p ) a\equiv b\;\;(\mathop{{\rm mod}}p)
  11. a b ( mod q ) a\equiv b\;\;(\mathop{{\rm mod}}q)
  12. a b ( mod p q ) a\equiv b\;\;(\mathop{{\rm mod}}pq)
  13. m e d m ( mod p q ) m^{e^{d}}\equiv m\;\;(\mathop{{\rm mod}}pq)
  14. e f 1 ( mod n - 1 ) ef\equiv 1\;\;(\mathop{{\rm mod}}n-1)
  15. m e f m k ( n - 1 ) m m ( mod n ) m^{e^{f}}\equiv m^{k(n-1)}m\neq m\;\;(\mathop{{\rm mod}}n)

Rufus_Bowen.html

  1. M M
  2. f f
  3. M M
  4. M M
  5. k a ka
  6. n n
  7. n n

Runge–Gross_theorem.html

  1. H ^ v ( t ) | Ψ ( t ) = i t | Ψ ( t ) . \hat{H}_{v}(t)|\Psi(t)\rangle=i\frac{\partial}{\partial t}|\Psi(t)\rangle.
  2. ρ ( 𝐫 , t ) = N s 1 s N d 𝐫 2 d 𝐫 N | Ψ ( 𝐫 1 , s 1 , 𝐫 2 , s 2 , , 𝐫 N , s N , t ) | 2 . \rho(\mathbf{r},t)=N\sum_{s_{1}}\cdots\sum_{s_{N}}\int\ \mathrm{d}\mathbf{r}_{% 2}\ \cdots\int\ \mathrm{d}\mathbf{r}_{N}\ |\Psi(\mathbf{r}_{1},s_{1},\mathbf{r% }_{2},s_{2},...,\mathbf{r}_{N},s_{N},t)|^{2}.
  3. v ( 𝐫 , t ) + c ( t ) e - i c ( t ) | Ψ ( t ) ρ ( 𝐫 , t ) . v(\mathbf{r},t)+c(t)\rightarrow e^{-ic(t)}|\Psi(t)\rangle\rightarrow\rho(% \mathbf{r},t).
  4. ρ ( 𝐫 , t ) = ρ [ v , Ψ 0 ] ( 𝐫 , t ) v ( 𝐫 , t ) = v [ ρ , Ψ 0 ] ( 𝐫 , t ) \rho(\mathbf{r},t)=\rho[v,\Psi_{0}](\mathbf{r},t)\leftrightarrow v(\mathbf{r},% t)=v[\rho,\Psi_{0}](\mathbf{r},t)
  5. u k ( 𝐫 ) k t k ( v ( 𝐫 , t ) - v ( 𝐫 , t ) ) | t = t 0 u_{k}(\mathbf{r})\equiv\left.\frac{\partial^{k}}{\partial t^{k}}\big(v(\mathbf% {r},t)-v^{\prime}(\mathbf{r},t)\big)\right|_{t=t_{0}}
  6. i 𝐣 ( 𝐫 , t ) t = Ψ ( t ) | [ 𝐣 ^ ( 𝐫 ) , H ^ v ( t ) ] | Ψ ( t ) . i\frac{\partial\mathbf{j}(\mathbf{r},t)}{\partial t}=\langle\Psi(t)|[\hat{% \mathbf{j}}(\mathbf{r}),\hat{H}_{v}(t)]|\Psi(t)\rangle.
  7. i t ( 𝐣 ( 𝐫 , t ) - 𝐣 ( 𝐫 , t ) ) | t = t 0 = Ψ ( t 0 ) | [ 𝐣 ^ ( 𝐫 ) , H ^ v ( t 0 ) - H ^ v ( t 0 ) ] | Ψ ( t 0 ) , = Ψ ( t 0 ) | [ 𝐣 ^ ( 𝐫 ) , V ^ ( t 0 ) - V ^ ( t 0 ) ] | Ψ ( t 0 ) , = i ρ ( 𝐫 , t 0 ) ( v ( 𝐫 , t 0 ) - v ( 𝐫 , t 0 ) ) . \begin{aligned}\displaystyle i\left.\frac{\partial}{\partial t}\big(\mathbf{j}% (\mathbf{r},t)-\mathbf{j}^{\prime}(\mathbf{r},t)\big)\right|_{t=t_{0}}&% \displaystyle=\langle\Psi(t_{0})|[\hat{\mathbf{j}}(\mathbf{r}),\hat{H}_{v}(t_{% 0})-\hat{H}_{v^{\prime}}(t_{0})]|\Psi(t_{0})\rangle,\\ &\displaystyle=\langle\Psi(t_{0})|[\hat{\mathbf{j}}(\mathbf{r}),\hat{V}(t_{0})% -\hat{V}^{\prime}(t_{0})]|\Psi(t_{0})\rangle,\\ &\displaystyle=i\rho(\mathbf{r},t_{0})\nabla\big(v(\mathbf{r},t_{0})-v^{\prime% }(\mathbf{r},t_{0})\big).\end{aligned}
  8. i k + 1 k + 1 t k + 1 ( 𝐣 ( 𝐫 , t ) - 𝐣 ( 𝐫 , t ) ) | t = t 0 = i ρ ( 𝐫 , t ) i k k t k ( v ( 𝐫 , t ) - v ( 𝐫 , t ) ) | t = t 0 , i^{k+1}\left.\frac{\partial^{k+1}}{\partial t^{k+1}}\big(\mathbf{j}(\mathbf{r}% ,t)-\mathbf{j}^{\prime}(\mathbf{r},t)\big)\right|_{t=t_{0}}=i\rho(\mathbf{r},t% )\nabla i^{k}\left.\frac{\partial^{k}}{\partial t^{k}}\big(v(\mathbf{r},t)-v^{% \prime}(\mathbf{r},t)\big)\right|_{t=t_{0}},
  9. ρ ( 𝐫 , t ) t + 𝐣 ( 𝐫 , t ) = 0. \frac{\partial\rho(\mathbf{r},t)}{\partial t}+\nabla\cdot\mathbf{j}(\mathbf{r}% ,t)=0.
  10. k + 2 t k + 2 ( ρ ( 𝐫 , t ) - ρ ( 𝐫 , t ) ) | t = t 0 = - k + 1 t k + 1 ( 𝐣 ( 𝐫 , t ) - 𝐣 ( 𝐫 , t ) ) | t = t 0 , = - [ ρ ( 𝐫 , t 0 ) k t k ( v ( 𝐫 , t 0 ) - v ( 𝐫 , t 0 ) ) | t = t 0 ] , = - [ ρ ( 𝐫 , t 0 ) u k ( 𝐫 ) ] . \begin{aligned}\displaystyle\left.\frac{\partial^{k+2}}{\partial t^{k+2}}(\rho% (\mathbf{r},t)-\rho^{\prime}(\mathbf{r},t))\right|_{t=t_{0}}&\displaystyle=-% \nabla\cdot\left.\frac{\partial^{k+1}}{\partial t^{k+1}}\big(\mathbf{j}(% \mathbf{r},t)-\mathbf{j}^{\prime}(\mathbf{r},t)\big)\right|_{t=t_{0}},\\ &\displaystyle=-\nabla\cdot[\rho(\mathbf{r},t_{0})\nabla\left.\frac{\partial^{% k}}{\partial t^{k}}\big(v(\mathbf{r},t_{0})-v^{\prime}(\mathbf{r},t_{0})\big)% \right|_{t=t_{0}}],\\ &\displaystyle=-\nabla\cdot[\rho(\mathbf{r},t_{0})\nabla u_{k}(\mathbf{r})].% \end{aligned}
  11. ( ρ ( 𝐫 , t 0 ) u k ( 𝐫 ) ) = 0 , \nabla\cdot(\rho(\mathbf{r},t_{0})\nabla u_{k}(\mathbf{r}))=0,
  12. 0 = d 𝐫 u k ( 𝐫 ) ( ρ ( 𝐫 , t 0 ) u k ( 𝐫 ) ) , = - d 𝐫 ρ ( 𝐫 , t 0 ) ( u k ( 𝐫 ) ) 2 + 1 2 d 𝐒 ρ ( 𝐫 , t 0 ) ( u k 2 ( 𝐫 ) ) . \begin{aligned}\displaystyle 0&\displaystyle=\int\mathrm{d}\mathbf{r}\ u_{k}(% \mathbf{r})\nabla\cdot(\rho(\mathbf{r},t_{0})\nabla u_{k}(\mathbf{r})),\\ &\displaystyle=-\int\mathrm{d}\mathbf{r}\ \rho(\mathbf{r},t_{0})(\nabla u_{k}(% \mathbf{r}))^{2}+\frac{1}{2}\int\mathrm{d}\mathbf{S}\cdot\rho(\mathbf{r},t_{0}% )(\nabla u_{k}^{2}(\mathbf{r})).\end{aligned}
  13. ρ ( 𝐫 , t 0 ) ( u k ( 𝐫 ) ) 2 = 0 , \rho(\mathbf{r},t_{0})(\nabla u_{k}(\mathbf{r}))^{2}=0,

Ruthenium(III)_acetylacetonate.html

  1. \overrightarrow{\leftarrow}

Ruze's_Equation.html

  1. G ( ϵ ) = G 0 e - ( 4 π ϵ λ ) 2 G\left(\epsilon\right)=G_{0}\,\,e^{-\left(\frac{4\pi\epsilon}{\lambda}\right)^% {2}}
  2. ϵ \displaystyle\epsilon
  3. λ \displaystyle\lambda
  4. G 0 \displaystyle G_{0}
  5. G ( ϵ ) = G 0 - 685.81 ( ϵ λ ) 2 G\left(\epsilon\right)=G_{0}\,-\,685.81\left(\frac{\epsilon}{\lambda}\right)^{2}
  6. - 685.81 = 10 log 10 ( e - ( 4 π ) 2 ) -685.81=10\log_{10}\left(e^{{-\left(4\pi\right)}^{2}}\right)
  7. G ( ϵ ) = G 0 e - ( 2 π ϵ λ ) 2 G\left(\epsilon\right)=G_{0}\,\,e^{-\left(\frac{2\pi\epsilon}{\lambda}\right)^% {2}}
  8. G ( ϵ ) = G 0 - 171.45 ( ϵ λ ) 2 G\left(\epsilon\right)=G_{0}\,-\,171.45\left(\frac{\epsilon}{\lambda}\right)^{2}
  9. ϵ \displaystyle\epsilon
  10. λ \displaystyle\lambda

RV_coefficient.html

  1. Σ X Y = E ( X T Y ) , \Sigma_{XY}=E(X^{T}Y)\,,
  2. COVV ( X , Y ) = T r ( Σ X Y Σ Y X ) . \mathrm{COVV}(X,Y)=Tr(\Sigma_{XY}\Sigma_{YX})\,.
  3. VAV ( X ) = T r ( Σ X X 2 ) . \mathrm{VAV}(X)=Tr(\Sigma_{XX}^{2})\,.
  4. RV ( X , Y ) = COVV ( X , Y ) VAV ( X ) VAV ( Y ) . \mathrm{RV}(X,Y)=\frac{\mathrm{COVV}(X,Y)}{\sqrt{\mathrm{VAV}(X)\mathrm{VAV}(Y% )}}\,.

Rylands_Papyri.html

  1. 𝔓 \mathfrak{P}

Rømer's_determination_of_the_speed_of_light.html

  1. c / v {c}/{v}
  2. c / v {c}/{v}
  3. c v = P 2 π τ {c\over v}={P\over{2\pi\tau}}
  4. c / v {c}/{v}
  5. c / v {c}/{v}
  6. 54000 / 27706 {54000}/{27706}
  7. 377 / 1141 {377}/{1141}
  8. 2 π r / P 2π{r}/{P}
  9. r / c {r}/{c}
  10. 2 π τ c / P 2π{τc}/{P}
  11. c / v {c}/{v}

S2_(star).html

  1. 15 M 15M_{\odot}

Sagitta_(geometry).html

  1. s s
  2. r r
  3. \ell
  4. s = r - r 2 - 2 s=r-\sqrt{r^{2}-\ell^{2}}
  5. θ \theta
  6. s = r versin θ 2 = r ( 1 - cos θ 2 ) s=r\operatorname{versin}\frac{\theta}{2}=r\left(1-\cos\frac{\theta}{2}\right)
  7. r 2 = 2 + ( r - s ) 2 r^{2}=\ell^{2}+(r-s)^{2}
  8. r = s 2 + 2 2 s r=\frac{s^{2}+\ell^{2}}{2s}
  9. = 2 r s - s 2 \ell=\sqrt{2rs-s^{2}}
  10. s 2 2 r s\approx\frac{\ell^{2}}{2r}
  11. a + s 2 r a\approx\ell+\frac{s^{2}}{r}
  12. a a

Sagitta_(optics).html

  1. S = r 2 2 × ROC S=\frac{r^{2}}{2\times\,\text{ROC}}
  2. S = r 2 4 × focal length S=\frac{r^{2}}{4\times\,\text{focal length}}

Sakuma–Hattori_equation.html

  1. S ( T ) = C exp ( c 2 λ x T ) - 1 S(T)=\frac{C}{\exp\left(\frac{c_{2}}{\lambda_{x}T}\right)-1}
  2. C C
  3. c 2 c_{2}
  4. λ x \lambda_{x}
  5. T T
  6. λ x = A + B T \lambda_{x}=A+\frac{B}{T}
  7. S ( T ) = C exp ( c 2 A T + B ) - 1 S(T)=\frac{C}{\exp\left(\frac{c_{2}}{AT+B}\right)-1}
  8. T = c 2 A ln ( C S + 1 ) - B A T=\frac{c_{2}}{A\ln\left(\frac{C}{S}+1\right)}-\frac{B}{A}
  9. d S d T = [ S ( T ) ] 2 A c 2 C ( A T + B ) 2 exp ( c 2 A T + B ) \frac{dS}{dT}=\left[S(T)\right]^{2}\frac{Ac_{2}}{C\left(AT+B\right)^{2}}\exp% \left(\frac{c_{2}}{AT+B}\right)
  10. S ( T ) = c 1 λ 5 [ exp ( c 2 λ T ) - 1 ] S(T)=\frac{c_{1}}{\lambda^{5}\left[\exp\left(\frac{c_{2}}{\lambda T}\right)-1% \right]}
  11. S ( T ) = λ 1 λ 2 c 1 λ 5 [ exp ( c 2 λ T ) - 1 ] d λ S(T)=\int_{\lambda_{1}}^{\lambda_{2}}\frac{c_{1}}{\lambda^{5}\left[\exp\left(% \frac{c_{2}}{\lambda T}\right)-1\right]}d\lambda
  12. 0 λ 2 c 1 λ 5 [ exp ( c 2 λ T ) - 1 ] d λ = c 1 ( T c 2 ) 4 c 2 / ( λ 2 T ) x 3 exp ( x ) - 1 d x \int_{0}^{\lambda_{2}}\frac{c_{1}}{\lambda^{5}[\exp(\frac{c_{2}}{\lambda T})-1% ]}d\lambda=c_{1}(\frac{T}{c_{2}})^{4}\int_{c_{2}/(\lambda_{2}T)}^{\infty}\frac% {x^{3}}{\exp(x)-1}dx
  13. λ = c 2 / ( x T ) \lambda=c_{2}/(xT)
  14. d λ = - c 2 / ( x 2 T ) d x d\lambda=-c_{2}/(x^{2}T)dx
  15. J ( c ) c x 3 exp x - 1 d x = c x 3 exp ( - x ) 1 - exp ( - x ) d x = c n 1 x 3 exp ( - n x ) d x J(c)\equiv\int_{c}^{\infty}\frac{x^{3}}{\exp x-1}dx=\int_{c}^{\infty}\frac{x^{% 3}\exp(-x)}{1-\exp(-x)}dx=\int_{c}^{\infty}\sum_{n\geq 1}x^{3}\exp(-nx)dx
  16. = n 1 exp ( - n c ) ( n c ) 3 + 3 ( n c ) 2 + 6 n c + 6 n 4 =\sum_{n\geq 1}\exp(-nc)\frac{(nc)^{3}+3(nc)^{2}+6nc+6}{n^{4}}
  17. S ( T ) = C exp ( c 2 A T + B ) - 1 S(T)=\frac{C}{\exp\left(\frac{c_{2}}{AT+B}\right)-1}
  18. S ( T ) = C exp ( A T 2 + B 2 T ) - 1 S(T)=\frac{C}{\exp\left(\frac{A}{T^{2}}+\frac{B}{2T}\right)-1}
  19. S ( T ) = C exp ( - c 2 A T + B ) S(T)=C\exp\left(\frac{-c_{2}}{AT+B}\right)
  20. S ( T ) = C T A exp ( B T ) - 1 S(T)=\frac{CT^{A}}{\exp\left(\frac{B}{T}\right)-1}
  21. S ( T ) = C T A exp ( - B T ) S(T)=CT^{A}{\exp\left(\frac{-B}{T}\right)}
  22. S ( T ) = C exp ( c 2 A T ) - 1 S(T)=\frac{C}{\exp\left(\frac{c_{2}}{AT}\right)-1}
  23. S ( T ) = C ( 1 + A T ) - B S(T)=C\left(1+\frac{A}{T}\right)-B
  24. S ( T ) = C exp ( - c 2 A T ) S(T)=C\exp\left(\frac{-c_{2}}{AT}\right)
  25. S ( T ) = C exp ( - A T + B T 2 ) S(T)=C\exp\left(\frac{-A}{T}+\frac{B}{T^{2}}\right)
  26. S ( T ) = C T A S(T)=CT^{A}

Salter's_duck.html

  1. 0.16 < r λ < 0.2 0.16<\frac{r}{\lambda}<0.2
  2. R = ( υ n - u n ) 2 ds u n 2 ds \text{R}\,\!=\int\limits\big(\upsilon_{n}-\mathit{u_{n}}\big)^{2}\text{ds}\,\!% \int\limits\mathit{u_{n}}^{2}\text{ds}\,\!
  3. η = ( 1 - R m i n ) × 1 ( - 4 π d λ ) \eta=\big(1-\mathit{R}_{min}\big)\times 1\big(\frac{-4\pi d}{\lambda}\big)

Sample_matrix_inversion.html

  1. x ( k ) , k = 1 , 2 , , K - 1 x(k),k=1,2,\dots,K-1
  2. R ^ u ( k ) = ( 1 / k ) ( x ( k ) x H ( k ) ) . \hat{R}_{u}(k)=(1/k)\sum(x(k)x^{H}(k)).

Sample_maximum_and_minimum.html

  1. { X 1 , , X n } , \{X_{1},\dots,X_{n}\},
  2. X n + 1 , X_{n+1},
  3. 1 / ( n + 1 ) 1/(n+1)
  4. 1 / ( n + 1 ) 1/(n+1)
  5. ( n - 1 ) / ( n + 1 ) (n-1)/(n+1)
  6. X n + 1 X_{n+1}
  7. { X 1 , , X n } . \{X_{1},\dots,X_{n}\}.
  8. ( n - 1 ) / ( n + 1 ) (n-1)/(n+1)
  9. 1 , 2 , , N 1,2,\dots,N
  10. M , M + 1 , , N M,M+1,\dots,N
  11. k + 1 k m - 1 \frac{k+1}{k}m-1

Samplesort.html

  1. O ( n / P + l o g ( P ) ) O(n/P+log(P))
  2. P P
  3. l o g ( P ) log(P)
  4. n / P * l o g ( P ) n/P*log(P)
  5. n / P n/P
  6. O ( n / P ) O(n/P)

Samuelson's_inequality.html

  1. x ¯ = x 1 + + x n n \overline{x}=\frac{x_{1}+\cdots+x_{n}}{n}
  2. s = 1 n i = 1 n ( x i - x ¯ ) 2 s=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}
  3. x ¯ - s n - 1 x i x ¯ + s n - 1 for i = 1 , , n . \overline{x}-s\sqrt{n-1}\leq x_{i}\leq\overline{x}+s\sqrt{n-1}\qquad\,\text{% for }i=1,\dots,n.
  4. a 0 x n + a 1 x n - 1 + + a n - 1 x + a n = 0 a_{0}x^{n}+a_{1}x^{n-1}+\cdots+a_{n-1}x+a_{n}=0
  5. a 0 = 1 a_{0}=1
  6. t 1 = x i t_{1}=\sum x_{i}
  7. t 2 = x i 2 t_{2}=\sum x_{i}^{2}
  8. a 1 = - x i = - t 1 a_{1}=-\sum x_{i}=-t_{1}
  9. a 2 = x i x j = t 1 2 - t 2 2 where i < j a_{2}=\sum x_{i}x_{j}=\frac{t_{1}^{2}-t_{2}}{2}\qquad\,\text{ where }i<j
  10. t 2 = a 1 2 - 2 a 2 t_{2}=a_{1}^{2}-2a_{2}
  11. - a 1 / n ± b n - 1 -a_{1}/n\pm b\sqrt{n-1}
  12. b = n t 2 - t 1 n = n a 1 2 + a 1 - 2 n a 2 n b=\frac{\sqrt{nt_{2}-t_{1}}}{n}=\frac{\sqrt{na_{1}^{2}+a_{1}-2na_{2}}}{n}
  13. - a 1 n -\tfrac{a_{1}}{n}
  14. a 1 a_{1}
  15. a 2 a_{2}

Sardinas–Patterson_algorithm.html

  1. { a 1 , b 011 , c 01110 , d 1110 , e 10011 } \{\,a\mapsto 1,b\mapsto 011,c\mapsto 01110,d\mapsto 1110,e\mapsto 10011\,\}
  2. x 1 x_{1}
  3. y 1 y_{1}
  4. x 1 x_{1}
  5. y 1 y_{1}
  6. x 1 w = y 1 x_{1}w=y_{1}
  7. w w
  8. x 1 = 011 x_{1}=011
  9. y 1 = 01110 y_{1}=01110
  10. w = 10 w=10
  11. x 2 , , x p x_{2},\ldots,x_{p}
  12. y 2 , , y q y_{2},\ldots,y_{q}
  13. x 2 x p = w y 2 y q x_{2}\cdots x_{p}=wy_{2}\cdots y_{q}
  14. x = x 1 x 2 x p x=x_{1}x_{2}\cdots x_{p}
  15. y 1 y 2 y q y_{1}y_{2}\cdots y_{q}
  16. w = 10 w=10
  17. N - 1 D N^{-1}D
  18. N - 1 D = { y x y D and x N } N^{-1}D=\{\,y\mid xy\in D~{}\textrm{ and }~{}x\in N\,\}
  19. C C
  20. i = 1 i=1
  21. S i S_{i}
  22. S i S_{i}
  23. S 1 = C - 1 C { ε } S_{1}=C^{-1}C\setminus\{\varepsilon\}
  24. ε \varepsilon
  25. S i + 1 = C - 1 S i S i - 1 C S_{i+1}=C^{-1}S_{i}\cup S_{i}^{-1}C
  26. i 1 i\geq 1
  27. S i S_{i}
  28. i i
  29. S i S_{i}
  30. S i S_{i}
  31. S j S_{j}
  32. j < i j<i
  33. S i S_{i}
  34. S i S_{i}

Satisfiability.html

  1. a a
  2. b b
  3. R ( a 0 , a 0 ) R(a_{0},a_{0})
  4. R ( a 0 , a 1 ) R(a_{0},a_{1})
  5. x y ( R ( x , y ) z R ( y , z ) ) \forall xy(R(x,y)\rightarrow\exists zR(y,z))
  6. x y z ( R ( y , x ) R ( z , x ) x = z ) ) \forall xyz(R(y,x)\wedge R(z,x)\rightarrow x=z))
  7. R ( a 0 , a 0 ) , R ( a 0 , a 1 ) , R ( a 1 , a 2 ) , R(a_{0},a_{0}),R(a_{0},a_{1}),R(a_{1},a_{2}),\ldots
  8. R ( a , b ) R(a,b)
  9. R R
  10. a 0 a_{0}

Savilian_Professor_of_Geometry.html

  1. \infty
  2. \infty

Scale-free_ideal_gas.html

  1. F ( k , v ) = N Ω k 2 exp [ - ( v / k - w ¯ ) 2 / 2 σ w 2 ] 2 π σ w , F(k,v)=\frac{N}{\Omega k^{2}}\frac{\exp\left[-(v/k-\overline{w})^{2}/2\sigma_{% w}^{2}\right]}{\sqrt{2\pi}\sigma_{w}},
  2. w ¯ = v / k \overline{w}=\langle v/k\rangle
  3. σ w \sigma_{w}
  4. S = N κ { ln Ω N 2 π σ w H + 3 2 } , S=N\kappa\left\{\ln\frac{\Omega}{N}\frac{\sqrt{2\pi}\sigma_{w}}{H^{\prime}}+% \frac{3}{2}\right\},
  5. κ \kappa
  6. H = 1 / M Δ τ H^{\prime}=1/M\Delta\tau
  7. Δ τ \Delta\tau

Scale_(descriptive_set_theory).html

  1. A X = X 0 × X 1 × X m - 1 A\subseteq X=X_{0}\times X_{1}\times\ldots X_{m-1}
  2. ( ϕ n ) n < ω (\phi_{n})_{n<\omega}
  3. n x ( φ n ( x ) φ n ( y ) S ( n , x , y ) T ( n , x , y ) ) \forall n\forall x(\varphi_{n}(x)\leq\varphi_{n}(y)\iff S(n,x,y)\iff T(n,x,y))

Scanning_SQUID_microscope.html

  1. d B = μ 0 4 π I d l × r r 2 . d\vec{B}=\frac{\mu_{0}}{4\pi}\frac{Id\vec{l}\times\vec{r}}{r^{2}}\,.

Scattering_rate.html

  1. H 0 H_{0}
  2. H 1 H_{1}
  3. H H
  4. H = H 0 + H 1 H=H_{0}+H_{1}
  5. H 0 | k = E ( k ) | k H_{0}|k\rangle=E(k)|k\rangle
  6. | k ( t ) I = e i H 0 t / | k ( t ) S = k c k ( t ) | k |k(t)\rangle_{I}=e^{iH_{0}t/\hbar}|k(t)\rangle_{S}=\sum_{k^{\prime}}c_{k^{% \prime}}(t)|k^{\prime}\rangle
  7. i t | k ( t ) I = H 1 I | k ( t ) I i\hbar\frac{\partial}{\partial t}|k(t)\rangle_{I}=H_{1I}|k(t)\rangle_{I}
  8. H H
  9. H 1 I H_{1I}
  10. c k ( t ) = δ k , k - i 0 t d t k | H 1 ( t ) | k e - i ( E k - E k ) t / c_{k^{\prime}}(t)=\delta_{k,k^{\prime}}-\frac{i}{\hbar}\int_{0}^{t}dt^{\prime}% \;\langle k^{\prime}|H_{1}(t^{\prime})|k\rangle\,e^{-i(E_{k}-E_{k^{\prime}})t^% {\prime}/\hbar}
  11. c k ( 0 ) = δ k , k c_{k^{\prime}}^{(0)}=\delta_{k,k^{\prime}}
  12. c k ( 1 ) = - i 0 t d t k | H 1 ( t ) | k e - i ( E k - E k ) t / c_{k^{\prime}}^{(1)}=-\frac{i}{\hbar}\int_{0}^{t}dt^{\prime}\;\langle k^{% \prime}|H_{1}(t^{\prime})|k\rangle\,e^{-i(E_{k}-E_{k^{\prime}})t^{\prime}/\hbar}
  13. | k |k^{\prime}\rangle
  14. | c k ( t ) | 2 |c_{k^{\prime}}(t)|^{2}
  15. c k ( 1 ) c_{k^{\prime}}^{(1)}
  16. c k ( 1 ) = k | H 1 | k E k - E k ( 1 - e i ( E k - E k ) t / ) c_{k^{\prime}}^{(1)}=\frac{\langle\ k^{\prime}|H_{1}|k\rangle}{E_{k^{\prime}}-% E_{k}}(1-e^{i(E_{k^{\prime}}-E_{k})t/\hbar})
  17. | c k ( t ) | 2 = | k | H 1 | k | 2 s i n 2 ( E k - E k 2 t ) ( E k - E k 2 ) 2 1 2 |c_{k^{\prime}}(t)|^{2}=|\langle\ k^{\prime}|H_{1}|k\rangle|^{2}\frac{sin^{2}(% \frac{E_{k^{\prime}}-E_{k}}{2\hbar}t)}{(\frac{E_{k^{\prime}}-E_{k}}{2\hbar})^{% 2}}\frac{1}{\hbar^{2}}
  18. lim α 1 π s i n 2 ( α x ) α x 2 = δ ( x ) \lim_{\alpha\rightarrow\infty}\frac{1}{\pi}\frac{sin^{2}(\alpha x)}{\alpha x^{% 2}}=\delta(x)
  19. k k
  20. k k^{\prime}
  21. P ( k , k ) = 2 π | k | H 1 | k | 2 δ ( E k - E k ) P(k,k^{\prime})=\frac{2\pi}{\hbar}|\langle\ k^{\prime}|H_{1}|k\rangle|^{2}% \delta(E_{k^{\prime}}-E_{k})
  22. E k E_{k}
  23. E k E_{k^{\prime}}
  24. δ \delta
  25. w ( k ) = k P ( k , k ) = 2 π k | k | H 1 | k | 2 δ ( E k - E k ) w(k)=\sum_{k^{\prime}}P(k,k^{\prime})=\frac{2\pi}{\hbar}\sum_{k^{\prime}}|% \langle\ k^{\prime}|H_{1}|k\rangle|^{2}\delta(E_{k^{\prime}}-E_{k})
  26. w ( k ) = 2 π L 3 ( 2 π ) 3 d 3 k | k | H 1 | k | 2 δ ( E k - E k ) w(k)=\frac{2\pi}{\hbar}\frac{L^{3}}{(2\pi)^{3}}\int d^{3}k^{\prime}|\langle\ k% ^{\prime}|H_{1}|k\rangle|^{2}\delta(E_{k^{\prime}}-E_{k})

Scenario_optimization.html

  1. N N
  2. N N
  3. R δ ( x ) R_{\delta}(x)
  4. x x
  5. δ \delta
  6. N N
  7. δ 1 , , δ N \delta_{1},\dots,\delta_{N}
  8. δ i \delta_{i}
  9. max x min i = 1 , , N R δ i ( x ) . ( 1 ) \max_{x}\min_{i=1,\dots,N}R_{\delta_{i}}(x).\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)
  10. x x^{\ast}
  11. R R^{\ast}
  12. R R^{\ast}
  13. N N
  14. ϵ \epsilon
  15. R R^{\ast}
  16. 1 - ϵ 1-\epsilon

Schauder_estimates.html

  1. f C ( Ω ) f\in C(\Omega)
  2. | f | 0 ; Ω = sup x Ω | f ( x ) | |f|_{0;\Omega}=\sup_{x\in\Omega}|f(x)|
  3. α \alpha
  4. f C α ( Ω ) f\in C^{\alpha}(\Omega)
  5. [ f ] 0 , α ; Ω = sup x , y Ω | f ( x ) - f ( y ) | | x - y | α . [f]_{0,\alpha;\Omega}=\sup_{x,y\in\Omega}\frac{|f(x)-f(y)|}{|x-y|^{\alpha}}.
  6. | f | 0 , α ; Ω = | f | 0 ; Ω + [ f ] 0 , α ; Ω = sup x Ω | f ( x ) | + sup x , y Ω | f ( x ) - f ( y ) | | x - y | α . |f|_{0,\alpha;\Omega}=|f|_{0;\Omega}+[f]_{0,\alpha;\Omega}=\sup_{x\in\Omega}|f% (x)|+\sup_{x,y\in\Omega}\frac{|f(x)-f(y)|}{|x-y|^{\alpha}}.
  7. C k ( Ω ) C^{k}(\Omega)
  8. | u | k ; Ω = | β | k sup x Ω | D β u ( x ) | |u|_{k;\Omega}=\sum_{|\beta|\leq k}\sup_{x\in\Omega}|D^{\beta}u(x)|
  9. β \beta
  10. α \alpha
  11. [ u ] k , α ; Ω = sup | β | = k x , y Ω | D β u ( x ) - D β u ( y ) | | x - y | α [u]_{k,\alpha;\Omega}=\sup_{\stackrel{x,y\in\Omega}{|\beta|=k}}\frac{|D^{\beta% }u(x)-D^{\beta}u(y)|}{|x-y|^{\alpha}}
  12. | u | k , α ; Ω = | u | k ; Ω + [ u ] k , α ; Ω = | β | k sup x Ω | D β u ( x ) | + sup | β | = k x , y Ω | D β u ( x ) - D β u ( y ) | | x - y | α . |u|_{k,\alpha;\Omega}=|u|_{k;\Omega}+[u]_{k,\alpha;\Omega}=\sum_{|\beta|\leq k% }\sup_{x\in\Omega}|D^{\beta}u(x)|+\sup_{\stackrel{x,y\in\Omega}{|\beta|=k}}% \frac{|D^{\beta}u(x)-D^{\beta}u(y)|}{|x-y|^{\alpha}}.
  13. d x = d ( x , Ω ) d_{x}=d(x,\partial\Omega)
  14. d x , y = min ( d x , d y ) d_{x,y}=\min(d_{x},d_{y})
  15. | u | k , α ; Ω * = | u | k ; Ω * + [ u ] k , α ; Ω * = | β | k sup x Ω | d x | β | D β u ( x ) | + sup | β | = k x , y Ω d x , y k + α | D β u ( x ) - D β u ( y ) | | x - y | α |u|^{*}_{k,\alpha;\Omega}=|u|^{*}_{k;\Omega}+[u]^{*}_{k,\alpha;\Omega}=\sum_{|% \beta|\leq k}\sup_{x\in\Omega}|d_{x}^{|\beta|}D^{\beta}u(x)|+\sup_{\stackrel{x% ,y\in\Omega}{|\beta|=k}}d_{x,y}^{k+\alpha}\frac{|D^{\beta}u(x)-D^{\beta}u(y)|}% {|x-y|^{\alpha}}
  16. | u | k , α ; Ω ( m ) = | u | k ; Ω ( m ) + [ u ] k , α ; Ω ( m ) = | β | k sup x Ω | d x | β | + m D β u ( x ) | + sup | β | = k x , y Ω d x , y m + k + α | D β u ( x ) - D β u ( y ) | | x - y | α . |u|^{(m)}_{k,\alpha;\Omega}=|u|^{(m)}_{k;\Omega}+[u]^{(m)}_{k,\alpha;\Omega}=% \sum_{|\beta|\leq k}\sup_{x\in\Omega}|d_{x}^{|\beta|+m}D^{\beta}u(x)|+\sup_{% \stackrel{x,y\in\Omega}{|\beta|=k}}d_{x,y}^{m+k+\alpha}\frac{|D^{\beta}u(x)-D^% {\beta}u(y)|}{|x-y|^{\alpha}}.
  17. u C 2 , α ( Ω ) u\in C^{2,\alpha}(\Omega)
  18. Ω \Omega
  19. i , j a i , j ( x ) D i D j u ( x ) + i b i ( x ) D i u ( x ) + c ( x ) u ( x ) = f ( x ) \sum_{i,j}a_{i,j}(x)D_{i}D_{j}u(x)+\sum_{i}b_{i}(x)D_{i}u(x)+c(x)u(x)=f(x)
  20. f C α ( Ω ) f\in C^{\alpha}(\Omega)
  21. λ > 0 \lambda>0
  22. a i , j a_{i,j}
  23. a i , j ( x ) ξ i ξ j λ | ξ | 2 \sum a_{i,j}(x)\xi_{i}\xi_{j}\geq\lambda|\xi|^{2}
  24. x Ω , ξ n x\in\Omega,\xi\in\mathbb{R}^{n}
  25. Λ \Lambda
  26. | a i , j | 0 , α ; Ω , | b i | 0 , α ; Ω ( 1 ) , | c | 0 , α ; Ω ( 2 ) Λ . |a_{i,j}|_{0,\alpha;\Omega},|b_{i}|^{(1)}_{0,\alpha;\Omega},|c|^{(2)}_{0,% \alpha;\Omega}\leq\Lambda.
  27. C 2 , α C^{2,\alpha}
  28. | u | 2 , α ; Ω * C ( n , α , λ , Λ ) ( | u | 0 , Ω + | f | 0 , α ; Ω ( 2 ) ) . |u|^{*}_{2,\alpha;\Omega}\leq C(n,\alpha,\lambda,\Lambda)(|u|_{0,\Omega}+|f|^{% (2)}_{0,\alpha;\Omega}).
  29. Ω \Omega
  30. C 2 , α C^{2,\alpha}
  31. C 2 , α C^{2,\alpha}
  32. ϕ ( x ) \phi(x)
  33. C 2 , α C^{2,\alpha}
  34. | u | 2 , α ; Ω C ( n , α , λ , Λ , Ω ) ( | u | 0 , Ω + | f | 0 , α ; Ω + | ϕ | 2 , α ; Ω ) . |u|_{2,\alpha;\Omega}\leq C(n,\alpha,\lambda,\Lambda,\Omega)(|u|_{0,\Omega}+|f% |_{0,\alpha;\Omega}+|\phi|_{2,\alpha;\partial\Omega}).

Schlieren_imaging.html

  1. v ( x , y ) = 2 π κ λ p ( x , y , z ) d z v(x,y)=\frac{2\pi\kappa}{\lambda}\int{p(x,y,z)}\,dz
  2. κ \kappa
  3. λ \lambda
  4. p ( x , y , z ) p(x,y,z)

Schoenflies_problem.html

  1. C 2 C\subset\mathbb{R}^{2}
  2. f : 2 2 f:\mathbb{R}^{2}\to\mathbb{R}^{2}
  3. f ( C ) f(C)
  4. X = ψ i X i . \displaystyle{X=\sum\psi_{i}\cdot X_{i}.}

Schreier_coset_graph.html

  1. p : K G p:K\rightarrow G

Schur's_lemma_(from_Riemannian_geometry).html

  1. ( M n , g ) (M^{n},g)
  2. n 3 n\geq 3
  3. f : M f:M\rightarrow\mathbb{R}
  4. sect ( Π p ) = f ( p ) \mathrm{sect}(\Pi_{p})=f(p)
  5. Π p T p M \Pi_{p}\subset T_{p}M
  6. p M , p\in M,
  7. f f
  8. M M
  9. f : M f:M\rightarrow\mathbb{R}
  10. Ric ( X p ) = f ( p ) X p \mathrm{Ric}(X_{p})=f(p)X_{p}
  11. X p T p M X_{p}\in T_{p}M
  12. p M , p\in M,
  13. f f
  14. n 3 n\geq 3
  15. Π p T p M \Pi_{p}\subset T_{p}M
  16. T p M T_{p}M

Schur_algebra.html

  1. S k ( n , r ) S_{k}(n,r)
  2. k k
  3. n , r 0 n,r\geq 0
  4. k [ x i j ] k[x_{ij}]
  5. k k
  6. n 2 n^{2}
  7. x i j x_{ij}
  8. n n
  9. A k ( n , r ) A_{k}(n,r)
  10. r r
  11. A k ( n , r ) A_{k}(n,r)
  12. r r
  13. x i j x_{ij}
  14. k [ x i j ] = r 0 A k ( n , r ) . k[x_{ij}]=\bigoplus_{r\geq 0}A_{k}(n,r).
  15. k [ x i j ] k[x_{ij}]
  16. Δ \Delta
  17. ε \varepsilon
  18. Δ ( x i j ) = l x i l x l j , ε ( x i j ) = δ i j \Delta(x_{ij})=\textstyle\sum_{l}x_{il}\otimes x_{lj},\quad\varepsilon(x_{ij})% =\delta_{ij}\quad
  19. k [ x i j ] k[x_{ij}]
  20. A k ( n , r ) A_{k}(n,r)
  21. k [ x i j ] k[x_{ij}]
  22. r r
  23. S k ( n , r ) = Hom k ( A k ( n , r ) , k ) S_{k}(n,r)=\mathrm{Hom}_{k}(A_{k}(n,r),k)
  24. S k ( n , r ) S_{k}(n,r)
  25. A k ( n , r ) A_{k}(n,r)
  26. A A
  27. Δ ( a ) = a i b i \Delta(a)=\textstyle\sum a_{i}\otimes b_{i}
  28. f f
  29. g g
  30. A A
  31. a f ( a i ) g ( b i ) . \textstyle a\mapsto\sum f(a_{i})g(b_{i}).
  32. A A
  33. S k ( n , r ) S_{k}(n,r)
  34. V = k n V=k^{n}
  35. n n
  36. k k
  37. V r = V V ( r factors ) . V^{\otimes r}=V\otimes\cdots\otimes V\quad(r\,\text{ factors}).\,
  38. 𝔖 r \mathfrak{S}_{r}
  39. r r
  40. S k ( n , r ) End 𝔖 r ( V r ) . S_{k}(n,r)\cong\mathrm{End}_{\mathfrak{S}_{r}}(V^{\otimes r}).
  41. S k ( n , r ) S_{k}(n,r)
  42. S k ( n , r ) S_{k}(n,r)
  43. k k
  44. ( n 2 + r - 1 r ) {\textstyle\left({{n^{2}+r-1}\atop{r}}\right)}
  45. S k ( n , r ) S_{k}(n,r)
  46. λ \lambda
  47. λ \lambda
  48. r r
  49. n n
  50. S k ( n , r ) S_{k}(n,r)
  51. GL n ( k ) \mathrm{GL}_{n}(k)
  52. GL n ( k ) \mathrm{GL}_{n}(k)
  53. V = k n V=k^{n}
  54. S k ( n , r ) S ( n , r ) k S_{k}(n,r)\cong S_{\mathbb{Z}}(n,r)\otimes_{\mathbb{Z}}k
  55. k k

Schur–Horn_theorem.html

  1. 𝐝 = { d i } i = 1 N \mathbf{d}=\{d_{i}\}_{i=1}^{N}
  2. λ = { λ i } i = 1 N \mathbf{\lambda}=\{\lambda_{i}\}_{i=1}^{N}
  3. n \mathbb{R}^{n}
  4. { d i } i = 1 N \{d_{i}\}_{i=1}^{N}
  5. { λ i } i = 1 N \{\lambda_{i}\}_{i=1}^{N}
  6. i = 1 n d i i = 1 n λ i n = 1 , 2 , , N \sum_{i=1}^{n}d_{i}\leq\sum_{i=1}^{n}\lambda_{i}\qquad n=1,2,\ldots,N
  7. i = 1 N d i = i = 1 N λ i . \sum_{i=1}^{N}d_{i}=\sum_{i=1}^{N}\lambda_{i}.
  8. x ~ = ( x 1 , x 2 , , x n ) n \tilde{x}=(x_{1},x_{2},\ldots,x_{n})\in\mathbb{R}^{n}
  9. 𝒦 x ~ \mathcal{K}_{\tilde{x}}
  10. { ( x π ( 1 ) , x π ( 2 ) , , x π ( n ) ) n : π S n } \{(x_{\pi(1)},x_{\pi(2)},\ldots,x_{\pi(n)})\in\mathbb{R}^{n}:\pi\in S_{n}\}
  11. S n S_{n}
  12. { 1 , 2 , , n } \{1,2,\ldots,n\}
  13. n \mathbb{R}^{n}
  14. x 1 x 2 x n , y 1 y 2 y n x_{1}\geq x_{2}\geq\cdots\geq x_{n},y_{1}\geq y_{2}\geq\cdots\geq y_{n}
  15. x 1 + x 2 + + x n = y 1 + y 2 + + y n , x_{1}+x_{2}+\cdots+x_{n}=y_{1}+y_{2}+\cdots+y_{n},
  16. ( y 1 , y 2 , , y n ) ( = y ~ ) 𝒦 x ~ (y_{1},y_{2},\cdots,y_{n})(=\tilde{y})\in\mathcal{K}_{\tilde{x}}
  17. y 1 x 1 , y 1 + y 2 x 1 + x 2 , , y 1 + y 2 + + y n - 1 x 1 + x 2 + + x n y_{1}\leq x_{1},y_{1}+y_{2}\leq x_{1}+x_{2},\ldots,y_{1}+y_{2}+\cdots+y_{n-1}% \leq x_{1}+x_{2}+\cdots+x_{n}
  18. ( x 1 ( 1 ) , x 2 ( 1 ) , , x n ( 1 ) ) ( = x ~ 1 ) , , ( x 1 ( n ) , x 2 ( n ) , , x n ( n ) ) ( = x ~ n ) (x_{1}^{(1)},x_{2}^{(1)},\cdots,x_{n}^{(1)})(=\tilde{x}_{1}),\ldots,(x_{1}^{(n% )},x_{2}^{(n)},\ldots,x_{n}^{(n)})(=\tilde{x}_{n})
  19. 𝒦 x ~ \mathcal{K}_{\tilde{x}}
  20. x ~ 1 = x ~ , x ~ n = y ~ , \tilde{x}_{1}=\tilde{x},\tilde{x}_{n}=\tilde{y},
  21. x ~ k + 1 = t x ~ k + ( 1 - t ) τ ( x k ~ ) \tilde{x}_{k+1}=t\tilde{x}_{k}+(1-t)\tau(\tilde{x_{k}})
  22. k k
  23. { 1 , 2 , , n - 1 } \{1,2,\ldots,n-1\}
  24. τ \tau
  25. S n S_{n}
  26. t t
  27. [ 0 , 1 ] [0,1]
  28. k k
  29. 𝐝 = { d i } i = 1 N \mathbf{d}=\{d_{i}\}_{i=1}^{N}
  30. λ = { λ i } i = 1 N \mathbf{\lambda}=\{\lambda_{i}\}_{i=1}^{N}
  31. { d i } i = 1 N \{d_{i}\}_{i=1}^{N}
  32. { λ i } i = 1 N \{\lambda_{i}\}_{i=1}^{N}
  33. ( d 1 , d 2 , , d n ) (d_{1},d_{2},\ldots,d_{n})
  34. ( λ 1 , λ 2 , , λ n ) (\lambda_{1},\lambda_{2},\ldots,\lambda_{n})
  35. 𝐝 \mathbf{d}
  36. λ \mathbf{\lambda}
  37. A ( = a j k ) A(=a_{jk})
  38. n × n n\times n
  39. { λ i } i = 1 n \{\lambda_{i}\}_{i=1}^{n}
  40. A A
  41. a ~ \tilde{a}
  42. n \mathbb{R}^{n}
  43. ( λ 1 , λ 2 , , λ n ) (\lambda_{1},\lambda_{2},\ldots,\lambda_{n})
  44. λ ~ \tilde{\lambda}
  45. Λ \Lambda
  46. λ 1 , λ 2 , , λ n \lambda_{1},\lambda_{2},\ldots,\lambda_{n}
  47. \Rightarrow
  48. A A
  49. U Λ U - 1 U\Lambda U^{-1}
  50. U U
  51. a i i = j = 1 n λ j | u i j | 2 , i = 1 , 2 , , n a_{ii}=\sum_{j=1}^{n}\lambda_{j}|u_{ij}|^{2},\;i=1,2,\ldots,n
  52. S = ( s i j ) S=(s_{ij})
  53. s i j = | u i j | 2 s_{ij}=|u_{ij}|^{2}
  54. U U
  55. S S
  56. a ~ = S λ ~ \tilde{a}=S\tilde{\lambda}
  57. S S
  58. a ~ \tilde{a}
  59. λ ~ \tilde{\lambda}
  60. \Leftarrow
  61. a ~ \tilde{a}
  62. { λ i } i = 1 n \{\lambda_{i}\}_{i=1}^{n}
  63. t a ~ + ( 1 - t ) τ ( a ~ ) t\tilde{a}+(1-t)\tau(\tilde{a})
  64. τ \tau
  65. S n S_{n}
  66. ξ \xi
  67. 1 1
  68. ξ a j k ¯ = - ξ a j k \overline{\xi a_{jk}}=-\xi a_{jk}
  69. U U
  70. ξ t , t \xi\sqrt{t},\sqrt{t}
  71. j , j j,j
  72. k , k k,k
  73. - 1 - t 2 , ξ 1 - t 2 -\sqrt{1-t^{2}},\xi\sqrt{1-t^{2}}
  74. j , k j,k
  75. k , j k,j
  76. 1 1
  77. j , j j,j
  78. k , k k,k
  79. 0
  80. U A U - 1 UAU^{-1}
  81. t a j j + ( 1 - t ) a k k ta_{jj}+(1-t)a_{kk}
  82. j , j j,j
  83. ( 1 - t ) a j j + t a k k (1-t)a_{jj}+ta_{kk}
  84. k , k k,k
  85. a l l a_{ll}
  86. l , l l,l
  87. l j , k l\neq j,k
  88. τ \tau
  89. { 1 , 2 , , n } \{1,2,\ldots,n\}
  90. j j
  91. k k
  92. U A U - 1 UAU^{-1}
  93. t a ~ + ( 1 - t ) τ ( a ~ ) t\tilde{a}+(1-t)\tau(\tilde{a})
  94. Λ \Lambda
  95. { λ i } i = 1 n \{\lambda_{i}\}_{i=1}^{n}
  96. ( λ 1 , λ 2 , , λ n ) (\lambda_{1},\lambda_{2},\ldots,\lambda_{n})
  97. 𝒰 ( n ) \mathcal{U}(n)
  98. n × n n\times n
  99. 𝔲 ( n ) \mathfrak{u}(n)
  100. 𝔲 ( n ) * \mathfrak{u}(n)^{*}
  101. ( n ) \mathcal{H}(n)
  102. Ψ : ( n ) 𝔲 ( n ) * \Psi:\mathcal{H}(n)\rightarrow\mathfrak{u}(n)^{*}
  103. Ψ ( A ) ( B ) = tr ( i A B ) \Psi(A)(B)=\mathrm{tr}(iAB)
  104. A ( n ) , B 𝔲 ( n ) A\in\mathcal{H}(n),B\in\mathfrak{u}(n)
  105. 𝒰 ( n ) \mathcal{U}(n)
  106. ( n ) \mathcal{H}(n)
  107. 𝔲 ( n ) * \mathfrak{u}(n)^{*}
  108. Ψ \Psi
  109. 𝒰 ( n ) \mathcal{U}(n)
  110. U 𝒰 ( n ) U\in\mathcal{U}(n)
  111. λ ~ = ( λ 1 , λ 2 , , λ n ) n \tilde{\lambda}=(\lambda_{1},\lambda_{2},\ldots,\lambda_{n})\in\mathbb{R}^{n}
  112. Λ ( n ) \Lambda\in\mathcal{H}(n)
  113. λ ~ \tilde{\lambda}
  114. 𝒪 λ ~ \mathcal{O}_{\tilde{\lambda}}
  115. Λ \Lambda
  116. 𝒰 ( n ) \mathcal{U}(n)
  117. 𝒰 ( n ) \mathcal{U}(n)
  118. Ψ \Psi
  119. 𝒪 λ ~ \mathcal{O}_{\tilde{\lambda}}
  120. 𝒪 λ ~ \mathcal{O}_{\tilde{\lambda}}
  121. 𝒰 ( n ) \mathcal{U}(n)
  122. 𝕋 \mathbb{T}
  123. 𝒰 ( n ) \mathcal{U}(n)
  124. 1 1
  125. 𝔱 \mathfrak{t}
  126. 𝕋 \mathbb{T}
  127. 𝔱 * \mathfrak{t}^{*}
  128. Ψ \Psi
  129. 𝔱 \mathfrak{t}
  130. 𝔱 * \mathfrak{t}^{*}
  131. 𝔱 𝔲 ( n ) \mathfrak{t}\hookrightarrow\mathfrak{u}(n)
  132. Φ : ( n ) 𝔲 ( n ) * 𝔱 * \Phi:\mathcal{H}(n)\cong\mathfrak{u}(n)^{*}\rightarrow\mathfrak{t}^{*}
  133. A A
  134. A A
  135. 𝒪 λ ~ \mathcal{O}_{\tilde{\lambda}}
  136. 𝕋 \mathbb{T}
  137. Φ \Phi
  138. Φ ( 𝒪 λ ~ ) \Phi(\mathcal{O}_{\tilde{\lambda}})
  139. A ( n ) A\in\mathcal{H}(n)
  140. 𝕋 \mathbb{T}
  141. A A
  142. 𝒪 λ ~ \mathcal{O}_{\tilde{\lambda}}
  143. λ 1 , λ 2 , , λ n \lambda_{1},\lambda_{2},\ldots,\lambda_{n}
  144. Φ ( 𝒪 λ ~ ) \Phi(\mathcal{O}_{\tilde{\lambda}})

Science_and_technology_in_Iran.html

  1. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 \begin{matrix}&&&&&1\\ &&&&1&&1\\ &&&1&&2&&1\\ &&1&&3&&3&&1\\ &1&&4&&6&&4&&1\end{matrix}

Science_and_technology_in_Russia.html

  1. E = k m c 2 E=kmc^{2}

Searle's_bar_method.html

  1. ( Δ Q Δ t ) bar = - k A Δ T bar L \left(\frac{\Delta Q}{\Delta t}\right)_{\mathrm{bar}}=-kA\frac{\Delta T_{% \mathrm{bar}}}{L}
  2. ( Δ Q Δ t ) water = C w Δ m Δ t Δ T water \left(\frac{\Delta Q}{\Delta t}\right)_{\mathrm{water}}=C_{\mathrm{w}}\frac{% \Delta m}{\Delta t}\Delta T_{\mathrm{water}}
  3. ( Δ Q Δ t ) bar = ( Δ Q Δ t ) water \left(\frac{\Delta Q}{\Delta t}\right)_{\mathrm{bar}}=\left(\frac{\Delta Q}{% \Delta t}\right)_{\mathrm{water}}
  4. k = - C w L A Δ m Δ t Δ T water Δ T bar k=-C_{\mathrm{w}}\frac{L}{A}\frac{\Delta m}{\Delta t}\frac{\Delta T_{\mathrm{% water}}}{\Delta T_{\mathrm{bar}}}

Secondary_plot_(kinetics).html

  1. 1 v 0 \displaystyle\frac{1}{v_{0}}
  2. y-intercept = K M B v max [ B ] + 1 v max \displaystyle\mbox{y-intercept}~{}=\frac{K_{M}^{B}}{v_{\max}{[}B{]}}+\frac{1}{% v_{\max}}
  3. K M B K_{M}^{B}
  4. K M B K_{M}^{B}
  5. v max v_{\max}
  6. v max v_{\max}
  7. apparent K m = K m × ( 1 + [ I ] K I ) \displaystyle\mbox{apparent }~{}K_{m}=K_{m}\times\left(1+\frac{[I]}{K_{I}}\right)
  8. slope = K m v max × ( 1 + [ I ] K I ) \displaystyle\mbox{slope}~{}=\frac{K_{m}}{v_{\max}}\times\left(1+\frac{[I]}{K_% {I}}\right)

Secondary_vector_bundle_structure.html

  1. ( E , p , M ) (E,p,M)
  2. p : E M p:E→M
  3. ( T E , E , T M , M ) (TE,E,TM,M)
  4. T E = T T M TE=TTM
  5. T M TM
  6. ( E , p , M ) (E,p,M)
  7. N N
  8. X X
  9. T M TM
  10. p : E M p:E→M
  11. 2 N 2N
  12. + * : T ( E × E ) T E , λ * : T E T E +_{*}:T(E\times E)\to TE,\qquad\lambda_{*}:TE\to TE
  13. + : E × E E , λ : E E +:E\times E\to E,\qquad\lambda:E\to E
  14. ( U , φ ) (U,φ)
  15. M M
  16. { ψ : W φ ( U ) × 𝐑 N ψ ( v k e k | x ) := ( x 1 , , x n , v 1 , , v N ) \begin{cases}\psi:W\to\varphi(U)\times\mathbf{R}^{N}\\ \psi\left(v^{k}e_{k}|_{x}\right):=\left(x^{1},\ldots,x^{n},v^{1},\ldots,v^{N}% \right)\end{cases}
  17. E E
  18. p * ( X k x k | v + Y v | v ) = X k x k | p ( v ) , p_{*}\left(X^{k}\frac{\partial}{\partial x^{k}}\Bigg|_{v}+Y^{\ell}\frac{% \partial}{\partial v^{\ell}}\Bigg|_{v}\right)=X^{k}\frac{\partial}{\partial x^% {k}}\Bigg|_{p(v)},
  19. X X
  20. p * - 1 ( X ) = { X k x k | v + Y v | v : v E x ; Y 1 , , Y N 𝐑 } . p^{-1}_{*}(X)=\left\{X^{k}\frac{\partial}{\partial x^{k}}\Bigg|_{v}+Y^{\ell}% \frac{\partial}{\partial v^{\ell}}\Bigg|_{v}\ :\ v\in E_{x};Y^{1},\ldots,Y^{N}% \in\mathbf{R}\right\}.
  21. χ ( X k x k | v + Y v | v ) = ( X k x k | p ( v ) , ( v 1 , , v N , Y 1 , , Y N ) ) \chi\left(X^{k}\frac{\partial}{\partial x^{k}}\Bigg|_{v}+Y^{\ell}\frac{% \partial}{\partial v^{\ell}}\Bigg|_{v}\right)=\left(X^{k}\frac{\partial}{% \partial x^{k}}\Bigg|_{p(v)},\left(v^{1},\ldots,v^{N},Y^{1},\ldots,Y^{N}\right% )\right)
  22. ( X k x k | v + Y v | v ) + * ( X k x k | w + Z v | w ) = X k x k | v + w + ( Y + Z ) v | v + w \left(X^{k}\frac{\partial}{\partial x^{k}}\Bigg|_{v}+Y^{\ell}\frac{\partial}{% \partial v^{\ell}}\Bigg|_{v}\right)+_{*}\left(X^{k}\frac{\partial}{\partial x^% {k}}\Bigg|_{w}+Z^{\ell}\frac{\partial}{\partial v^{\ell}}\Bigg|_{w}\right)=X^{% k}\frac{\partial}{\partial x^{k}}\Bigg|_{v+w}+(Y^{\ell}+Z^{\ell})\frac{% \partial}{\partial v^{\ell}}\Bigg|_{v+w}
  23. λ * ( X k x k | v + Y v | v ) = X k x k | λ v + λ Y v | λ v , \lambda_{*}\left(X^{k}\frac{\partial}{\partial x^{k}}\Bigg|_{v}+Y^{\ell}\frac{% \partial}{\partial v^{\ell}}\Bigg|_{v}\right)=X^{k}\frac{\partial}{\partial x^% {k}}\Bigg|_{\lambda v}+\lambda Y^{\ell}\frac{\partial}{\partial v^{\ell}}\Bigg% |_{\lambda v},
  24. T E = H E V E TE=HE⊕VE
  25. ( E , p , M ) (E,p,M)
  26. { κ : T v E E p ( v ) κ ( X ) := vl v - 1 ( vpr X ) \begin{cases}\kappa:T_{v}E\to E_{p(v)}\\ \kappa(X):=\operatorname{vl}_{v}^{-1}(\operatorname{vpr}X)\end{cases}
  27. { : T M × Γ ( E ) Γ ( E ) X v := κ ( v * X ) \begin{cases}\nabla:TM\times\Gamma(E)\to\Gamma(E)\\ \nabla_{X}v:=\kappa(v_{*}X)\end{cases}
  28. Γ ( E ) Γ(E)
  29. X + Y v = X v + Y v λ X v = λ X v X ( v + w ) = X v + X w X ( λ v ) = λ X v X ( f v ) = X [ f ] v + f X v \begin{aligned}\displaystyle\nabla_{X+Y}v&\displaystyle=\nabla_{X}v+\nabla_{Y}% v\\ \displaystyle\nabla_{\lambda X}v&\displaystyle=\lambda\nabla_{X}v\\ \displaystyle\nabla_{X}(v+w)&\displaystyle=\nabla_{X}v+\nabla_{X}w\\ \displaystyle\nabla_{X}(\lambda v)&\displaystyle=\lambda\nabla_{X}v\\ \displaystyle\nabla_{X}(fv)&\displaystyle=X[f]v+f\nabla_{X}v\end{aligned}
  30. ( T E , p < s u b > , T M ) (TE,p<sub>∗,TM)

Security_of_cryptographic_hash_functions.html

  1. n n
  2. n n
  3. F 2 p + 1 F_{2p+1}
  4. 2 n / 2 2^{n/2}
  5. 2 3 n / 2 2^{3n/2}
  6. 2 3 n 2^{3n}
  7. 2 n 2n

Segre_class.html

  1. E E
  2. M M
  3. s ( E ) s(E)
  4. c ( E ) c(E)
  5. c ( E ) = 1 + c 1 ( E ) + c 2 ( E ) + c(E)=1+c_{1}(E)+c_{2}(E)+\cdots\,
  6. s ( E ) = 1 + s 1 ( E ) + s 2 ( E ) + s(E)=1+s_{1}(E)+s_{2}(E)+\cdots\,
  7. c 1 ( E ) = - s 1 ( E ) , c 2 ( E ) = s 1 ( E ) 2 - s 2 ( E ) , , c n ( E ) = - s 1 ( E ) c n - 1 ( E ) - s 2 ( E ) c n - 2 ( E ) - - s n ( E ) c_{1}(E)=-s_{1}(E),\quad c_{2}(E)=s_{1}(E)^{2}-s_{2}(E),\quad\dots,\quad c_{n}% (E)=-s_{1}(E)c_{n-1}(E)-s_{2}(E)c_{n-2}(E)-\cdots-s_{n}(E)
  8. x 1 , , x k x_{1},\dots,x_{k}
  9. i Ω 2 π \frac{i\Omega}{2\pi}
  10. Ω \Omega
  11. E E
  12. c ( E ) = i = 1 k ( 1 + x i ) = c 0 + c 1 + + c k c(E)=\prod_{i=1}^{k}(1+x_{i})=c_{0}+c_{1}+\cdots+c_{k}\,
  13. c i c_{i}
  14. i i
  15. x 1 , , x k x_{1},\dots,x_{k}
  16. E E^{\vee}
  17. - x 1 , , - x k -x_{1},\dots,-x_{k}
  18. s ( E ) = i = 1 k 1 1 - x i = s 0 + s 1 + s(E)=\prod_{i=1}^{k}\frac{1}{1-x_{i}}=s_{0}+s_{1}+\cdots
  19. x 1 , x k x_{1},\dots x_{k}
  20. s i ( E ) s_{i}(E^{\vee})
  21. x 1 , x k x_{1},\dots x_{k}

Segre_cubic.html

  1. x 0 + x 1 + x 2 + x 3 + x 4 + x 5 = 0 \displaystyle x_{0}+x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=0
  2. x 0 3 + x 1 3 + x 2 3 + x 3 3 + x 4 3 + x 5 3 = 0. \displaystyle x_{0}^{3}+x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3}+x_{5}^{3}=0.

Segre_surface.html

  1. x 1 x 2 x 3 + x 2 x 3 x 4 + x 3 x 4 x 5 + x 4 x 5 x 1 + x 5 x 1 x 2 = 0. x_{1}x_{2}x_{3}+x_{2}x_{3}x_{4}+x_{3}x_{4}x_{5}+x_{4}x_{5}x_{1}+x_{5}x_{1}x_{2% }=0.\,

Seismic_noise.html

  1. f 0 = V s 4 H f_{0}=\frac{V_{s}}{4H}

Selected_reaction_monitoring.html

  1. A B C D + A B + C D + ABCD^{+}\to AB+CD^{+}
  2. A B C D + A B + C D + C + D + ABCD^{+}\to AB+CD^{+}\to C+D^{+}
  3. A B C D + A B + C D + ABCD^{+}\to AB+CD^{+}
  4. A B C D + A B + + C D ABCD^{+}\to AB^{+}+CD

Self-financing_portfolio.html

  1. h i ( t ) h_{i}(t)
  2. t t
  3. S i ( t ) S_{i}(t)
  4. V ( t ) = i = 1 n h i ( t ) S i ( t ) . V(t)=\sum_{i=1}^{n}h_{i}(t)S_{i}(t).
  5. ( h 1 ( t ) , , h n ( t ) ) (h_{1}(t),\dots,h_{n}(t))
  6. d V ( t ) = i = 1 n h i ( t ) d S i ( t ) . dV(t)=\sum_{i=1}^{n}h_{i}(t)dS_{i}(t).
  7. ( Ω , , { t } t = 0 T , P ) (\Omega,\mathcal{F},\{\mathcal{F}_{t}\}_{t=0}^{T},P)
  8. K t K_{t}
  9. L d p ( K t ) = { X L d p ( T ) : X K t P - a . s . } L_{d}^{p}(K_{t})=\{X\in L_{d}^{p}(\mathcal{F}_{T}):X\in K_{t}\;P-a.s.\}
  10. ( H t ) t = 0 T (H_{t})_{t=0}^{T}
  11. t { 0 , 1 , , T } t\in\{0,1,\dots,T\}
  12. H t - H t - 1 - K t P - a . s . H_{t}-H_{t-1}\in-K_{t}\;P-a.s.
  13. H - 1 = 0 H_{-1}=0
  14. H τ - K 0 - k = 1 τ L d p ( K k ) H_{\tau}\in-K_{0}-\sum_{k=1}^{\tau}L_{d}^{p}(K_{k})
  15. Δ t 0 \Delta t\to 0

Self-similarity_matrix.html

  1. V = ( v 1 , v 2 , , v n ) V=(v_{1},v_{2},\ldots,v_{n})
  2. v i v_{i}
  3. S ( j , k ) = s ( v j , v k ) j , k ( 1 , , n ) S(j,k)=s(v_{j},v_{k})\quad j,k\in(1,\ldots,n)
  4. s ( v j , v k ) s(v_{j},v_{k})
  5. s ( v j , v k ) = v j v k s(v_{j},v_{k})=v_{j}\cdot v_{k}

Self-Similarity_of_Network_Data_Analysis.html

  1. X X
  2. μ \mu
  3. σ 2 \sigma^{2}
  4. γ ( t ) \gamma(t)
  5. γ ( t ) \gamma(t)
  6. γ ( t ) t - β L ( t ) \gamma(t)\rightarrow t^{-\beta}L(t)
  7. t t\to\infty
  8. 0 < β < 1 0<\beta<1
  9. L ( t ) L(t)
  10. lim t L ( t x ) L ( t ) = 1 \lim_{t\to\infty}\frac{L(tx)}{L(t)}=1
  11. x > 0 x>0
  12. L ( t ) = c o n s t L(t)=const
  13. L ( t ) = log ( t ) L(t)=\log(t)
  14. X k ( m ) = 1 m H ( X k m - m + 1 + + X k m ) X_{k}^{(m)}=\frac{1}{m^{H}}(X_{km-m+1}+\cdot\cdot\cdot+X_{km})
  15. k = 1 , 2 , 3 , k=1,2,3,\ldots
  16. m m
  17. m m
  18. X X
  19. H H
  20. X k ( m ) X_{k}^{(m)}
  21. X X
  22. H H
  23. 1 2 < H < 1 \frac{1}{2}<H<1
  24. X ( t ) = B H ( t + 1 ) - B H ( t ) , t 1 X(t)=B_{H}(t+1)-B_{H}(t),~{}\forall t\geq 1
  25. B H ( ) B_{H}(\cdot)
  26. X X
  27. H H
  28. X k ( m ) X_{k}^{(m)}
  29. X X
  30. X X
  31. H H
  32. γ ( m ) ( t ) 1 2 [ ( t + 1 ) 2 H - 2 t 2 H + ( t - 1 ) 2 H ] \gamma^{(m)}(t)\to\frac{1}{2}[(t+1)^{2H}-2t^{2H}+(t-1)^{2H}]
  33. m m\to\infty
  34. t = 1 , 2 , 3 , ~{}\forall t=1,2,3,\ldots
  35. X ( t ) X(t)
  36. μ \mu
  37. σ 2 \sigma^{2}
  38. t t
  39. γ ( t ) = cov ( X ( h ) , X ( h + t ) ) σ 2 = E [ ( X ( h ) - μ ) ( X ( h + t ) - μ ) ] σ 2 \gamma(t)={\mathrm{cov}(X(h),X(h+t))\over\sigma^{2}}={E[(X(h)-\mu)(X(h+t)-\mu)% ]\over\sigma^{2}}
  40. t = 0 | γ ( t ) | = \sum_{t=0}^{\infty}|\gamma(t)|=\infty
  41. γ ( t ) t - β L ( t ) \gamma(t)\rightarrow t^{-\beta}L(t)
  42. t t\to\infty
  43. γ ( t ) t - β L ( t ) \gamma(t)\rightarrow t^{-\beta}L(t)
  44. X X
  45. f t ( w ) = t = 0 γ ( t ) e i w t f_{t}(w)=\sum_{t=0}^{\infty}\gamma(t)e^{iwt}
  46. w - γ L ( w ) w^{-\gamma}L(w)
  47. w 0 w\to 0
  48. 0 < γ < 1 0<\gamma<1
  49. L L
  50. X ( m ) = 1 m ( X 1 + + X m ) X^{(m)}=\frac{1}{m}(X_{1}+\cdot\cdot\cdot+X_{m})
  51. γ ( t ) t - β L ( t ) \gamma(t)\rightarrow t^{-\beta}L(t)
  52. t t\to\infty
  53. V a r ( X ( m ) ) a m - β Var(X^{(m)})\to am^{-\beta}
  54. m m\to\infty
  55. a a
  56. X ( 1 ) , , X ( n ) X_{(1)},\ldots,X_{(n)}
  57. Y ( n ) = i = 1 n X ( i ) Y_{(n)}=\sum_{i=1}^{n}X_{(i)}
  58. X ( i ) X_{(i)}
  59. S 2 ( n ) = 1 n i = 1 n X ( i ) 2 - ( 1 n ) 2 Y n 2 S^{2}(n)=\frac{1}{n}\sum_{i=1}^{n}X^{2}_{(i)}-(\frac{1}{n})^{2}Y^{2}_{n}
  60. R S ( n ) = 1 S ( n ) [ max 0 t n ( Y t - t n Y n ) - min 0 t n ( Y t - t n Y n ) ] \frac{R}{S}(n)=\frac{1}{S(n)}[\max_{0\leq t\leq n}(Y_{t}-\frac{t}{n}Y_{n})-% \min_{0\leq t\leq n}(Y_{t}-\frac{t}{n}Y_{n})]
  61. X ( i ) X_{(i)}
  62. E ( R S ( n ) ) C H × n H E(\frac{R}{S}(n))\to C_{H}\times n^{H}
  63. l o g R S ( n ) = l o g ( C H ) + H l o g ( n ) + ϵ n log\frac{R}{S}(n)=log(C_{H})+Hlog(n)+\epsilon_{n}
  64. ϵ n N ( 0 , σ 2 ) \epsilon_{n}\thicksim N(0,\sigma^{2})
  65. N N
  66. k k
  67. N k \frac{N}{k}
  68. R S ( n ) \frac{R}{S}(n)
  69. k k
  70. l o g ( n ) , l o g R S ( n ) log(n),log\frac{R}{S}(n)
  71. k k
  72. n n
  73. H H
  74. V a r ( X ¯ n ) c n 2 H - 2 , c > 0 Var(\bar{X}_{n})\to cn^{2H-2},~{}\forall c>0
  75. X ¯ 1 , X ¯ 2 , , X ¯ m k \bar{X}_{1},\bar{X}_{2},\cdots,\bar{X}_{m_{k}}
  76. m k m_{k}
  77. k k
  78. X ¯ ( k ) = 1 m k i = 1 m k X ¯ i ( k ) \bar{X}(k)=\frac{1}{m_{k}}\sum_{i=1}^{m_{k}}\bar{X}_{i}(k)
  79. S 2 ( k ) = 1 m k - 1 i = 1 m k ( X ¯ i ( k ) - X ¯ ( k ) ) 2 S^{2}(k)=\frac{1}{m_{k}-1}\sum_{i=1}^{m_{k}}(\bar{X}_{i}(k)-\bar{X}(k))^{2}
  80. log S 2 ( k ) \log S^{2}(k)
  81. log k \log k
  82. k k
  83. 2 H - 2 2H-2
  84. H ^ = 1 + 1 2 ( s l o p e ) . \hat{H}=1+\frac{1}{2}(slope).
  85. X X
  86. X X
  87. X X
  88. f x ( w ; θ ) = σ ϵ 2 f x ( w ; ( 1 , η ) ) f_{x}(w;\theta)=\sigma_{\epsilon}^{2}f_{x}(w;(1,\eta))
  89. θ = ( σ ϵ 2 , η ) = ( σ ϵ 2 , H , θ 3 , , θ k ) , H = γ + 1 2 \theta=(\sigma_{\epsilon}^{2},\eta)=(\sigma_{\epsilon}^{2},H,\theta_{3},\ldots% ,\theta_{k}),H=\frac{\gamma+1}{2}
  90. θ 3 , , θ k \theta_{3},\ldots,\theta_{k}
  91. X j = i = 1 k α i X j - i + ϵ j X_{j}=\sum_{i=1}^{k}\alpha_{i}X_{j-i}+\epsilon_{j}
  92. V a r ( ϵ j ) = σ ϵ 2 Var(\epsilon_{j})=\sigma_{\epsilon}^{2}
  93. η ^ \hat{\eta}
  94. η \eta
  95. Q ( η ) = - π π I ( w ) f ( w ; ( 1 , η ) ) d w Q(\eta)=\int_{-\pi}^{\pi}\frac{I(w)}{f(w;(1,\eta))}\,dw
  96. ( 2 π n ) - 1 | j = 1 n X j e i w j | 2 (2\pi n)^{-1}|\sum_{j=1}^{n}X_{j}e^{iwj}|^{2}
  97. σ ^ 2 = - π π I ( w ) f ( w ; ( 1 , η ^ ) ) d w \hat{\sigma}^{2}=\int_{-\pi}^{\pi}\frac{I(w)}{f(w;(1,\hat{\eta}))}\,dw
  98. n 1 / 2 ( θ ^ - θ ) n^{1/2}(\hat{\theta}-\theta)
  99. X j X_{j}
  100. H H
  101. I n ( w ) I_{n}(w)
  102. | λ | 1 - 2 H |\lambda|^{1-2H}
  103. log ( I n ( w ) ) \log(I_{n}(w))
  104. log ( w ) \log(w)
  105. log ( I n ( w ) ) \log(I_{n}(w))
  106. log ( w ) \log(w)
  107. β ^ \hat{\beta}
  108. 1 - 2 H 1-2H
  109. H ^ \hat{H}
  110. X X
  111. X X
  112. w - γ L ( w ) w^{-\gamma}L(w)
  113. w w\to\infty
  114. m - H L - 1 2 ( m ) i = ( j - 1 ) m + 1 m k ( X i - E ( | X i | ) ) , j = 1 , 2 , , [ n m ] m^{-H}L^{-\frac{1}{2}}(m)\sum_{i=(j-1)m+1}^{m}k(X_{i}-E(|X_{i}|)),~{}j=1,2,% \ldots,[\tfrac{n}{m}]
  115. m m\to\infty

Selfridge–Conway_discrete_procedure.html

  1. 2 n - 2 + 1 2^{n-2}+1
  2. 1 / 2 n - 1 1/2^{n-1}

Semi-Lagrangian_scheme.html

  1. F F
  2. D F D t = F t + ( 𝐯 ) F , \frac{DF}{Dt}=\frac{\partial F}{\partial t}+(\mathbf{v}\cdot\vec{\nabla})F,
  3. F F
  4. 𝐯 \mathbf{v}
  5. F F
  6. D 𝐕 D t = 𝐒 ( 𝐕 ) , \frac{D\mathbf{V}}{Dt}=\mathbf{S}(\mathbf{V}),
  7. 𝐕 \mathbf{V}
  8. 𝐒 ( 𝐕 ) \mathbf{S}(\mathbf{V})

Semi-major_axis.html

  1. b = a 1 - e 2 , b=a\sqrt{1-e^{2}},\,
  2. = a ( 1 - e 2 ) , \ell=a(1-e^{2}),\,
  3. a = b 2 . a\ell=b^{2}.\,
  4. a a\,\!
  5. b b\,\!
  6. r ( 1 + e cos θ ) = . r(1+e\cos\theta)=\ell.\,
  7. r = 1 - e r={\ell\over{1-e}}\,\!
  8. r = 1 + e r={\ell\over{1+e}}\,\!
  9. ( for θ = π and θ = 0 ) (\,\text{for }\theta=\pi\,\,\text{ and }\theta=0)
  10. a = 1 - e 2 . a={\ell\over 1-e^{2}}.\,
  11. ( x - h ) 2 a 2 - ( y - k ) 2 b 2 = 1. \frac{\left(x-h\right)^{2}}{a^{2}}-\frac{\left(y-k\right)^{2}}{b^{2}}=1.
  12. a = 1 - e 2 . a={\ell\over 1-e^{2}}.
  13. T = 2 π a 3 μ T=2\pi\sqrt{a^{3}\over\mu}
  14. μ \mu
  15. H = a μ ( 1 - e 2 ) H=\sqrt{a\cdot\mu\cdot(1-e^{2})}
  16. μ \mu
  17. T 2 a 3 T^{2}\propto a^{3}\,
  18. T 2 = 4 π 2 G ( M + m ) a 3 T^{2}=\frac{4\pi^{2}}{G(M+m)}a^{3}\,
  19. b = a 1 - e 2 b=a\sqrt{1-e^{2}}\,\!
  20. a ( 1 + e 2 2 ) . a\left(1+\frac{e^{2}}{2}\right).\,
  21. a = - μ 2 ε a=-{\mu\over{2\varepsilon}}\,
  22. a = μ 2 ε a={\mu\over{2\varepsilon}}\,
  23. ε = v 2 2 - μ | 𝐫 | \varepsilon={v^{2}\over{2}}-{\mu\over\left|\mathbf{r}\right|}
  24. μ = G ( M + m ) \mu=G(M+m)\,
  25. 𝐫 \mathbf{r}\,
  26. ε \varepsilon

Semicomputable_function.html

  1. f : f:\mathbb{Q}\rightarrow\mathbb{R}
  2. f : f:\mathbb{Q}\rightarrow\mathbb{R}
  3. ϕ ( x , k ) : × \phi(x,k):\mathbb{Q}\times\mathbb{N}\rightarrow\mathbb{Q}
  4. x x
  5. f ( x ) f(x)
  6. k k
  7. lim k ϕ ( x , k ) = f ( x ) \lim_{k\rightarrow\infty}\phi(x,k)=f(x)
  8. k : ϕ ( x , k + 1 ) ϕ ( x , k ) \forall k\in\mathbb{N}:\phi(x,k+1)\leq\phi(x,k)
  9. f : f:\mathbb{Q}\rightarrow\mathbb{R}
  10. - f ( x ) -f(x)
  11. ϕ ( x , k ) \phi(x,k)
  12. lim k ϕ ( x , k ) = f ( x ) \lim_{k\rightarrow\infty}\phi(x,k)=f(x)
  13. k : ϕ ( x , k + 1 ) ϕ ( x , k ) \forall k\in\mathbb{N}:\phi(x,k+1)\geq\phi(x,k)

Semigroup_with_involution.html

  1. A ( A * A B B * ) B A(A^{*}A\wedge BB^{*})B
  2. Y = X X Y=X\sqcup X^{\dagger}
  3. \sqcup\,
  4. θ : X X \theta:X\rightarrow X^{\dagger}
  5. θ \theta
  6. : Y Y {}\dagger:Y\to Y
  7. θ \theta
  8. y = { θ ( y ) if y X θ - 1 ( y ) if y X y^{\dagger}=\begin{cases}\theta(y)&\,\text{if }y\in X\\ \theta^{-1}(y)&\,\text{if }y\in X^{\dagger}\end{cases}
  9. Y + Y^{+}\,
  10. Y Y\,
  11. Y + Y^{+}\,
  12. w = w 1 w 2 w k Y + w=w_{1}w_{2}\cdots w_{k}\in Y^{+}
  13. w i Y . w_{i}\in Y.
  14. \dagger
  15. Y Y
  16. : Y + Y + {}^{\dagger}:Y^{+}\rightarrow Y^{+}
  17. Y + Y^{+}\,
  18. w = w k w k - 1 w 2 w 1 . w^{\dagger}=w_{k}^{\dagger}w_{k-1}^{\dagger}\cdots w_{2}^{\dagger}w_{1}^{% \dagger}.
  19. Y + Y^{+}\,
  20. ( X X ) + (X\sqcup X^{\dagger})^{+}
  21. {}^{\dagger}\,
  22. X X^{\dagger}
  23. θ \theta
  24. Y + Y^{+}\,
  25. Y * = Y + { ε } Y^{*}=Y^{+}\cup\{\varepsilon\}
  26. ε \varepsilon\,
  27. Y * Y^{*}\,
  28. ε = ε \varepsilon^{\dagger}=\varepsilon
  29. θ \theta\,
  30. X X\,
  31. X X^{\dagger}\,
  32. Y + Y^{+}\,
  33. Y * Y^{*}\,
  34. S S\,
  35. Φ : X S \Phi:X\rightarrow S
  36. Φ ¯ : ( X X ) + S \overline{\Phi}:(X\sqcup X^{\dagger})^{+}\rightarrow S
  37. Φ = ι Φ ¯ \Phi=\iota\circ\overline{\Phi}
  38. ι : X ( X X ) + \iota:X\rightarrow(X\sqcup X^{\dagger})^{+}
  39. ( X X ) + (X\sqcup X^{\dagger})^{+}
  40. ( X X ) * (X\sqcup X^{\dagger})^{*}
  41. x x xx^{\dagger}
  42. x x x^{\dagger}x
  43. { ( y y , ε ) : y Y } \{(yy^{\dagger},\varepsilon):y\in Y\}
  44. ( ) = ) ( = ε ()=)(=\varepsilon
  45. { ( x x , ε ) : x X } \{(xx^{\dagger},\varepsilon):x\in X\}
  46. ( ) = ε ()=\varepsilon

Semigroup_with_two_elements.html

  1. \wedge
  2. \wedge
  3. \wedge
  4. \wedge
  5. \wedge
  6. \wedge
  7. S = { ( 1 0 0 1 ) , ( 1 0 0 0 ) } S=\left\{\begin{pmatrix}1&0\\ 0&1\end{pmatrix},\begin{pmatrix}1&0\\ 0&0\end{pmatrix}\right\}

Semilinear_transformation.html

  1. T : V W T\colon V\to W
  2. T ( v + v ) = T ( v ) + T ( v ) T(v+v^{\prime})=T(v)+T(v^{\prime})
  3. T ( λ v ) = λ θ T ( v ) , T(\lambda v)=\lambda^{\theta}T(v),
  4. λ θ \lambda^{\theta}
  5. λ \lambda
  6. Γ L ( V ) , \operatorname{\Gamma L}(V),
  7. q = p i , q=p^{i},
  8. 𝐙 / p 𝐙 . \mathbf{Z}/p\mathbf{Z}.
  9. θ \theta
  10. f : V W f\colon V\to W
  11. θ \theta
  12. x , y x,y
  13. l l
  14. f ( x + y ) = f ( x ) + f ( y ) , f(x+y)=f(x)+f(y),
  15. f ( l x ) = l θ f ( x ) , f(lx)=l^{\theta}f(x),
  16. l θ l^{\theta}
  17. l l
  18. θ . \theta.
  19. θ \theta
  20. θ \theta
  21. n θ f ( x ) = f ( n x ) = f ( x + + x ) = n f ( x ) n^{\theta}f(x)=f(nx)=f(x+\dots+x)=nf(x)
  22. ( l 1 + l 2 ) θ f ( x ) = f ( ( l 1 + l 2 ) x ) = f ( l 1 x ) + f ( l 2 x ) = ( l 1 θ + l 2 θ ) f ( x ) (l_{1}+l_{2})^{\theta}f(x)=f((l_{1}+l_{2})x)=f(l_{1}x)+f(l_{2}x)=(l_{1}^{% \theta}+l_{2}^{\theta})f(x)
  23. ( l 1 + l 2 ) θ = l 1 θ + l 2 θ . (l_{1}+l_{2})^{\theta}=l_{1}^{\theta}+l_{2}^{\theta}.
  24. ( l 1 l 2 ) θ f ( x ) = f ( l 1 l 2 x ) = l 1 θ f ( l 2 x ) = l 1 θ l 2 θ f ( x ) (l_{1}l_{2})^{\theta}f(x)=f(l_{1}l_{2}x)=l_{1}^{\theta}f(l_{2}x)=l_{1}^{\theta% }l_{2}^{\theta}f(x)
  25. θ \theta
  26. K = 𝐂 , V = 𝐂 n , K=\mathbf{C},V=\mathbf{C}^{n},
  27. e 1 , , e n . e_{1},\ldots,e_{n}.
  28. f : V V f\colon V\to V
  29. f ( i = 1 n z i e i ) = i = 1 n z ¯ i e i f\left(\sum_{i=1}^{n}z_{i}e_{i}\right)=\sum_{i=1}^{n}\bar{z}_{i}e_{i}
  30. K = G F ( q ) K=GF(q)
  31. q = p i , q=p^{i},
  32. l θ = l p . l^{\theta}=l^{p}.
  33. f : V W f\colon V\to W
  34. θ \theta
  35. f ~ ( i = 1 n l i e i ) := f ( i = 1 n l i θ e i ) \widetilde{f}\left(\sum_{i=1}^{n}l_{i}e_{i}\right):=f\left(\sum_{i=1}^{n}l_{i}% ^{\theta}e_{i}\right)
  36. Γ L ( V ) . \operatorname{\Gamma L}(V).
  37. Γ L ( V ) \operatorname{\Gamma L}(V)
  38. Γ L ( V ) = GL ( V ) Gal ( K / k ) \operatorname{\Gamma L}(V)=\operatorname{GL}(V)\rtimes\operatorname{Gal}(K/k)
  39. K / k . K/k.
  40. Γ L ( V ) \operatorname{\Gamma L}(V)
  41. b B l b b b B l b σ b \sum_{b\in B}l_{b}b\mapsto\sum_{b\in B}l_{b}^{\sigma}b
  42. σ Gal ( K / k ) . \sigma\in\operatorname{Gal}(K/k).
  43. Γ L ( V ) \operatorname{\Gamma L}(V)
  44. GL ( V ) Γ L ( V ) . \operatorname{GL}(V)\leq\operatorname{\Gamma L}(V).
  45. σ Gal ( K / k ) , \sigma\in\operatorname{Gal}(K/k),
  46. g : V V g\colon V\to V
  47. g ( b B l b b ) := b B f ( l b σ - 1 b ) = b B l b f ( b ) g\left(\sum_{b\in B}l_{b}b\right):=\sum_{b\in B}f\left(l_{b}^{\sigma^{-1}}b% \right)=\sum_{b\in B}l_{b}f(b)
  48. g GL ( V ) . g\in\operatorname{GL}(V).
  49. h := f g - 1 . h:=fg^{-1}.
  50. v = b B l b b v=\sum_{b\in B}l_{b}b
  51. h v = f g - 1 v = b B l b σ b hv=fg^{-1}v=\sum_{b\in B}l_{b}^{\sigma}b
  52. Γ L ( V ) = GL ( V ) Gal ( K / k ) . \operatorname{\Gamma L}(V)=\operatorname{GL}(V)\rtimes\operatorname{Gal}(K/k).
  53. Γ L ( V ) \operatorname{\Gamma L}(V)
  54. Γ L ( V ) \operatorname{\Gamma L}(V)
  55. P Γ L ( V ) , \operatorname{P\Gamma L}(V),
  56. f : V W f\colon V\to W
  57. f : P G ( V ) P G ( W ) . f\colon PG(V)\to PG(W).

Sensitivity_(control_systems).html

  1. M s M_{s}
  2. M s = max 0 ω < | S ( j ω ) | = max 0 ω < | 1 1 + G ( j ω ) C ( j ω ) | M_{s}=\max_{0\leq\omega<\infty}\left|S(j\omega)\right|=\max_{0\leq\omega<% \infty}\left|\frac{1}{1+G(j\omega)C(j\omega)}\right|
  3. G ( s ) G(s)
  4. C ( s ) C(s)
  5. S S
  6. | S | |S|
  7. | S ( j ω ) | |S(j\omega)|
  8. - 1 -1
  9. | S ( j ω ) | |S(j\omega)|
  10. M s M_{s}
  11. M s M_{s}
  12. - 1 -1
  13. M s M_{s}
  14. 1 M s \frac{1}{M_{s}}
  15. - 1 -1
  16. 1 M s \frac{1}{M_{s}}