wpmath0000008_3

Dermott's_law.html

  1. T ( n ) = T ( 0 ) C n T(n)=T(0)\cdot C^{n}
  2. n = 1 , 2 , 3 , 4 \scriptstyle n=1,2,3,4\ldots

Development_(topology).html

  1. X X
  2. X X
  3. F 1 , F 2 , F_{1},F_{2},\ldots
  4. X X
  5. C X C\subset X
  6. p p
  7. C C
  8. F j F_{j}
  9. F j F_{j}
  10. p p
  11. C C
  12. F 1 , F 2 , F_{1},F_{2},\ldots
  13. F i + 1 F i F_{i+1}\subset F_{i}
  14. i i
  15. F i + 1 F_{i+1}
  16. F i F_{i}
  17. i i

DFFITS.html

  1. DFFITS = y i ^ - y i ( i ) ^ s ( i ) h i i \,\text{DFFITS}={\widehat{y_{i}}-\widehat{y_{i(i)}}\over s_{(i)}\sqrt{h_{ii}}}
  2. y i ^ \widehat{y_{i}}
  3. y i ( i ) ^ \widehat{y_{i(i)}}
  4. s ( i ) s_{(i)}
  5. h i i h_{ii}
  6. h i i / ( 1 - h i i ) \sqrt{h_{ii}/(1-h_{ii})}
  7. h i i / ( 1 - h i i ) \sqrt{h_{ii}/(1-h_{ii})}
  8. p n - p p n \sqrt{p\over n-p}\approx\sqrt{p\over n}
  9. 2 p n 2\sqrt{p\over n}

Diachronics_of_plural_inflection_in_the_Gallo-Italian_languages.html

  1. \emptyset
  2. \emptyset

Diagnosis_(artificial_intelligence).html

  1. A b Ab\,
  2. ¬ A b ( S ) I n t 1 O b s 1 \neg Ab(S)\Rightarrow Int1\wedge Obs1
  3. A b ( S ) I n t 2 O b s 2 Ab(S)\Rightarrow Int2\wedge Obs2
  4. I n t 1 Int1\,
  5. O b s 1 Obs1\,
  6. I n t 2 Int2\,
  7. O b s 2 Obs2\,
  8. O b s Obs\,
  9. ¬ A b ( S ) \neg Ab(S)\,
  10. A b ( S ) Ab(S)\,

Diagonally_dominant_matrix.html

  1. | a i i | j i | a i j | for all i , |a_{ii}|\geq\sum_{j\neq i}|a_{ij}|\quad\,\text{for all }i,\,
  2. 𝐀 = [ 3 - 2 1 1 - 3 2 - 1 2 4 ] \mathbf{A}=\begin{bmatrix}3&-2&1\\ 1&-3&2\\ -1&2&4\end{bmatrix}
  3. | a 11 | | a 12 | + | a 13 | |a_{11}|\geq|a_{12}|+|a_{13}|
  4. | 3 | | - 2 | + | 1 | |3|\geq|-2|+|1|
  5. | a 22 | | a 21 | + | a 23 | |a_{22}|\geq|a_{21}|+|a_{23}|
  6. | - 3 | | 1 | + | 2 | |-3|\geq|1|+|2|
  7. | a 33 | | a 31 | + | a 32 | |a_{33}|\geq|a_{31}|+|a_{32}|
  8. | 4 | | - 1 | + | 2 | |4|\geq|-1|+|2|
  9. 𝐁 = [ - 2 2 1 1 3 2 1 - 2 0 ] \mathbf{B}=\begin{bmatrix}-2&2&1\\ 1&3&2\\ 1&-2&0\end{bmatrix}
  10. | b 11 | < | b 12 | + | b 13 | |b_{11}|<|b_{12}|+|b_{13}|
  11. | - 2 | < | 2 | + | 1 | |-2|<|2|+|1|
  12. | b 22 | | b 21 | + | b 23 | |b_{22}|\geq|b_{21}|+|b_{23}|
  13. | 3 | | 1 | + | 2 | |3|\geq|1|+|2|
  14. | b 33 | < | b 31 | + | b 32 | |b_{33}|<|b_{31}|+|b_{32}|
  15. | 0 | < | 1 | + | - 2 | |0|<|1|+|-2|
  16. | b 11 | |b_{11}|
  17. | b 33 | |b_{33}|
  18. 𝐂 = [ - 4 2 1 1 6 2 1 - 2 5 ] \mathbf{C}=\begin{bmatrix}-4&2&1\\ 1&6&2\\ 1&-2&5\end{bmatrix}
  19. | c 11 | | c 12 | + | c 13 | |c_{11}|\geq|c_{12}|+|c_{13}|
  20. | - 4 | > | 2 | + | 1 | |-4|>|2|+|1|
  21. | c 22 | | c 21 | + | c 23 | |c_{22}|\geq|c_{21}|+|c_{23}|
  22. | 6 | > | 1 | + | 2 | |6|>|1|+|2|
  23. | c 33 | | c 31 | + | c 32 | |c_{33}|\geq|c_{31}|+|c_{32}|
  24. | 5 | > | 1 | + | - 2 | |5|>|1|+|-2|
  25. A A
  26. D D
  27. A A
  28. A A
  29. D + I D+I
  30. M ( t ) = ( 1 - t ) ( D + I ) + t A M(t)=(1-t)(D+I)+tA
  31. A A
  32. det ( A ) 0 \mathrm{det}(A)\geq 0
  33. A A
  34. ( - 5 2 1 ) ( 1 1 0 1 1 0 1 0 1 ) ( - 5 2 1 ) = 10 - 5 5 < 0 \begin{pmatrix}-\sqrt{5}&2&1\end{pmatrix}\begin{pmatrix}1&1&0\\ 1&1&0\\ 1&0&1\end{pmatrix}\begin{pmatrix}-\sqrt{5}\\ 2\\ 1\end{pmatrix}=10-5\sqrt{5}<0
  35. A A
  36. x I xI
  37. x x
  38. q q
  39. q q

Diagram_(category_theory).html

  1. A A
  2. A ¯ \underline{A}
  3. J = 0 1 J=0\overrightarrow{\to}1
  4. f , g : X Y f,g\colon X\to Y
  5. f : X X f\colon X\to X
  6. \bullet\overrightarrow{\to}\bullet
  7. f , g : X Y f,g\colon X\to Y

Diaphoneme.html

  1. / / R P , G A k S S E , K A k v s . x / / \bigg/\bigg/\frac{RP,GA\qquad\mathrm{k}}{SSE,KA\qquad\mathrm{k}~{}vs.\mathrm{x% }}\bigg/\bigg/

Dichloramine.html

  1. NH 2 Cl + Cl 2 NHCl 2 + HCl \mathrm{NH_{2}Cl+Cl_{2}\longrightarrow NHCl_{2}+HCl}

Dickman_function.html

  1. ρ ( u ) \rho(u)
  2. u ρ ( u ) + ρ ( u - 1 ) = 0 u\rho^{\prime}(u)+\rho(u-1)=0\,
  3. ρ ( u ) = 1 \rho(u)=1
  4. a a
  5. Ψ ( x , x 1 / a ) x ρ ( a ) \Psi(x,x^{1/a})\sim x\rho(a)\,
  6. Ψ ( x , y ) \Psi(x,y)
  7. Ψ ( x , x 1 / a ) \Psi(x,x^{1/a})
  8. x ρ ( a ) x\rho(a)
  9. Ψ ( x , x 1 / a ) = x ρ ( a ) + O ( x / log x ) \Psi(x,x^{1/a})=x\rho(a)+O(x/\log x)
  10. log ρ \log\rho
  11. Ψ ( x , y ) = x u O ( - u ) \Psi(x,y)=xu^{O(-u)}
  12. ρ ( u ) u - u \rho(u)\approx u^{-u}
  13. ρ ( u ) u - u . \rho(u)\approx u^{-u}.\,
  14. ρ ( u ) 1 ξ 2 π u exp ( - u ξ + Ei ( ξ ) ) \rho(u)\sim\frac{1}{\xi\sqrt{2\pi u}}\cdot\exp(-u\xi+\operatorname{Ei}(\xi))
  15. e ξ - 1 = u ξ . e^{\xi}-1=u\xi.\,
  16. ρ ( x ) 1 / x ! . \rho(x)\leq 1/x!.
  17. u u
  18. ρ ( u ) \rho(u)
  19. × 10 - 1 \times 10^{-}1
  20. × 10 - 2 \times 10^{-}2
  21. × 10 - 3 \times 10^{-}3
  22. × 10 - 4 \times 10^{-}4
  23. × 10 - 5 \times 10^{-}5
  24. × 10 - 7 \times 10^{-}7
  25. × 10 - 8 \times 10^{-}8
  26. × 10 - 9 \times 10^{-}9
  27. × 10 - 11 \times 10^{-}11
  28. ρ n \rho_{n}
  29. ρ n ( u ) = ρ ( u ) \rho_{n}(u)=\rho(u)
  30. ρ ( u ) = 1 \rho(u)=1
  31. ρ ( u ) = 1 - log u \rho(u)=1-\log u
  32. ρ ( u ) = 1 - ( 1 - log ( u - 1 ) ) log ( u ) + Li 2 ( 1 - u ) + π 2 12 \rho(u)=1-(1-\log(u-1))\log(u)+\operatorname{Li}_{2}(1-u)+\frac{\pi^{2}}{12}
  33. ρ n \rho_{n}
  34. σ ( u , v ) \sigma(u,v)
  35. ρ ( u ) \rho(u)
  36. Ψ ( x , y , z ) \Psi(x,y,z)
  37. Ψ ( x , x 1 / a , x 1 / b ) x σ ( b , a ) . \Psi(x,x^{1/a},x^{1/b})\sim x\sigma(b,a).\,
  38. e - γ e^{-\gamma}

Dielectric_loss.html

  1. ϵ = ϵ - j ϵ ′′ \epsilon=\epsilon^{\prime}-j\epsilon^{\prime\prime}
  2. 𝐄 = 𝐄 o e j ω t \mathbf{E}=\mathbf{E}_{o}e^{j\omega t}
  3. × 𝐇 = j ω ϵ 𝐄 + ( ω ϵ ′′ + σ ) 𝐄 \nabla\times\mathbf{H}=j\omega\epsilon^{\prime}\mathbf{E}+(\omega\epsilon^{% \prime\prime}+\sigma)\mathbf{E}
  4. tan δ = ω ϵ ′′ + σ ω ϵ \tan\delta=\frac{\omega\epsilon^{\prime\prime}+\sigma}{\omega\epsilon^{\prime}}
  5. P = P o e - δ k z P=P_{o}e^{-\delta kz}
  6. P o P_{o}
  7. k = ω μ ϵ = 2 π λ k=\omega\sqrt{\mu\epsilon^{\prime}}=\frac{2\pi}{\lambda}
  8. μ = μ - j μ ′′ \mu=\mu^{\prime}-j\mu^{\prime\prime}
  9. tan δ m = μ ′′ μ \tan\delta_{m}=\frac{\mu^{\prime\prime}}{\mu^{\prime}}
  10. tan δ e = ϵ ′′ ϵ \tan\delta_{e}=\frac{\epsilon^{\prime\prime}}{\epsilon^{\prime}}
  11. ESR = σ ϵ ω 2 C \mathrm{ESR}=\frac{\sigma}{\epsilon^{\prime}\omega^{2}C}
  12. C C
  13. tan δ = ESR | X c | = ω C ESR = σ ϵ ω \tan\delta=\frac{\mathrm{ESR}}{|X_{c}|}=\omega C\cdot\mathrm{ESR}=\frac{\sigma% }{\epsilon^{\prime}\omega}
  14. tan δ = D F = 1 Q \tan\delta=DF=\frac{1}{Q}

Difference_density_map.html

  1. C d i f f m a p = ( m | F o b s | - D | F c a l c | ) e x p ( 2 π i ϕ c a l c ) C_{diffmap}=(m|F_{obs}|-D|F_{calc}|)exp(2\pi i\phi_{calc})

Differential_(infinitesimal).html

  1. d y = d y d x d x , \mathrm{d}y=\frac{\mathrm{d}y}{\mathrm{d}x}\,\mathrm{d}x,\,
  2. f ( x ) d x , \int f(x)\,\mathrm{d}x,\,
  3. | f ( x ) - f ( p ) - d f p ( x - p ) | < ε | x - p | . \left|f(x)-f(p)-\mathrm{d}f_{p}(x-p)\right|<\varepsilon\left|x-p\right|.\,
  4. d f p = j = 1 n D j f ( p ) ( d x j ) p . \mathrm{d}f_{p}=\sum_{j=1}^{n}D_{j}f(p)\,(\mathrm{d}x^{j})_{p}.\,
  5. d f = f x 1 d x 1 + f x 2 d x 2 + + f x n d x n . \mathrm{d}f=\frac{\partial f}{\partial x^{1}}\,\mathrm{d}x^{1}+\frac{\partial f% }{\partial x^{2}}\,\mathrm{d}x^{2}+\cdots+\frac{\partial f}{\partial x^{n}}\,% \mathrm{d}x^{n}.
  6. d f = d f d x d x \mathrm{d}f=\frac{\mathrm{d}f}{\mathrm{d}x}\mathrm{d}x

Differential_calculus_over_commutative_algebras.html

  1. M M
  2. \mathbb{R}
  3. A = C ( M ) , A=C^{\infty}(M),
  4. M M
  5. A A
  6. Γ \Gamma
  7. M M
  8. A A
  9. E M E\rightarrow M
  10. F M F\rightarrow M
  11. \mathbb{R}
  12. Δ : Γ ( E ) Γ ( F ) \Delta:\Gamma(E)\rightarrow\Gamma(F)
  13. f 0 , , f k A f_{0},\ldots,f_{k}\in A
  14. [ f k [ f k - 1 [ [ f 0 , Δ ] ] ] = 0 [f_{k}[f_{k-1}[\cdots[f_{0},\Delta]\cdots]]=0
  15. [ f , Δ ] : Γ ( E ) Γ ( F ) [f,\Delta]:\Gamma(E)\rightarrow\Gamma(F)
  16. [ f , Δ ] ( s ) = Δ ( f s ) - f Δ ( s ) . [f,\Delta](s)=\Delta(f\cdot s)-f\cdot\Delta(s).
  17. A A
  18. P P
  19. A A
  20. Q Q
  21. Diff k ( P , Q ) \mathrm{Diff}_{k}(P,Q)
  22. A A
  23. Diff k \mathrm{Diff}_{k}
  24. \mathbb{R}
  25. C ( M ) C^{\infty}(M)

Differential_ideal.html

  1. I Ω * ( M ) I\subset\Omega^{*}(M)
  2. f : m n f:\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}
  3. F r ( x , u , | I | u x I ) = 0 , 1 | I | k F^{r}(x,u,\frac{\partial^{|I|}u}{\partial x^{I}})=0,\quad 1\leq|I|\leq k
  4. Σ \Sigma
  5. Σ \Sigma
  6. I I\,
  7. a I a\in I
  8. b I b\in I
  9. b n I b^{n}\in I
  10. n > 0 n>0\,
  11. a a\,

Differential_variational_inequality.html

  1. u ( t ) K u(t)\in K
  2. v - u ( t ) , F ( t , x ( t ) , u ( t ) ) 0 \langle v-u(t),F(t,x(t),u(t))\rangle\geq 0
  3. v K v\in K
  4. d x d t = f ( t , x ( t ) , u ( t ) ) , x ( t 0 ) = x 0 . \frac{dx}{dt}=f(t,x(t),u(t)),\quad x(t_{0})=x_{0}.
  5. K u ( t ) F ( t , x ( t ) , u ( t ) ) K * . K\ni u(t)\quad\perp\quad F(t,x(t),u(t))\in K^{*}.\,
  6. r r
  7. m d 2 y d t 2 = - m g + N ( t ) m\frac{d^{2}y}{dt^{2}}=-mg+N(t)
  8. m m
  9. N ( t ) N(t)
  10. g g
  11. y ( t ) y(t)
  12. N ( t ) N(t)
  13. y ( t ) - r 0 y(t)-r\geq 0
  14. t t
  15. y ( t ) - r > 0 y(t)-r>0
  16. N ( t ) = 0 N(t)=0
  17. y ( t ) - r = 0 y(t)-r=0
  18. N ( t ) N(t)
  19. N ( t ) < 0 N(t)<0
  20. 0 y ( t ) - r N ( t ) 0. 0\leq y(t)-r\quad\perp\quad N(t)\geq 0.
  21. K = { z z 0 } K=\{\,z\mid z\geq 0\,\}
  22. K * = K K^{*}=K
  23. v ( t ) v(t)
  24. i ( t ) i(t)
  25. 0 v ( t ) i ( t ) 0 0\leq v(t)\quad\perp\quad i(t)\geq 0
  26. t t
  27. y ( t ) y(t)
  28. d y / d t ( t ) dy/dt(t)
  29. N ( t ) N(t)
  30. d 2 y / d t 2 = ( 1 / m ) [ - m g + N ( t ) ] d^{2}y/dt^{2}=(1/m)[-mg+N(t)]
  31. N ( t ) N(t)
  32. d 2 y / d t 2 = b ( t ) d^{2}y/dt^{2}=b(t)
  33. N ( t ) N(t)
  34. b ( t ) b(t)

Differentiation_in_Fréchet_spaces.html

  1. D F ( u ) v = lim τ 0 F ( u + v τ ) - F ( u ) τ DF(u)v=\lim_{\tau\rightarrow 0}\frac{F(u+v\tau)-F(u)}{\tau}
  2. D k + 1 F ( u ) { v 1 , v 2 , , v k + 1 } = lim τ 0 D k F ( u + τ v k + 1 ) { v 1 , , v k } - D k F ( u ) { v 1 , , v k } τ . D^{k+1}F(u)\{v_{1},v_{2},\dots,v_{k+1}\}=\lim_{\tau\rightarrow 0}\frac{D^{k}F(% u+\tau v_{k+1})\{v_{1},\dots,v_{k}\}-D^{k}F(u)\{v_{1},\dots,v_{k}\}}{\tau}.
  3. F ( b ) - F ( a ) = 0 1 D F ( a + ( b - a ) t ) ( b - a ) d t F(b)-F(a)=\int_{0}^{1}DF(a+(b-a)t)\cdot(b-a)dt
  4. F ( u + h ) = F ( u ) + D F ( u ) h + 1 2 ! D 2 F ( u ) { h , h } + + 1 ( k - 1 ) ! D k - 1 F ( u ) { h , h , , h } + R k F(u+h)=F(u)+DF(u)h+\frac{1}{2!}D^{2}F(u)\{h,h\}+\dots+\frac{1}{(k-1)!}D^{k-1}F% (u)\{h,h,\dots,h\}+R_{k}
  5. R k ( u , h ) = 1 ( k - 1 ) ! 0 1 ( 1 - t ) k - 1 D k F ( u + t h ) { h , h , , h } d t R_{k}(u,h)=\frac{1}{(k-1)!}\int_{0}^{1}(1-t)^{k-1}D^{k}F(u+th)\{h,h,\dots,h\}dt
  6. D k F ( u ) { h 1 , , h k } = D k F ( u ) { h σ ( 1 ) , , h σ ( k ) } D^{k}F(u)\{h_{1},...,h_{k}\}=D^{k}F(u)\{h_{\sigma(1)},\dots,h_{\sigma(k)}\}

Differentiator.html

  1. Y = X Z R Z R + Z C = X R R + 1 j ω C = X 1 1 + 1 j ω R C , Y=X\frac{Z_{R}}{Z_{R}+Z_{C}}=X\frac{R}{R+\frac{1}{j\omega C}}=X\frac{1}{1+% \frac{1}{j\omega RC}},
  2. X X
  3. Y Y
  4. Z R Z_{R}
  5. Z C Z_{C}
  6. K ( j ω ) = 1 1 + 1 j ω R C = 1 1 + ω 0 j ω , K(j\omega)=\frac{1}{1+\frac{1}{j\omega RC}}=\frac{1}{1+\frac{\omega_{0}}{j% \omega}},
  7. ω 0 = 1 R C . \omega_{0}=\frac{1}{RC}.
  8. H ( ω ) | K ( j ω ) | = 1 1 + ( ω 0 ω ) 2 , H(\omega)\triangleq|K(j\omega)|=\frac{1}{\sqrt{1+\left(\frac{\omega_{0}}{% \omega}\right)^{2}}},
  9. ϕ ( ω ) arg K ( j ω ) = arctan ω 0 ω , \phi(\omega)\triangleq\arg K(j\omega)=\arctan\frac{\omega_{0}}{\omega},
  10. ω 0 = R L \omega_{0}=\frac{R}{L}
  11. h ( t ) = - 1 { K ( p ) } = δ ( t ) - ω 0 e - ω 0 t = δ ( t ) - 1 τ e - t τ h(t)=\mathcal{L}^{-1}\left\{K(p)\right\}=\delta(t)-\omega_{0}e^{-\omega_{0}t}=% \delta(t)-\frac{1}{\tau}e^{-\frac{t}{\tau}}
  12. τ = 1 ω 0 \tau=\frac{1}{\omega_{0}}
  13. δ ( t ) \delta(t)
  14. V = V ( i n f i n i t y ) + [ ( V ( 0 + ) - V ( i n f i n i t y ) ] e - t τ V=V(infinity)+[(V(0+)-V(infinity)]e^{-\frac{t}{\tau}}
  15. I = C d V d t I=C\frac{dV}{dt}
  16. I = V R I=\frac{V}{R}
  17. V out = - R C d V in d t V_{\,\text{out}}=-{R}{C}\frac{dV_{\,\text{in}}}{dt}
  18. 2 π f C \frac{2}{πfC}
  19. R < s u b > f X c \frac{R<sub>f}{X_{c}}

Diffusion_creep.html

  1. J x = - D x Δ C Δ x J_{x}=-D_{x}\frac{\Delta C}{\Delta x}
  2. Δ C / Δ x {\Delta C}/{\Delta x}
  3. ϵ ˙ \dot{\epsilon}
  4. ϵ ˙ = A e - Q R T σ n d m \!\dot{\epsilon}=Ae^{\frac{-Q}{RT}}\frac{\sigma^{n}}{d^{m}}

Digital_scan_back.html

  1. 1 50 s × 10 , 000 pixels = 200 s \frac{1}{50}\mbox{ s}~{}\times 10,000\mbox{ pixels}~{}=200\mbox{ s}~{}
  2. 10 , 000 pixels × 10 , 000 pixels × 48 bpp × 1 byte 8 bits = 600 megabytes 10,000\mbox{ pixels}~{}\times 10,000\mbox{ pixels}~{}\times 48\mbox{ bpp}~{}% \times\frac{1\mbox{ byte}~{}}{8\mbox{ bits}~{}}=600\mbox{ megabytes}~{}

Digital_sundial.html

  1. L θ L_{\theta}
  2. θ [ 0 , π ) \theta\in[0,\pi)
  3. F 2 F\subset\mathbb{R}^{2}
  4. proj θ F \mathrm{proj}_{\theta}F
  5. L θ L_{\theta}
  6. G θ L θ G_{\theta}\subset L_{\theta}
  7. θ [ 0 , π ) \theta\in[0,\pi)
  8. θ G θ \bigcup_{\theta}G_{\theta}
  9. F 2 F\subset\mathbb{R}^{2}
  10. G θ proj θ F G_{\theta}\subset\mathrm{proj}_{\theta}F
  11. ( proj θ F ) G θ (\mathrm{proj}_{\theta}F)\setminus G_{\theta}
  12. θ [ 0 , π ) . \theta\in[0,\pi).
  13. G θ G_{\theta}

Digital_topology.html

  1. g = 1 + ( M 5 + 2 M 6 - M 3 ) / 8 , g=1+(M_{5}+2M_{6}-M_{3})/8,\!

Dilatometer.html

  1. α = 1 V ( V T ) p \alpha=\frac{1}{V}\biggl(\frac{\partial V}{\partial T}\biggr)_{p}

Diphtheria_toxin.html

  1. \rightleftharpoons

Directed_percolation.html

  1. p p\,
  2. t t
  3. t = 0 t=0
  4. N ( t ) N(t)
  5. d d\,
  6. d d c = 4 d\geq d_{c}=4\,
  7. d < 4 d<4\,
  8. d d
  9. d = 1 d=1\,
  10. d = 2 d=2\,
  11. d = 3 d=3\,
  12. d 4 d\geq 4\,
  13. β \beta\,
  14. 0.276486 ± 0.000008 0.276486\pm 0.000008
  15. 0.583 ± 0.003 0.583\pm 0.003
  16. 0.813 ± 0.009 0.813\pm 0.009
  17. 1 1\,
  18. ν \nu_{\perp}
  19. 1.096854 ± 0.000004 1.096854\pm 0.000004
  20. 0.733 ± 0.008 0.733\pm 0.008
  21. 0.584 ± 0.005 0.584\pm 0.005
  22. 1 / 2 1/2\,
  23. ν | | \nu_{||}\,
  24. 1.733847 ± 0.000006 1.733847\pm 0.000006
  25. 1.295 ± 0.006 1.295\pm 0.006
  26. 1.11 ± 0.01 1.11\pm 0.01
  27. 1 1\,

Directional_stability.html

  1. θ \theta
  2. ψ \psi
  3. M V d ψ d t = Y c o s ( θ - ψ ) MV\frac{d\psi}{dt}=Ycos(\theta-\psi)
  4. M V d ψ d t = Y MV\frac{d\psi}{dt}=Y
  5. I d 2 θ d t 2 = N I\frac{d^{2}\theta}{dt^{2}}=N
  6. ϕ ( f r o n t ) = θ - ψ - a V d θ d t \phi(front)=\theta-\psi-\frac{a}{V}\frac{d\theta}{dt}
  7. ϕ ( r e a r ) = θ - ψ + b V d θ d t \phi(rear)=\theta-\psi+\frac{b}{V}\frac{d\theta}{dt}
  8. Y = 2 k ( ϕ ( f r o n t ) + ϕ ( r e a r ) ) = 4 k ( θ - ψ ) + 2 k ( b - a ) V d θ d t Y=2k(\phi(front)+\phi(rear))=4k(\theta-\psi)+2k\frac{(b-a)}{V}\frac{d\theta}{dt}
  9. N = 2 k ( a ϕ ( f r o n t ) - b ϕ ( r e a r ) ) = 2 k ( a - b ) ( θ - ψ ) - 2 k ( a 2 + b 2 ) V d θ d t N=2k(a\phi(front)-b\phi(rear))=2k(a-b)(\theta-\psi)-2k\frac{(a^{2}+b^{2})}{V}% \frac{d\theta}{dt}
  10. ω \omega
  11. d ω d t = 2 k ( a - b ) I ( θ - ψ ) - 2 k ( a 2 + b 2 ) V I ω \frac{d\omega}{dt}=2k\frac{(a-b)}{I}(\theta-\psi)-2k\frac{(a^{2}+b^{2})}{VI}\omega
  12. d θ d t = ω \frac{d\theta}{dt}=\omega
  13. d ψ d t = 4 k M V ( θ - ψ ) + 2 k ( b - a ) M V 2 ω \frac{d\psi}{dt}=\frac{4k}{MV}(\theta-\psi)+2k\frac{(b-a)}{MV^{2}}\omega
  14. θ - ψ = β \theta-\psi=\beta
  15. d ω d t = 2 k ( a - b ) I β - 2 k ( a 2 + b 2 ) V I ω \frac{d\omega}{dt}=2k\frac{(a-b)}{I}\beta-2k\frac{(a^{2}+b^{2})}{VI}\omega
  16. d β d t = - 4 k M V β + ( 1 - 2 k ) ( b - a ) M V 2 ω \frac{d\beta}{dt}=-\frac{4k}{MV}\beta+(1-2k)\frac{(b-a)}{MV^{2}}\omega
  17. ω \omega
  18. β \beta
  19. d 2 β d t 2 + ( 4 k M V + 2 k ( a 2 + b 2 ) V I ) d β d t + ( 4 k 2 ( a + b ) 2 M V 2 I + 2 k ( b - a ) I ) β = 0 \frac{d^{2}\beta}{dt^{2}}+(\frac{4k}{MV}+\frac{2k(a^{2}+b^{2})}{VI})\frac{d% \beta}{dt}+(\frac{4k^{2}(a+b)^{2}}{MV^{2}I}+\frac{2k(b-a)}{I})\beta=0
  20. d β d t \frac{d\beta}{dt}
  21. β \beta
  22. ( 4 k M V + 2 k ( a 2 + b 2 ) V I ) (\frac{4k}{MV}+\frac{2k(a^{2}+b^{2})}{VI})
  23. ( 4 k 2 ( a + b ) 2 M I V 2 + 2 k ( b - a ) I ) (\frac{4k^{2}(a+b)^{2}}{MIV^{2}}+\frac{2k(b-a)}{I})
  24. ( b > a ) (b>a)
  25. V 2 = 2 k ( a + b ) 2 M ( a - b ) V^{2}=\frac{2k(a+b)^{2}}{M(a-b)}
  26. Y = 2 k ( ϕ ( f r o n t ) ) = 2 k ( θ - ψ ) - 2 k a V d θ d t Y=2k(\phi(front))=2k(\theta-\psi)-2k\frac{a}{V}\frac{d\theta}{dt}
  27. N = 2 k ( a ϕ ( f r o n t ) ) = 2 k a ( θ - ψ ) - 2 k a 2 V d θ d t N=2k(a\phi(front))=2ka(\theta-\psi)-2k\frac{a^{2}}{V}\frac{d\theta}{dt}
  28. d 2 β d t 2 + ( 2 k M V + 2 k a 2 V I ) d β d t - ( 2 k a I ) β = 0 \frac{d^{2}\beta}{dt^{2}}+(\frac{2k}{MV}+\frac{2ka^{2}}{VI})\frac{d\beta}{dt}-% (\frac{2ka}{I})\beta=0
  29. β \beta
  30. Y = 2 k ( ϕ ( r e a r ) ) = 2 k ( θ - ψ ) + 2 k b V d θ d t Y=2k(\phi(rear))=2k(\theta-\psi)+2k\frac{b}{V}\frac{d\theta}{dt}
  31. N = - 2 k ( b ϕ ( r e a r ) ) = - 2 k b ( θ - ψ ) - 2 k b 2 V d θ d t N=-2k(b\phi(rear))=-2kb(\theta-\psi)-2k\frac{b^{2}}{V}\frac{d\theta}{dt}
  32. d 2 β d t 2 + ( 2 k M V + 2 k b 2 V I ) d β d t + ( 2 k b I ) β = 0 \frac{d^{2}\beta}{dt^{2}}+(\frac{2k}{MV}+\frac{2kb^{2}}{VI})\frac{d\beta}{dt}+% (\frac{2kb}{I})\beta=0
  33. β \beta
  34. Y = 2 k ( ϕ ( f r o n t ) + ϕ ( r e a r ) ) = 4 k ( θ - ψ ) + 2 k ( b - a ) V d θ d t + 2 k η Y=2k(\phi(front)+\phi(rear))=4k(\theta-\psi)+2k\frac{(b-a)}{V}\frac{d\theta}{% dt}+2k\eta
  35. η \eta
  36. N = 2 k ( a ϕ ( f r o n t ) - b ϕ ( r e a r ) ) = 2 k ( a - b ) ( θ - ψ ) - 2 k ( a 2 + b 2 ) V d θ d t + 2 k a η N=2k(a\phi(front)-b\phi(rear))=2k(a-b)(\theta-\psi)-2k\frac{(a^{2}+b^{2})}{V}% \frac{d\theta}{dt}+2ka\eta
  37. d 2 β d t 2 + ( 4 k M V + 2 k ( a 2 + b 2 ) V I ) d β d t + ( 4 k 2 ( a + b ) 2 M V 2 I + 2 k ( b - a ) I ) β = - 2 k M V d η d t + ( 2 k a I - 4 k 2 b ( a + b ) I M V 2 ) η \frac{d^{2}\beta}{dt^{2}}+(\frac{4k}{MV}+\frac{2k(a^{2}+b^{2})}{VI})\frac{d% \beta}{dt}+(\frac{4k^{2}(a+b)^{2}}{MV^{2}I}+\frac{2k(b-a)}{I})\beta=-\frac{2k}% {MV}\frac{d\eta}{dt}+(\frac{2ka}{I}-\frac{4k^{2}b(a+b)}{IMV^{2}})\eta
  38. β \beta
  39. η \eta
  40. ( 2 k a I - 4 k 2 b ( a + b ) I M V 2 ) (\frac{2ka}{I}-\frac{4k^{2}b(a+b)}{IMV^{2}})
  41. V 2 = 2 k b ( a + b ) M a V^{2}=\frac{2kb(a+b)}{Ma}

Dirichlet_density.html

  1. lim s 1 + p A 1 p s log ( 1 s - 1 ) \lim_{s\rightarrow 1^{+}}{\sum_{p\in A}{1\over p^{s}}\over\log(\frac{1}{s-1})}
  2. p A 1 1 - p - s \prod_{p\in A}{1\over 1-p^{-s}}

Discontinuous_Deformation_Analysis.html

  1. k / M \sqrt{k/M}
  2. k k
  3. M M

Discrete_Poisson_equation.html

  1. Δ x = Δ y \Delta x=\Delta y
  2. ( 2 u ) i j = 1 Δ x 2 ( u i + 1 , j + u i - 1 , j + u i , j + 1 + u i , j - 1 - 4 u i j ) = g i j ({\nabla}^{2}u)_{ij}=\frac{1}{\Delta x^{2}}(u_{i+1,j}+u_{i-1,j}+u_{i,j+1}+u_{i% ,j-1}-4u_{ij})=g_{ij}
  3. 2 i m - 1 2\leq i\leq m-1
  4. 2 j n - 1 2\leq j\leq n-1
  5. u = [ u 11 , u 21 , , u m 1 , u 12 , u 22 , , u m 2 , , u m n ] T \vec{u}=\begin{bmatrix}u_{11},u_{21},\ldots,u_{m1},u_{12},u_{22},\ldots,u_{m2}% ,\ldots,u_{mn}\end{bmatrix}^{T}
  6. A u = b A\vec{u}=\vec{b}
  7. A = [ D - I 0 0 0 0 - I D - I 0 0 0 0 - I D - I 0 0 0 0 - I D - I 0 0 0 - I D - I 0 0 - I D ] , A=\begin{bmatrix}~{}D&-I&~{}0&~{}0&~{}0&\ldots&~{}0\\ -I&~{}D&-I&~{}0&~{}0&\ldots&~{}0\\ ~{}0&-I&~{}D&-I&~{}0&\ldots&~{}0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ ~{}0&\ldots&~{}0&-I&~{}D&-I&~{}0\\ ~{}0&\ldots&\ldots&~{}0&-I&~{}D&-I\\ ~{}0&\ldots&\ldots&\ldots&~{}0&-I&~{}D\end{bmatrix},
  8. I I
  9. D D
  10. D = [ 4 - 1 0 0 0 0 - 1 4 - 1 0 0 0 0 - 1 4 - 1 0 0 0 0 - 1 4 - 1 0 0 0 - 1 4 - 1 0 0 - 1 4 ] , D=\begin{bmatrix}~{}4&-1&~{}0&~{}0&~{}0&\ldots&~{}0\\ -1&~{}4&-1&~{}0&~{}0&\ldots&~{}0\\ ~{}0&-1&~{}4&-1&~{}0&\ldots&~{}0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\ ~{}0&\ldots&~{}0&-1&~{}4&-1&~{}0\\ ~{}0&\ldots&\ldots&~{}0&-1&~{}4&-1\\ ~{}0&\ldots&\ldots&\ldots&~{}0&-1&~{}4\end{bmatrix},
  11. b \vec{b}
  12. b = - Δ x 2 [ g 11 , g 21 , , g m 1 , g 12 , g 22 , , g m 2 , , g m n ] T . \vec{b}=-\Delta x^{2}\begin{bmatrix}g_{11},g_{21},\ldots,g_{m1},g_{12},g_{22},% \ldots,g_{m2},\ldots,g_{mn}\end{bmatrix}^{T}.
  13. u i j u_{ij}
  14. D D
  15. m m
  16. u u
  17. [ u 1 j , u 2 j , , u i - 1 , j , u i j , u i + 1 , j , , u m j ] T \begin{bmatrix}u_{1j},&u_{2j},&\ldots,&u_{i-1,j},&u_{ij},&u_{i+1,j},&\ldots,&u% _{mj}\end{bmatrix}^{T}
  18. I I
  19. D D
  20. m m
  21. u u
  22. [ u 1 , j - 1 , u 2 , j - 1 , , u i - 1 , j - 1 , u i , j - 1 , u i + 1 , j - 1 , , u m , j - 1 ] T \begin{bmatrix}u_{1,j-1},&u_{2,j-1},&\ldots,&u_{i-1,j-1},&u_{i,j-1},&u_{i+1,j-% 1},&\ldots,&u_{m,j-1}\end{bmatrix}^{T}
  23. [ u 1 , j + 1 , u 2 , j + 1 , , u i - 1 , j + 1 , u i , j + 1 , u i + 1 , j + 1 , , u m , j + 1 ] T \begin{bmatrix}u_{1,j+1},&u_{2,j+1},&\ldots,&u_{i-1,j+1},&u_{i,j+1},&u_{i+1,j+% 1},&\ldots,&u_{m,j+1}\end{bmatrix}^{T}
  24. n n
  25. m m
  26. A A
  27. u u
  28. I I
  29. D D
  30. 2 i m - 1 2\leq i\leq m-1
  31. 2 j n - 1 2\leq j\leq n-1
  32. D D
  33. I I
  34. m = 5 m=5
  35. n = 5 n=5
  36. [ U ] = [ u 22 , u 32 , u 42 , u 23 , u 33 , u 43 , u 24 , u 34 , u 44 ] T \begin{bmatrix}U\end{bmatrix}=\begin{bmatrix}u_{22},u_{32},u_{42},u_{23},u_{33% },u_{43},u_{24},u_{34},u_{44}\end{bmatrix}^{T}
  37. A = [ 4 - 1 0 - 1 0 0 0 0 0 - 1 4 - 1 0 - 1 0 0 0 0 0 - 1 4 0 0 - 1 0 0 0 - 1 0 0 4 - 1 0 - 1 0 0 0 - 1 0 - 1 4 - 1 0 - 1 0 0 0 - 1 0 - 1 4 0 0 - 1 0 0 0 - 1 0 0 4 - 1 0 0 0 0 0 - 1 0 - 1 4 - 1 0 0 0 0 0 - 1 0 - 1 4 ] A=\begin{bmatrix}~{}4&-1&~{}0&-1&~{}0&~{}0&~{}0&~{}0&~{}0\\ -1&~{}4&-1&~{}0&-1&~{}0&~{}0&~{}0&~{}0\\ ~{}0&-1&~{}4&~{}0&~{}0&-1&~{}0&~{}0&~{}0\\ -1&~{}0&~{}0&~{}4&-1&~{}0&-1&~{}0&~{}0\\ ~{}0&-1&~{}0&-1&~{}4&-1&~{}0&-1&~{}0\\ ~{}0&~{}0&-1&~{}0&-1&~{}4&~{}0&~{}0&-1\\ ~{}0&~{}0&~{}0&-1&~{}0&~{}0&~{}4&-1&~{}0\\ ~{}0&~{}0&~{}0&~{}0&-1&~{}0&-1&~{}4&-1\\ ~{}0&~{}0&~{}0&~{}0&~{}0&-1&~{}0&-1&~{}4\end{bmatrix}
  38. b = [ - Δ x 2 g 22 + u 12 + u 21 - Δ x 2 g 32 + u 31 - Δ x 2 g 42 + u 52 + u 41 - Δ x 2 g 23 + u 13 - Δ x 2 g 33 - Δ x 2 g 43 + u 53 - Δ x 2 g 24 + u 14 + u 25 - Δ x 2 g 34 + u 35 - Δ x 2 g 44 + u 54 + u 45 ] . \vec{b}=\left[\begin{array}[]{l}-\Delta x^{2}g_{22}+u_{12}+u_{21}\\ -\Delta x^{2}g_{32}+u_{31}\\ -\Delta x^{2}g_{42}+u_{52}+u_{41}\\ -\Delta x^{2}g_{23}+u_{13}\\ -\Delta x^{2}g_{33}\\ -\Delta x^{2}g_{43}+u_{53}\\ -\Delta x^{2}g_{24}+u_{14}+u_{25}\\ -\Delta x^{2}g_{34}+u_{35}\\ -\Delta x^{2}g_{44}+u_{54}+u_{45}\end{array}\right].
  39. u u
  40. D D
  41. I I
  42. D = [ 4 - 1 0 - 1 4 - 1 0 - 1 4 ] D=\begin{bmatrix}~{}4&-1&~{}0\\ -1&~{}4&-1\\ ~{}0&-1&~{}4\\ \end{bmatrix}
  43. - I = [ - 1 0 0 0 - 1 0 0 0 - 1 ] . -I=\begin{bmatrix}-1&~{}0&~{}0\\ ~{}0&-1&~{}0\\ ~{}0&~{}0&-1\end{bmatrix}.
  44. [ A ] \begin{bmatrix}A\end{bmatrix}
  45. [ U ] \begin{bmatrix}U\end{bmatrix}
  46. O ( n ) O(n)
  47. v x x + v y y + v z z = 0 \frac{\partial v_{x}}{\partial x}+\frac{\partial v_{y}}{\partial y}+\frac{% \partial v_{z}}{\partial z}=0
  48. v x v_{x}
  49. x x
  50. v y v_{y}
  51. y y
  52. v z v_{z}
  53. z z
  54. 2 p = f ( ν , V ) \nabla^{2}p=f(\nu,V)
  55. ν \nu
  56. V V

Discrete_series_representation.html

  1. ρ ( g ) v , w \langle\rho(g)\cdot v,w\rangle\,
  2. d ρ ( g ) v , w ρ ( g ) x , y ¯ d g = v , x w , y ¯ d\int\langle\rho(g)\cdot v,w\rangle\overline{\langle\rho(g)\cdot x,y\rangle}dg% =\langle v,x\rangle\overline{\langle w,y\rangle}
  3. ( - 1 ) dim ( G ) - dim ( K ) 2 w W K det ( w ) e w ( v ) ( v , α ) > 0 ( e α 2 - e - α 2 ) (-1)^{\frac{\dim(G)-\dim(K)}{2}}{\sum_{w\in W_{K}}\det(w)e^{w(v)}\over\prod_{(% v,\alpha)>0}\left(e^{\frac{\alpha}{2}}-e^{-\frac{\alpha}{2}}\right)}

Discrete_tomography.html

  1. m 3 m\geq 3
  2. m = 2 m=2
  3. m 3 m\geq 3
  4. m = 2 m=2
  5. k k
  6. ( k - 1 ) (k-1)
  7. k 3 k\geq 3

Discriminant_function_analysis.html

  1. j j
  2. j \mathbb{R}_{j}
  3. x j x\in\mathbb{R}_{j}
  4. x j x\in j
  5. j \mathbb{R}_{j}
  6. π i f i ( x ) \pi_{i}f_{i}(x)
  7. f i ( x ) f_{i}(x)

Discriminant_validity.html

  1. r x y r_{xy}
  2. r x x r_{xx}
  3. r y y r_{yy}
  4. r x y r x x r y y \cfrac{r_{xy}}{\sqrt{r_{xx}\cdot r_{yy}}}
  5. 0.30 0.47 * 0.52 = 0.607 \cfrac{0.30}{\sqrt{0.47*0.52}}=0.607

Discrimination_testing.html

  1. p = 0.5 p=0.5
  2. p = 0.5 p=0.5
  3. p = 1 / 3 p=1/3

Disgregation.html

  1. d Q T = 0 \int\frac{dQ}{T}=0
  2. d Q T 0 \int\frac{dQ}{T}\geq 0

Disintegration_theorem.html

  1. E L x μ ( E ) = 0. E\subseteq L_{x}\implies\mu(E)=0.
  2. μ ( E ) = [ 0 , 1 ] μ x ( E L x ) d x \mu(E)=\int_{[0,1]}\mu_{x}(E\cap L_{x})\,\mathrm{d}x
  3. ν \nu
  4. ν \nu
  5. ν \nu
  6. x μ x x\mapsto\mu_{x}
  7. x μ x ( B ) x\mapsto\mu_{x}(B)
  8. ν \nu
  9. μ x ( Y π - 1 ( x ) ) = 0 , \mu_{x}\left(Y\setminus\pi^{-1}(x)\right)=0,
  10. Y f ( y ) d μ ( y ) = X π - 1 ( x ) f ( y ) d μ x ( y ) d ν ( x ) . \int_{Y}f(y)\,\mathrm{d}\mu(y)=\int_{X}\int_{\pi^{-1}(x)}f(y)\,\mathrm{d}\mu_{% x}(y)\mathrm{d}\nu(x).
  11. μ ( E ) = X μ x ( E ) d ν ( x ) . \mu(E)=\int_{X}\mu_{x}\left(E\right)\,\mathrm{d}\nu(x).
  12. { μ x 1 } x 1 X 1 \{\mu_{x_{1}}\}_{x_{1}\in X_{1}}
  13. μ = X 1 μ x 1 μ ( π 1 - 1 ( d x 1 ) ) = X 1 μ x 1 d ( π 1 ) * ( μ ) ( x 1 ) , \mu=\int_{X_{1}}\mu_{x_{1}}\,\mu\left(\pi_{1}^{-1}(\mathrm{d}x_{1})\right)=% \int_{X_{1}}\mu_{x_{1}}\,\mathrm{d}(\pi_{1})_{*}(\mu)(x_{1}),
  14. X 1 × X 2 f ( x 1 , x 2 ) μ ( d x 1 , d x 2 ) = X 1 ( X 2 f ( x 1 , x 2 ) μ ( d x 2 | x 1 ) ) μ ( π 1 - 1 ( d x 1 ) ) \int_{X_{1}\times X_{2}}f(x_{1},x_{2})\,\mu(\mathrm{d}x_{1},\mathrm{d}x_{2})=% \int_{X_{1}}\left(\int_{X_{2}}f(x_{1},x_{2})\mu(\mathrm{d}x_{2}|x_{1})\right)% \mu\left(\pi_{1}^{-1}(\mathrm{d}x_{1})\right)
  15. μ ( A × B ) = A μ ( B | x 1 ) μ ( π 1 - 1 ( d x 1 ) ) . \mu(A\times B)=\int_{A}\mu\left(B|x_{1}\right)\,\mu\left(\pi_{1}^{-1}(\mathrm{% d}x_{1})\right).
  16. E ( f | π 1 ) ( x 1 ) = X 2 f ( x 1 , x 2 ) μ ( d x 2 | x 1 ) , \operatorname{E}(f|\pi_{1})(x_{1})=\int_{X_{2}}f(x_{1},x_{2})\mu(\mathrm{d}x_{% 2}|x_{1}),
  17. μ ( A × B | π 1 ) ( x 1 ) = 1 A ( x 1 ) μ ( B | x 1 ) . \mu(A\times B|\pi_{1})(x_{1})=1_{A}(x_{1})\cdot\mu(B|x_{1}).

Dispersion_point.html

  1. X { p } X\setminus\{p\}
  2. X { p } X\setminus\{p\}
  3. X { p } X\setminus\{p\}

Displacement_field_(mechanics).html

  1. R 0 : Ω P \vec{R}_{0}:\Omega\rightarrow P
  2. R 0 \vec{R}_{0}
  3. Ω \Omega
  4. P P
  5. R 1 \vec{R}_{1}
  6. u = R 1 - R 0 \vec{u}=\vec{R}_{1}-\vec{R}_{0}
  7. u \vec{u}

Displacement_operator.html

  1. D ^ ( α ) = exp ( α a ^ - α a ^ ) \hat{D}(\alpha)=\exp\left(\alpha\hat{a}^{\dagger}-\alpha^{\ast}\hat{a}\right)
  2. α \alpha
  3. α * \alpha^{*}
  4. a ^ \hat{a}
  5. a ^ \hat{a}^{\dagger}
  6. α \alpha
  7. D ^ ( α ) | 0 = | α \hat{D}(\alpha)|0\rangle=|\alpha\rangle
  8. | α |\alpha\rangle
  9. D ^ ( α ) D ^ ( α ) = D ^ ( α ) D ^ ( α ) = 1 ^ \hat{D}(\alpha)\hat{D}^{\dagger}(\alpha)=\hat{D}^{\dagger}(\alpha)\hat{D}(% \alpha)=\hat{1}
  10. 1 ^ \hat{1}
  11. D ^ ( α ) = D ^ ( - α ) \hat{D}^{\dagger}(\alpha)=\hat{D}(-\alpha)
  12. - α -\alpha
  13. D ^ ( α ) a ^ D ^ ( α ) = a ^ + α \hat{D}^{\dagger}(\alpha)\hat{a}\hat{D}(\alpha)=\hat{a}+\alpha
  14. D ^ ( α ) a ^ D ^ ( α ) = a ^ - α \hat{D}(\alpha)\hat{a}\hat{D}^{\dagger}(\alpha)=\hat{a}-\alpha
  15. e α a ^ - α * a ^ e β a ^ - β * a ^ = e ( α + β ) a ^ - ( β * + α * ) a ^ e ( α β * - α * β ) / 2 . e^{\alpha\hat{a}^{\dagger}-\alpha^{*}\hat{a}}e^{\beta\hat{a}^{\dagger}-\beta^{% *}\hat{a}}=e^{(\alpha+\beta)\hat{a}^{\dagger}-(\beta^{*}+\alpha^{*})\hat{a}}e^% {(\alpha\beta^{*}-\alpha^{*}\beta)/2}.
  16. D ^ ( α ) D ^ ( β ) = e ( α β * - α * β ) / 2 D ^ ( α + β ) \hat{D}(\alpha)\hat{D}(\beta)=e^{(\alpha\beta^{*}-\alpha^{*}\beta)/2}\hat{D}(% \alpha+\beta)
  17. e ( α β * - α * β ) / 2 e^{(\alpha\beta^{*}-\alpha^{*}\beta)/2}
  18. D ^ ( α ) = e - 1 2 | α | 2 e + α a ^ e - α * a ^ \hat{D}(\alpha)=e^{-\frac{1}{2}|\alpha|^{2}}e^{+\alpha\hat{a}^{\dagger}}e^{-% \alpha^{*}\hat{a}}
  19. D ^ ( α ) = e + 1 2 | α | 2 e - α * a ^ e + α a ^ \hat{D}(\alpha)=e^{+\frac{1}{2}|\alpha|^{2}}e^{-\alpha^{*}\hat{a}}e^{+\alpha% \hat{a}^{\dagger}}
  20. A ^ ψ = d 𝐤 ψ ( 𝐤 ) a ^ ( 𝐤 ) \hat{A}_{\psi}^{\dagger}=\int d\mathbf{k}\psi(\mathbf{k})\hat{a}^{\dagger}(% \mathbf{k})
  21. 𝐤 \mathbf{k}
  22. ω 𝐤 \omega_{\mathbf{k}}
  23. | 𝐤 | = ω 𝐤 / c |\mathbf{k}|=\omega_{\mathbf{k}}/c
  24. D ^ ψ ( α ) = exp ( α A ^ ψ - α A ^ ψ ) \hat{D}_{\psi}(\alpha)=\exp\left(\alpha\hat{A}_{\psi}^{\dagger}-\alpha^{\ast}% \hat{A}_{\psi}\right)
  25. | α ψ D ^ ψ ( α ) | 0 |\alpha_{\psi}\rangle\equiv\hat{D}_{\psi}(\alpha)|0\rangle

Distance-regular_graph.html

  1. { - c 1 c d - 1 c d a 0 a 1 a d - 1 a d b 0 b 1 b d - 1 - } , \left\{\begin{matrix}-&c_{1}&\cdots&c_{d-1}&c_{d}\\ a_{0}&a_{1}&\cdots&a_{d-1}&a_{d}\\ b_{0}&b_{1}&\cdots&b_{d-1}&-\end{matrix}\right\},
  2. B := ( a 0 b 0 0 0 0 c 1 a 1 b 1 0 0 0 c 2 a 2 0 0 0 0 0 a d - 1 b d - 1 0 0 0 c d a d ) , B:=\begin{pmatrix}a_{0}&b_{0}&0&\cdots&0&0\\ c_{1}&a_{1}&b_{1}&\cdots&0&0\\ 0&c_{2}&a_{2}&\cdots&0&0\\ \vdots&\vdots&\vdots&&\vdots&\vdots\\ 0&0&0&\cdots&a_{d-1}&b_{d-1}\\ 0&0&0&\cdots&c_{d}&a_{d}\end{pmatrix},
  3. A A i = a i A i + b i A i + 1 + c i A i - 1 . AA_{i}=a_{i}A_{i}+b_{i}A_{i+1}+c_{i}A_{i-1}.
  4. A i A j = k = 0 d p i j k A k . A_{i}A_{j}=\sum_{k=0}^{d}p_{ij}^{k}A_{k}.

Distance_decay.html

  1. I = c o n s t . × d - 2 I=const.\times d^{-2}
  2. I 1 / d 2 I\propto 1/d^{2}
  3. I e - d I\propto e^{-d}

Distance_measures_(cosmology).html

  1. z z
  2. a = 1 / ( 1 + z ) a=1/(1+z)
  3. t t
  4. η \eta
  5. E ( z ) = Ω m ( 1 + z ) 3 + Ω k ( 1 + z ) 2 + Ω Λ E(z)=\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{k}(1+z)^{2}+\Omega_{\Lambda}}
  6. d H = c / H 0 d_{H}=c/H_{0}
  7. Ω m \Omega_{m}
  8. Ω Λ \Omega_{\Lambda}
  9. Ω k = 1 - Ω m - Ω Λ \Omega_{k}=1-\Omega_{m}-\Omega_{\Lambda}
  10. H 0 H_{0}
  11. c c
  12. z z
  13. d C ( z ) = d H 0 z d z E ( z ) d_{C}(z)=d_{H}\int_{0}^{z}\frac{dz^{\prime}}{E(z^{\prime})}
  14. d M ( z ) = { d H Ω k sinh ( Ω k d C ( z ) / d H ) for Ω k > 0 d C ( z ) for Ω k = 0 d H | Ω k | sin ( | Ω k | d C ( z ) / d H ) for Ω k < 0 d_{M}(z)=\left\{\begin{array}[]{ll}\frac{d_{H}}{\sqrt{\Omega_{k}}}\sinh\left(% \sqrt{\Omega_{k}}d_{C}(z)/d_{H}\right)&\,\text{for }\Omega_{k}>0\\ d_{C}(z)&\,\text{for }\Omega_{k}=0\\ \frac{d_{H}}{\sqrt{|\Omega_{k}}|}\sin\left(\sqrt{|\Omega_{k}|}d_{C}(z)/d_{H}% \right)&\,\text{for }\Omega_{k}<0\end{array}\right.
  15. d A ( z ) = d M ( z ) 1 + z d_{A}(z)=\frac{d_{M}(z)}{1+z}
  16. d L ( z ) = ( 1 + z ) d M ( z ) d_{L}(z)=(1+z)d_{M}(z)
  17. d T ( z ) = d H 0 z d z ( 1 + z ) E ( z ) d_{T}(z)=d_{H}\int_{0}^{z}\frac{dz^{\prime}}{(1+z^{\prime})E(z^{\prime})}
  18. Ω k 0 \Omega_{k}\to 0
  19. Ω Λ = 0.732 \Omega_{\Lambda}=0.732
  20. Ω matter = 0.266 \Omega_{\rm matter}=0.266
  21. Ω radiation = 0.266 / 3454 \Omega_{\rm radiation}=0.266/3454
  22. Ω k \Omega_{k}
  23. Ω Λ = 0.732 \Omega_{\Lambda}=0.732
  24. Ω matter = 0.266 \Omega_{\rm matter}=0.266
  25. Ω radiation = 0.266 / 3454 \Omega_{\rm radiation}=0.266/3454
  26. Ω k \Omega_{k}
  27. χ \chi
  28. r r
  29. z z
  30. δ θ \delta\theta
  31. δ θ d M ( z ) \delta\theta d_{M}(z)
  32. d M d_{M}
  33. x x
  34. z z
  35. δ θ \delta\theta
  36. d A ( z ) = x / δ θ d_{A}(z)=x/\delta\theta
  37. L L
  38. S S
  39. d L ( z ) = L / 4 π S d_{L}(z)=\sqrt{L/4\pi S}
  40. d L ( z ) d_{L}(z)
  41. d L = ( 1 + z ) 2 d A d_{L}=(1+z)^{2}d_{A}

Distributed_key_generation.html

  1. O ( n t ) O(nt)
  2. n n
  3. t t
  4. O ( l o g 3 n ) O(log^{3}n)

Distributed_minimum_spanning_tree.html

  1. O ( V log V ) O(V\log V)
  2. V V
  3. O ( V ) O(V)
  4. O ( V log * V + D ) O(\sqrt{V}\log^{*}V+D)
  5. Ω ( V log V ) . \Omega\left({\frac{\sqrt{V}}{\log V}}\right).
  6. G ( V , E ) G(V,E)
  7. V V
  8. E E
  9. O ( log V ) O(\log V)
  10. O ( log n ) O(\log n)
  11. O ( D + L log n ) O(D+L\log n)
  12. L L

Distributed_System_Security_Architecture.html

  1. A C D C : B A\rightarrow CDC:B
  2. C D C A : c e r t i f i c a t e ( B , C A ) CDC\rightarrow A:certificate(B,CA)
  3. A B : A , { T A , A } K A B , { L , A , P A } S A , { { K A B } P B } S A A\rightarrow B:A,\{T_{A},A\}K_{AB},\{L,A,P^{\prime}_{A}\}S_{A},\{\{K_{AB}\}P_{% B}\}S^{\prime}_{A}
  4. B C D C : A B\rightarrow CDC:A
  5. C D C B : c e r t i f i c a t e ( A , C A ) CDC\rightarrow B:certificate(A,CA)
  6. B A : { T A + 1 } K A B B\rightarrow A:\{T_{A}+1\}K_{AB}

Distributive_category.html

  1. A , B , C A,B,C
  2. [ 𝑖𝑑 A × ι 1 , 𝑖𝑑 A × ι 2 ] : A × B + A × C A × ( B + C ) [\mathit{id}_{A}\times\iota_{1},\mathit{id}_{A}\times\iota_{2}]:A\times B+A% \times C\to A\times(B+C)
  3. A A
  4. 0 A × 0 0\to A\times 0
  5. A A
  6. A × - A\times-
  7. f f
  8. f f
  9. A × - A\times-

Distributive_law_between_monads.html

  1. ( S , μ S , η S ) (S,\mu^{S},\eta^{S})
  2. ( T , μ T , η T ) (T,\mu^{T},\eta^{T})
  3. l : T S S T l:TS\to ST
  4. S μ T μ S T T S l T S\mu^{T}\cdot\mu^{S}TT\cdot SlT
  5. η S T η T \eta^{S}T\cdot\eta^{T}

Divided_power_structure.html

  1. x n / n ! x^{n}/n!
  2. n ! n!
  3. γ n : I A \gamma_{n}:I\to A
  4. γ 0 ( x ) = 1 \gamma_{0}(x)=1
  5. γ 1 ( x ) = x \gamma_{1}(x)=x
  6. x I x\in I
  7. γ n ( x ) I \gamma_{n}(x)\in I
  8. γ n ( x + y ) = i = 0 n γ n - i ( x ) γ i ( y ) \gamma_{n}(x+y)=\sum_{i=0}^{n}\gamma_{n-i}(x)\gamma_{i}(y)
  9. x , y I x,y\in I
  10. γ n ( λ x ) = λ n γ n ( x ) \gamma_{n}(\lambda x)=\lambda^{n}\gamma_{n}(x)
  11. λ A , x I \lambda\in A,x\in I
  12. γ m ( x ) γ n ( x ) = ( ( m , n ) ) γ m + n ( x ) \gamma_{m}(x)\gamma_{n}(x)=((m,n))\gamma_{m+n}(x)
  13. x I x\in I
  14. ( ( m , n ) ) = ( m + n ) ! m ! n ! ((m,n))=\frac{(m+n)!}{m!n!}
  15. γ n ( γ m ( x ) ) = C n , m γ m n ( x ) \gamma_{n}(\gamma_{m}(x))=C_{n,m}\gamma_{mn}(x)
  16. x I x\in I
  17. C n , m = ( m n ) ! ( m ! ) n n ! C_{n,m}=\frac{(mn)!}{(m!)^{n}n!}
  18. γ n ( x ) \gamma_{n}(x)
  19. x [ n ] x^{[n]}
  20. x := [ x , x 2 2 , , x n n ! , ] [ x ] \mathbb{Z}\langle{x}\rangle:=\mathbb{Z}[x,\frac{x^{2}}{2},\ldots,\frac{x^{n}}{% n!},\ldots]\subset\mathbb{Q}[x]
  21. \mathbb{Z}
  22. γ n ( x ) = 1 n ! x n \gamma_{n}(x)=\frac{1}{n!}\cdot x^{n}
  23. x n = n ! γ n ( x ) x^{n}=n!\gamma_{n}(x)
  24. p > 0 p>0
  25. I p = 0 I^{p}=0
  26. γ n ( x ) = 1 n ! x n \gamma_{n}(x)=\frac{1}{n!}x^{n}
  27. n p n\geq p
  28. I p I^{p}
  29. x p x^{p}
  30. x I x\in I
  31. S M S^{\cdot}M
  32. ( S M ) ˇ = H o m A ( S M , A ) (S^{\cdot}M)\check{~{}}=Hom_{A}(S^{\cdot}M,A)
  33. Γ A ( M ˇ ) \Gamma_{A}(\check{M})
  34. A x 1 , x 2 , , x n A\langle x_{1},x_{2},\ldots,x_{n}\rangle
  35. x 1 , x 2 , , x n , x_{1},x_{2},\ldots,x_{n},
  36. c x 1 [ i 1 ] x 2 [ i 2 ] x n [ i n ] cx_{1}^{[i_{1}]}x_{2}^{[i_{2}]}\cdots x_{n}^{[i_{n}]}
  37. c A c\in A
  38. Γ A ( M ) , \Gamma_{A}(M),
  39. Γ + ( M ) \Gamma_{+}(M)
  40. M Γ + ( M ) . M\to\Gamma_{+}(M).

Divisia_monetary_aggregates_index.html

  1. M t = j = 1 n x j t , M_{t}=\sum_{j=1}^{n}x_{jt},
  2. x j t x_{jt}
  3. n n
  4. M t M_{t}
  5. log M t D - log M t - 1 D = j = 1 n s j t * ( log x j t - log x j , t - 1 ) , \log M_{t}^{D}-\log M_{t-1}^{D}=\sum_{j=1}^{n}s_{jt}^{*}(\log x_{jt}-\log x_{j% ,t-1}),
  6. s j t * = 1 2 ( s j t + s j , t - 1 ) , s_{jt}^{*}=\frac{1}{2}(s_{jt}+s_{j,t-1}),
  7. j = 1 , , n j=1,...,n
  8. s j t = π j t x j t k = 1 n π k t x k t , s_{jt}=\frac{\pi_{jt}x_{jt}}{\sum_{k=1}^{n}\pi_{kt}x_{kt}},
  9. j j
  10. t t
  11. π j t \pi_{jt}
  12. j j
  13. π j t = R t - r j t 1 + R t , \pi_{jt}=\frac{R_{t}-r_{jt}}{1+R_{t}},
  14. j j
  15. r j t r_{jt}
  16. j j
  17. R t R_{t}
  18. M t D M_{t}^{D}
  19. M t M_{t}

Divisor_summatory_function.html

  1. D ( x ) = n x d ( n ) = j , k j k x 1 D(x)=\sum_{n\leq x}d(n)=\sum_{j,k\atop jk\leq x}1
  2. d ( n ) = σ 0 ( n ) = j , k j k = n 1 d(n)=\sigma_{0}(n)=\sum_{j,k\atop jk=n}1
  3. D k ( x ) = n x d k ( n ) = m n x d k - 1 ( n ) D_{k}(x)=\sum_{n\leq x}d_{k}(n)=\sum_{mn\leq x}d_{k-1}(n)
  4. O ( x ) O(\sqrt{x})
  5. D ( x ) = k = 1 x x k = 2 k = 1 u x k - u 2 D(x)=\sum_{k=1}^{x}\left\lfloor\frac{x}{k}\right\rfloor=2\sum_{k=1}^{u}\left% \lfloor\frac{x}{k}\right\rfloor-u^{2}
  6. u = x u=\left\lfloor\sqrt{x}\right\rfloor
  7. D ( x ) = x log x + x ( 2 γ - 1 ) + Δ ( x ) D(x)=x\log x+x(2\gamma-1)+\Delta(x)
  8. γ \gamma
  9. Δ ( x ) = O ( x ) . \Delta(x)=O\left(\sqrt{x}\right).
  10. O O
  11. θ \theta
  12. Δ ( x ) = O ( x θ + ϵ ) \Delta(x)=O\left(x^{\theta+\epsilon}\right)
  13. ϵ > 0 \epsilon>0
  14. O ( x 1 / 3 log x ) . O(x^{1/3}\log x).
  15. inf θ 1 / 4 \inf\theta\geq 1/4
  16. K K
  17. Δ ( x ) > K x 1 / 4 \Delta(x)>Kx^{1/4}
  18. Δ ( x ) < - K x 1 / 4 \Delta(x)<-Kx^{1/4}
  19. inf θ 33 / 100. \inf\theta\leq 33/100.
  20. inf θ 27 / 82. \inf\theta\leq 27/82.
  21. inf θ 15 / 46. \inf\theta\leq 15/46.
  22. inf θ 12 / 37 \inf\theta\leq 12/37
  23. inf θ 346 / 1067 \inf\theta\leq 346/1067
  24. inf θ 35 / 108 \inf\theta\leq 35/108
  25. inf θ 7 / 22. \inf\theta\leq 7/22.
  26. inf θ 131 / 416. \inf\theta\leq 131/416.
  27. inf θ \inf\theta
  28. Δ ( x ) / x 1 / 4 \Delta(x)/x^{1/4}
  29. D k ( x ) = x P k ( log x ) + Δ k ( x ) D_{k}(x)=xP_{k}(\log x)+\Delta_{k}(x)\,
  30. P k P_{k}
  31. k - 1 k-1
  32. Δ k ( x ) = O ( x 1 - 1 / k log k - 2 x ) \Delta_{k}(x)=O\left(x^{1-1/k}\log^{k-2}x\right)
  33. k 2 k\geq 2
  34. k = 2 k=2
  35. k k
  36. α k \alpha_{k}
  37. Δ k ( x ) = O ( x α k + ε ) \Delta_{k}(x)=O\left(x^{\alpha_{k}+\varepsilon}\right)
  38. ε > 0 \varepsilon>0
  39. α 2 \alpha_{2}
  40. θ \theta
  41. α 2 131 416 , \alpha_{2}\leq\frac{131}{416}\ ,
  42. α 3 43 96 , \alpha_{3}\leq\frac{43}{96}\ ,
  43. α k 3 k - 4 4 k ( 4 k 8 ) , \alpha_{k}\leq\frac{3k-4}{4k}\quad(4\leq k\leq 8)\ ,
  44. α 9 35 54 , α 10 41 60 , α 11 7 10 , \alpha_{9}\leq\frac{35}{54}\ ,\quad\alpha_{10}\leq\frac{41}{60}\ ,\quad\alpha_% {11}\leq\frac{7}{10}\quad\ ,
  45. α k k - 2 k + 2 ( 12 k 25 ) , \alpha_{k}\leq\frac{k-2}{k+2}\quad(12\leq k\leq 25)\ ,
  46. α k k - 1 k + 4 ( 26 k 50 ) , \alpha_{k}\leq\frac{k-1}{k+4}\quad(26\leq k\leq 50)\ ,
  47. α k 31 k - 98 32 k ( 51 k 57 ) , \alpha_{k}\leq\frac{31k-98}{32k}\quad(51\leq k\leq 57)\ ,
  48. α k 7 k - 34 7 k ( k 58 ) . \alpha_{k}\leq\frac{7k-34}{7k}\quad(k\geq 58)\ .
  49. α k = k - 1 2 k . \alpha_{k}=\frac{k-1}{2k}\ .
  50. D ( x ) = 1 2 π i c - i c + i ζ 2 ( w ) x w w d w D(x)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\zeta^{2}(w)\frac{x^{w}}{w}\,dw
  51. c > 1 c>1
  52. ζ ( s ) \zeta(s)
  53. Δ ( x ) = 1 2 π i c - i c + i ζ 2 ( w ) x w w d w \Delta(x)=\frac{1}{2\pi i}\int_{c^{\prime}-i\infty}^{c^{\prime}+i\infty}\zeta^% {2}(w)\frac{x^{w}}{w}\,dw
  54. 0 < c < 1 0<c^{\prime}<1
  55. D ( x ) D(x)
  56. w = 1 w=1
  57. D k ( x ) = 1 2 π i c - i c + i ζ k ( w ) x w w d w D_{k}(x)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\zeta^{k}(w)\frac{x^{w}}{% w}\,dw
  58. Δ k ( x ) \Delta_{k}(x)
  59. k 2 k\geq 2

DNA-directed_DNA_polymerase.html

  1. \rightleftharpoons

DNA_ligase_(NAD+).html

  1. \rightleftharpoons

DNA_supercoil.html

  1. L k o = b p / 10.4 Lk_{o}=bp/10.4
  2. L k = T w + W r Lk=Tw+Wr
  3. Δ L k = L k - L k o \Delta{Lk=Lk-Lk_{o}}
  4. Δ G / N = 10 R T σ 2 {\Delta G/N=10RT\sigma^{2}}
  5. Δ G / N = 700 K c a l / b p * ( Δ L k / N ) {\Delta G/N=700Kcal/bp*(\Delta Lk/N)}

Domain_coloring.html

  1. f : f:\mathbb{R}\rightarrow{}\mathbb{R}
  2. f ( x ) = x 2 f(x)=x^{2}
  3. g : g:\mathbb{C}\rightarrow{}\mathbb{C}
  4. z = r e i θ z=re^{i\theta}
  5. θ \theta
  6. r = | z | r=|z|
  7. w = sin ( z ) w=\sin(z)
  8. - 2 π -2\pi
  9. 2 π 2\pi
  10. - 1.5 -1.5
  11. 1.5 1.5

Dominance_(economics).html

  1. D = i = 1 n - 1 ( s i - s i + 1 ) 2 D=\sum_{i=1}^{n-1}(s_{i}-s_{i+1})^{2}
  2. s 1 s i s i + 1 s n s_{1}\geq...\geq s_{i}\geq s_{i+1}\geq...\geq s_{n}
  3. I D = i = 1 n h i 2 ID=\sum_{i=1}^{n}h_{i}^{2}
  4. h i = s i 2 / H H I . h_{i}=\left.s_{i}^{2}\right/HHI.
  5. A I = i = 1 n ( s i - 1 n ) 2 / n . AI=\left.\sum_{i=1}^{n}\left(s_{i}-{1\over n}\right)^{2}\right/n.

Doppler_echocardiography.html

  1. X 0 ( i , j ) X_{0}(i,j)
  2. X 1 ( i + α , j + β ) X_{1}(i+\alpha,j+\beta)
  3. ( α , β ) (\alpha,\beta)
  4. D ( α , β ) = i = 1 j = 1 | X 0 ( i , j ) - X 1 ( i + α , j + β ) | D(\alpha,\beta)=\sum_{i=1}\sum_{j=1}|X_{0}(i,j)-X_{1}(i+\alpha,j+\beta)|
  5. D ( α , β ) = i = 1 j = 1 ( X 0 ( i , j ) - X 1 ( i + α , j + β ) ) 2 D(\alpha,\beta)=\sum_{i=1}\sum_{j=1}(X_{0}(i,j)-X_{1}(i+\alpha,j+\beta))^{2}
  6. ρ ( α , β ) = i = 1 j = 1 ( X 0 ( i , j ) - X 0 ¯ ) ( X 1 ( i + α , j + β ) - X 1 ¯ ) ( i = 1 j = 1 ( X 0 ( i , j ) - X 0 ¯ ) 2 ) ( i = 1 j = 1 ( X 1 ( i + α , j + β ) - X 1 ¯ ) 2 ) \rho(\alpha,\beta)=\frac{\sum_{i=1}\sum_{j=1}(X_{0}(i,j)-\bar{X_{0}})(X_{1}(i+% \alpha,j+\beta)-\bar{X_{1}})}{\sqrt{(\sum_{i=1}\sum_{j=1}(X_{0}(i,j)-\bar{X_{0% }})^{2})(\sum_{i=1}\sum_{j=1}(X_{1}(i+\alpha,j+\beta)-\bar{X_{1}})^{2})}}
  7. X 0 ¯ \bar{X_{0}}
  8. X 1 ¯ \bar{X_{1}}
  9. X 0 ( i , j ) X_{0}(i,j)
  10. X 1 ( i , j ) X_{1}(i,j)
  11. ( α , β ) (\alpha,\beta)
  12. k i n t = k s - ( R 12 ( k s + 1 ) - R 12 ( k s - 1 ) ) 2 ( R 12 ( k s + 1 ) - 2 R 12 ( k s ) + R 12 ( k s - 1 ) ) k_{int}=k_{s}-\frac{(R_{12}(k_{s}+1)-R_{12}(k_{s}-1))}{2(R_{12}(k_{s}+1)-2R_{1% 2}(k_{s})+R_{12}(k_{s}-1))}
  13. R 12 R_{12}
  14. k s k_{s}
  15. k i n t k_{int}
  16. R u p R_{up}
  17. R d o w n R_{down}
  18. R l a t e r a l = R u p * R d o w n * ; R a x i a l = R u p * R d o w n R_{lateral}=R_{up}*R_{down}^{*};R_{axial}=R_{up}*R_{down}
  19. R d o w n * R_{down}^{*}
  20. R d o w n R_{down}

Double_knitting.html

  1. n n
  2. n n

Doubling_time.html

  1. T d = log ( 2 ) log ( 1 + r 100 ) T_{d}=\frac{\log(2)}{\log(1+\frac{r}{100})}
  2. 70 r \frac{70}{r}
  3. N ( t ) = C 2 t / d N(t)=C2^{t/d}
  4. T d = ( t 2 - t 1 ) * log ( 2 ) log ( q 2 q 1 ) . T_{d}=(t_{2}-t_{1})*\frac{\log(2)}{\log(\frac{q_{2}}{q_{1}})}.

Dowling_geometry.html

  1. p L ( y ) = ( y - 1 ) ( y - m - 1 ) ( y - [ n - 1 ] m - 1 ) , p_{L}(y)=(y-1)(y-m-1)\cdots(y-[n-1]m-1),

Drinfeld_module.html

  1. a 0 + a 1 τ + a 2 τ 2 + a_{0}+a_{1}\tau+a_{2}\tau^{2}+\cdots
  2. τ a = a p τ \tau a=a^{p}\tau
  3. a 0 x 1 + a 1 x p + a 2 x p 2 + = a 0 τ 0 + a 1 τ + a 2 τ 2 + a_{0}x^{1}+a_{1}x^{p}+a_{2}x^{p^{2}}+\cdots=a_{0}\tau^{0}+a_{1}\tau+a_{2}\tau^% {2}+\cdots\,
  4. \infty
  5. \infty
  6. \infty
  7. F q [ t ] F_{q}[t]
  8. ι : A L \iota:A\to L
  9. ϕ : A L { τ } \phi:A\to L\{\tau\}
  10. ϕ \phi
  11. d : L { τ } L , a 0 + a 1 τ + a 0 d:L\{\tau\}\to L,\,a_{0}+a_{1}\tau+\cdots\mapsto a_{0}
  12. ι : A L \iota:A\to L
  13. d ϕ = ι d\circ\phi=\iota
  14. ϕ \phi

Drinker_paradox.html

  1. x P . [ D ( x ) y P . D ( y ) ] . \exists x\in P.\ [D(x)\rightarrow\forall y\in P.\ D(y)].\,
  2. ¬ [ x . [ D ( x ) y . D ( y ) ] ] \neg[\exists x.\ [D(x)\rightarrow\forall y.\ D(y)]]\,
  3. x y . [ D ( x ) ¬ D ( y ) ] \forall x\exists y.\ [D(x)\wedge\neg D(y)]\,
  4. x . [ D ( x ) ¬ D ( f ( x ) ) ] \forall x.\ [D(x)\wedge\neg D(f(x))]\,
  5. D ( x ) D(x)
  6. ¬ D ( f ( x ) ) \neg D(f(x))
  7. { d , f ( d ) , f ( f ( d ) ) , f ( f ( f ( d ) ) ) , } \{d,f(d),f(f(d)),f(f(f(d))),\ldots\}
  8. x . [ D ( x ) ¬ D ( f ( x ) ) ] D ( d ) ¬ D ( f ( d ) ) E ¬ D ( f ( d ) ) E x . [ D ( x ) ¬ D ( f ( x ) ) ] D ( f ( d ) ) ¬ D ( f ( f ( d ) ) ) E D ( f ( d ) ) E E \cfrac{\cfrac{\cfrac{\forall x.\ [D(x)\wedge\neg D(f(x))]\,}{D(d)\wedge\neg D(% f(d))}\forall_{E}}{\neg D(f(d))}\wedge_{E}\qquad\cfrac{\cfrac{\forall x.\ [D(x% )\wedge\neg D(f(x))]\,}{D(f(d))\wedge\neg D(f(f(d)))}\forall_{E}}{D(f(d))}% \wedge_{E}}{\bot}\ \Rightarrow_{E}
  9. [ D ( d ) ¬ D ( f ( d ) ) ] [D(d)\wedge\neg D(f(d))]
  10. ¬ D ( f ( d ) ) \neg D(f(d))
  11. [ D ( f ( d ) ) ¬ D ( f ( f ( d ) ) ) ] [D(f(d))\wedge\neg D(f(f(d)))]
  12. D ( f ( d ) ) D(f(d))
  13. ¬ D ( f ( d ) ) \neg D(f(d))
  14. D ( f ( d ) ) D(f(d))
  15. D ( d ) ¬ D ( f ( d ) ) D(d)\wedge\neg D(f(d))
  16. D ( d ) ¬ D ( f ( d ) ) D ( f ( d ) ) ¯ ¬ D ( f ( f ( d ) ) ) D(d)\wedge\underline{\neg D(f(d))\wedge D(f(d))}\wedge\neg D(f(f(d)))
  17. A B A\rightarrow B
  18. A B A\rightarrow B
  19. D ( x ) D(x)
  20. x D ( x ) \forall xD(x)
  21. D ( x ) D(x)
  22. x x
  23. x P . [ D ( x ) y P . D ( y ) ] \exists x\in P.\ [D(x)\rightarrow\forall y\in P.\ D(y)]\,

Dual_abelian_variety.html

  1. t T t\in T
  2. f : E E f:E\rightarrow E^{\prime}
  3. n n
  4. f ^ : E E \hat{f}:E^{\prime}\rightarrow E
  5. f f ^ = [ n ] . f\circ\hat{f}=[n].
  6. [ n ] [n]
  7. n n
  8. e n e e\mapsto ne
  9. n 2 . n^{2}.
  10. E Div ( E ) 0 Div ( E ) 0 E E^{\prime}\rightarrow\mbox{Div}~{}^{0}(E^{\prime})\to\mbox{Div}~{}^{0}(E)% \rightarrow E\,
  11. Div 0 {\mathrm{Div}}^{0}
  12. E Div 0 ( E ) E\rightarrow{\mbox{Div}~{}}^{0}(E)
  13. P P - O P\to P-O
  14. O O
  15. E E
  16. Div 0 ( E ) E {\mbox{Div}~{}}^{0}(E)\rightarrow E\,
  17. n P P n P P . \sum n_{P}P\to\sum n_{P}P.
  18. f f ^ = [ n ] f\circ\hat{f}=[n]
  19. f f
  20. E Div 0 ( E ) Div 0 ( E ) E E\rightarrow{\mbox{Div}~{}}^{0}(E)\to{\mbox{Div}~{}}^{0}(E^{\prime})\to E^{% \prime}\,
  21. f f
  22. n n
  23. f * f * f_{*}f^{*}
  24. n n
  25. Div 0 ( E ) . {\mbox{Div}~{}}^{0}(E^{\prime}).
  26. Pic 0 {\mathrm{Pic}}^{0}
  27. Div 0 . {\mbox{Div}~{}}^{0}.
  28. E Div 0 ( E ) E\rightarrow{\mbox{Div}~{}}^{0}(E)
  29. E Pic 0 ( E ) . E\to{\mbox{Pic}~{}}^{0}(E).
  30. E Pic 0 ( E ) Pic 0 ( E ) E E^{\prime}\to{\mbox{Pic}~{}}^{0}(E^{\prime})\to{\mbox{Pic}~{}}^{0}(E)\to E\,
  31. f f ^ = [ n ] f\circ\hat{f}=[n]
  32. f ^ f = [ n ] . \hat{f}\circ f=[n].
  33. ϕ = f ^ f . \phi=\hat{f}\circ f.
  34. ϕ f ^ = f ^ [ n ] = [ n ] f ^ . \phi\circ\hat{f}=\hat{f}\circ[n]=[n]\circ\hat{f}.
  35. f ^ \hat{f}
  36. ϕ = [ n ] . \phi=[n].

Dual_object.html

  1. X X
  2. ( 𝐂 , , I , α , λ , ρ ) (\mathbf{C},\otimes,I,\alpha,\lambda,\rho)
  3. X * X^{*}
  4. X X
  5. η : I X X * \eta:I\to X\otimes X^{*}
  6. ε : X * X I \varepsilon:X^{*}\otimes X\to I
  7. X X
  8. X * X^{*}

Duality_(electrical_circuits).html

  1. v = i R i = v G v=iR\iff i=vG\,
  2. i C = C d d t v C v L = L d d t i L i_{C}=C\frac{d}{dt}v_{C}\iff v_{L}=L\frac{d}{dt}i_{L}
  3. v C ( t ) = V 0 + 1 C 0 t i C ( τ ) d τ i L ( t ) = I 0 + 1 L 0 t v L ( τ ) d τ v_{C}(t)=V_{0}+{1\over C}\int_{0}^{t}i_{C}(\tau)\,d\tau\iff i_{L}(t)=I_{0}+{1% \over L}\int_{0}^{t}v_{L}(\tau)\,d\tau
  4. v R 1 = v R 1 R 1 + R 2 i G 1 = i G 1 G 1 + G 2 v_{R_{1}}=v\frac{R_{1}}{R_{1}+R_{2}}\iff i_{G_{1}}=i\frac{G_{1}}{G_{1}+G_{2}}
  5. Z R = R Y G = G Z_{R}=R\iff Y_{G}=G
  6. Z G = 1 G Y R = 1 R Z_{G}={1\over G}\iff Y_{R}={1\over R}
  7. Z C = 1 C s Y L = 1 L s Z_{C}={1\over Cs}\iff Y_{L}={1\over Ls}
  8. Z L = L s Y c = C s Z_{L}=Ls\iff Y_{c}=Cs

Duplication_and_elimination_matrices.html

  1. [ a b b d ] \left[\begin{smallmatrix}a&b\\ b&d\end{smallmatrix}\right]
  2. [ 1 0 0 0 1 0 0 1 0 0 0 1 ] [ a b d ] = [ a b b d ] \begin{bmatrix}1&0&0\\ 0&1&0\\ 0&1&0\\ 0&0&1\end{bmatrix}\begin{bmatrix}a\\ b\\ d\end{bmatrix}=\begin{bmatrix}a\\ b\\ b\\ d\end{bmatrix}
  3. [ a b c d ] \left[\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right]
  4. [ 1 0 0 0 0 1 0 0 0 0 0 1 ] [ a c b d ] = [ a c d ] \begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&0&1\end{bmatrix}\begin{bmatrix}a\\ c\\ b\\ d\end{bmatrix}=\begin{bmatrix}a\\ c\\ d\end{bmatrix}

Dvoretzky–Kiefer–Wolfowitz_inequality.html

  1. F n ( x ) = 1 n i = 1 n 𝟏 { X i x } , x . F_{n}(x)=\frac{1}{n}\sum_{i=1}^{n}\mathbf{1}_{\{X_{i}\leq x\}},\qquad x\in% \mathbb{R}.
  2. Pr ( sup x ( F n ( x ) - F ( x ) ) > ε ) e - 2 n ε 2 for every ε 1 2 n ln 2 , \Pr\Bigl(\sup_{x\in\mathbb{R}}\bigl(F_{n}(x)-F(x)\bigr)>\varepsilon\Bigr)\leq e% ^{-2n\varepsilon^{2}}\qquad\,\text{for every }\varepsilon\geq\sqrt{\tfrac{1}{2% n}\ln 2},
  3. Pr ( sup x | F n ( x ) - F ( x ) | > ε ) 2 e - 2 n ε 2 for every ε > 0. \Pr\Bigl(\sup_{x\in\mathbb{R}}|F_{n}(x)-F(x)|>\varepsilon\Bigr)\leq 2e^{-2n% \varepsilon^{2}}\qquad\,\text{for every }\varepsilon>0.
  4. sup x | F n ( x ) - F ( x ) | = d sup x | G n ( F ( x ) ) - F ( x ) | sup 0 t 1 | G n ( t ) - t | , \sup_{x\in\mathbb{R}}|F_{n}(x)-F(x)|\stackrel{d}{=}\sup_{x\in\mathbb{R}}|G_{n}% (F(x))-F(x)|\leq\sup_{0\leq t\leq 1}|G_{n}(t)-t|,

Dynamic_light_scattering.html

  1. g 2 ( q ; τ ) = I ( t ) I ( t + τ ) I ( t ) 2 g^{2}(q;\tau)=\frac{\langle I(t)I(t+\tau)\rangle}{\langle I(t)\rangle^{2}}
  2. g 2 ( q ; τ ) g^{2}(q;\tau)
  3. q q
  4. τ \tau
  5. I I
  6. g 1 ( q ; τ ) g^{1}(q;\tau)
  7. g 2 ( q ; τ ) = 1 + β [ g 1 ( q ; τ ) ] 2 g^{2}(q;\tau)=1+\beta\left[g^{1}(q;\tau)\right]^{2}
  8. g 1 ( q ; τ ) = exp ( - Γ τ ) \ g^{1}(q;\tau)=\exp(-\Gamma\tau)\,
  9. Γ = q 2 D t \ \Gamma=q^{2}D_{t}\,
  10. q = 4 π n 0 λ sin ( θ 2 ) \ q=\frac{4\pi n_{0}}{\lambda}\sin\left(\frac{\theta}{2}\right)
  11. θ \theta
  12. g 1 ( q ; τ ) = i = 1 n G i ( Γ i ) exp ( - Γ i τ ) = G ( Γ ) exp ( - Γ τ ) d Γ . g^{1}(q;\tau)=\sum_{i=1}^{n}G_{i}(\Gamma_{i})\exp(-\Gamma_{i}\tau)=\int G(% \Gamma)\exp(-\Gamma\tau)\,d\Gamma.
  13. g 1 ( q ; τ ) g^{1}(q;\tau)
  14. g 1 ( q ; τ ) = exp ( - Γ ¯ τ ) ( 1 + μ 2 2 ! τ 2 - μ 3 3 ! τ 3 + ) \ g^{1}(q;\tau)=\exp\left(-\bar{\Gamma}\tau\right)\left(1+\frac{\mu_{2}}{2!}% \tau^{2}-\frac{\mu_{3}}{3!}\tau^{3}+\cdots\right)
  15. Γ ¯ \scriptstyle\bar{\Gamma}
  16. μ 2 / Γ ¯ 2 \scriptstyle\mu_{2}/\bar{\Gamma}^{2}
  17. Γ ¯ = q 2 D z \ \bar{\Gamma}=q^{2}D_{z}\,
  18. τ \ \tau
  19. Γ ¯ \scriptstyle\bar{\Gamma}
  20. A B = 5 4 4 \Mu p + 2 \Nu \Mu l \Mu p + \Mu l \Mu p - \Nu + \Mu l \frac{A}{B}=\frac{5}{4}\frac{4\Mu_{p}+2\Nu\Mu_{l}\Mu_{p}+\Mu_{l}}{\Mu_{p}-\Nu+% \Mu_{l}}
  21. A B \frac{A}{B}
  22. \Mu p \Mu_{p}
  23. \Mu l \Mu_{l}

Dynamic_pressure.html

  1. q = 1 2 ρ v 2 , q=\tfrac{1}{2}\,\rho\,v^{2},
  2. v = 2 q ρ v=\sqrt{2q\over\rho}
  3. q q\;
  4. ρ \rho\;
  5. v v\;
  6. v v
  7. v v
  8. q q
  9. q q
  10. p s = ρ m R T , p_{s}=\rho_{m}\,R\,T,\,
  11. a a
  12. M M
  13. a = γ R T m m a=\sqrt{\gamma\,R\,T\over m_{m}}
  14. M = v a , M=\frac{v}{a},
  15. q = 1 2 ρ v 2 q=\tfrac{1}{2}\,\rho\,v^{2}
  16. q = 1 2 γ p s M 2 , q=\tfrac{1}{2}\,\gamma\,p_{s}\,M^{2},
  17. p s p_{s}\;
  18. ρ m \rho_{m}\;
  19. m m m_{m}\;
  20. ρ = ρ m m m \rho\ =\rho_{m}m_{m}\;
  21. R R\;
  22. T T\;
  23. M M\;
  24. γ \gamma\;
  25. v v\;
  26. a a\;

Dynamical_system_(definition).html

  1. Φ : U T × M M \Phi:U\subset T\times M\to M
  2. I ( x ) = { t T : ( t , x ) U } I(x)=\{t\in T:(t,x)\in U\}\,
  3. Φ ( 0 , x ) = x \Phi(0,x)=x\,
  4. Φ ( t 2 , Φ ( t 1 , x ) ) = Φ ( t 1 + t 2 , x ) , \Phi(t_{2},\Phi(t_{1},x))=\Phi(t_{1}+t_{2},x),\,
  5. t 1 , t 2 , t 1 + t 2 I ( x ) \,t_{1},t_{2},t_{1}+t_{2}\in I(x)\,
  6. Φ x ( t ) := Φ ( t , x ) \Phi_{x}(t):=\Phi(t,x)\,
  7. Φ t ( x ) := Φ ( t , x ) \Phi^{t}(x):=\Phi(t,x)\,
  8. Φ x : I ( x ) M \Phi_{x}:I(x)\to M
  9. γ x := { Φ ( t , x ) : t I ( x ) } \gamma_{x}:=\{\Phi(t,x):t\in I(x)\}
  10. Φ ( t , x ) S . \Phi(t,x)\in S.
  11. ϕ n = ϕ ϕ ϕ \scriptstyle\phi^{n}=\phi\circ\phi\circ\ldots\circ\phi
  12. s y m b o l x ˙ = s y m b o l v ( t , s y m b o l x ) \dot{symbol{x}}=symbol{v}(t,symbol{x})
  13. s y m b o l x | < m t p l > t = 0 = s y m b o l x 0 symbol{x}|_{<}mtpl>{{t=0}}=symbol{x}_{0}
  14. s y m b o l x ˙ \scriptstyle{\dot{symbol{x}}}
  15. s y m b o l < m t p l > x ( t ) = Φ ( t , s y m b o l x 0 ) symbol<mtpl>{{x}}(t)=\Phi(t,symbol{{x}}_{0})
  16. s y m b o l x ˙ - s y m b o l v ( t , s y m b o l x ) = 0 < m t p l > G ( t , Φ ( t , s y m b o l x 0 ) ) = 0 \dot{symbol{x}}-symbol{v}(t,symbol{x})=0\qquad\Leftrightarrow\qquad\mathfrak{<% }mtpl>{{G}}\left(t,\Phi(t,symbol{{x}}_{0})\right)=0
  17. 𝔊 : ( T × M ) M 𝐂 \scriptstyle\mathfrak{G}:{{(T\times M)}^{M}}\to\mathbf{C}

E-function.html

  1. f ( x ) = n = 0 c n x n n ! f(x)=\sum_{n=0}^{\infty}c_{n}\frac{x^{n}}{n!}
  2. | c n | ¯ = O ( n n ε ) \overline{\left|c_{n}\right|}=O\left(n^{n\varepsilon}\right)
  3. q n = O ( n n ε ) q_{n}=O\left(n^{n\varepsilon}\right)
  4. y i = j = 1 n f i j ( x ) y j ( 1 i n ) y^{\prime}_{i}=\sum_{j=1}^{n}f_{ij}(x)y_{j}\quad(1\leq i\leq n)

Early_effect.html

  1. I C I_{\mathrm{C}}
  2. β F \beta_{\mathrm{F}}
  3. I C = I S ( 1 + V CE V A ) e V BE V T β F = β F0 ( 1 + V CE V A ) \begin{aligned}\displaystyle I_{\mathrm{C}}&\displaystyle=I_{\mathrm{S}}\left(% 1+\frac{V_{\mathrm{CE}}}{V_{\mathrm{A}}}\right)e^{\frac{V_{\mathrm{BE}}}{V_{% \mathrm{T}}}}\\ \displaystyle\beta_{\mathrm{F}}&\displaystyle=\beta_{\mathrm{F0}}\left(1+\frac% {V_{\mathrm{CE}}}{V_{\mathrm{A}}}\right)\end{aligned}
  4. V CE V_{\mathrm{CE}}
  5. V T V_{\mathrm{T}}
  6. kT / q \mathrm{kT/q}
  7. V A V_{\mathrm{A}}
  8. β F0 \beta_{\mathrm{F0}}
  9. r O = V A + V C E I C V A I C r_{O}=\frac{V_{A}+V_{CE}}{I_{C}}\approx\frac{V_{A}}{I_{C}}
  10. V C B V_{CB}
  11. r O = V A + V C B I C r_{O}=\frac{V_{A}+V_{CB}}{I_{C}}
  12. r O r_{O}
  13. V C B V_{CB}
  14. r O = 1 + λ V D S λ I D = 1 I D ( 1 λ + V D S ) r_{O}=\frac{1+\lambda V_{DS}}{\lambda I_{D}}=\frac{1}{I_{D}}\left(\frac{1}{% \lambda}+V_{DS}\right)
  15. V D S V_{DS}
  16. I D I_{D}
  17. λ \lambda
  18. d 2 Δ p B ( x ) d x 2 = Δ p B ( x ) L B 2 \frac{d^{2}\Delta p_{\,\text{B}}(x)}{dx^{2}}=\frac{\Delta p_{\,\text{B}}(x)}{L% _{\,\text{B}}^{2}}
  19. C 1 C_{1}
  20. C 2 C_{2}
  21. Δ p B ( x ) = C 1 e x L B + C 2 e - x L B \Delta p_{\,\text{B}}(x)=C_{1}e^{\frac{x}{L_{\,\text{B}}}}+C_{2}e^{-\frac{x}{L% _{\,\text{B}}}}
  22. 0
  23. 0 0^{\prime}
  24. 0 ′′ 0^{\prime\prime}
  25. Δ n B ( x ′′ ) = A 1 e x ′′ L B + A 2 e - x ′′ L B Δ n c ( x ) = B 1 e x L B + B 2 e - x L B \begin{aligned}\displaystyle\Delta n_{\,\text{B}}(x^{\prime\prime})&% \displaystyle=A_{1}e^{\frac{x^{\prime\prime}}{L_{\,\text{B}}}}+A_{2}e^{-\frac{% x^{\prime\prime}}{L_{\,\text{B}}}}\\ \displaystyle\Delta n_{\,\text{c}}(x^{\prime})&\displaystyle=B_{1}e^{\frac{x^{% \prime}}{L_{\,\text{B}}}}+B_{2}e^{-\frac{x^{\prime}}{L_{\,\text{B}}}}\end{aligned}
  26. Δ n E ( 0 ′′ ) = n E O ( e 1 k T q V EB - 1 ) \Delta n_{\,\text{E}}(0^{\prime\prime})=n_{\,\text{E}O}\left(e^{\frac{1}{kT}qV% _{\,\text{EB}}}-1\right)
  27. A 1 A_{1}
  28. B 1 B_{1}
  29. x ′′ 0 x^{\prime\prime}\rightarrow 0
  30. x 0 x^{\prime}\rightarrow 0
  31. Δ n E ( x ′′ ) 0 Δ n c ( x ) 0 \begin{aligned}\displaystyle\Delta n_{\,\text{E}}(x^{\prime\prime})&% \displaystyle\rightarrow 0\\ \displaystyle\Delta n_{\,\text{c}}(x^{\prime})&\displaystyle\rightarrow 0\end{aligned}
  32. A 1 = B 1 = 0 A_{1}=B_{1}=0
  33. Δ n E ( 0 ′′ ) \Delta n_{\,\text{E}}(0^{\prime\prime})
  34. Δ n c ( 0 ) \Delta n_{\,\text{c}}(0^{\prime})
  35. A 2 A_{2}
  36. B 2 B_{2}
  37. Δ n E ( x ′′ ) = n E 0 ( e q V EB k T - 1 ) e - x ′′ L E Δ n C ( x ) = n C 0 ( e q V CB k T - 1 ) e - x L C \begin{aligned}\displaystyle\Delta n_{\,\text{E}}(x^{\prime\prime})&% \displaystyle=n_{\,\text{E}0}\left(e^{\frac{qV_{\,\text{EB}}}{kT}}-1\right)e^{% -\frac{x^{\prime\prime}}{L_{\,\text{E}}}}\\ \displaystyle\Delta n_{\,\text{C}}(x^{\prime})&\displaystyle=n_{\,\text{C}0}% \left(e^{\frac{qV_{\,\text{CB}}}{kT}}-1\right)e^{-\frac{x^{\prime}}{L_{\,\text% {C}}}}\end{aligned}
  38. I E n I_{\,\text{E}n}
  39. I C n I_{\,\text{C}n}
  40. I E n = - q A D E d Δ E ( x ′′ ) d x | x ′′ = 0 ′′ I C n = - q A D C L C n C 0 ( e q V CB k T - 1 ) \begin{aligned}\displaystyle I_{\,\text{E}n}&\displaystyle=\left.-qAD_{\,\text% {E}}\frac{d\Delta_{\,\text{E}}(x^{\prime\prime})}{dx}\right|_{x^{\prime\prime}% =0^{\prime\prime}}\\ \displaystyle I_{\,\text{C}n}&\displaystyle=-qA\frac{D_{\,\text{C}}}{L_{\,% \text{C}}}n_{\,\text{C}0}\left(e^{\frac{qV_{\,\text{CB}}}{kT}}-1\right)\end{aligned}
  41. Δ p B ( x ) \Delta p_{\,\text{B}}(x)
  42. x x
  43. Δ p B ( x ) = D 1 x + D 2 \Delta p_{\,\text{B}}(x)=D_{1}x+D_{2}
  44. Δ p B \Delta p_{\,\text{B}}
  45. Δ p B ( 0 ) = D 2 Δ p B ( W ) = D 1 W + Δ p B ( 0 ) \begin{aligned}\displaystyle\Delta p_{\,\text{B}}(0)&\displaystyle=D_{2}\\ \displaystyle\Delta p_{\,\text{B}}(W)&\displaystyle=D_{1}W+\Delta p_{\,\text{B% }}(0)\end{aligned}
  46. Δ p B ( x ) = - 1 W [ Δ p B ( 0 ) - Δ p B ( W ) ] x + Δ p B ( 0 ) \Delta p\text{B}(x)=-\frac{1}{W}\left[\Delta p\text{B}(0)-\Delta p\text{B}(W)% \right]x+\Delta p\text{B}(0)
  47. I E p I_{\,\text{E}p}
  48. I E p ( 0 ) = - q A D B d Δ p B d x | x = 0 I E p ( 0 ) = q A D B W [ Δ p B ( 0 ) - Δ p B ( W ) ] \begin{aligned}\displaystyle I_{\,\text{E}p}(0)&\displaystyle=-qAD_{\,\text{B}% }\frac{d\Delta p\text{B}}{dx}|_{x=0}\\ \displaystyle I_{\,\text{E}p}(0)&\displaystyle=\frac{qAD\text{B}}{W}\left[% \Delta p\text{B}(0)-\Delta p\text{B}(W)\right]\end{aligned}
  49. I E p I_{\,\text{E}p}
  50. I E n I_{\,\text{E}n}
  51. Δ p B ( 0 ) \Delta p_{\,\text{B}}(0)
  52. Δ p B ( W ) \Delta p_{\,\text{B}}(W)
  53. Δ p B ( W ) = p B 0 e q V CB k T Δ p B ( 0 ) = p B 0 e q V EB k T I E = q A [ ( D E n E 0 L E + D B p B 0 W ) ( e q V EB k T - 1 ) - D B W p B 0 ( e q V CB k T - 1 ) ] \begin{aligned}\displaystyle\Delta p_{\,\text{B}}(W)&\displaystyle=p_{\,\text{% B}0}e^{\frac{qV\text{CB}}{kT}}\\ \displaystyle\Delta p_{\,\text{B}}(0)&\displaystyle=p_{\,\text{B}0}e^{\frac{qV% \text{EB}}{kT}}\\ \displaystyle I_{\,\text{E}}&\displaystyle=qA\left[\left(\frac{D\text{E}n_{\,% \text{E}0}}{L\text{E}}+\frac{D\text{B}p_{\,\text{B}0}}{W}\right)\left(e^{\frac% {qV\text{EB}}{kT}}-1\right)-\frac{D_{\,\text{B}}}{W}p_{\,\text{B}0}\left(e^{% \frac{qV_{\,\text{CB}}}{kT}}-1\right)\right]\end{aligned}
  54. I C p ( W ) = I E p ( 0 ) I C = I C p ( W ) + I C n ( 0 ) I C = q A [ D B W p B 0 ( e q V EB k T - 1 ) - ( D C n C 0 L C + D B p B 0 W ) ( e q V CB k T - 1 ) ] \begin{aligned}\displaystyle I_{\,\text{C}p}(W)&\displaystyle=I_{\,\text{E}p}(% 0)\\ \displaystyle I_{\,\text{C}}&\displaystyle=I_{\,\text{C}p}(W)+I_{\,\text{C}n}(% 0^{\prime})\\ \displaystyle I_{\,\text{C}}&\displaystyle=qA\left[\frac{D\text{B}}{W}p_{\,% \text{B}0}\left(e^{\frac{qV\text{EB}}{kT}}-1\right)-\left(\frac{D\text{C}n_{\,% \text{C}0}}{L\text{C}}+\frac{D\text{B}p_{\,\text{B}0}}{W}\right)\left(e^{\frac% {qV\text{CB}}{kT}}-1\right)\right]\end{aligned}
  55. I B = I E - I C I B = q A [ D E L E n E 0 ( e q V EB k T - 1 ) + D C L C n C 0 ( e q V CB k T - 1 ) ] \begin{aligned}\displaystyle I_{\,\text{B}}&\displaystyle=I_{\,\text{E}}-I_{\,% \text{C}}\\ \displaystyle I_{\,\text{B}}&\displaystyle=qA\left[\frac{D\text{E}}{L\text{E}}% n_{\,\text{E}0}\left(e^{\frac{qV\text{EB}}{kT}}-1\right)+\frac{D\text{C}}{L% \text{C}}n_{\,\text{C}0}\left(e^{\frac{qV\text{CB}}{kT}}-1\right)\right]\end{aligned}

Early_ITU_model.html

  1. L = 0.2 f 0.3 d 0.6 L=0.2\,f^{0.3}\,d^{0.6}

Earnings_response_coefficient.html

  1. U R = a + b ( ern - u ) + e UR=a+b(\,\text{ern}-u)+e

Ebullioscopic_constant.html

  1. Δ T = i K b m \mathrm{\Delta T=i\cdot K_{b}\cdot m}
  2. i \mathrm{i}
  3. K b = RT b 2 / 1000 L v \mathrm{K_{b}=RT_{b}^{2}/1000L_{v}}
  4. R \mathrm{R}
  5. T b \mathrm{T_{b}}
  6. L v \mathrm{L_{v}}
  7. Δ T \Delta\mathrm{T}

Ecosystem_model.html

  1. d X d t = α . X - β . X . Y \frac{dX}{dt}=\alpha.X-\beta.X.Y
  2. d Y d t = γ . β . X . Y - δ . Y \frac{dY}{dt}=\gamma.\beta.X.Y-\delta.Y
  3. X X
  4. Y Y
  5. α \alpha
  6. β \beta
  7. Y Y
  8. X X
  9. γ \gamma
  10. Y Y
  11. δ \delta

Eddy_diffusion.html

  1. m 2 s - 1 m^{2}s^{-1}

Edmund_Clifton_Stoner.html

  1. ϵ ( k ) = ϵ 0 ( k ) - I n - n n \epsilon_{\uparrow}(k)=\epsilon_{0}(k)-I\frac{n_{\uparrow}-n_{\downarrow}}{n}
  2. ϵ ( k ) = ϵ 0 ( k ) + I n - n n \epsilon_{\downarrow}(k)=\epsilon_{0}(k)+I\frac{n_{\uparrow}-n_{\downarrow}}{n}
  3. ϵ 0 ( k ) \epsilon_{0}(k)
  4. ϵ \epsilon_{\uparrow}
  5. ϵ \epsilon_{\downarrow}
  6. I I
  7. n = n + n n=n_{\uparrow}+n_{\downarrow}
  8. k k
  9. k k

Effective_number_of_bits.html

  1. ENOB = SINAD - 1.76 6.02 \mathrm{ENOB}=\frac{\mathrm{SINAD}-1.76}{6.02}
  2. 6.02 20 log 10 2 6.02\approx 20\log_{10}2

Effective_porosity.html

  1. V c l V_{cl}
  2. V s h V_{sh}
  3. Effective porosity = Total porosity - CBW \,\text{Effective porosity}=\,\text{Total porosity}-\,\text{CBW}
  4. ϕ e 1 \phi_{e1}
  5. ϕ e 2 \phi_{e2}
  6. ϕ e 3 \phi_{e3}
  7. ϕ e 4 \phi_{e4}
  8. ϕ e 5 \phi_{e5}
  9. ϕ e 6 \phi_{e6}
  10. CBW = ϕ t SF Qv \,\text{CBW}=\phi_{t}\cdot\,\text{SF}\cdot\,\text{Qv}
  11. ϕ t \phi_{t}
  12. SF \,\text{SF}
  13. Qv \,\text{Qv}
  14. SF \,\text{SF}
  15. 0.6425 S - 0.5 + 0.22 0.6425\cdot S^{-0.5}+0.22
  16. S S

Effective_radius.html

  1. R e R_{e}
  2. R e R_{e}
  3. R 4 \sqrt[4]{R}
  4. I ( R ) = I e e - 7.67 ( R R e 4 - 1 ) I(R)=I_{e}\cdot e^{-7.67\left(\sqrt[4]{\frac{R}{R_{e}}}-1\right)}
  5. I e I_{e}
  6. R = R e R=R_{e}
  7. R = 0 R=0
  8. I ( R = 0 ) = I e e 7.67 2000 I e I(R=0)=I_{e}\cdot e^{7.67}\approx 2000\cdot I_{e}
  9. 2000 I e 2000\cdot I_{e}

Egli_model.html

  1. P R 50 = 0.668 G B G M [ h B h M d 2 ] 2 [ 40 f ] 2 P T P_{R50}=0.668G_{B}G_{M}\left[\frac{h_{B}h_{M}}{d^{2}}\right]^{2}\left[{\frac{4% 0}{f}}\right]^{2}P_{T}
  2. P R 50 P_{R50}
  3. P T P_{T}
  4. G B G_{B}
  5. G M G_{M}
  6. h B h_{B}
  7. h M h_{M}
  8. d d
  9. f f

Ehresmann_connection.html

  1. γ ~ ( t ) \tilde{\gamma}(t)
  2. γ ~ ( 0 ) = e \tilde{\gamma}(0)=e
  3. π ( γ ~ ( t ) ) = γ ( t ) . \pi(\tilde{\gamma}(t))=\gamma(t).
  4. γ ~ ( t ) H γ ~ ( t ) . \tilde{\gamma}^{\prime}(t)\in H_{\tilde{\gamma}(t)}.
  5. X ~ T e E \tilde{X}\in T_{e}E
  6. R = 1 2 [ v , v ] R=\tfrac{1}{2}[v,v]
  7. R ( X , Y ) = v ( [ ( id - v ) X , ( id - v ) Y ] ) R(X,Y)=v\left([(\mathrm{id}-v)X,(\mathrm{id}-v)Y]\right)
  8. R ( X , Y ) = [ X H , Y H ] V R\left(X,Y\right)=\left[X_{H},Y_{H}\right]_{V}
  9. [ v , R ] = 0 \left[v,R\right]=0
  10. H e g = d ( R g ) e ( H e ) H_{eg}=\mathrm{d}(R_{g})_{e}(H_{e})
  11. d ( R g ) e \mathrm{d}(R_{g})_{e}
  12. V e V_{e}
  13. ι : V e 𝔤 \iota\colon V_{e}\to\mathfrak{g}
  14. R h * ω = Ad ( h - 1 ) ω R_{h}^{*}\omega=\hbox{Ad}(h^{-1})\omega
  15. σ : E × M E E \sigma:E\times_{M}E\to E
  16. H λ e = d ( S λ ) e ( H e ) H_{\lambda e}=\mathrm{d}(S_{\lambda})_{e}(H_{e})
  17. d σ ( H H ) = H d\sigma(H\boxtimes H)=H
  18. H H H\boxtimes H
  19. E × M E E\times_{M}E

Ehrling's_lemma.html

  1. x Y ε x X + C ( ε ) x Z \|x\|_{Y}\leq\varepsilon\|x\|_{X}+C(\varepsilon)\|x\|_{Z}
  2. : H k ( Ω ) 𝐑 : u u := | α | k D α u L 2 ( Ω ) 2 \|\cdot\|:H^{k}(\Omega)\to\mathbf{R}:u\mapsto\|u\|:=\sqrt{\sum_{|\alpha|\leq k% }\|\mathrm{D}^{\alpha}u\|_{L^{2}(\Omega)}^{2}}
  3. : H k ( Ω ) 𝐑 : u u := u L 1 ( Ω ) 2 + | α | = k D α u L 2 ( Ω ) 2 . \|\cdot\|^{\prime}:H^{k}(\Omega)\to\mathbf{R}:u\mapsto\|u\|^{\prime}:=\sqrt{\|% u\|_{L^{1}(\Omega)}^{2}+\sum_{|\alpha|=k}\|\mathrm{D}^{\alpha}u\|_{L^{2}(% \Omega)}^{2}}.

EigenTrust.html

  1. s i j = sat ( i , j ) - unsat ( i , j ) s_{ij}=\operatorname{sat}(i,j)-\operatorname{unsat}(i,j)
  2. c i j = max ( s i j , 0 ) j max ( s i j , 0 ) c_{ij}=\frac{\max(s_{ij},0)}{\sum_{j}\max(s_{ij},0)}
  3. t i k = j c i j c j k t_{ik}=\sum_{j}c_{ij}c_{jk}
  4. t ¯ i \bar{t}_{i}
  5. t i k t_{ik}
  6. t ¯ i = C T c ¯ i . \bar{t}_{i}=C^{T}\bar{c}_{i}.\,
  7. t ¯ = ( C T ) x c ¯ i . \bar{t}=(C^{T})^{x}\bar{c}_{i}.\,
  8. t ¯ i \bar{t}_{i}
  9. t ¯ i \bar{t}_{i}
  10. t ¯ i \bar{t}_{i}
  11. c ¯ i \bar{c}_{i}
  12. e ¯ \bar{e}
  13. t ¯ 0 = e ¯ ; \bar{t}_{0}=\bar{e};
  14. t ¯ ( k + 1 ) = C T t ¯ ( k ) ; \bar{t}^{(k+1)}=C^{T}\bar{t}^{(k)};
  15. δ = || t ( k + 1 ) - t ( k ) || ; {\delta}=||t^{(k+1)}-t^{(k)}||;
  16. δ < error ; {\delta}<\mathrm{error};

Eight_foot_pitch.html

  1. f = v 2 l f=\frac{v}{2l}

Einstein_radius.html

  1. α 1 = 4 G c 2 M b 1 \alpha_{1}=\frac{4G}{c^{2}}\frac{M}{b_{1}}
  2. α 1 ( θ 1 ) = 4 G c 2 M θ 1 1 D L \alpha_{1}(\theta_{1})=\frac{4G}{c^{2}}\frac{M}{\theta_{1}}\frac{1}{D_{\rm L}}
  3. θ 1 D S = θ S D S + α 1 D LS \theta_{1}\;D_{\rm S}=\theta_{\rm S}\;D_{\rm S}+\alpha_{1}\;D_{\rm LS}
  4. α 1 ( θ 1 ) = D S D LS ( θ 1 - θ S ) \alpha_{1}(\theta_{1})=\frac{D_{\rm S}}{D_{\rm LS}}(\theta_{1}-\theta_{\rm S})
  5. θ 1 - θ S = 4 G c 2 M θ 1 D LS D S D L \theta_{1}-\theta_{\rm S}=\frac{4G}{c^{2}}\;\frac{M}{\theta_{1}}\;\frac{D_{\rm LS% }}{D_{\rm S}D_{\rm L}}
  6. θ E = ( 4 G M c 2 D LS D L D S ) 1 / 2 \theta_{E}=\left(\frac{4GM}{c^{2}}\;\frac{D_{\rm LS}}{D_{\rm L}D_{\rm S}}% \right)^{1/2}
  7. θ 1 = θ S + θ E 2 θ 1 \theta_{1}=\theta_{\rm S}+\frac{\theta_{E}^{2}}{\theta_{1}}
  8. θ E = ( M 10 11.09 M ) 1 / 2 ( D L D S / D LS Gpc ) - 1 / 2 arcsec \theta_{E}=\left(\frac{M}{10^{11.09}M_{\bigodot}}\right)^{1/2}\left(\frac{D_{% \rm L}D_{\rm S}/D_{\rm LS}}{\rm{Gpc}}\right)^{-1/2}\rm{arcsec}
  9. θ 2 D S = - θ S D S + α 2 D LS \theta_{2}\;D_{\rm S}=-\;\theta_{\rm S}\;D_{\rm S}+\alpha_{2}\;D_{\rm LS}
  10. θ 2 + θ S = 4 G c 2 M θ 2 D LS D S D L \theta_{2}+\theta_{\rm S}=\frac{4G}{c^{2}}\;\frac{M}{\theta_{2}}\;\frac{D_{\rm LS% }}{D_{\rm S}D_{\rm L}}
  11. θ 2 = - θ S + θ E 2 θ 2 \theta_{2}=-\;\theta_{\rm S}+\frac{\theta_{E}^{2}}{\theta_{2}}

Ekman_transport.html

  1. A z A_{z}\,\!
  2. f f\,\!
  3. 1 ρ τ x z = - f v , \frac{1}{\rho}\frac{\partial\tau_{x}}{\partial z}=-fv,\,
  4. 1 ρ τ y z = f u , \frac{1}{\rho}\frac{\partial\tau_{y}}{\partial z}=fu,\,
  5. τ \tau\,\!
  6. ρ \rho\,\!
  7. u u\,\!
  8. v v\,\!
  9. τ x = - M y f , \tau_{x}=-M_{y}f,\,
  10. τ y = M x f , \tau_{y}=M_{x}f,\,
  11. M x = 0 z ρ u d z , M_{x}=\int^{z}_{0}\rho udz,\,
  12. M y = 0 z ρ v d z . M_{y}=\int^{z}_{0}\rho vdz.\,
  13. M x M_{x}\,\!
  14. M y M_{y}\,\!
  15. τ x z = ρ A z 2 u z 2 , \frac{\partial\tau_{x}}{\partial z}=\rho A_{z}\frac{\partial^{2}u}{\partial z^% {2}},\,\!
  16. τ y z = ρ A z 2 v z 2 , \frac{\partial\tau_{y}}{\partial z}=\rho A_{z}\frac{\partial^{2}v}{\partial z^% {2}},\,\!
  17. A z A_{z}\,\!
  18. A z 2 u z 2 = - f v , A_{z}\frac{\partial^{2}u}{\partial z^{2}}=-fv,\,\!
  19. A z 2 v z 2 = f u . A_{z}\frac{\partial^{2}v}{\partial z^{2}}=fu.\,\!
  20. ( u , v ) 0 {(u,v)\to 0}
  21. z , {z\to\infty},
  22. z = 0 z=0\,\!
  23. u E = ± V 0 cos ( π 4 + π D E z ) exp ( π D E z ) , v E = V 0 sin ( π 4 + π D E z ) exp ( π D E z ) , \begin{aligned}\displaystyle u_{E}&\displaystyle=\pm V_{0}\cos\left(\frac{\pi}% {4}+\frac{\pi}{D_{E}}z\right)\exp\left(\frac{\pi}{D_{E}}z\right),\\ \displaystyle v_{E}&\displaystyle=V_{0}\sin\left(\frac{\pi}{4}+\frac{\pi}{D_{E% }}z\right)\exp\left(\frac{\pi}{D_{E}}z\right),\end{aligned}
  24. u E u_{E}\,\!
  25. v E v_{E}\,\!
  26. V 0 = 2 π τ y η D E ρ | f | ; V_{0}=\frac{\sqrt{2}\pi\tau_{y\eta}}{D_{E}\rho|f|};\,\!
  27. τ y η \tau_{y\eta}\,\!
  28. D E = π ( 2 A z | f | ) 1 / 2 D_{E}=\pi\left(\frac{2A_{z}}{|f|}\right)^{1/2}\,\!

Elasticity_of_a_function.html

  1. E f ( a ) = a f ( a ) f ( a ) Ef(a)=\frac{a}{f(a)}f^{\prime}(a)
  2. = lim x a f ( x ) - f ( a ) x - a a f ( a ) = lim x a f ( x ) - f ( a ) f ( a ) a x - a = lim x a 1 - f ( x ) f ( a ) 1 - x a % Δ f ( a ) % Δ a =\lim_{x\to a}\frac{f(x)-f(a)}{x-a}\frac{a}{f(a)}=\lim_{x\to a}\frac{f(x)-f(a)% }{f(a)}\frac{a}{x-a}=\lim_{x\to a}\frac{1-\frac{f(x)}{f(a)}}{1-\frac{x}{a}}% \approx\frac{\%\Delta f(a)}{\%\Delta a}
  3. E f ( x ) = d log f ( x ) d log x . Ef(x)=\frac{d\log f(x)}{d\log x}.
  4. f ( x ) f(x)
  5. x x
  6. ( a , f ( a ) ) (a,f(a))
  7. α \alpha
  8. f ( x ) = C x α f(x)=Cx^{\alpha}
  9. C > 0 C>0
  10. E ( f ( x ) g ( x ) ) = E f ( x ) + E g ( x ) E(f(x)\cdot g(x))=Ef(x)+Eg(x)
  11. E f ( x ) g ( x ) = E f ( x ) - E g ( x ) E\frac{f(x)}{g(x)}=Ef(x)-Eg(x)
  12. E ( f ( x ) + g ( x ) ) = f ( x ) E ( f ( x ) ) + g ( x ) E ( g ( x ) ) f ( x ) + g ( x ) E(f(x)+g(x))=\frac{f(x)\cdot E(f(x))+g(x)\cdot E(g(x))}{f(x)+g(x)}
  13. E ( f ( x ) - g ( x ) ) = f ( x ) E ( f ( x ) ) - g ( x ) E ( g ( x ) ) f ( x ) - g ( x ) E(f(x)-g(x))=\frac{f(x)\cdot E(f(x))-g(x)\cdot E(g(x))}{f(x)-g(x)}
  14. D f ( x ) = E f ( x ) f ( x ) x Df(x)=\frac{Ef(x)\cdot f(x)}{x}
  15. E ( a ) = 0 E(a)=0
  16. E ( a f ( x ) ) = E f ( x ) E(a\cdot f(x))=Ef(x)
  17. E ( b x a ) = a E(bx^{a})=a
  18. S f ( x ) = 1 f ( x ) f ( x ) = d ln f ( x ) d x Sf(x)=\frac{1}{f(x)}f^{\prime}(x)=\frac{d\ln f(x)}{dx}
  19. d f ( x ) d ln ( x ) = d f ( x ) d x x \frac{df(x)}{d\ln(x)}=\frac{df(x)}{dx}x

Elasticity_of_substitution.html

  1. U ( c 1 , c 2 ) U(c_{1},c_{2})
  2. E 21 = d ln ( c 2 / c 1 ) d ln ( M R S 12 ) = d ln ( c 2 / c 1 ) d ln ( U c 1 / U c 2 ) = d ( c 2 / c 1 ) c 2 / c 1 d ( U c 1 / U c 2 ) U c 1 / U c 2 = d ( c 2 / c 1 ) c 2 / c 1 d ( p 1 / p 2 ) p 1 / p 2 E_{21}=\frac{d\ln(c_{2}/c_{1})}{d\ln(MRS_{12})}=\frac{d\ln(c_{2}/c_{1})}{d\ln(% U_{c_{1}}/U_{c_{2}})}=\frac{\frac{d(c_{2}/c_{1})}{c_{2}/c_{1}}}{\frac{d(U_{c_{% 1}}/U_{c_{2}})}{U_{c_{1}}/U_{c_{2}}}}=\frac{\frac{d(c_{2}/c_{1})}{c_{2}/c_{1}}% }{\frac{d(p_{1}/p_{2})}{p_{1}/p_{2}}}
  3. M R S MRS
  4. M R S 12 = p 1 / p 2 MRS_{12}=p_{1}/p_{2}
  5. E 21 = E 12 E_{21}=E_{12}
  6. E 21 = d ln ( c 2 / c 1 ) d ln ( U c 1 / U c 2 ) = d ( - ln ( c 1 / c 2 ) ) d ( - ln ( U c 2 / U c 1 ) ) = d ln ( c 1 / c 2 ) d ln ( U c 2 / U c 1 ) = E 12 E_{21}=\frac{d\ln(c_{2}/c_{1})}{d\ln(U_{c_{1}}/U_{c_{2}})}=\frac{d\left(-\ln(c% _{1}/c_{2})\right)}{d\left(-\ln(U_{c_{2}}/U_{c_{1}})\right)}=\frac{d\ln(c_{1}/% c_{2})}{d\ln(U_{c_{2}}/U_{c_{1}})}=E_{12}
  7. E 21 = d ln ( c 2 / c 1 ) d ln ( M R S 12 ) = - d ln ( c 2 / c 1 ) d ln ( M R S 21 ) = - d ln ( c 2 / c 1 ) d ln ( U c 2 / U c 1 ) = - d ( c 2 / c 1 ) c 2 / c 1 d ( U c 2 / U c 1 ) U c 2 / U c 1 = - d ( c 2 / c 1 ) c 2 / c 1 d ( p 2 / p 1 ) p 2 / p 1 E_{21}=\frac{d\ln(c_{2}/c_{1})}{d\ln(MRS_{12})}=-\frac{d\ln(c_{2}/c_{1})}{d\ln% (MRS_{21})}=-\frac{d\ln(c_{2}/c_{1})}{d\ln(U_{c_{2}}/U_{c_{1}})}=-\frac{\frac{% d(c_{2}/c_{1})}{c_{2}/c_{1}}}{\frac{d(U_{c_{2}}/U_{c_{1}})}{U_{c_{2}}/U_{c_{1}% }}}=-\frac{\frac{d(c_{2}/c_{1})}{c_{2}/c_{1}}}{\frac{d(p_{2}/p_{1})}{p_{2}/p_{% 1}}}
  8. t t
  9. t + 1 t+1
  10. f ( x 1 , x 2 ) f(x_{1},x_{2})
  11. σ 21 = d ln ( x 2 / x 1 ) d ln M R T S 12 = d ln ( x 2 / x 1 ) d ln ( d f d x 1 / d f d x 2 ) = d ( x 2 / x 1 ) x 2 / x 1 d ( d f d x 1 / d f d x 2 ) d f d x 1 / d f d x 2 = - d ( x 2 / x 1 ) x 2 / x 1 d ( d f d x 2 / d f d x 1 ) d f d x 2 / d f d x 1 \sigma_{21}=\frac{d\ln(x_{2}/x_{1})}{d\ln MRTS_{12}}=\frac{d\ln(x_{2}/x_{1})}{% d\ln(\frac{df}{dx_{1}}/\frac{df}{dx_{2}})}=\frac{\frac{d(x_{2}/x_{1})}{x_{2}/x% _{1}}}{\frac{d(\frac{df}{dx_{1}}/\frac{df}{dx_{2}})}{\frac{df}{dx_{1}}/\frac{% df}{dx_{2}}}}=-\frac{\frac{d(x_{2}/x_{1})}{x_{2}/x_{1}}}{\frac{d(\frac{df}{dx_% {2}}/\frac{df}{dx_{1}})}{\frac{df}{dx_{2}}/\frac{df}{dx_{1}}}}
  12. M R T S MRTS
  13. f ( x 1 , x 2 ) = x 1 a x 2 1 - a f(x_{1},x_{2})=x_{1}^{a}x_{2}^{1-a}
  14. M R T S 12 = a 1 - a x 2 x 1 MRTS_{12}=\frac{a}{1-a}\frac{x_{2}}{x_{1}}
  15. a 1 - a x 2 x 1 = θ \frac{a}{1-a}\frac{x_{2}}{x_{1}}=\theta
  16. x 2 x 1 = 1 - a a θ \frac{x_{2}}{x_{1}}=\frac{1-a}{a}\theta
  17. σ 21 = d ln ( x 2 x 1 ) d ln M R T S 12 = d ln ( x 2 x 1 ) d ln ( a 1 - a x 2 x 1 ) = d ln ( 1 - a a θ ) d ln ( θ ) = d 1 - a a θ d θ θ 1 - a a θ = 1 \sigma_{21}=\frac{d\ln(\frac{x_{2}}{x_{1}})}{d\ln MRTS_{12}}=\frac{d\ln(\frac{% x_{2}}{x_{1}})}{d\ln(\frac{a}{1-a}\frac{x_{2}}{x_{1}})}=\frac{d\ln(\frac{1-a}{% a}\theta)}{d\ln(\theta)}=\frac{d\frac{1-a}{a}\theta}{d\theta}\frac{\theta}{% \frac{1-a}{a}\theta}=1
  18. S 21 S_{21}
  19. c 2 c_{2}
  20. c 1 c_{1}
  21. S 21 p 2 c 2 p 1 c 1 S_{21}\equiv\frac{p_{2}c_{2}}{p_{1}c_{1}}
  22. p 2 / p 1 p_{2}/p_{1}
  23. d S 21 d ( p 2 / p 1 ) = c 2 c 1 + p 2 p 1 d ( c 2 / c 1 ) d ( p 2 / p 1 ) = c 2 c 1 [ 1 + d ( c 2 / c 1 ) d ( p 2 / p 1 ) p 2 / p 1 c 2 / c 1 ] = c 2 c 1 ( 1 - E 21 ) \frac{dS_{21}}{d\left(p_{2}/p_{1}\right)}=\frac{c_{2}}{c_{1}}+\frac{p_{2}}{p_{% 1}}\cdot\frac{d\left(c_{2}/c_{1}\right)}{d\left(p_{2}/p_{1}\right)}=\frac{c_{2% }}{c_{1}}\left[1+\frac{d\left(c_{2}/c_{1}\right)}{d\left(p_{2}/p_{1}\right)}% \cdot\frac{p_{2}/p_{1}}{c_{2}/c_{1}}\right]=\frac{c_{2}}{c_{1}}\left(1-E_{21}\right)
  24. c 2 c_{2}
  25. c 2 c_{2}
  26. c 2 c_{2}
  27. c 2 c_{2}
  28. c 2 c_{2}
  29. c 2 c_{2}
  30. c 2 c_{2}
  31. c 2 c_{2}
  32. c 2 c_{2}
  33. c 2 c_{2}
  34. c 2 c_{2}
  35. c 2 c_{2}
  36. c 1 c_{1}
  37. d ( x 2 / x 1 ) x 2 / x 1 = d log ( x 2 / x 1 ) = d log x 2 - d log x 1 = - ( d log x 1 - d log x 2 ) = - d log ( x 1 / x 2 ) = - d ( x 1 / x 2 ) x 1 / x 2 \ \frac{d(x_{2}/x_{1})}{x_{2}/x_{1}}=d\log(x_{2}/x_{1})=d\log x_{2}-d\log x_{1% }=-(d\log x_{1}-d\log x_{2})=-d\log(x_{1}/x_{2})=-\frac{d(x_{1}/x_{2})}{x_{1}/% x_{2}}
  38. σ = - d ( c 1 / c 2 ) d M R S M R S c 1 / c 2 = - d log ( c 1 / c 2 ) d log M R S \ \sigma=-\frac{d(c_{1}/c_{2})}{dMRS}\frac{MRS}{c_{1}/c_{2}}=-\frac{d\log(c_{1% }/c_{2})}{d\log MRS}

Electric_clock.html

  1. v = 120 f p v=\frac{120f}{p}\,

Electrically_short.html

  1. 2 π h λ 2\pi h\over\lambda
  2. 1 \scriptstyle\ll 1

Electromechanical_coupling_coefficient.html

  1. 𝒦 - 2 = 1 energy converted per input energy \mathcal{K}^{-2}=\frac{1}{\mbox{ energy converted per input energy}~{}}

Electron_temperature.html

  1. E = ( 3 / 2 ) k B T \langle E\rangle=(3/2)\langle k_{B}T\rangle

Electronic_filter_topology.html

  1. H ( s ) = V o V i = - 1 A s 2 + B s + C = K ω 0 2 s 2 + ω 0 Q s + ω 0 2 H(s)=\frac{V_{o}}{V_{i}}=-\frac{1}{As^{2}+Bs+C}=\frac{K{\omega_{0}}^{2}}{s^{2}% +\frac{\omega_{0}}{Q}s+{\omega_{0}}^{2}}
  2. A = ( R 1 R 3 C 2 C 5 ) A=(R_{1}R_{3}C_{2}C_{5})\,
  3. B = R 3 C 5 + R 1 C 5 + R 1 R 3 C 5 / R 4 B=R_{3}C_{5}+R_{1}C_{5}+R_{1}R_{3}C_{5}/R_{4}\,
  4. C = R 1 / R 4 C=R_{1}/R_{4}\,
  5. Q = R 3 R 4 C 2 C 5 ( R 4 + R 3 + | K | R 3 ) C 5 Q=\frac{\sqrt{R_{3}R_{4}C_{2}C_{5}}}{(R_{4}+R_{3}+|K|R_{3})C_{5}}
  6. K = - R 4 / R 1 K=-R_{4}/R_{1}\,
  7. ω 0 = 2 π f 0 = 1 / R 3 R 4 C 2 C 5 \omega_{0}=2\pi f_{0}=1/\sqrt{R_{3}R_{4}C_{2}C_{5}}
  8. H ( s ) = G lpf ω 0 2 s 2 + ω 0 Q s + ω 0 2 H(s)=\frac{G_{\mathrm{lpf}}{\omega_{0}}^{2}}{s^{2}+\frac{\omega_{0}}{Q}s+{% \omega_{0}}^{2}}
  9. G lpf = R 2 / R 1 G_{\mathrm{lpf}}=R_{2}/R_{1}
  10. H ( s ) = G bpf ω 0 Q s s 2 + ω 0 Q s + ω 0 2 H(s)=\frac{G_{\mathrm{bpf}}\frac{\omega_{0}}{Q}s}{s^{2}+\frac{\omega_{0}}{Q}s+% {\omega_{0}}^{2}}
  11. G bpf = - R 3 / R 1 G_{\mathrm{bpf}}=-R_{3}/R_{1}
  12. ω 0 = 1 / R 2 R 4 C 1 C 2 \omega_{0}=1/\sqrt{R_{2}R_{4}C_{1}C_{2}}
  13. Q = R 3 2 C 1 R 2 R 4 C 2 Q=\sqrt{\frac{{R_{3}}^{2}C_{1}}{R_{2}R_{4}C_{2}}}
  14. B = ω 0 / Q B=\omega_{0}/Q
  15. ζ = 1 / 2 Q \zeta=1/2Q

Electrostatic_force_microscope.html

  1. F e l e c t r o s t a t i c = 1 2 C z Δ V 2 F_{electrostatic}=\frac{1}{2}\frac{\partial C}{\partial z}\Delta V^{2}
  2. C / z {∂C}/{∂z}

Elementary_matrix.html

  1. R i R j R_{i}\leftrightarrow R_{j}
  2. k R i R i , where k 0 kR_{i}\rightarrow R_{i},\ \mbox{where }~{}k\neq 0
  3. R i + k R j R i , where i j R_{i}+kR_{j}\rightarrow R_{i},\mbox{where }~{}i\neq j
  4. T i , j = [ 1 0 1 1 0 1 ] T_{i,j}=\begin{bmatrix}1&&&&&&&\\ &\ddots&&&&&&\\ &&0&&1&&\\ &&&\ddots&&&&\\ &&1&&0&&\\ &&&&&&\ddots&\\ &&&&&&&1\end{bmatrix}\quad
  5. T i ( m ) = [ 1 1 m 1 1 ] T_{i}(m)=\begin{bmatrix}1&&&&&&\\ &\ddots&&&&&\\ &&1&&&&\\ &&&m&&&\\ &&&&1&&\\ &&&&&\ddots&\\ &&&&&&1\end{bmatrix}\quad
  6. T i , j ( m ) = [ 1 1 m 1 1 ] T_{i,j}(m)=\begin{bmatrix}1&&&&&&&\\ &\ddots&&&&&&\\ &&1&&&&&\\ &&&\ddots&&&&\\ &&m&&1&&\\ &&&&&&\ddots&\\ &&&&&&&1\end{bmatrix}

Elementary_reaction.html

  1. A products. \mbox{A}~{}\rightarrow\mbox{products.}~{}
  2. d [ A ] d t = - k [ A ] . \frac{d[\mbox{A}~{}]}{dt}=-k[\mbox{A}~{}].
  3. A + B products. \mbox{A + B}~{}\rightarrow\mbox{products.}~{}
  4. d [ A ] d t = d [ B ] d t = - k [ A ] [ B ] . \frac{d[\mbox{A}~{}]}{dt}=\frac{d[\mbox{B}~{}]}{dt}=-k[\mbox{A}~{}][\mbox{B}~{% }].

Elkies_trinomial_curves.html

  1. y 2 = x ( 81 x 5 + 396 x 4 + 738 x 3 + 660 x 2 + 269 x + 48 ) y^{2}=x(81x^{5}+396x^{4}+738x^{3}+660x^{2}+269x+48)
  2. 37 2 x 7 - 28 x + 3 2 37^{2}x^{7}-28x+3^{2}
  3. ( 499 2 / 113 ) x 7 - 212 x + 3 4 (499^{2}/113)x^{7}-212x+3^{4}
  4. y 2 = 2 x 6 + 4 x 5 + 36 x 4 + 16 x 3 - 45 x 2 + 190 x + 1241 y^{2}=2x^{6}+4x^{5}+36x^{4}+16x^{3}-45x^{2}+190x+1241

Elliptic_rational_functions.html

  1. R n ( ξ , x ) cd ( n K ( 1 / L n ) K ( 1 / ξ ) cd - 1 ( x , 1 / ξ ) , 1 / L n ) R_{n}(\xi,x)\equiv\mathrm{cd}\left(n\frac{K(1/L_{n})}{K(1/\xi)}\,\mathrm{cd}^{% -1}(x,1/\xi),1/L_{n}\right)
  2. L n ( ξ ) = R n ( ξ , ξ ) L_{n}(\xi)=R_{n}(\xi,\xi)
  3. R n ( ξ , x ) R_{n}(\xi,x)
  4. | x | ξ |x|\geq\xi
  5. R n ( ξ , x ) = r 0 i = 1 n ( x - x i ) i = 1 n ( x - x p i ) R_{n}(\xi,x)=r_{0}\,\frac{\prod_{i=1}^{n}(x-x_{i})}{\prod_{i=1}^{n}(x-x_{pi})}
  6. x i x_{i}
  7. x p i x_{pi}
  8. r 0 r_{0}
  9. R n ( ξ , 1 ) = 1 R_{n}(\xi,1)=1
  10. R n ( ξ , x ) = r 0 x i = 1 n - 1 ( x - x i ) i = 1 n - 1 ( x - x p i ) R_{n}(\xi,x)=r_{0}\,x\,\frac{\prod_{i=1}^{n-1}(x-x_{i})}{\prod_{i=1}^{n-1}(x-x% _{pi})}
  11. R n 2 ( ξ , x ) 1 R_{n}^{2}(\xi,x)\leq 1
  12. | x | 1 |x|\leq 1\,
  13. R n 2 ( ξ , x ) = 1 R_{n}^{2}(\xi,x)=1
  14. | x | = 1 |x|=1\,
  15. R n 2 ( ξ , - x ) = R n 2 ( ξ , x ) R_{n}^{2}(\xi,-x)=R_{n}^{2}(\xi,x)
  16. R n 2 ( ξ , x ) > 1 R_{n}^{2}(\xi,x)>1
  17. x > 1 x>1\,
  18. R n ( ξ , 1 ) = 1 R_{n}(\xi,1)=1\,
  19. R m ( R n ( ξ , ξ ) , R n ( ξ , x ) ) = R m n ( ξ , x ) R_{m}(R_{n}(\xi,\xi),R_{n}(\xi,x))=R_{m\cdot n}(\xi,x)\,
  20. R n R_{n}
  21. R n R_{n}
  22. R 2 R_{2}
  23. R 3 R_{3}
  24. R n R_{n}
  25. n = 2 a 3 b n=2^{a}3^{b}
  26. R n R_{n}
  27. R n R_{n}
  28. L m n ( ξ ) = L m ( L n ( ξ ) ) L_{m\cdot n}(\xi)=L_{m}(L_{n}(\xi))
  29. T n ( x ) T_{n}(x)
  30. lim ξ = R n ( ξ , x ) = T n ( x ) \lim_{\xi=\rightarrow\,\infty}R_{n}(\xi,x)=T_{n}(x)\,
  31. R n ( ξ , - x ) = R n ( ξ , x ) R_{n}(\xi,-x)=R_{n}(\xi,x)\,
  32. R n ( ξ , - x ) = - R n ( ξ , x ) R_{n}(\xi,-x)=-R_{n}(\xi,x)\,
  33. R n ( ξ , x ) R_{n}(\xi,x)
  34. ± 1 \pm 1
  35. - 1 x 1 -1\leq x\leq 1
  36. 1 / R n ( ξ , x ) 1/R_{n}(\xi,x)
  37. - 1 / ξ x 1 / ξ -1/\xi\leq x\leq 1/\xi
  38. ± 1 / L n ( ξ ) \pm 1/L_{n}(\xi)
  39. R n ( ξ , ξ / x ) = R n ( ξ , ξ ) R n ( ξ , x ) R_{n}(\xi,\xi/x)=\frac{R_{n}(\xi,\xi)}{R_{n}(\xi,x)}\,
  40. x p i x z i = ξ x_{pi}x_{zi}=\xi\,
  41. x n i ( ξ ) x_{ni}(\xi)
  42. x n i x_{ni}
  43. ξ \xi
  44. cd ( ( 2 m - 1 ) K ( 1 / z ) , 1 z ) = 0 \mathrm{cd}\left((2m-1)K\left(1/z\right),\frac{1}{z}\right)=0\,
  45. n K ( 1 / L n ) K ( 1 / ξ ) cd - 1 ( x m , 1 / ξ ) = ( 2 m - 1 ) K ( 1 / L n ) n\frac{K(1/L_{n})}{K(1/\xi)}\mathrm{cd}^{-1}(x_{m},1/\xi)=(2m-1)K(1/L_{n})
  46. x m = cd ( K ( 1 / ξ ) 2 m - 1 n , 1 ξ ) . x_{m}=\mathrm{cd}\left(K(1/\xi)\,\frac{2m-1}{n},\frac{1}{\xi}\right).
  47. R m R_{m}
  48. R n R_{n}
  49. R m n R_{m\cdot n}
  50. 2 i 3 j 2^{i}3^{j}
  51. R 8 ( ξ , x ) R_{8}(\xi,x)
  52. X n R n ( ξ , x ) L n R n ( ξ , ξ ) t n 1 - 1 / L n 2 . X_{n}\equiv R_{n}(\xi,x)\qquad L_{n}\equiv R_{n}(\xi,\xi)\qquad t_{n}\equiv% \sqrt{1-1/L_{n}^{2}}.
  53. R 2 ( ξ , x ) = ( t + 1 ) x 2 - 1 ( t - 1 ) x 2 + 1 R_{2}(\xi,x)=\frac{(t+1)x^{2}-1}{(t-1)x^{2}+1}
  54. t 1 - 1 / ξ 2 t\equiv\sqrt{1-1/\xi^{2}}
  55. L 2 = 1 + t 1 - t , L 4 = 1 + t 2 1 - t 2 , L 8 = 1 + t 4 1 - t 4 L_{2}=\frac{1+t}{1-t},\qquad L_{4}=\frac{1+t_{2}}{1-t_{2}},\qquad L_{8}=\frac{% 1+t_{4}}{1-t_{4}}
  56. X 2 = ( t + 1 ) x 2 - 1 ( t - 1 ) x 2 + 1 , X 4 = ( t 2 + 1 ) X 2 2 - 1 ( t 2 - 1 ) X 2 2 + 1 , X 8 = ( t 4 + 1 ) X 4 2 - 1 ( t 4 - 1 ) X 4 2 + 1 . X_{2}=\frac{(t+1)x^{2}-1}{(t-1)x^{2}+1},\qquad X_{4}=\frac{(t_{2}+1)X_{2}^{2}-% 1}{(t_{2}-1)X_{2}^{2}+1},\qquad X_{8}=\frac{(t_{4}+1)X_{4}^{2}-1}{(t_{4}-1)X_{% 4}^{2}+1}.
  57. x = 1 ± 1 + t ( 1 - X 2 1 + X 2 ) , X 2 = 1 ± 1 + t 2 ( 1 - X 4 1 + X 4 ) , X 4 = 1 ± 1 + t 4 ( 1 - X 8 1 + X 8 ) . x=\frac{1}{\pm\sqrt{1+t\,\left(\frac{1-X_{2}}{1+X_{2}}\right)}},\qquad X_{2}=% \frac{1}{\pm\sqrt{1+t_{2}\,\left(\frac{1-X_{4}}{1+X_{4}}\right)}},\qquad X_{4}% =\frac{1}{\pm\sqrt{1+t_{4}\,\left(\frac{1-X_{8}}{1+X_{8}}\right)}}.\qquad
  58. R 8 ( ξ , x ) R_{8}(\xi,x)
  59. X 8 = 0 X_{8}=0
  60. X 4 X_{4}
  61. X 4 X_{4}
  62. X 2 X_{2}
  63. R 8 ( ξ , x ) R_{8}(\xi,x)
  64. t n t_{n}
  65. R 1 ( ξ , x ) = x R_{1}(\xi,x)=x\,
  66. R 2 ( ξ , x ) = ( t + 1 ) x 2 - 1 ( t - 1 ) x 2 + 1 R_{2}(\xi,x)=\frac{(t+1)x^{2}-1}{(t-1)x^{2}+1}
  67. t 1 - 1 ξ 2 t\equiv\sqrt{1-\frac{1}{\xi^{2}}}
  68. R 3 ( ξ , x ) = x ( 1 - x p 2 ) ( x 2 - x z 2 ) ( 1 - x z 2 ) ( x 2 - x p 2 ) R_{3}(\xi,x)=x\,\frac{(1-x_{p}^{2})(x^{2}-x_{z}^{2})}{(1-x_{z}^{2})(x^{2}-x_{p% }^{2})}
  69. G 4 ξ 2 + ( 4 ξ 2 ( ξ 2 - 1 ) ) 2 / 3 G\equiv\sqrt{4\xi^{2}+(4\xi^{2}(\xi^{2}\!-\!1))^{2/3}}
  70. x p 2 2 ξ 2 G 8 ξ 2 ( ξ 2 + 1 ) + 12 G ξ 2 - G 3 - G 3 x_{p}^{2}\equiv\frac{2\xi^{2}\sqrt{G}}{\sqrt{8\xi^{2}(\xi^{2}\!+\!1)+12G\xi^{2% }-G^{3}}-\sqrt{G^{3}}}
  71. x z 2 = ξ 2 / x p 2 x_{z}^{2}=\xi^{2}/x_{p}^{2}
  72. R 4 ( ξ , x ) = R 2 ( R 2 ( ξ , ξ ) , R 2 ( ξ , x ) ) = ( 1 + t ) ( 1 + t ) 2 x 4 - 2 ( 1 + t ) ( 1 + t ) x 2 + 1 ( 1 + t ) ( 1 - t ) 2 x 4 - 2 ( 1 + t ) ( 1 - t ) x 2 + 1 R_{4}(\xi,x)=R_{2}(R_{2}(\xi,\xi),R_{2}(\xi,x))=\frac{(1+t)(1+\sqrt{t})^{2}x^{% 4}-2(1+t)(1+\sqrt{t})x^{2}+1}{(1+t)(1-\sqrt{t})^{2}x^{4}-2(1+t)(1-\sqrt{t})x^{% 2}+1}
  73. R 6 ( ξ , x ) = R 3 ( R 2 ( ξ , ξ ) , R 2 ( ξ , x ) ) R_{6}(\xi,x)=R_{3}(R_{2}(\xi,\xi),R_{2}(\xi,x))\,
  74. n = 2 i 3 j n=2^{i}\,3^{j}
  75. L 1 ( ξ ) = ξ L_{1}(\xi)=\xi\,
  76. L 2 ( ξ ) = 1 + t 1 - t = ( ξ + ξ 2 - 1 ) 2 L_{2}(\xi)=\frac{1+t}{1-t}=\left(\xi+\sqrt{\xi^{2}-1}\right)^{2}
  77. L 3 ( ξ ) = ξ 3 ( 1 - x p 2 ξ 2 - x p 2 ) 2 L_{3}(\xi)=\xi^{3}\left(\frac{1-x_{p}^{2}}{\xi^{2}-x_{p}^{2}}\right)^{2}
  78. L 4 ( ξ ) = ( ξ + ( ξ 2 - 1 ) 1 / 4 ) 4 ( ξ + ξ 2 - 1 ) 2 L_{4}(\xi)=\left(\sqrt{\xi}+(\xi^{2}-1)^{1/4}\right)^{4}\left(\xi+\sqrt{\xi^{2% }-1}\right)^{2}
  79. L 6 ( ξ ) = L 3 ( L 2 ( ξ ) ) L_{6}(\xi)=L_{3}(L_{2}(\xi))\,
  80. x n j x_{nj}
  81. x 11 = 0 x_{11}=0\,
  82. x 21 = ξ 1 - t x_{21}=\xi\sqrt{1-t}\,
  83. x 22 = - x 21 x_{22}=-x_{21}\,
  84. x 31 = x z x_{31}=x_{z}\,
  85. x 32 = 0 x_{32}=0\,
  86. x 33 = - x 31 x_{33}=-x_{31}\,
  87. x 41 = ξ ( 1 - t ) ( 1 + t - t ( t + 1 ) ) x_{41}=\xi\sqrt{\left(1-\sqrt{t}\right)\left(1+t-\sqrt{t(t+1)}\right)}\,
  88. x 42 = ξ ( 1 - t ) ( 1 + t + t ( t + 1 ) ) x_{42}=\xi\sqrt{\left(1-\sqrt{t}\right)\left(1+t+\sqrt{t(t+1)}\right)}\,
  89. x 43 = - x 42 x_{43}=-x_{42}\,
  90. x 44 = - x 41 x_{44}=-x_{41}\,
  91. x p , n i x_{p,ni}
  92. x p , n i = ξ / ( x n i ) x_{p,ni}=\xi/(x_{ni})

Elliptic_unit.html

  1. Θ 𝐚 = α - 12 Δ E N 𝐚 - 1 𝐚 P = 0 , P 0 ( x - x ( P ) ) - 6 . \Theta_{\mathbf{a}}=\alpha^{-12}\Delta_{E}^{N\mathbf{a}-1}\prod_{\mathbf{a}P=0% ,P\neq 0}(x-x(P))^{-6}\ .
  2. 𝐛 Q = 0 Θ 𝐚 ( P + R ) = Θ 𝐚 ( β P ) . \prod_{\mathbf{b}Q=0}\Theta_{\mathbf{a}}(P+R)=\Theta_{\mathbf{a}}(\beta P)\ .

Embedded_atom_model.html

  1. i i\!
  2. E i = F α ( i j ρ β ( r i j ) ) + 1 2 i j ϕ α β ( r i j ) E_{i}=F_{\alpha}\left(\sum_{i\neq j}\rho_{\beta}(r_{ij})\right)+\frac{1}{2}% \sum_{i\neq j}\phi_{\alpha\beta}(r_{ij})
  3. r i j r_{ij}\!
  4. i i\!
  5. j j\!
  6. ϕ α β \phi_{\alpha\beta}
  7. ρ β \rho_{\beta}\!
  8. j j\!
  9. β \beta\!
  10. i i\!
  11. F F\!
  12. i i\!
  13. α \alpha\!

Empirical_measure.html

  1. P P
  2. X 1 , X 2 , , X n X_{1},X_{2},\dots,X_{n}
  3. P P
  4. F F
  5. X 1 , X 2 , X_{1},X_{2},\dots
  6. P n ( A ) = 1 n i = 1 n I A ( X i ) = 1 n i = 1 n δ X i ( A ) P_{n}(A)={1\over n}\sum_{i=1}^{n}I_{A}(X_{i})=\frac{1}{n}\sum_{i=1}^{n}\delta_% {X_{i}}(A)
  7. I A I_{A}
  8. δ X \delta_{X}
  9. ( P n ( c ) ) c 𝒞 \bigl(P_{n}(c)\bigr)_{c\in\mathcal{C}}
  10. 𝒞 \mathcal{C}
  11. P n P_{n}
  12. f : S f:S\to\mathbb{R}
  13. f P n f = S f d P n = 1 n i = 1 n f ( X i ) f\mapsto P_{n}f=\int_{S}f\,dP_{n}=\frac{1}{n}\sum_{i=1}^{n}f(X_{i})
  14. f f
  15. P n f P_{n}f
  16. 𝔼 f \mathbb{E}f
  17. 1 n 𝔼 ( f - 𝔼 f ) 2 \frac{1}{n}\mathbb{E}(f-\mathbb{E}f)^{2}
  18. P n f P_{n}f
  19. 𝔼 f \mathbb{E}f
  20. f f
  21. 𝒞 \mathcal{C}
  22. \mathcal{F}
  23. c 𝒞 c\in\mathcal{C}
  24. f f\in\mathcal{F}
  25. P n - P 𝒞 = sup c 𝒞 | P n ( c ) - P ( c ) | 0 , \|P_{n}-P\|_{\mathcal{C}}=\sup_{c\in\mathcal{C}}|P_{n}(c)-P(c)|\to 0,
  26. P n - P = sup f | P n f - 𝔼 f | 0. \|P_{n}-P\|_{\mathcal{F}}=\sup_{f\in\mathcal{F}}|P_{n}f-\mathbb{E}f|\to 0.
  27. X 1 , , X n X_{1},\dots,X_{n}
  28. F n ( x ) = P n ( ( - , x ] ) = P n I ( - , x ] . F_{n}(x)=P_{n}((-\infty,x])=P_{n}I_{(-\infty,x]}.
  29. 𝒞 = { ( - , x ] : x } . \mathcal{C}=\{(-\infty,x]:x\in\mathbb{R}\}.
  30. 𝒞 \mathcal{C}
  31. sup F F n ( x ) - F ( x ) 0 \sup_{F}\|F_{n}(x)-F(x)\|_{\infty}\to 0

EMule.html

  1. Ratio 1 = 2 Uploaded    Total Downloaded    Total \textrm{Ratio}_{1}=\frac{2\cdot\textrm{Uploaded \,\, Total}}{\textrm{% Downloaded \,\, Total}}
  2. Ratio 2 = Uploaded    Total + 2 \textrm{Ratio}_{2}=\sqrt{\textrm{Uploaded \,\, Total}+2}

En_(Lie_algebra).html

  1. [ 2 - 1 0 - 1 2 0 0 0 2 ] \left[\begin{smallmatrix}2&-1&0\\ -1&2&0\\ 0&0&2\end{smallmatrix}\right]
  2. [ 2 - 1 0 0 - 1 2 - 1 0 0 - 1 2 - 1 0 0 - 1 2 ] \left[\begin{smallmatrix}2&-1&0&0\\ -1&2&-1&0\\ 0&-1&2&-1\\ 0&0&-1&2\end{smallmatrix}\right]
  3. [ 2 - 1 0 0 0 - 1 2 - 1 0 0 0 - 1 2 - 1 - 1 0 0 - 1 2 0 0 0 - 1 0 2 ] \left[\begin{smallmatrix}2&-1&0&0&0\\ -1&2&-1&0&0\\ 0&-1&2&-1&-1\\ 0&0&-1&2&0\\ 0&0&-1&0&2\end{smallmatrix}\right]
  4. [ 2 - 1 0 0 0 0 - 1 2 - 1 0 0 0 0 - 1 2 - 1 0 - 1 0 0 - 1 2 - 1 0 0 0 0 - 1 2 0 0 0 - 1 0 0 2 ] \left[\begin{smallmatrix}2&-1&0&0&0&0\\ -1&2&-1&0&0&0\\ 0&-1&2&-1&0&-1\\ 0&0&-1&2&-1&0\\ 0&0&0&-1&2&0\\ 0&0&-1&0&0&2\end{smallmatrix}\right]
  5. [ 2 - 1 0 0 0 0 0 - 1 2 - 1 0 0 0 0 0 - 1 2 - 1 0 0 - 1 0 0 - 1 2 - 1 0 0 0 0 0 - 1 2 - 1 0 0 0 0 0 - 1 2 0 0 0 - 1 0 0 0 2 ] \left[\begin{smallmatrix}2&-1&0&0&0&0&0\\ -1&2&-1&0&0&0&0\\ 0&-1&2&-1&0&0&-1\\ 0&0&-1&2&-1&0&0\\ 0&0&0&-1&2&-1&0\\ 0&0&0&0&-1&2&0\\ 0&0&-1&0&0&0&2\end{smallmatrix}\right]
  6. [ 2 - 1 0 0 0 0 0 0 - 1 2 - 1 0 0 0 0 0 0 - 1 2 - 1 0 0 0 - 1 0 0 - 1 2 - 1 0 0 0 0 0 0 - 1 2 - 1 0 0 0 0 0 0 - 1 2 - 1 0 0 0 0 0 0 - 1 2 0 0 0 - 1 0 0 0 0 2 ] \left[\begin{smallmatrix}2&-1&0&0&0&0&0&0\\ -1&2&-1&0&0&0&0&0\\ 0&-1&2&-1&0&0&0&-1\\ 0&0&-1&2&-1&0&0&0\\ 0&0&0&-1&2&-1&0&0\\ 0&0&0&0&-1&2&-1&0\\ 0&0&0&0&0&-1&2&0\\ 0&0&-1&0&0&0&0&2\end{smallmatrix}\right]
  7. E ~ 8 {\tilde{E}}_{8}
  8. [ 2 - 1 0 0 0 0 0 0 0 - 1 2 - 1 0 0 0 0 0 0 0 - 1 2 - 1 0 0 0 0 - 1 0 0 - 1 2 - 1 0 0 0 0 0 0 0 - 1 2 - 1 0 0 0 0 0 0 0 - 1 2 - 1 0 0 0 0 0 0 0 - 1 2 - 1 0 0 0 0 0 0 0 - 1 2 0 0 0 - 1 0 0 0 0 0 2 ] \left[\begin{smallmatrix}2&-1&0&0&0&0&0&0&0\\ -1&2&-1&0&0&0&0&0&0\\ 0&-1&2&-1&0&0&0&0&-1\\ 0&0&-1&2&-1&0&0&0&0\\ 0&0&0&-1&2&-1&0&0&0\\ 0&0&0&0&-1&2&-1&0&0\\ 0&0&0&0&0&-1&2&-1&0\\ 0&0&0&0&0&0&-1&2&0\\ 0&0&-1&0&0&0&0&0&2\end{smallmatrix}\right]
  9. [ 2 - 1 0 0 0 0 0 0 0 0 - 1 2 - 1 0 0 0 0 0 0 0 0 - 1 2 - 1 0 0 0 0 0 - 1 0 0 - 1 2 - 1 0 0 0 0 0 0 0 0 - 1 2 - 1 0 0 0 0 0 0 0 0 - 1 2 - 1 0 0 0 0 0 0 0 0 - 1 2 - 1 0 0 0 0 0 0 0 0 - 1 2 - 1 0 0 0 0 0 0 0 0 - 1 2 0 0 0 - 1 0 0 0 0 0 0 2 ] \left[\begin{smallmatrix}2&-1&0&0&0&0&0&0&0&0\\ -1&2&-1&0&0&0&0&0&0&0\\ 0&-1&2&-1&0&0&0&0&0&-1\\ 0&0&-1&2&-1&0&0&0&0&0\\ 0&0&0&-1&2&-1&0&0&0&0\\ 0&0&0&0&-1&2&-1&0&0&0\\ 0&0&0&0&0&-1&2&-1&0&0\\ 0&0&0&0&0&0&-1&2&-1&0\\ 0&0&0&0&0&0&0&-1&2&0\\ 0&0&-1&0&0&0&0&0&0&2\end{smallmatrix}\right]

End_(topology).html

  1. \scriptstyle\mathbb{R}
  2. n \scriptstyle\mathbb{R}^{n}
  3. n K \scriptstyle\mathbb{R}^{n}\setminus K
  4. 2 \scriptstyle\mathbb{R}^{2}
  5. + X \scriptstyle\mathbb{R}^{+}\to X
  6. \scriptstyle\mathbb{N}

Energy_conversion_efficiency.html

  1. η = P out P in \eta=\frac{P_{\mathrm{out}}}{P_{\mathrm{in}}}

Energy_Slave.html

  1. L w = L n + L d + C E i Lw=Ln+Ld+CEi
  2. L w - L n - L d = C E i Lw-Ln-Ld=CEi
  3. L w = 10 L a b o r e r s 0.6 P h L a b o r e r = 6 P h Lw=10\,Laborers\cdot 0.6\,\frac{Ph}{Laborer}=6\,Ph
  4. L n = 10 L a b o r e r s 0.1 P h L a b o r e r = 1 P h Ln=10\,Laborers\cdot 0.1\,\frac{Ph}{Laborer}=1\,Ph
  5. L d = 1 L a b o r e r 0.5 P h L a b o r e r = 0.5 P h Ld=1\,Laborer\cdot 0.5\,\frac{Ph}{Laborer}=0.5\,Ph
  6. C E i = 6 P h - 1 P h - 0.5 P h = 4.5 P h CEi=6Ph-1Ph-0.5Ph=4.5Ph
  7. E s {}_{Es}
  8. E i * {}_{Ei^{*}}
  9. E s * E i {}_{\frac{Es^{*}}{Ei}}
  10. E s = E i * E s * E i Es=Ei^{*}\cdot\frac{Es^{*}}{Ei}
  11. 7.5 E s c a l o r i e s E s {}_{7.5\,Es\cdot\frac{calories}{Es}}

Engine_efficiency.html

  1. η = w o r k d o n e h e a t a b s o r b e d = Q 1 - Q 2 Q 1 \eta=\frac{work\ done}{heat\ absorbed}=\frac{Q1-Q2}{Q1}
  2. Q 1 Q1
  3. Q 1 - Q 2 Q1-Q2

Ensemble_interpretation.html

  1. | y |y\rangle
  2. Q \operatorname{Q}
  3. q q
  4. Q | y = q | y \operatorname{Q}|y\rangle=q|y\rangle
  5. | ψ = a | χ 1 + b | χ 2 |\psi\rangle=a|\chi_{1}\rangle+b|\chi_{2}\rangle
  6. 𝒫 ( χ 1 ) \mathcal{P}(\chi_{1})
  7. 𝒫 ( χ 2 ) \mathcal{P}(\chi_{2})
  8. a | χ 1 𝒫 ( χ 1 ) a|\chi_{1}\rangle\rightarrow\mathcal{P}(\chi_{1})
  9. a | χ 1 χ 1 a|\chi_{1}\rangle\rightarrow\chi_{1}
  10. | ψ = | 1 + | 2 + | 3 + | 4 + | 5 + | 6 6 |\psi\rangle=\frac{|1\rangle+|2\rangle+|3\rangle+|4\rangle+|5\rangle+|6\rangle% }{\sqrt{6}}

Entropy_(order_and_disorder).html

  1. δ Q T 0 \int\frac{\delta Q}{T}\geq 0

Entropy_and_life.html

  1. S = d Q τ S=\int{dQ\over\tau}
  2. S S
  3. d Q dQ
  4. τ \tau
  5. Δ G Δ H - T Δ S \Delta G\equiv\Delta H-T\Delta S\,

Enzyme_inhibitor.html

  1. V = V m a x [ S ] α K m + α [ S ] = ( 1 / α ) V m a x [ S ] ( α / α ) K m + [ S ] V=\frac{V_{max}[S]}{\alpha K_{m}+\alpha^{\prime}[S]}=\frac{(1/\alpha^{\prime})% V_{max}[S]}{(\alpha/\alpha^{\prime})K_{m}+[S]}
  2. α = 1 + [ I ] K i \alpha=1+\frac{[I]}{K_{i}}
  3. α = 1 + [ I ] K i . \alpha^{\prime}=1+\frac{[I]}{K_{i}^{\prime}}.
  4. V max 1 + [ I ] K i \cfrac{V_{\max}}{1+\cfrac{[I]}{K_{i}}}
  5. V max [ I ] + K i K i \cfrac{V_{\max}}{\cfrac{[I]+K_{i}}{K_{i}}}
  6. V max [ I ] + K i [ I ] + K i - [ I ] \cfrac{V_{\max}}{\cfrac{[I]+K_{i}}{[I]+K_{i}-[I]}}
  7. V max 1 1 - [ I ] [ I ] + K i \cfrac{V_{\max}}{\cfrac{1}{1-\cfrac{[I]}{[I]+K_{i}}}}
  8. V max - V max [ I ] [ I ] + K i V_{\max}-V_{\max}\cfrac{[I]}{[I]+K_{i}}
  9. [ S ] [ S ] + K m \cfrac{[S]}{[S]+K_{m}}
  10. [ I ] [ I ] + K i \cfrac{[I]}{[I]+K_{i}}
  11. V max - Δ V max [ I ] [ I ] + K i V_{\max}-\Delta V_{\max}\cfrac{[I]}{[I]+K_{i}}
  12. V max 1 - ( V max 1 - V max 2 ) [ I ] [ I ] + K i V_{\max}1-(V_{\max}1-V_{\max}2)\cfrac{[I]}{[I]+K_{i}}
  13. V max 1 - ( V max 1 - V max 2 ) [ X ] [ X ] + K x V_{\max}1-(V_{\max}1-V_{\max}2)\cfrac{[X]}{[X]+K_{x}}
  14. K m 1 - ( K m 1 - K m 2 ) [ X ] [ X ] + K x K_{m}1-(K_{m}1-K_{m}2)\cfrac{[X]}{[X]+K_{x}}

Epigraph_(mathematics).html

  1. epi f = { ( x , μ ) : x n , μ , μ f ( x ) } n + 1 . \mbox{epi}~{}f=\{(x,\mu)\,:\,x\in\mathbb{R}^{n},\,\mu\in\mathbb{R},\,\mu\geq f% (x)\}\subseteq\mathbb{R}^{n+1}.
  2. epi f S = { ( x , μ ) : x n , μ , μ > f ( x ) } n + 1 . \mbox{epi}~{}_{S}f=\{(x,\mu)\,:\,x\in\mathbb{R}^{n},\,\mu\in\mathbb{R},\,\mu>f% (x)\}\subseteq\mathbb{R}^{n+1}.
  3. n \mathbb{R}^{n}

Epsilon_Eridani_in_fiction.html

  1. M \begin{smallmatrix}M_{\odot}\end{smallmatrix}

Epstein–Zin_preferences.html

  1. U t U_{t}
  2. { c t , c t + 1 , c t + 2 , } \{c_{t},c_{t+1},c_{t+2},...\}
  3. U t = [ ( 1 - β ) c t ρ + β μ t ( U t + 1 ) ρ ] 1 / ρ , U_{t}=[(1-\beta)c_{t}^{\rho}+\beta\mu_{t}(U_{t+1})^{\rho}]^{1/\rho},
  4. μ t ( ) \mu_{t}()
  5. 0 < β < 1 0<\beta<1
  6. 1 / β - 1 1/\beta-1
  7. ρ < 1 \rho<1
  8. 1 / ( 1 - ρ ) 1/(1-\rho)
  9. μ t ( U t + 1 ) = [ E t U t + 1 α ] 1 / α \mu_{t}(U_{t+1})=[E_{t}U_{t+1}^{\alpha}]^{1/\alpha}
  10. E t E_{t}
  11. U t + 1 U_{t+1}
  12. α < 1 \alpha<1
  13. α \alpha
  14. α = ρ \alpha=\rho
  15. ρ \rho
  16. α \alpha

Equating_coefficients.html

  1. 1 x ( x - 1 ) ( x - 2 ) , \frac{1}{x(x-1)(x-2)},\,
  2. A x + B x - 1 + C x - 2 , \frac{A}{x}+\frac{B}{x-1}+\frac{C}{x-2},\,
  3. A ( x - 1 ) ( x - 2 ) + B x ( x - 2 ) + C x ( x - 1 ) = 1 , A(x-1)(x-2)+Bx(x-2)+Cx(x-1)=1,\,
  4. ( A + B + C ) x 2 - ( 3 A + 2 B + C ) x + 2 A = 1. (A+B+C)x^{2}-(3A+2B+C)x+2A=1.\,
  5. A + B + C = 0 , A+B+C=0,\,
  6. 3 A + 2 B + C = 0 , 3A+2B+C=0,\,
  7. 2 A = 1. 2A=1.\,
  8. A = 1 2 , B = - 1 , C = 1 2 . A=\frac{1}{2},\,B=-1,\,C=\frac{1}{2}.\,
  9. a + b c \sqrt{a+b\sqrt{c}\ }
  10. a + b c = d + e . \sqrt{a+b\sqrt{c}\ }=\sqrt{d}+\sqrt{e}.
  11. a + b c = d + e + 2 d e . a+b\sqrt{c}=d+e+2\sqrt{de}.
  12. a = d + e , a=d+e,
  13. b c = 2 d e b\sqrt{c}=2\sqrt{de}
  14. b 2 c = 4 d e . b^{2}c=4de.
  15. e = a + a 2 - b 2 c 2 , e=\frac{a+\sqrt{a^{2}-b^{2}c}}{2},
  16. d = a - a 2 - b 2 c 2 , d=\frac{a-\sqrt{a^{2}-b^{2}c}}{2},
  17. a 2 - b 2 c \sqrt{a^{2}-b^{2}c}
  18. x - 2 y + 1 = 0 , x-2y+1=0,
  19. 3 x + 5 y - 8 = 0 , 3x+5y-8=0,
  20. 4 x + 3 y - 7 = 0. 4x+3y-7=0.
  21. a ( x - 2 y + 1 ) + b ( 3 x + 5 y - 8 ) = 4 x + 3 y - 7. a(x-2y+1)+b(3x+5y-8)=4x+3y-7.
  22. a + 3 b = 4 , a+3b=4,
  23. - 2 a + 5 b = 3 , -2a+5b=3,
  24. a - 8 b = - 7. a-8b=-7.
  25. a + b i c + d i = e + f i , \frac{a+bi}{c+di}=e+fi,
  26. ( c e - f d ) + ( e d + c f ) i = a + b i . (ce-fd)+(ed+cf)i=a+bi.
  27. c e - f d = a , ce-fd=a,
  28. e d + c f = b . ed+cf=b.
  29. e = a c + b d c 2 + d 2 and f = b c - a d c 2 + d 2 . e=\frac{ac+bd}{c^{2}+d^{2}}\quad\quad\,\text{and}\quad\quad f=\frac{bc-ad}{c^{% 2}+d^{2}}.

Equations_defining_abelian_varieties.html

  1. L n , L^{n},

Equilibrium_unfolding.html

  1. k f k_{f}
  2. k u k_{u}
  3. U N U\rightarrow N
  4. N U N\rightarrow U
  5. K e q = def k u k f = [ U ] e q [ N ] e q K_{eq}\ \stackrel{\mathrm{def}}{=}\ \frac{k_{u}}{k_{f}}=\frac{\left[U\right]_{% eq}}{\left[N\right]_{eq}}
  6. Δ G o \Delta G^{o}
  7. Δ G o = - R T ln K e q \Delta G^{o}=-RT\ln K_{eq}
  8. R R
  9. T T
  10. Δ G o \Delta G^{o}
  11. Δ G o \Delta G^{o}
  12. k f k_{f}
  13. k u k_{u}
  14. Δ G o \Delta G^{o}
  15. p N p_{N}
  16. p U p_{U}
  17. p N = 1 1 + e - Δ G / R T p_{N}=\frac{1}{1+e^{-\Delta G/RT}}
  18. p U = e - Δ G / R T 1 + e - Δ G / R T p_{U}=\frac{e^{-\Delta G/RT}}{1+e^{-\Delta G/RT}}
  19. Δ n \Delta n
  20. Δ G = Δ G w - R T Δ n ln ( 1 + k [ D ] ) \Delta G=\Delta G_{w}-RT\Delta n\ln\left(1+k[D]\right)
  21. Δ G w \Delta G_{w}
  22. Δ G w \Delta G_{w}
  23. Δ n \Delta n
  24. Δ G = Δ G w - R T Δ n ln ( 1 + ( K - 1 ) X D ) \Delta G=\Delta G_{w}-RT\Delta n\ln\left(1+(K-1)X_{D}\right)
  25. K K
  26. X d X_{d}
  27. M - 1 M^{-1}
  28. M - 1 M^{-1}
  29. Δ G = m ( [ D ] 1 / 2 - [ D ] ) \Delta G=m\left([D]_{1/2}-[D]\right)
  30. m m
  31. [ D ] 1 / 2 \left[D\right]_{1/2}
  32. p N = p U = 1 / 2 p_{N}=p_{U}=1/2
  33. Δ G \Delta G
  34. m m
  35. [ D ] 1 / 2 \left[D\right]_{1/2}
  36. Δ G \Delta G
  37. m m
  38. [ D ] 1 / 2 \left[D\right]_{1/2}
  39. p N p_{N}
  40. p U p_{U}
  41. p N p_{N}
  42. p U p_{U}
  43. A A
  44. A N A_{N}
  45. A U A_{U}
  46. A = A N p N + A U p U A=A_{N}p_{N}+A_{U}p_{U}
  47. A A
  48. A N A_{N}
  49. A U A_{U}
  50. Δ G \Delta G
  51. A N A_{N}
  52. A U A_{U}
  53. A A
  54. Δ H \Delta H
  55. Δ S \Delta S
  56. Δ G \Delta G
  57. Δ C p \Delta C_{p}
  58. Δ H ( T ) = Δ H ( T d ) + T d T Δ C p d T \ \Delta H(T)=\Delta H(T_{d})+\int_{T_{d}}^{T}\Delta C_{p}dT
  59. Δ H ( T ) = Δ H ( T d ) + Δ C p [ T - T d ] \ \Delta H(T)=\Delta H(T_{d})+\Delta C_{p}[T-T_{d}]
  60. Δ S ( T ) = Δ H ( T d ) T d + T d T Δ C p d l n T \ \Delta S(T)=\frac{\Delta H(T_{d})}{T_{d}}+\int_{T_{d}}^{T}\Delta C_{p}dlnT
  61. Δ S ( T ) = Δ H ( T d ) T d + Δ C p l n T T d \ \Delta S(T)=\frac{\Delta H(T_{d})}{T_{d}}+\Delta C_{p}ln\frac{T}{T_{d}}
  62. Δ G ( T ) = Δ H - T Δ S \ \Delta G(T)=\Delta H-T\Delta S
  63. Δ G ( T ) = Δ H ( T d ) T d - T T d + T d T Δ C p d T - T T d T Δ C p d l n T \ \Delta G(T)=\Delta H(T_{d})\frac{T_{d}-T}{T_{d}}+\int_{T_{d}}^{T}\Delta C_{p% }dT-T\int_{T_{d}}^{T}\Delta C_{p}dlnT
  64. Δ G ( T ) = Δ H ( T d ) ( 1 - T T d ) - Δ C p [ T d - T + T l n ( T T d ) ] \ \Delta G(T)=\Delta H(T_{d})(1-\frac{T}{T_{d}})-\Delta C_{p}[T_{d}-T+Tln(% \frac{T}{T_{d}})]
  65. Δ H \ \Delta H
  66. Δ S \ \Delta S
  67. Δ G \ \Delta G
  68. T \ T
  69. T d \ T_{d}
  70. Δ C p \ \Delta C_{p}
  71. Δ C p \ \Delta C_{p}
  72. Δ C p \ \Delta C_{p}
  73. Δ H ( T d ) \ \Delta H(T_{d})
  74. T d \ T_{d}
  75. p H \ pH
  76. Δ C p \ \Delta C_{p}
  77. Δ C p \ \Delta C_{p}
  78. Δ C p \ \Delta C_{p}
  79. Δ A S A = A S A u n f o l d e d - A S A n a t i v e \ \Delta ASA=ASA_{unfolded}-ASA_{native}
  80. A S A n a t i v e \ ASA_{native}
  81. A S A u n f o l d e d ASA_{unfolded}
  82. A S A u n f o l d e d = a p o l a r × A S A p o l a r + a a r o m a t i c × A S A a r o m a t i c + a n o n - p o l a r × A S A n o n - p o l a r \ ASA_{unfolded}=a_{polar}\times ASA_{polar}+a_{aromatic}\times ASA_{aromatic}% +a_{non-polar}\times ASA_{non-polar}
  83. Δ A S A \ \Delta ASA
  84. Δ C p \ \Delta C_{p}
  85. Δ C p = 0.61 * Δ A S A \ \Delta C_{p}=0.61*\Delta ASA
  86. A p e a k \ A_{peak}
  87. Δ H v H ( T ) = - R d l n K d T - 1 \ \Delta H_{vH}(T)=-R\frac{dlnK}{dT^{-1}}
  88. T = T d \ T=T_{d}
  89. Δ H v H ( T d ) \ \Delta H_{vH}(T_{d})
  90. Δ H v H ( T d ) = R T d 2 Δ C p m a x A p e a k \ \Delta H_{vH}(T_{d})=\frac{RT_{d}^{2}\Delta C_{p}^{max}}{A_{peak}}
  91. A p e a k = Δ H v H ( T d ) \ A_{peak}=\Delta H_{vH}(T_{d})
  92. Δ C p m a x \ \Delta C_{p}^{max}

Equivalence_(measure_theory).html

  1. μ ν μ ν μ . \mu\sim\nu\iff\mu\ll\nu\ll\mu.
  2. μ ( A ) = 0 ν ( A ) = 0 \mu(A)=0\iff\nu(A)=0

Equivalence_of_metrics.html

  1. X X
  2. d 1 d_{1}
  3. d 2 d_{2}
  4. X X
  5. d 1 d_{1}
  6. d 2 d_{2}
  7. X X
  8. A X A\subseteq X
  9. d 1 d_{1}
  10. d 2 d_{2}
  11. x X x\in X
  12. r > 0 r>0
  13. r , r ′′ > 0 r^{\prime},r^{\prime\prime}>0
  14. B r ( x ; d 1 ) B r ( x ; d 2 ) B_{r^{\prime}}(x;d_{1})\subseteq B_{r}(x;d_{2})
  15. B r ′′ ( x ; d 2 ) B r ( x ; d 1 ) . B_{r^{\prime\prime}}(x;d_{2})\subseteq B_{r}(x;d_{1}).
  16. I : X X I:X\to X
  17. ( d 1 , d 2 ) (d_{1},d_{2})
  18. ( d 2 , d 1 ) (d_{2},d_{1})
  19. f : + f:\mathbb{R}_{+}\to\mathbb{R}
  20. d 2 = f d 1 d_{2}=f\circ d_{1}
  21. x X x\in X
  22. α \alpha
  23. β \beta
  24. y X y\in X
  25. α d 1 ( x , y ) d 2 ( x , y ) β d 1 ( x , y ) . \alpha d_{1}(x,y)\leq d_{2}(x,y)\leq\beta d_{1}(x,y).
  26. d 1 d_{1}
  27. d 2 d_{2}
  28. α \alpha
  29. β \beta
  30. x , y X x,y\in X
  31. α d 1 ( x , y ) d 2 ( x , y ) β d 1 ( x , y ) . \alpha d_{1}(x,y)\leq d_{2}(x,y)\leq\beta d_{1}(x,y).
  32. X X
  33. X X

Equivalent_temperature.html

  1. T e T + L v c p d r T_{e}\approx T+\frac{L_{v}}{c_{pd}}r
  2. L v \,L_{v}
  3. c p d \,c_{pd}
  4. \approx

Erdős–Kac_theorem.html

  1. ω ( n ) - log log n log log n \frac{\omega(n)-\log\log n}{\sqrt{\log\log n}}
  2. log log n \sqrt{\log\log n}
  3. Φ ( a , b ) \Phi(a,b)
  4. Φ ( a , b ) = 1 2 π a b e - t 2 / 2 d t . \Phi(a,b)=\frac{1}{\sqrt{2\pi}}\int_{a}^{b}e^{-t^{2}/2}\,dt.
  5. f ( p 1 a 1 p k a k ) = f ( p 1 ) + + f ( p k ) \scriptstyle f(p_{1}^{a_{1}}\cdots p_{k}^{a_{k}})=f(p_{1})+\cdots+f(p_{k})
  6. | f ( p ) | 1 \scriptstyle|f(p)|\leq 1
  7. lim x ( 1 x # { n x : a f ( n ) - A ( n ) B ( n ) b } ) = Φ ( a , b ) \lim_{x\rightarrow\infty}\left(\frac{1}{x}\cdot\#\left\{n\leq x:a\leq\frac{f(n% )-A(n)}{B(n)}\leq b\right\}\right)=\Phi(a,b)
  8. A ( n ) = p n f ( p ) p , B ( n ) = p n f ( p ) 2 p . A(n)=\sum_{p\leq n}\frac{f(p)}{p},\qquad B(n)=\sqrt{\sum_{p\leq n}\frac{f(p)^{% 2}}{p}}.
  9. n p n_{p}
  10. I n 2 + I n 3 + I n 5 + I n 7 + I_{n_{2}}+I_{n_{3}}+I_{n_{5}}+I_{n_{7}}+\ldots
  11. n n
  12. n n
  13. n n
  14. 10 100 10^{100}
  15. O ( 1 log ( log ( n ) ) ) O\left(\frac{1}{\sqrt{\log(\log(n))}}\right)

Ergodicity.html

  1. n n
  2. ( X , Σ , μ ) (X,\;\Sigma,\;\mu\,)
  3. T : X X T:X\to X
  4. μ \mu
  5. μ \mu
  6. E Σ E\in\Sigma
  7. T - 1 ( E ) = E T^{-1}(E)=E\,
  8. μ ( E ) = 0 \mu(E)=0\,
  9. μ ( E ) = 1 \mu(E)=1\,
  10. E Σ E\in\Sigma
  11. μ ( T - 1 ( E ) E ) = 0 \mu(T^{-1}(E)\bigtriangleup E)=0
  12. μ ( E ) = 0 \mu(E)=0
  13. μ ( E ) = 1 \mu(E)=1\,
  14. \bigtriangleup
  15. E Σ E\in\Sigma
  16. μ ( n = 1 T - n E ) = 1 \mu(\bigcup_{n=1}^{\infty}T^{-n}E)=1
  17. μ ( ( T - n E ) H ) > 0 \mu((T^{-n}E)\cap H)>0
  18. f : X f:X\to\mathbb{R}
  19. f T = f f\circ T=f
  20. μ ( T t ( A ) A ) = 0 \mu(T^{t}(A)\bigtriangleup A)=0
  21. i i

Ergun_equation.html

  1. f p = 150 G r p + 1.75 f_{p}=\frac{150}{Gr_{p}}+1.75
  2. f p f_{p}
  3. G r p Gr_{p}
  4. f p = Δ p L D p ρ v s 2 ( ϵ 3 1 - ϵ ) f_{p}=\frac{\Delta p}{L}\frac{D_{p}}{\rho v_{s}^{2}}\left(\frac{\epsilon^{3}}{% 1-\epsilon}\right)
  5. G r p = ρ v s D p ( 1 - ϵ ) μ Gr_{p}=\frac{\rho v_{s}D_{p}}{(1-\epsilon)\mu}
  6. G r p Gr_{p}
  7. Δ p \Delta p
  8. L L
  9. D p D_{p}
  10. ρ \rho
  11. μ \mu
  12. v s v_{s}
  13. ϵ \epsilon
  14. Δ p = 150 μ L D p 2 ( 1 - ϵ ) 2 ϵ 3 v s + 1.75 L ρ D p ( 1 - ϵ ) ϵ 3 v s 2 \Delta p=\frac{150\mu~{}L}{D_{p}^{2}}~{}\frac{(1-\epsilon)^{2}}{\epsilon^{3}}v% _{s}+\frac{1.75~{}L~{}\rho}{D_{p}}~{}\frac{(1-\epsilon)}{\epsilon^{3}}v_{s}^{2}

Erland_Samuel_Bring.html

  1. x 5 + p x + q = 0 x^{5}+px+q=0

Error_exponent.html

  1. M = 2 n R M=2^{nR}\;
  2. y 1 n y_{1}^{n}
  3. P error 1 2 = x 1 n ( 2 ) Q ( x 1 n ( 2 ) ) 1 ( p ( y 1 n | x 1 n ( 2 ) ) > p ( y 1 n | x 1 n ( 1 ) ) ) P_{\mathrm{error}\ 1\to 2}=\sum_{x_{1}^{n}(2)}Q(x_{1}^{n}(2))1(p(y_{1}^{n}|x_{% 1}^{n}(2))>p(y_{1}^{n}|x_{1}^{n}(1)))
  4. 1 ( p ( y 1 n | x 1 n ( 2 ) ) > p ( y 1 n | x 1 n ( 1 ) ) 1(p(y_{1}^{n}|x_{1}^{n}(2))>p(y_{1}^{n}|x_{1}^{n}(1))
  5. ( p ( y 1 n | x 1 n ( 2 ) ) p ( y 1 n | x 1 n ( 1 ) ) ) s \left(\frac{p(y_{1}^{n}|x_{1}^{n}(2))}{p(y_{1}^{n}|x_{1}^{n}(1))}\right)^{s}
  6. s > 0 s>0\;
  7. P error 1 2 x 1 n ( 2 ) Q ( x 1 n ( 2 ) ) ( p ( y 1 n | x 2 n ( 2 ) ) p ( y 1 n | x 1 n ( 1 ) ) ) s . P_{\mathrm{error}\ 1\to 2}\leq\sum_{x_{1}^{n}(2)}Q(x_{1}^{n}(2))\left(\frac{p(% y_{1}^{n}|x_{2}^{n}(2))}{p(y_{1}^{n}|x_{1}^{n}(1))}\right)^{s}.
  8. P error 1 any M ρ x 1 n Q ( x 1 n ) ( p ( y 1 n | x 2 n ) p ( y 1 n | x 1 n ( 1 ) ) ) s ρ . P_{\mathrm{error}\ 1\to\mathrm{any}}\leq M^{\rho}\sum_{x_{1}^{n}}Q(x_{1}^{n})% \left(\frac{p(y_{1}^{n}|x_{2}^{n})}{p(y_{1}^{n}|x_{1}^{n}(1))}\right)^{s\rho}.
  9. X 1 n ( 1 ) , y 1 n X_{1}^{n}(1),y_{1}^{n}
  10. P error 1 any M ρ y 1 n ( x 1 n ( 1 ) Q ( x 1 n ( 1 ) ) [ p ( y 1 n | x 1 n ( 1 ) ) ] 1 - s ρ ) ( x 1 n Q ( x 1 n ) [ p ( y 1 n | x 1 n ) ] s ) ρ . P_{\mathrm{error}\ 1\to\mathrm{any}}\leq M^{\rho}\sum_{y_{1}^{n}}\left(\sum_{x% _{1}^{n}(1)}Q(x_{1}^{n}(1))[p(y_{1}^{n}|x_{1}^{n}(1))]^{1-s\rho}\right)\left(% \sum_{x_{1}^{n}}Q(x_{1}^{n})[p(y_{1}^{n}|x_{1}^{n})]^{s}\right)^{\rho}.
  11. s = 1 - s ρ s=1-s\rho
  12. x 1 n x_{1}^{n}
  13. P error 1 any M ρ y 1 n ( x 1 n Q ( x 1 n ) [ p ( y 1 n | x 1 n ) ] 1 1 + ρ ) 1 + ρ . P_{\mathrm{error}\ 1\to\mathrm{any}}\leq M^{\rho}\sum_{y_{1}^{n}}\left(\sum_{x% _{1}^{n}}Q(x_{1}^{n})[p(y_{1}^{n}|x_{1}^{n})]^{\frac{1}{1+\rho}}\right)^{1+% \rho}.
  14. P error 1 any M ρ i = 1 n y i ( x i Q i ( x i ) [ p i ( y i | x i ) ] 1 1 + ρ ) 1 + ρ P_{\mathrm{error}\ 1\to\mathrm{any}}\leq M^{\rho}\prod_{i=1}^{n}\sum_{y_{i}}% \left(\sum_{x_{i}}Q_{i}(x_{i})[p_{i}(y_{i}|x_{i})]^{\frac{1}{1+\rho}}\right)^{% 1+\rho}
  15. P error 1 any M ρ ( y ( x Q ( x ) [ p ( y | x ) ] 1 1 + ρ ) 1 + ρ ) n . P_{\mathrm{error}\ 1\to\mathrm{any}}\leq M^{\rho}\left(\sum_{y}\left(\sum_{x}Q% (x)[p(y|x)]^{\frac{1}{1+\rho}}\right)^{1+\rho}\right)^{n}.
  16. E o ( ρ , Q ) = - ln ( y ( x Q ( x ) [ p ( y | x ) ] 1 1 + ρ ) 1 + ρ ) , E_{o}(\rho,Q)=-\ln\left(\sum_{y}\left(\sum_{x}Q(x)[p(y|x)]^{\frac{1}{1+\rho}}% \right)^{1+\rho}\right),
  17. P error exp ( - n ( E o ( ρ , Q ) - ρ R ) ) . P_{\mathrm{error}}\leq\exp(-n(E_{o}(\rho,Q)-\rho R)).
  18. ρ \rho
  19. E r ( R ) = max Q max ρ ε [ 0 , 1 ] E o ( ρ , Q ) - ρ R . E_{r}(R)=\max_{Q}\max_{\rho\varepsilon[0,1]}E_{o}(\rho,Q)-\rho R.\;
  20. ε > 0 \varepsilon>0
  21. X X
  22. n n
  23. n n
  24. X 1 : n X^{1:n}
  25. n . ( H ( X ) + ε ) n.(H(X)+\varepsilon)
  26. X 1 : n X^{1:n}
  27. 1 - ε 1-\varepsilon
  28. M = e n R M=e^{nR}\,\!
  29. M = m M=m\,
  30. X 1 n X_{1}^{n}
  31. P ( X 1 n | A m ) P(X_{1}^{n}|A_{m})
  32. A m A_{m}\,
  33. m m
  34. X 1 n X_{1}^{n}
  35. m m
  36. P ( X 1 n ) P(X_{1}^{n})
  37. X 1 n ( 1 ) X_{1}^{n}(1)
  38. 1 1
  39. X 1 n ( 2 ) X_{1}^{n}(2)
  40. X 1 n ( 1 ) X_{1}^{n}(1)\,
  41. P ( X 1 n ( 2 ) ) > P ( X 1 n ( 1 ) ) P(X_{1}^{n}(2))>P(X_{1}^{n}(1))
  42. S i S_{i}\,
  43. X 1 n ( i ) X_{1}^{n}(i)
  44. P ( S i ) = P ( X 1 n ( i ) ) . P(S_{i})=P(X_{1}^{n}(i))\,.
  45. P ( E ) = i P ( E | S i ) P ( S i ) . P(E)=\sum_{i}P(E|S_{i})P(S_{i})\,.
  46. P ( E | S i ) P(E|S_{i})\,
  47. A i A_{i^{\prime}}\,
  48. X 1 n ( i ) X_{1}^{n}(i^{\prime})
  49. X 1 n ( i ) X_{1}^{n}(i)
  50. P ( X 1 n ( i ) ) P ( X 1 n ( i ) ) P(X_{1}^{n}(i^{\prime}))\geq P(X_{1}^{n}(i))
  51. X i , i X_{i,i^{\prime}}\,
  52. i i\,
  53. i i^{\prime}\,
  54. P ( A i ) = P ( X i , i P ( X 1 n ( i ) ) P ( X 1 n ( i ) ) ) P(A_{i^{\prime}})=P\left(X_{i,i^{\prime}}\bigcap P(X_{1}^{n}(i^{\prime})\right% )\geq P(X_{1}^{n}(i)))\,
  55. P ( X i , i ) = 1 M P(X_{i,i^{\prime}})=\frac{1}{M}\,
  56. P ( A i ) = 1 M P ( P ( X 1 n ( i ) ) P ( X 1 n ( i ) ) ) . P(A_{i^{\prime}})=\frac{1}{M}P(P(X_{1}^{n}(i^{\prime}))\geq P(X_{1}^{n}(i)))\,.
  57. [ P ( P ( X 1 n ( i ) ) P ( X 1 n ( i ) ) ) ] ( P ( X 1 n ( i ) ) P ( X 1 n ( i ) ) ) s \left[P(P(X_{1}^{n}(i^{\prime}))\geq P(X_{1}^{n}(i)))\right]\leq\left(\frac{P(% X_{1}^{n}(i^{\prime}))}{P(X_{1}^{n}(i))}\right)^{s}\,
  58. s > 0 . s>0\,.
  59. P ( P ( X 1 n ( i ) ) > P ( X 1 n ( i ) ) ) P(P(X_{1}^{n}(i^{\prime}))>P(X_{1}^{n}(i)))\,
  60. 1 1\,
  61. 0 0\,
  62. P ( X 1 n ( i ) ) P ( X 1 n ( i ) ) , P(X_{1}^{n}(i^{\prime}))\geq P(X_{1}^{n}(i))\,,
  63. P ( X 1 n ( i ) ) P ( X 1 n ( i ) ) 1 \frac{P(X_{1}^{n}(i^{\prime}))}{P(X_{1}^{n}(i))}\geq 1\,
  64. ( P ( X 1 n ( i ) ) P ( X 1 n ( i ) ) ) s 0 \left(\frac{P(X_{1}^{n}(i^{\prime}))}{P(X_{1}^{n}(i))}\right)^{s}\geq 0\,
  65. ρ [ 0 , 1 ] \rho\in[0,1]\,
  66. P ( E | S i ) P ( i i A i ) ( i i P ( A i ) ) ρ ( 1 M i i ( P ( X 1 n ( i ) ) P ( X 1 n ( i ) ) ) s ) ρ . P(E|S_{i})\leq P(\bigcup_{i\neq i^{\prime}}A_{i^{\prime}})\leq\left(\sum_{i% \neq i^{\prime}}P(A_{i^{\prime}})\right)^{\rho}\leq\left(\frac{1}{M}\sum_{i% \neq i^{\prime}}\left(\frac{P(X_{1}^{n}(i^{\prime}))}{P(X_{1}^{n}(i))}\right)^% {s}\right)^{\rho}\,.
  67. P ( E ) P(E)\,
  68. P ( E ) = i P ( E | S i ) P ( S i ) i P ( X 1 n ( i ) ) ( 1 M i ( P ( X 1 n ( i ) ) P ( X 1 n ( i ) ) ) s ) ρ . P(E)=\sum_{i}P(E|S_{i})P(S_{i})\leq\sum_{i}P(X_{1}^{n}(i))\left(\frac{1}{M}% \sum_{i^{\prime}}\left(\frac{P(X_{1}^{n}(i^{\prime}))}{P(X_{1}^{n}(i))}\right)% ^{s}\right)^{\rho}\,.
  69. i i^{\prime}\,
  70. P ( E ) 1 M ρ i P ( X 1 n ( i ) ) 1 - s ρ ( i P ( X 1 n ( i ) ) s ) ρ . P(E)\leq\frac{1}{M^{\rho}}\sum_{i}P(X_{1}^{n}(i))^{1-s\rho}\left(\sum_{i^{% \prime}}P(X_{1}^{n}(i^{\prime}))^{s}\right)^{\rho}\,.
  71. 1 - s ρ = s 1-s\rho=s\,
  72. s = 1 1 + ρ . s=\frac{1}{1+\rho}\,.
  73. s s\,
  74. i i^{\prime}\,
  75. P ( E ) 1 M ρ ( i P ( X 1 n ( i ) ) 1 1 + ρ ) 1 + ρ . P(E)\leq\frac{1}{M^{\rho}}\left(\sum_{i}P(X_{1}^{n}(i))^{\frac{1}{1+\rho}}% \right)^{1+\rho}\,.
  76. M = e n R M=e^{nR}\,\!
  77. X 1 n ( i ) X_{1}^{n}(i)\,
  78. P ( E ) exp ( - n [ ρ R - ln ( x i P ( x i ) 1 1 + ρ ) ( 1 + ρ ) ] ) . P(E)\leq\exp\left(-n\left[\rho R-\ln\left(\sum_{x_{i}}P(x_{i})^{\frac{1}{1+% \rho}}\right)(1+\rho)\right]\right).
  79. ρ \rho\,
  80. E 0 ( ρ ) = ln ( x i P ( x i ) 1 1 + ρ ) ( 1 + ρ ) , E_{0}(\rho)=\ln\left(\sum_{x_{i}}P(x_{i})^{\frac{1}{1+\rho}}\right)(1+\rho)\,,
  81. E r ( R ) = max ρ [ 0 , 1 ] [ ρ R - E 0 ( ρ ) ] . E_{r}(R)=\max_{\rho\in[0,1]}\left[\rho R-E_{0}(\rho)\right].\,

Essential_matrix.html

  1. 3 × 3 3\times 3
  2. 𝐄 \mathbf{E}
  3. 𝐲 \mathbf{y}
  4. 𝐲 \mathbf{y}^{\prime}
  5. ( 𝐲 ) 𝐄 𝐲 = 0 (\mathbf{y}^{\prime})^{\top}\,\mathbf{E}\,\mathbf{y}=0
  6. 𝐲 \mathbf{y}
  7. 𝐲 \mathbf{y}^{\prime}
  8. 𝐄 \mathbf{E}
  9. 𝐄 \mathbf{E}
  10. ( x 1 , x 2 , x 3 ) (x_{1},x_{2},x_{3})
  11. ( x 1 , x 2 , x 3 ) (x^{\prime}_{1},x^{\prime}_{2},x^{\prime}_{3})
  12. ( y 1 y 2 ) = 1 x 3 ( x 1 x 2 ) \begin{pmatrix}y_{1}\\ y_{2}\end{pmatrix}=\frac{1}{x_{3}}\begin{pmatrix}x_{1}\\ x_{2}\end{pmatrix}
  13. ( y 1 y 2 ) = 1 x 3 ( x 1 x 2 ) \begin{pmatrix}y^{\prime}_{1}\\ y^{\prime}_{2}\end{pmatrix}=\frac{1}{x^{\prime}_{3}}\begin{pmatrix}x^{\prime}_% {1}\\ x^{\prime}_{2}\end{pmatrix}
  14. ( y 1 y 2 1 ) = 1 x 3 ( x 1 x 2 x 3 ) \begin{pmatrix}y_{1}\\ y_{2}\\ 1\end{pmatrix}=\frac{1}{x_{3}}\begin{pmatrix}x_{1}\\ x_{2}\\ x_{3}\end{pmatrix}
  15. ( y 1 y 2 1 ) = 1 x 3 ( x 1 x 2 x 3 ) \begin{pmatrix}y^{\prime}_{1}\\ y^{\prime}_{2}\\ 1\end{pmatrix}=\frac{1}{x^{\prime}_{3}}\begin{pmatrix}x^{\prime}_{1}\\ x^{\prime}_{2}\\ x^{\prime}_{3}\end{pmatrix}
  16. 𝐲 = 1 x 3 𝐱 ~ \mathbf{y}=\frac{1}{x_{3}}\,\tilde{\mathbf{x}}
  17. 𝐲 = 1 x 3 𝐱 ~ \mathbf{y}^{\prime}=\frac{1}{x^{\prime}_{3}}\,\tilde{\mathbf{x}}^{\prime}
  18. 𝐲 \mathbf{y}
  19. 𝐲 \mathbf{y}^{\prime}
  20. 𝐱 ~ \tilde{\mathbf{x}}
  21. 𝐱 ~ \tilde{\mathbf{x}}^{\prime}
  22. 𝐱 ~ = 𝐑 ( 𝐱 ~ - 𝐭 ) \tilde{\mathbf{x}}^{\prime}=\mathbf{R}\,(\tilde{\mathbf{x}}-\mathbf{t})
  23. 𝐑 \mathbf{R}
  24. 3 × 3 3\times 3
  25. 𝐭 \mathbf{t}
  26. 𝐄 = 𝐑 [ 𝐭 ] × \mathbf{E}=\mathbf{R}\,[\mathbf{t}]_{\times}
  27. [ 𝐭 ] × [\mathbf{t}]_{\times}
  28. 𝐭 \mathbf{t}
  29. 𝐄 \mathbf{E}
  30. ( 𝐱 ~ ) T 𝐄 𝐱 ~ = ( 1 ) ( 𝐱 ~ - 𝐭 ) T 𝐑 T 𝐑 [ 𝐭 ] × 𝐱 ~ = ( 2 ) ( 𝐱 ~ - 𝐭 ) T [ 𝐭 ] × 𝐱 ~ = ( 3 ) 0 (\tilde{\mathbf{x}}^{\prime})^{T}\,\mathbf{E}\,\tilde{\mathbf{x}}\,\stackrel{(% 1)}{=}\,(\tilde{\mathbf{x}}-\mathbf{t})^{T}\,\mathbf{R}^{T}\,\mathbf{R}\,[% \mathbf{t}]_{\times}\,\tilde{\mathbf{x}}\,\stackrel{(2)}{=}\,(\tilde{\mathbf{x% }}-\mathbf{t})^{T}\,[\mathbf{t}]_{\times}\,\tilde{\mathbf{x}}\,\stackrel{(3)}{% =}\,0
  31. 𝐱 ~ \tilde{\mathbf{x}}^{\prime}
  32. 𝐱 ~ \tilde{\mathbf{x}}
  33. 𝐄 \mathbf{E}
  34. 𝐑 \mathbf{R}
  35. 𝐭 \mathbf{t}
  36. 𝐑 T 𝐑 = 𝐈 \mathbf{R}^{T}\,\mathbf{R}=\mathbf{I}
  37. 𝐑 \mathbf{R}
  38. x 3 x_{3}
  39. x 3 x^{\prime}_{3}
  40. 0 = ( 𝐱 ~ ) T 𝐄 𝐱 ~ = 1 x 3 ( 𝐱 ~ ) T 𝐄 1 x 3 𝐱 ~ = ( 𝐲 ) T 𝐄 𝐲 0=(\tilde{\mathbf{x}}^{\prime})^{T}\,\mathbf{E}\,\tilde{\mathbf{x}}=\frac{1}{x% ^{\prime}_{3}}(\tilde{\mathbf{x}}^{\prime})^{T}\,\mathbf{E}\,\frac{1}{x_{3}}% \tilde{\mathbf{x}}=(\mathbf{y}^{\prime})^{T}\,\mathbf{E}\,\mathbf{y}
  41. 3 × 3 3\times 3
  42. 3 × 3 3\times 3
  43. 𝐄 \mathbf{E}
  44. 𝐄 \mathbf{E}
  45. 𝐄 \mathbf{E}
  46. 𝐄 \mathbf{E}
  47. 𝐄 \mathbf{E}
  48. 𝐄 = 𝐑 [ 𝐭 ] × \mathbf{E}=\mathbf{R}\,[\mathbf{t}]_{\times}
  49. 𝐄 \mathbf{E}
  50. det 𝐄 = 0 \det\mathbf{E}=0
  51. 2 𝐄𝐄 T 𝐄 - tr ( 𝐄𝐄 T ) 𝐄 = 0. 2\mathbf{E}\mathbf{E}^{T}\mathbf{E}-\operatorname{tr}(\mathbf{E}\mathbf{E}^{T}% )\mathbf{E}=0.
  52. 𝐑 \mathbf{R}
  53. 𝐭 \mathbf{t}
  54. 𝐄 \mathbf{E}
  55. 𝐑 \mathbf{R}
  56. 𝐭 \mathbf{t}
  57. 𝐄 \mathbf{E}
  58. 𝐑 \mathbf{R}
  59. 𝐭 \mathbf{t}
  60. 𝐄 \mathbf{E}
  61. 𝐄 = 𝐔 𝚺 𝐕 T \mathbf{E}=\mathbf{U}\,\mathbf{\Sigma}\,\mathbf{V}^{T}
  62. 𝐔 \mathbf{U}
  63. 𝐕 \mathbf{V}
  64. 3 × 3 3\times 3
  65. 𝚺 \mathbf{\Sigma}
  66. 3 × 3 3\times 3
  67. 𝚺 = ( s 0 0 0 s 0 0 0 0 ) \mathbf{\Sigma}=\begin{pmatrix}s&0&0\\ 0&s&0\\ 0&0&0\end{pmatrix}
  68. 𝚺 \mathbf{\Sigma}
  69. 𝐄 \mathbf{E}
  70. 𝐖 = ( 0 - 1 0 1 0 0 0 0 1 ) \mathbf{W}=\begin{pmatrix}0&-1&0\\ 1&0&0\\ 0&0&1\end{pmatrix}
  71. 𝐖 - 1 = 𝐖 T = ( 0 1 0 - 1 0 0 0 0 1 ) \mathbf{W}^{-1}=\mathbf{W}^{T}=\begin{pmatrix}0&1&0\\ -1&0&0\\ 0&0&1\end{pmatrix}
  72. [ 𝐭 ] × = 𝐔 𝐖 𝚺 𝐔 T [\mathbf{t}]_{\times}=\mathbf{U}\,\mathbf{W}\,\mathbf{\Sigma}\,\mathbf{U}^{T}
  73. 𝐑 = 𝐔 𝐖 - 1 𝐕 T \mathbf{R}=\mathbf{U}\,\mathbf{W}^{-1}\,\mathbf{V}^{T}
  74. 𝚺 \mathbf{\Sigma}
  75. [ 𝐭 ] × = 𝐔 𝐙 𝐔 T [\mathbf{t}]_{\times}=\mathbf{U}\,\mathbf{Z}\,\mathbf{U}^{T}
  76. 𝐙 = ( 0 1 0 - 1 0 0 0 0 0 ) \mathbf{Z}=\begin{pmatrix}0&1&0\\ -1&0&0\\ 0&0&0\end{pmatrix}
  77. 𝐑 \mathbf{R}
  78. [ 𝐭 ] × [\mathbf{t}]_{\times}
  79. 𝐑 [ 𝐭 ] × = 𝐔 𝐖 - 1 𝐕 T 𝐕 𝐖 𝚺 𝐕 T = 𝐔 𝚺 𝐕 T = 𝐄 \mathbf{R}\,[\mathbf{t}]_{\times}=\mathbf{U}\,\mathbf{W}^{-1}\,\mathbf{V}^{T}% \,\mathbf{V}\,\mathbf{W}\,\mathbf{\Sigma}\,\mathbf{V}^{T}=\mathbf{U}\,\mathbf{% \Sigma}\,\mathbf{V}^{T}=\mathbf{E}
  80. [ 𝐭 ] × [\mathbf{t}]_{\times}
  81. 𝐭 \mathbf{t}
  82. 𝐖 𝚺 = ( 0 - s 0 s 0 0 0 0 0 ) \mathbf{W}\,\mathbf{\Sigma}=\begin{pmatrix}0&-s&0\\ s&0&0\\ 0&0&0\end{pmatrix}
  83. 𝐖 𝚺 \mathbf{W}\,\mathbf{\Sigma}
  84. ( 𝐖 𝚺 ) T = - 𝐖 𝚺 (\mathbf{W}\,\mathbf{\Sigma})^{T}=-\mathbf{W}\,\mathbf{\Sigma}
  85. [ 𝐭 ] × [\mathbf{t}]_{\times}
  86. ( [ 𝐭 ] × ) T = 𝐔 ( 𝐖 𝚺 ) T 𝐔 T = - 𝐔 𝐖 𝚺 𝐔 T = - [ 𝐭 ] × ([\mathbf{t}]_{\times})^{T}=\mathbf{U}\,(\mathbf{W}\,\mathbf{\Sigma})^{T}\,% \mathbf{U}^{T}=-\mathbf{U}\,\mathbf{W}\,\mathbf{\Sigma}\,\mathbf{U}^{T}=-[% \mathbf{t}]_{\times}
  87. [ 𝐭 ] × [\mathbf{t}]_{\times}
  88. 𝐭 \mathbf{t}
  89. 𝐑 \mathbf{R}
  90. 𝐑 \mathbf{R}
  91. det ( 𝐑 ) = ± 1 \det(\mathbf{R})=\pm 1
  92. det ( 𝐑 ) = 1 \det(\mathbf{R})=1
  93. 𝐄 \mathbf{E}
  94. 𝐄 \mathbf{E}
  95. 𝐑 \mathbf{R}
  96. 𝐭 \mathbf{t}
  97. 𝐄 \mathbf{E}
  98. 𝐄 \mathbf{E}
  99. 𝐭 \mathbf{t}
  100. 𝐄 \mathbf{E}
  101. 𝐄 𝐭 = 𝐑 [ 𝐭 ] × 𝐭 = 𝟎 \mathbf{E}\,\mathbf{t}=\mathbf{R}\,[\mathbf{t}]_{\times}\,\mathbf{t}=\mathbf{0}
  102. 𝐭 \mathbf{t}
  103. 𝐭 ^ \hat{\mathbf{t}}
  104. 𝐄 \mathbf{E}
  105. 𝐭 ^ \hat{\mathbf{t}}
  106. - 𝐭 ^ -\hat{\mathbf{t}}
  107. 𝐄 \mathbf{E}
  108. 𝐖 \mathbf{W}
  109. 𝐖 - 1 \mathbf{W}^{-1}
  110. 𝐑 \mathbf{R}
  111. 𝐭 \mathbf{t}
  112. 𝐄 \mathbf{E}
  113. 𝐭 \mathbf{t}
  114. s > 0 s>0
  115. 𝐑 \mathbf{R}
  116. 𝐭 \mathbf{t}
  117. 𝐄 \mathbf{E}
  118. 𝐄 \mathbf{E}
  119. 𝐭 ^ \hat{\mathbf{t}}
  120. 𝐄 \mathbf{E}
  121. ( x 1 , x 2 , x 3 ) (x_{1},x_{2},x_{3})
  122. ( y 1 , y 2 ) (y_{1},y_{2})
  123. ( y 1 , y 2 ) (y^{\prime}_{1},y^{\prime}_{2})
  124. 𝐫 k \mathbf{r}_{k}
  125. 𝐑 \mathbf{R}
  126. 𝐑 = ( - 𝐫 1 - - 𝐫 2 - - 𝐫 3 - ) \mathbf{R}=\begin{pmatrix}-\mathbf{r}_{1}-\\ -\mathbf{r}_{2}-\\ -\mathbf{r}_{3}-\end{pmatrix}
  127. y 1 = x 1 x 3 = 𝐫 1 ( 𝐱 ~ - 𝐭 ) 𝐫 3 ( 𝐱 ~ - 𝐭 ) = 𝐫 1 ( 𝐲 - 𝐭 / x 3 ) 𝐫 3 ( 𝐲 - 𝐭 / x 3 ) y^{\prime}_{1}=\frac{x^{\prime}_{1}}{x^{\prime}_{3}}=\frac{\mathbf{r}_{1}\cdot% (\tilde{\mathbf{x}}-\mathbf{t})}{\mathbf{r}_{3}\cdot(\tilde{\mathbf{x}}-% \mathbf{t})}=\frac{\mathbf{r}_{1}\cdot(\mathbf{y}-\mathbf{t}/x_{3})}{\mathbf{r% }_{3}\cdot(\mathbf{y}-\mathbf{t}/x_{3})}
  128. x 3 = ( 𝐫 1 - y 1 𝐫 3 ) 𝐭 ( 𝐫 1 - y 1 𝐫 3 ) 𝐲 x_{3}=\frac{(\mathbf{r}_{1}-y^{\prime}_{1}\,\mathbf{r}_{3})\cdot\mathbf{t}}{(% \mathbf{r}_{1}-y^{\prime}_{1}\,\mathbf{r}_{3})\cdot\mathbf{y}}
  129. x 3 x_{3}
  130. ( x 1 x 2 ) = x 3 ( y 1 y 2 ) \begin{pmatrix}x_{1}\\ x_{2}\end{pmatrix}=x_{3}\begin{pmatrix}y_{1}\\ y_{2}\end{pmatrix}
  131. y 2 y^{\prime}_{2}
  132. x 3 x_{3}
  133. x 3 = ( 𝐫 2 - y 2 𝐫 3 ) 𝐭 ( 𝐫 2 - y 2 𝐫 3 ) 𝐲 x_{3}=\frac{(\mathbf{r}_{2}-y^{\prime}_{2}\,\mathbf{r}_{3})\cdot\mathbf{t}}{(% \mathbf{r}_{2}-y^{\prime}_{2}\,\mathbf{r}_{3})\cdot\mathbf{y}}
  134. x 3 x_{3}
  135. x 3 x_{3}
  136. 𝐭 \mathbf{t}

Essentially_unique.html

  1. { 1 , 2 , 3 } \{1,2,3\}
  2. { a , b , c } \{a,b,c\}
  3. { 1 < 2 < 3 } \{1<2<3\}
  4. { a < b < c } \{a<b<c\}

Estimation_lemma.html

  1. f f
  2. Γ Γ
  3. | f ( z ) | |f(z)|
  4. M M
  5. z z
  6. Γ Γ
  7. | Γ f ( z ) d z | M l ( Γ ) , \left|\int_{\Gamma}f(z)\,dz\right|\leq M\,l(\Gamma),
  8. l ( Γ ) l(Γ)
  9. Γ Γ
  10. M := max z Γ | f ( z ) | M:=\max_{z\in\Gamma}|f(z)|
  11. | f ( z ) | |f(z)|
  12. | f ( z ) | |f(z)|
  13. | f ( z ) | |f(z)|
  14. f ( z ) f(z)
  15. | Γ f ( z ) d z | = | α β f ( γ ( t ) ) γ ( t ) d t | α β | f ( γ ( t ) ) | | γ ( t ) | d t M α β | γ ( t ) | d t = M l ( Γ ) \biggl|\int_{\Gamma}f(z)\,dz\biggr|=\biggl|\int_{\alpha}^{\beta}f(\gamma(t))% \gamma^{\prime}(t)\,dt\biggr|\leq\int_{\alpha}^{\beta}\left|f(\gamma(t))\right% |\left|\gamma^{\prime}(t)\right|\,dt\leq M\int_{\alpha}^{\beta}\left|\gamma^{% \prime}(t)\right|\,dt=M\,l(\Gamma)
  16. | z | |z|
  17. | Γ 1 ( z 2 + 1 ) 2 d z | , \biggl|\int_{\Gamma}\frac{1}{(z^{2}+1)^{2}}\,dz\biggr|,
  18. Γ Γ
  19. | z | = a |z|=a
  20. a > 1 a>1
  21. a a
  22. l ( Γ ) = 1 2 ( 2 π a ) = π a . l(\Gamma)=\frac{1}{2}(2\pi a)=\pi a.
  23. M M
  24. | z | = a |z|=a
  25. | z | 2 = | z 2 | = | z 2 + 1 - 1 | | z 2 + 1 | + 1 , |z|^{2}=|z^{2}|=|z^{2}+1-1|\leq|z^{2}+1|+1,
  26. | z 2 + 1 | | z | 2 - 1 = a 2 - 1 > 0 |z^{2}+1|\geq|z|^{2}-1=a^{2}-1>0
  27. | z | = a > 1 |z|=a>1
  28. Γ Γ
  29. | 1 ( z 2 + 1 ) 2 | 1 ( a 2 - 1 ) 2 . \left|\frac{1}{(z^{2}+1)^{2}}\right|\leq\frac{1}{(a^{2}-1)^{2}}.
  30. | Γ 1 ( z 2 + 1 ) 2 d z | π a ( a 2 - 1 ) 2 . \biggl|\int_{\Gamma}\frac{1}{(z^{2}+1)^{2}}\,dz\biggr|\leq\frac{\pi a}{(a^{2}-% 1)^{2}}.

Ethynyl_radical.html

  1. μ \mu
  2. γ \gamma
  3. γ \gamma
  4. γ \gamma
  5. γ \gamma

Euclidean_distance_matrix.html

  1. x 1 , x 2 , , x n x_{1},x_{2},\ldots,x_{n}
  2. A = ( a i j ) ; a i j = || x i - x j || 2 2 \begin{array}[]{rll}A&=&(a_{ij});\\ a_{ij}&=&||x_{i}-x_{j}||_{2}^{2}\end{array}
  3. a i j a_{ij}
  4. a i j = a j i a_{ij}=a_{ji}
  5. a i j a i k + a k j \sqrt{a_{ij}}\leq\sqrt{a_{ik}}+\sqrt{a_{kj}}
  6. a i j 0 a_{ij}\geq 0
  7. n ( n - 1 ) / 2 n(n-1)/2
  8. x 1 , x 2 , , x n x_{1},x_{2},\ldots,x_{n}

Euler–Maruyama_method.html

  1. d X t = a ( X t ) d t + b ( X t ) d W t , \mathrm{d}X_{t}=a(X_{t})\,\mathrm{d}t+b(X_{t})\,\mathrm{d}W_{t},
  2. Δ t > 0 \Delta t>0
  3. 0 = τ 0 < τ 1 < < τ N = T and Δ t = T / N ; 0=\tau_{0}<\tau_{1}<\cdots<\tau_{N}=T\,\text{ and }\Delta t=T/N;
  4. Y n + 1 = Y n + a ( Y n ) Δ t + b ( Y n ) Δ W n , \,Y_{n+1}=Y_{n}+a(Y_{n})\,\Delta t+b(Y_{n})\,\Delta W_{n},
  5. Δ W n = W τ n + 1 - W τ n . \Delta W_{n}=W_{\tau_{n+1}}-W_{\tau_{n}}.
  6. Δ t \Delta t
  7. d Y t = θ ( μ - Y t ) d t + σ d W t , Y 0 = I C . dY_{t}=\theta\cdot(\mu-Y_{t})\,dt+\sigma\,dW_{t},\;\;Y_{0}=IC.

European_Union_roaming_regulations.html

  1. x = ( 1 + VAT 100 % ) E C rate x=\left(1+\frac{\,\text{VAT}}{100\%}\right)\cdot EC_{\,\text{rate}}
  2. x = ( 1 + VAT 100 % ) E C rate E x rate x=\left(1+\frac{\,\text{VAT}}{100\%}\right)\cdot EC\text{rate}\cdot Ex\text{rate}
  3. E C rate EC_{\,\text{rate}}
  4. VAT {\,\text{VAT}}
  5. E x rate Ex_{\,\text{rate}}

Exceptional_object.html

  1. M 24 M_{24}
  2. M 24 M_{24}
  3. M 24 M_{24}
  4. n 3 , n 6 n\geq 3,n\neq 6
  5. A n A_{n}
  6. S n S_{n}
  7. S n S_{n}
  8. n = 6 , n=6,
  9. S 6 S_{6}
  10. A 6 A_{6}
  11. C 2 C_{2}
  12. C 2 × C 2 C_{2}\times C_{2}
  13. A 6 PSL ( 2 , 9 ) . A_{6}\cong\operatorname{PSL}(2,9).
  14. S 3 S_{3}
  15. n = 2 k - 2 n=2^{k}-2
  16. n = 2 , 6 , 14 , 30 , 62 , n=2,6,14,30,62,

Exchange_matrix.html

  1. J 2 = ( 0 1 1 0 ) ; J 3 = ( 0 0 1 0 1 0 1 0 0 ) ; J n = ( 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 ) . J_{2}=\begin{pmatrix}0&1\\ 1&0\end{pmatrix};\quad J_{3}=\begin{pmatrix}0&0&1\\ 0&1&0\\ 1&0&0\end{pmatrix};\quad J_{n}=\begin{pmatrix}0&0&\cdots&0&0&1\\ 0&0&\cdots&0&1&0\\ 0&0&\cdots&1&0&0\\ \vdots&\vdots&&\vdots&\vdots&\vdots\\ 0&1&\cdots&0&0&0\\ 1&0&\cdots&0&0&0\end{pmatrix}.
  2. J i , j = { 1 , j = n - i + 1 0 , j n - i + 1 J_{i,j}=\begin{cases}1,&j=n-i+1\\ 0,&j\neq n-i+1\\ \end{cases}

Exchangeable_random_variables.html

  1. X σ ( 1 ) , X σ ( 2 ) , X σ ( 3 ) , X_{\sigma(1)},X_{\sigma(2)},X_{\sigma(3)},\dots
  2. X = ( X 1 , X 2 , X 3 , ) {X}=(X_{1},X_{2},X_{3},...)
  3. F X F_{X}
  4. F X ( x ) = lim n 1 n i = 1 n I ( X i x ) . F_{X}(x)=\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}I(X_{i}\leq x).
  5. X {X}
  6. X | F X {X}|F_{X}
  7. F X F_{X}
  8. Pr ( X 1 x 1 , X 2 x 2 , , X n x n ) = i = 1 n F ( x i | θ ) d P ( F ) . \Pr(X_{1}\leq x_{1},X_{2}\leq x_{2},...,X_{n}\leq x_{n})=\int\prod_{i=1}^{n}F(% x_{i}|\theta)\,dP(F).
  9. F X F_{X}
  10. θ \theta
  11. p ( X 1 , X 2 , , X n ) = i = 1 n p ( X i | θ ) d P ( θ ) . p(X_{1},X_{2},...,X_{n})=\int\prod_{i=1}^{n}p(X_{i}|\theta)\,dP(\theta).
  12. X 1 , X 2 , X 3 , X_{1},X_{2},X_{3},...
  13. cov ( X i , X j ) = var ( E ( X i | F X ) ) = var ( E ( X i | θ ) ) 0 for i j . \operatorname{cov}(X_{i},X_{j})=\operatorname{var}(\operatorname{E}(X_{i}|F_{X% }))=\operatorname{var}(\operatorname{E}(X_{i}|\theta))\geq 0\quad\,\text{for }% i\neq j.
  14. X 1 , X 2 , , X n X_{1},X_{2},...,X_{n}
  15. σ 2 = var ( X i ) \sigma^{2}=\operatorname{var}(X_{i})
  16. cov ( X i , X j ) - σ 2 n - 1 for i j . \operatorname{cov}(X_{i},X_{j})\geq-\frac{\sigma^{2}}{n-1}\quad\,\text{for }i% \neq j.
  17. 0 \displaystyle 0
  18. ( X , Y ) (X,Y)
  19. μ = 0 \mu=0
  20. σ x = σ y = 1 \sigma_{x}=\sigma_{y}=1
  21. ρ ( - 1 , 1 ) \rho\in(-1,1)
  22. X X
  23. Y Y
  24. ρ = 0 \rho=0
  25. p ( x , y ) = p ( y , x ) exp [ - 1 2 ( 1 - ρ 2 ) ( x 2 + y 2 - 2 ρ x y ) ] . p(x,y)=p(y,x)\propto\exp\left[-\frac{1}{2(1-\rho^{2})}(x^{2}+y^{2}-2\rho xy)% \right].
  26. q = 1 - p q=1-p
  27. p = 1 / 2 , p=1/2,

Excitation_filter.html

  1. E = h c / λ E=hc/\lambda

Excitation_temperature.html

  1. T ex T_{\rm ex}
  2. n u n l = g u g l exp ( - Δ E k T ex ) , \frac{n_{\rm u}}{n_{\rm l}}=\frac{g_{\rm u}}{g_{\rm l}}\exp{(-\frac{\Delta E}{% kT_{\rm ex}})},

Excluded_point_topology.html

  1. X \ { p } X\backslash\{p\}

Expander_mixing_lemma.html

  1. S , T S,T
  2. G G
  3. n n
  4. S S
  5. T T
  6. d | S | | T | / n d|S||T|/n
  7. G = ( V , E ) G=(V,E)
  8. λ ( 0 , 1 ) \lambda\in(0,1)
  9. S , T V S,T\subseteq V
  10. E ( S , T ) = | { ( x , y ) S × T : x y E ( G ) } | E(S,T)=|\{(x,y)\in S\times T:xy\in E(G)\}|
  11. | E ( S , T ) - d | S | | T | n | d λ | S | | T | . \left|E(S,T)-\frac{d|S||T|}{n}\right|\leq d\lambda\sqrt{|S||T|}\,.
  12. A A
  13. G G
  14. U V U\subseteq V
  15. | U = v U | v 𝐑 n |U\rangle=\sum_{v\in U}|v\rangle\in\mathbf{R}^{n}
  16. | v |v\rangle
  17. 𝐑 n \mathbf{R}^{n}
  18. v t h v^{th}
  19. A | V = d | V A|V\rangle=d|V\rangle
  20. S S
  21. T T
  22. E ( S , T ) = S | A | T E(S,T)=\langle S|A|T\rangle
  23. | S |S\rangle
  24. | T |T\rangle
  25. | V |V\rangle
  26. | S = | S | n | V + | S |S\rangle=\frac{|S|}{n}|V\rangle+|S^{\prime}\rangle
  27. | T = | T | n | V + | T |T\rangle=\frac{|T|}{n}|V\rangle+|T^{\prime}\rangle
  28. S | V = T | V = 0 \langle S^{\prime}|V\rangle=\langle T^{\prime}|V\rangle=0
  29. S | A | T = | S | | T | n 2 V | A | V + S | A | T \langle S|A|T\rangle=\frac{|S||T|}{n^{2}}\langle V|A|V\rangle+\langle S^{% \prime}|A|T^{\prime}\rangle
  30. V | A | V = d | V 2 = d n \langle V|A|V\rangle=d\||V\rangle\|^{2}=dn
  31. | S | A | T | d λ | S | T d λ | S | T = d λ | S | | T | |\langle S^{\prime}|A|T^{\prime}\rangle|\leq d\lambda\||S^{\prime}\rangle\|\||% T^{\prime}\rangle\|\leq d\lambda\||S\rangle\|\||T\rangle\|=d\lambda\sqrt{|S||T|}
  32. S , T V S,T\subseteq V
  33. | E ( S , T ) - d | S | | T | n | d λ | S | | T | |E(S,T)-\frac{d|S||T|}{n}|\leq d\lambda\sqrt{|S||T|}
  34. O ( d λ ( 1 + log ( 1 / λ ) ) ) O(d\lambda(1+\log(1/\lambda)))

Expander_walk_sampling.html

  1. G = ( V , E ) G=(V,E)
  2. λ \lambda
  3. n n
  4. G G
  5. f : V [ 0 , 1 ] f:V\rightarrow[0,1]
  6. G G
  7. μ = E [ f ] \mu=E[f]
  8. f f
  9. μ = 1 n v V f ( v ) \mu=\frac{1}{n}\sum_{v\in V}f(v)
  10. Y 0 , Y 1 , , Y k Y_{0},Y_{1},\ldots,Y_{k}
  11. k k
  12. G G
  13. Y 0 Y_{0}
  14. γ > 0 \gamma>0
  15. Pr [ 1 k i = 0 k f ( Y i ) - μ > γ ] e - Ω ( γ 2 ( 1 - λ ) k ) . \Pr\left[\frac{1}{k}\sum_{i=0}^{k}f(Y_{i})-\mu>\gamma\right]\leq e^{-\Omega(% \gamma^{2}(1-\lambda)k)}.
  16. Ω \Omega
  17. 1 / 10 \geq 1/10
  18. Pr [ 1 k i = 0 k f ( Y i ) - μ < - γ ] e - Ω ( γ 2 ( 1 - λ ) k ) . \Pr\left[\frac{1}{k}\sum_{i=0}^{k}f(Y_{i})-\mu<-\gamma\right]\leq e^{-\Omega(% \gamma^{2}(1-\lambda)k)}.
  19. k k
  20. f f
  21. k log n k\log n
  22. log n + O ( k ) \log n+O(k)

Expansive.html

  1. ( X , d ) (X,d)
  2. f : X X f\colon X\to X
  3. ε 0 > 0 , \varepsilon_{0}>0,
  4. x y x\neq y
  5. X X
  6. n n
  7. d ( f n ( x ) , f n ( y ) ) ε 0 d(f^{n}(x),f^{n}(y))\geq\varepsilon_{0}
  8. n n
  9. f f
  10. X X
  11. d d^{\prime}
  12. d d
  13. f f
  14. ( X , d ) (X,d)
  15. f f
  16. ( X , d ) (X,d)
  17. f : X X f\colon X\to X
  18. X X
  19. ε 0 \varepsilon_{0}
  20. x y x\neq y
  21. X X
  22. n n\in\mathbb{N}
  23. d ( f n ( x ) , f n ( y ) ) ε 0 d(f^{n}(x),f^{n}(y))\geq\varepsilon_{0}
  24. ϵ > 0 \epsilon>0
  25. δ > 0 \delta>0
  26. N > 0 N>0
  27. x , y x,y
  28. X X
  29. d ( x , y ) > ϵ d(x,y)>\epsilon
  30. n n\in\mathbb{Z}
  31. | n | N |n|\leq N
  32. d ( f n ( x ) , f n ( y ) ) > c - δ , d(f^{n}(x),f^{n}(y))>c-\delta,
  33. c c
  34. f f
  35. X X
  36. f f
  37. X X

Exposure_assessment.html

  1. E = t 1 t 2 C ( t ) d t E=\int_{t_{1}}^{t_{2}}C(t)\,dt
  2. E = s u m ( t 1 t 2 C ( t ) d t t y t z C ( t ) d t ) E=sum(\int_{t_{1}}^{t_{2}}C(t)\,dt...\int_{t_{y}}^{t_{z}}C(t)\,dt)
  3. L A D D = ( C o n t a m i n a n t C o n c e n t r a t i o n ) ( I n t a k e R a t e ) ( E x p o s u r e D u r a t i o n ) / ( B o d y W e i g h t ) ( A v e r a g e L i f e t i m e ) LADD=(ContaminantConcentration)(IntakeRate)(ExposureDuration)/(BodyWeight)(AverageLifetime)

Exposure_range.html

  1. log 2 ( c ) \log_{2}(c)

Extended_finite-state_machine.html

  1. M = ( I , O , S , D , F , U , T ) M=(I,O,S,D,F,U,T)
  2. D 1 × × D n D_{1}\times\ldots\times D_{n}
  3. f i : D { 0 , 1 } f_{i}:D\rightarrow\{0,1\}
  4. u i : D D u_{i}:D\rightarrow D
  5. T : S × F × I S × U × O T:S\times F\times I\rightarrow S\times U\times O

Extended_precision.html

  1. ( - 1 ) < s u p > s × < v a r > m < / v a r > × 2 16382 (-1)<sup>s×<var>m</var>×2^{−16382}
  2. x y = 2 y log 2 x x^{y}=2^{\,y\ \log_{2}\,x}
  3. 2 ( - 1 ) s × E × M 2^{(-1)^{s}\,\times\,E}\,\times\,M
  4. s s
  5. E E
  6. M M
  7. 1 M Align l t ; 2 1≤M&lt;2
  8. log 2 ( 2 ( - 1 ) s × E × M ) = ( - 1 ) s × E × log 2 2 + log 2 M = ± E + log 2 M \log_{2}(2^{(-1)^{s}\,\times\,E}\,\times\,M)=(-1)^{s}\,\times\,E\,\times\,\log% _{2}2\,+\,\log_{2}M=\pm\,E\,+\,\log_{2}M
  9. 1 M Align l t ; 2 1≤M&lt;2
  10. 2 \sqrt{2}
  11. 2 4 \sqrt[4]{2}
  12. I I
  13. F F
  14. I I
  15. F F
  16. 2 intermediate result = 2 I + F = 2 I 2 F 2^{\mathrm{intermediate\ result}}=2^{I+F}=2^{I}\,2^{F}
  17. 2 intermediate result = 2 I + F = 2 I + ( 1 - 1 ) + F = 2 ( I - 1 ) + ( 1 + F ) = 2 I - 1 2 1 + F 2^{\mathrm{intermediate\ result}}=2^{I+F}=2^{I\,+\,(1-1)\,+\,F}=2^{(I-1)\,+\,(% 1+F)}=2^{I-1}\,2^{1+F}
  18. I I
  19. I - 1 I-1
  20. 2 < s u p > F 2<sup>F

Extendible_hashing.html

  1. h ( k ) h(k)
  2. h ( k 1 ) h(k_{1})
  3. h ( k 2 ) h(k_{2})
  4. h ( k 3 ) h(k_{3})
  5. h ( k 4 ) h(k_{4})
  6. h ( k 4 ) h(k_{4})
  7. h ( k 2 ) h(k2)
  8. h ( k 2 ) h(k2)
  9. h ( k 4 ) h(k_{4})
  10. h ( k 2 ) h(k2)
  11. h ( k 4 ) h(k_{4})

Extensional_tectonics.html

  1. β \beta
  2. β = t 0 t 1 \beta=\frac{t_{0}}{t_{1}}
  3. t 0 t_{0}
  4. t 1 t_{1}

Extra_special_group.html

  1. M ( p ) = a , b , c : a p = b p = 1 , c p = 1 , b a = a b c , c a = a c , c b = b c M(p)=\langle a,b,c:a^{p}=b^{p}=1,c^{p}=1,ba=abc,ca=ac,cb=bc\rangle
  2. N ( p ) = a , b , c : a p = b p = c , c p = 1 , b a = a b c , c a = a c , c b = b c N(p)=\langle a,b,c:a^{p}=b^{p}=c,c^{p}=1,ba=abc,ca=ac,cb=bc\rangle

Eye_movement_in_music_reading.html

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

Étale_topology.html

  1. 𝒪 X , x \mathcal{O}_{X,x}
  2. 𝒪 X , x sh \mathcal{O}_{X,x}\text{sh}

Faber–Jackson_relation.html

  1. L L
  2. σ \sigma
  3. L σ γ L\propto\sigma^{\gamma}
  4. γ \gamma
  5. R R
  6. M M
  7. U = - α G M 2 R U=-\alpha\frac{GM^{2}}{R}
  8. α = 3 5 \alpha\ =\frac{3}{5}
  9. σ \sigma
  10. 3 σ 2 = V 2 3\sigma^{2}=V^{2}
  11. K = 1 2 M V 2 K=\frac{1}{2}MV^{2}
  12. K = 3 2 M σ 2 K=\frac{3}{2}M\sigma^{2}
  13. 2 K + U = 0 2K+U=0
  14. σ 2 = 1 5 G M R . \sigma^{2}=\frac{1}{5}\frac{GM}{R}.
  15. M / L M/L
  16. M L M\propto L
  17. R R
  18. σ 2 \sigma^{2}
  19. R L G σ 2 . R\propto\frac{LG}{\sigma^{2}}.
  20. B = L / ( 4 π R 2 ) B=L/(4\pi R^{2})
  21. L = 4 π R 2 B . L=4\pi R^{2}B.
  22. R R
  23. L L
  24. L 4 π ( L G σ 2 ) 2 B L\propto 4\pi\left(\frac{LG}{\sigma^{2}}\right)^{2}B
  25. L σ 4 4 π G 2 B , L\propto\frac{\sigma^{4}}{4\pi G^{2}B},
  26. L σ 4 . L\propto\sigma^{4}.
  27. M V = - 23 M_{V}=-23
  28. L σ 3.1 L\propto\sigma^{3.1}
  29. L σ 15.0 L\propto\sigma^{15.0}
  30. r g r_{g}

Factor_base.html

  1. x ± y x\neq\pm y
  2. x 2 y 2 ( mod n ) x^{2}\equiv y^{2}\;\;(\mathop{{\rm mod}}n)
  3. x 2 ( mod n ) and y 2 ( mod n ) x^{2}\;\;(\mathop{{\rm mod}}n)\,\text{ and }y^{2}\;\;(\mathop{{\rm mod}}n)
  4. x 2 ( mod n ) x^{2}\;\;(\mathop{{\rm mod}}n)
  5. x 2 ( mod n ) x^{2}\;\;(\mathop{{\rm mod}}n)
  6. x 2 y 2 ( mod n ) x^{2}\equiv y^{2}\;\;(\mathop{{\rm mod}}n)
  7. n = 1 n \textstyle n=1\cdot n

Factor_of_automorphy.html

  1. G G
  2. X X
  3. G G
  4. X X
  5. f f
  6. f ( g . x ) = j g ( x ) f ( x ) f(g.x)=j_{g}(x)f(x)
  7. j g ( x ) j_{g}(x)
  8. G G
  9. f f
  10. j j
  11. j j
  12. G G
  13. Γ \Gamma
  14. G G
  15. Γ \Gamma
  16. G / Γ G/\Gamma
  17. Γ \Gamma

Factorization_system.html

  1. f = m e f=m\circ e
  2. e E e\in E
  3. m M m\in M
  4. u u
  5. v v
  6. v m e = m e u vme=m^{\prime}e^{\prime}u
  7. e , e E e,e^{\prime}\in E
  8. m , m M m,m^{\prime}\in M
  9. w w
  10. e e
  11. m m
  12. e m e\downarrow m
  13. u u
  14. v v
  15. v e = m u ve=mu
  16. w w
  17. H = { e | h H , e h } H^{\uparrow}=\{e\quad|\quad\forall h\in H,e\downarrow h\}
  18. H = { m | h H , h m } . H^{\downarrow}=\{m\quad|\quad\forall h\in H,h\downarrow m\}.
  19. E M E\cap M
  20. E M E\subset M^{\uparrow}
  21. M E . M\subset E^{\downarrow}.
  22. ( E , M ) (E,M)
  23. f = m e f=m\circ e
  24. e E e\in E
  25. m M . m\in M.
  26. E = M E=M^{\uparrow}
  27. M = E . M=E^{\downarrow}.
  28. f = m e f=m\circ e
  29. e E e\in E
  30. m M m\in M

Family_of_curves.html

  1. r ( θ ) = e 1 + e cos θ r(\theta)={e\over 1+e\cos\theta}

Faulhaber's_formula.html

  1. k = 1 n k p = 1 p + 2 p + 3 p + + n p \sum_{k=1}^{n}k^{p}=1^{p}+2^{p}+3^{p}+\cdots+n^{p}
  2. k = 1 n k p = 1 p + 1 j = 0 p ( - 1 ) j ( p + 1 j ) B j n p + 1 - j , where B 1 = - 1 2 . \sum_{k=1}^{n}k^{p}={1\over p+1}\sum_{j=0}^{p}(-1)^{j}{p+1\choose j}B_{j}n^{p+% 1-j},\qquad\mbox{where}~{}~{}B_{1}=-\frac{1}{2}.
  3. ( n + 1 ) k + 1 - 1 = m = 1 n ( ( m + 1 ) k + 1 - m k + 1 ) = p = 0 k ( k + 1 p ) ( 1 p + 2 p + + n p ) (n+1)^{k+1}-1=\sum_{m=1}^{n}\left((m+1)^{k+1}-m^{k+1}\right)=\sum_{p=0}^{k}{% \left({{k+1}\atop{p}}\right)}(1^{p}+2^{p}+\dots+n^{p})
  4. 1 + 2 + 3 + + n = n ( n + 1 ) 2 = n 2 + n 2 1+2+3+\cdots+n={n(n+1)\over 2}={n^{2}+n\over 2}
  5. 1 2 + 2 2 + 3 2 + + n 2 = n ( n + 1 ) ( 2 n + 1 ) 6 = 2 n 3 + 3 n 2 + n 6 1^{2}+2^{2}+3^{2}+\cdots+n^{2}={n(n+1)(2n+1)\over 6}={2n^{3}+3n^{2}+n\over 6}
  6. 1 3 + 2 3 + 3 3 + + n 3 = ( n ( n + 1 ) 2 ) 2 = n 4 + 2 n 3 + n 2 4 1^{3}+2^{3}+3^{3}+\cdots+n^{3}=\left({n(n+1)\over 2}\right)^{2}={n^{4}+2n^{3}+% n^{2}\over 4}
  7. 1 4 + 2 4 + 3 4 + + n 4 = n ( n + 1 ) ( 2 n + 1 ) ( 3 n 2 + 3 n - 1 ) 30 = 6 n 5 + 15 n 4 + 10 n 3 - n 30 \begin{aligned}\displaystyle 1^{4}+2^{4}+3^{4}+\cdots+n^{4}&\displaystyle={n(n% +1)(2n+1)(3n^{2}+3n-1)\over 30}\\ &\displaystyle={6n^{5}+15n^{4}+10n^{3}-n\over 30}\end{aligned}
  8. 1 5 + 2 5 + 3 5 + + n 5 = n 2 ( n + 1 ) 2 ( 2 n 2 + 2 n - 1 ) 12 = 2 n 6 + 6 n 5 + 5 n 4 - n 2 12 \begin{aligned}\displaystyle 1^{5}+2^{5}+3^{5}+\cdots+n^{5}&\displaystyle={n^{% 2}(n+1)^{2}(2n^{2}+2n-1)\over 12}\\ &\displaystyle={2n^{6}+6n^{5}+5n^{4}-n^{2}\over 12}\end{aligned}
  9. 1 6 + 2 6 + 3 6 + + n 6 = n ( n + 1 ) ( 2 n + 1 ) ( 3 n 4 + 6 n 3 - 3 n + 1 ) 42 = 6 n 7 + 21 n 6 + 21 n 5 - 7 n 3 + n 42 \begin{aligned}\displaystyle 1^{6}+2^{6}+3^{6}+\cdots+n^{6}&\displaystyle={n(n% +1)(2n+1)(3n^{4}+6n^{3}-3n+1)\over 42}\\ &\displaystyle={6n^{7}+21n^{6}+21n^{5}-7n^{3}+n\over 42}\end{aligned}
  10. S p ( n ) = k = 1 n k p , S_{p}(n)=\sum_{k=1}^{n}k^{p},
  11. p 0. p\geq 0.
  12. z z
  13. G ( z , n ) = p = 0 S p ( n ) 1 p ! z p . G(z,n)=\sum_{p=0}^{\infty}S_{p}(n)\frac{1}{p!}z^{p}.
  14. G ( z , n ) = p = 0 k = 1 n 1 p ! ( k z ) p = k = 1 n e k z = e z . 1 - e n z 1 - e z , = 1 - e n z e - z - 1 . \begin{aligned}\displaystyle G(z,n)=&\displaystyle\sum_{p=0}^{\infty}\sum_{k=1% }^{n}\frac{1}{p!}(kz)^{p}=\sum_{k=1}^{n}e^{kz}=e^{z}.\frac{1-e^{nz}}{1-e^{z}},% \\ \displaystyle=&\displaystyle\frac{1-e^{nz}}{e^{-z}-1}.\end{aligned}
  15. z z
  16. z z
  17. B j ( x ) B_{j}(x)
  18. z e z x e z - 1 = j = 0 B j ( x ) z j j ! , \frac{ze^{zx}}{e^{z}-1}=\sum_{j=0}^{\infty}B_{j}(x)\frac{z^{j}}{j!},
  19. B j = B j ( 0 ) B_{j}=B_{j}(0)
  20. B 1 = - 1 2 B_{1}=-\frac{1}{2}
  21. G ( z , n ) = \displaystyle G(z,n)=
  22. B j = 0 B_{j}=0
  23. j > 1 j>1
  24. B 1 = 1 2 B_{1}=\frac{1}{2}
  25. ( - 1 ) j (-1)^{j}
  26. k = 1 n k p = k = 0 p ( - 1 ) p - k k + 1 ( p k ) B p - k n k + 1 . \sum_{k=1}^{n}k^{p}=\sum_{k=0}^{p}{(-1)^{p-k}\over k+1}{p\choose k}B_{p-k}n^{k% +1}.
  27. G ( z , n ) G(z,n)
  28. G ( z , n ) = e ( n + 1 ) z e z - 1 - e z e z - 1 = j = 0 ( B j ( n + 1 ) - B j ( 1 ) ) z j - 1 j ! , \begin{aligned}\displaystyle G(z,n)=&\displaystyle\frac{e^{(n+1)z}}{e^{z}-1}-% \frac{e^{z}}{e^{z}-1}\\ \displaystyle=&\displaystyle\sum_{j=0}^{\infty}\left(B_{j}(n+1)-B_{j}(1)\right% )\frac{z^{j-1}}{j!},\end{aligned}
  29. k = 1 n k p = 1 p + 1 ( B p + 1 ( n + 1 ) - B p + 1 ( 1 ) ) . \sum_{k=1}^{n}k^{p}=\frac{1}{p+1}\left(B_{p+1}(n+1)-B_{p+1}(1)\right).
  30. B k = - k ζ ( 1 - k ) B_{k}=-k\zeta(1-k)
  31. k = 1 n k p = n p + 1 p + 1 - j = 0 p - 1 ( p j ) ζ ( - j ) n p - j . \sum\limits_{k=1}^{n}k^{p}=\frac{n^{p+1}}{p+1}-\sum\limits_{j=0}^{p-1}{p% \choose j}\zeta(-j)n^{p-j}.
  32. G ( z , n ) G(z,n)
  33. n n
  34. ( z ) < 0 \Re(z)<0
  35. lim n G ( z , n ) = 1 e - z - 1 = j = 0 ( - 1 ) j B j z j - 1 j ! \lim_{n\rightarrow\infty}G(z,n)=\frac{1}{e^{-z}-1}=\sum_{j=0}^{\infty}(-1)^{j}% B_{j}\frac{z^{j-1}}{j!}
  36. k = 1 k p = ( - 1 ) p + 1 B p + 1 p + 1 . \sum_{k=1}^{\infty}k^{p}=\frac{(-1)^{p+1}B_{p+1}}{p+1}.
  37. ζ ( s ) = n = 1 1 n s \zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}
  38. s = - p < 0 s=-p<0
  39. ζ ( s ) \zeta(s)
  40. k = 1 n k p = 1 p + 1 j = 0 p ( p + 1 j ) B j n p + 1 - j = 1 p + 1 j = 0 p ( p + 1 j ) B j n p + 1 - j \sum_{k=1}^{n}k^{p}={1\over p+1}\sum_{j=0}^{p}{p+1\choose j}B_{j}n^{p+1-j}={1% \over p+1}\sum_{j=0}^{p}{p+1\choose j}B^{j}n^{p+1-j}
  41. = ( B + n ) p + 1 - B p + 1 p + 1 . ={(B+n)^{p+1}-B^{p+1}\over p+1}.
  42. T ( b j ) = B j . T(b^{j})=B_{j}.\,
  43. k = 1 n k p = 1 p + 1 j = 0 p ( p + 1 j ) B j n p + 1 - j = 1 p + 1 j = 0 p ( p + 1 j ) T ( b j ) n p + 1 - j \sum_{k=1}^{n}k^{p}={1\over p+1}\sum_{j=0}^{p}{p+1\choose j}B_{j}n^{p+1-j}={1% \over p+1}\sum_{j=0}^{p}{p+1\choose j}T(b^{j})n^{p+1-j}
  44. = 1 p + 1 T ( j = 0 p ( p + 1 j ) b j n p + 1 - j ) = T ( ( b + n ) p + 1 - b p + 1 p + 1 ) . ={1\over p+1}T\left(\sum_{j=0}^{p}{p+1\choose j}b^{j}n^{p+1-j}\right)=T\left({% (b+n)^{p+1}-b^{p+1}\over p+1}\right).
  45. 1 p + 2 p + 3 p + + n p 1^{p}+2^{p}+3^{p}+\cdots+n^{p}
  46. a = 1 + 2 + 3 + + n = n ( n + 1 ) 2 . a=1+2+3+\cdots+n=\frac{n(n+1)}{2}.
  47. 1 3 + 2 3 + 3 3 + + n 3 = a 2 ; 1^{3}+2^{3}+3^{3}+\cdots+n^{3}=a^{2};
  48. 1 5 + 2 5 + 3 5 + + n 5 = 4 a 3 - a 2 3 ; 1^{5}+2^{5}+3^{5}+\cdots+n^{5}={4a^{3}-a^{2}\over 3};
  49. 1 7 + 2 7 + 3 7 + + n 7 = 12 a 4 - 8 a 3 + 2 a 2 6 ; 1^{7}+2^{7}+3^{7}+\cdots+n^{7}={12a^{4}-8a^{3}+2a^{2}\over 6};
  50. 1 9 + 2 9 + 3 9 + + n 9 = 16 a 5 - 20 a 4 + 12 a 3 - 3 a 2 5 ; 1^{9}+2^{9}+3^{9}+\cdots+n^{9}={16a^{5}-20a^{4}+12a^{3}-3a^{2}\over 5};
  51. 1 11 + 2 11 + 3 11 + + n 11 = 32 a 6 - 64 a 5 + 68 a 4 - 40 a 3 + 10 a 2 6 . 1^{11}+2^{11}+3^{11}+\cdots+n^{11}={32a^{6}-64a^{5}+68a^{4}-40a^{3}+10a^{2}% \over 6}.
  52. 1 2 p + 1 + 2 2 p + 1 + 3 2 p + 1 + + n 2 p + 1 = 1 2 2 p + 2 ( 2 p + 2 ) q = 0 p ( 2 p + 2 2 q ) ( 2 - 2 2 q ) B 2 q [ ( 8 a + 1 ) p + 1 - q - 1 ] . \begin{aligned}\displaystyle 1^{2p+1}+2^{2p+1}&\displaystyle+3^{2p+1}+\cdots+n% ^{2p+1}\\ &\displaystyle=\frac{1}{2^{2p+2}(2p+2)}\sum_{q=0}^{p}{\left({{2p+2}\atop{2q}}% \right)}(2-2^{2q})~{}B_{2q}~{}\left[(8a+1)^{p+1-q}-1\right].\end{aligned}
  53. j > 1 j>1
  54. k = 1 n k 2 m + 1 = c 1 a 2 + c 2 a 3 + + c m a m + 1 \sum_{k=1}^{n}k^{2m+1}=c_{1}a^{2}+c_{2}a^{3}+\cdots+c_{m}a^{m+1}
  55. k = 1 n k 2 m = n + 1 / 2 2 m + 1 ( 2 c 1 a + 3 c 2 a 2 + + ( m + 1 ) c m a m ) . \sum_{k=1}^{n}k^{2m}=\frac{n+1/2}{2m+1}(2c_{1}a+3c_{2}a^{2}+\cdots+(m+1)c_{m}a% ^{m}).

Fáry's_theorem.html

  1. G G
  2. n n
  3. G G
  4. G G
  5. a , b , c a,b,c
  6. G G
  7. n n
  8. G G
  9. a b c abc
  10. n = 3 n=3
  11. a a
  12. b b
  13. c c
  14. G G
  15. G G
  16. G G
  17. 3 n 6 3n−6
  18. v v
  19. G G
  20. 6 d e g ( v ) 6−deg(v)
  21. 12 12
  22. G G
  23. v v
  24. a a
  25. b b
  26. c c
  27. $\mathbf{ }$
  28. v v
  29. G G
  30. v v
  31. G G
  32. a b c abc
  33. G G
  34. P P
  35. v v
  36. P P
  37. v v
  38. v v
  39. P P
  40. G , G,

Feedback_linearization.html

  1. x ˙ = f ( x ) + g ( x ) u ( 1 ) y = h ( x ) ( 2 ) \begin{aligned}\displaystyle\dot{x}&\displaystyle=f(x)+g(x)u&\displaystyle(1)% \\ \displaystyle y&\displaystyle=h(x)&\displaystyle(2)\end{aligned}
  2. x n x\in\mathbb{R}^{n}
  3. u p u\in\mathbb{R}^{p}
  4. y m y\in\mathbb{R}^{m}
  5. u = a ( x ) + b ( x ) v u=a(x)+b(x)v\,
  6. v v
  7. u u\in\mathbb{R}
  8. y y\in\mathbb{R}
  9. z = T ( x ) z=T(x)
  10. u = a ( x ) + b ( x ) v u=a(x)+b(x)v\,
  11. v v\in\mathbb{R}
  12. y y
  13. y y
  14. ( n - 1 ) (n-1)
  15. y ˙ = d h ( x ) d t = d h ( x ) d x x ˙ = d h ( x ) d x f ( x ) + d h ( x ) d x g ( x ) u \begin{aligned}\displaystyle\dot{y}=\frac{\operatorname{d}h(x)}{\operatorname{% d}t}&\displaystyle=\frac{\operatorname{d}h(x)}{\operatorname{d}x}\dot{x}\\ &\displaystyle=\frac{\operatorname{d}h(x)}{\operatorname{d}x}f(x)+\frac{% \operatorname{d}h(x)}{\operatorname{d}x}g(x)u\end{aligned}
  16. h ( x ) h(x)
  17. f ( x ) f(x)
  18. L f h ( x ) = d h ( x ) d x f ( x ) , L_{f}h(x)=\frac{\operatorname{d}h(x)}{\operatorname{d}x}f(x),
  19. h ( x ) h(x)
  20. g ( x ) g(x)
  21. L g h ( x ) = d h ( x ) d x g ( x ) . L_{g}h(x)=\frac{\operatorname{d}h(x)}{\operatorname{d}x}g(x).
  22. y ˙ \dot{y}
  23. y ˙ = L f h ( x ) + L g h ( x ) u \dot{y}=L_{f}h(x)+L_{g}h(x)u
  24. L f 2 h ( x ) = L f L f h ( x ) = d ( L f h ( x ) ) d x f ( x ) , L_{f}^{2}h(x)=L_{f}L_{f}h(x)=\frac{\operatorname{d}(L_{f}h(x))}{\operatorname{% d}x}f(x),
  25. L g L f h ( x ) = d ( L f h ( x ) ) d x g ( x ) . L_{g}L_{f}h(x)=\frac{\operatorname{d}(L_{f}h(x))}{\operatorname{d}x}g(x).
  26. y y
  27. ( n - 1 ) (n-1)
  28. u u
  29. r 𝕎 r\in\mathbb{W}
  30. x 0 x_{0}
  31. L g L f k h ( x ) = 0 x L_{g}L_{f}^{k}h(x)=0\qquad\forall x
  32. x 0 x_{0}
  33. k r - 2 k\leq r-2
  34. L g L f r - 1 h ( x 0 ) 0 L_{g}L_{f}^{r-1}h(x_{0})\neq 0
  35. y y
  36. y y
  37. u u
  38. n n
  39. n n
  40. y = h ( x ) y ˙ = L f h ( x ) y ¨ = L f 2 h ( x ) y ( n - 1 ) = L f n - 1 h ( x ) y ( n ) = L f n h ( x ) + L g L f n - 1 h ( x ) u \begin{aligned}\displaystyle y&\displaystyle=h(x)\\ \displaystyle\dot{y}&\displaystyle=L_{f}h(x)\\ \displaystyle\ddot{y}&\displaystyle=L_{f}^{2}h(x)\\ &\displaystyle\vdots\\ \displaystyle y^{(n-1)}&\displaystyle=L_{f}^{n-1}h(x)\\ \displaystyle y^{(n)}&\displaystyle=L_{f}^{n}h(x)+L_{g}L_{f}^{n-1}h(x)u\end{aligned}
  41. y ( n ) y^{(n)}
  42. n n
  43. y y
  44. n n
  45. L g L f i h ( x ) L_{g}L_{f}^{i}h(x)
  46. i = 1 , , n - 2 i=1,\dots,n-2
  47. u u
  48. ( n - 1 ) (n-1)
  49. T ( x ) T(x)
  50. ( n - 1 ) (n-1)
  51. z = T ( x ) = [ z 1 ( x ) z 2 ( x ) z n ( x ) ] = [ y y ˙ y ( n - 1 ) ] = [ h ( x ) L f h ( x ) L f n - 1 h ( x ) ] z=T(x)=\begin{bmatrix}z_{1}(x)\\ z_{2}(x)\\ \vdots\\ z_{n}(x)\end{bmatrix}=\begin{bmatrix}y\\ \dot{y}\\ \vdots\\ y^{(n-1)}\end{bmatrix}=\begin{bmatrix}h(x)\\ L_{f}h(x)\\ \vdots\\ L_{f}^{n-1}h(x)\end{bmatrix}
  52. x x
  53. z z
  54. z z
  55. z z
  56. { z ˙ 1 = L f h ( x ) = z 2 ( x ) z ˙ 2 = L f 2 h ( x ) = z 3 ( x ) z ˙ n = L f n h ( x ) + L g L f n - 1 h ( x ) u . \begin{cases}\dot{z}_{1}&=L_{f}h(x)=z_{2}(x)\\ \dot{z}_{2}&=L_{f}^{2}h(x)=z_{3}(x)\\ &\vdots\\ \dot{z}_{n}&=L_{f}^{n}h(x)+L_{g}L_{f}^{n-1}h(x)u\end{cases}.
  57. u = 1 L g L f n - 1 h ( x ) ( - L f n h ( x ) + v ) u=\frac{1}{L_{g}L_{f}^{n-1}h(x)}(-L_{f}^{n}h(x)+v)
  58. v v
  59. z 1 = y z_{1}=y
  60. { z ˙ 1 = z 2 z ˙ 2 = z 3 z ˙ n = v \begin{cases}\dot{z}_{1}&=z_{2}\\ \dot{z}_{2}&=z_{3}\\ &\vdots\\ \dot{z}_{n}&=v\end{cases}
  61. n n
  62. v v
  63. v = - K z , v=-Kz\qquad,
  64. z z
  65. y y
  66. ( n - 1 ) (n-1)
  67. z ˙ = A z \dot{z}=Az
  68. A = [ 0 1 0 0 0 0 1 0 0 0 0 1 - k 1 - k 2 - k 3 - k n ] . A=\begin{bmatrix}0&1&0&\ldots&0\\ 0&0&1&\ldots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\ldots&1\\ -k_{1}&-k_{2}&-k_{3}&\ldots&-k_{n}\end{bmatrix}.
  69. k k
  70. n n

Feige–Fiat–Shamir_identification_scheme.html

  1. s 1 , , s k s_{1},\cdots,s_{k}
  2. s i s_{i}
  3. n n
  4. v i s i 2 ( mod n ) v_{i}\equiv s_{i}^{2}\;\;(\mathop{{\rm mod}}n)
  5. n n
  6. p p
  7. q q
  8. s i s_{i}
  9. v i v_{i}
  10. s i s_{i}
  11. v i v_{i}
  12. r r
  13. s { - 1 , 1 } s\in\{-1,1\}
  14. x s r 2 ( mod n ) x\equiv s\cdot r^{2}\;\;(\mathop{{\rm mod}}n)
  15. x x
  16. a 1 , , a k a_{1},\cdots,a_{k}
  17. a i a_{i}
  18. y r s 1 a 1 s 2 a 2 s k a k ( mod n ) y\equiv rs_{1}^{a_{1}}s_{2}^{a_{2}}\cdots s_{k}^{a_{k}}\;\;(\mathop{{\rm mod}}n)
  19. y 2 ± x v 1 a 1 v 2 a 2 v k a k ( mod n ) . y^{2}\equiv\pm\,xv_{1}^{a_{1}}v_{2}^{a_{2}}\cdots v_{k}^{a_{k}}\;\;(\mathop{{% \rm mod}}n).
  20. r r
  21. a i a_{i}
  22. s i s_{i}
  23. v i v_{i}
  24. s s
  25. v i v_{i}
  26. s i s_{i}
  27. a i a_{i}
  28. y y
  29. x y 2 v 1 - a 1 v 2 - a 2 v k - a k ( mod n ) x\equiv y^{2}v_{1}^{-a_{1}}v_{2}^{-a_{2}}\cdots v_{k}^{-a_{k}}\;\;(\mathop{{% \rm mod}}n)
  30. x x
  31. a i a_{i}
  32. y y
  33. a i a_{i}
  34. 2 k 2^{k}
  35. t t
  36. 2 k t 2^{kt}
  37. k = 5 k=5
  38. t = 4 t=4

Fejér's_theorem.html

  1. s n ( x ) = k = - n n c k e i k x , s_{n}(x)=\sum_{k=-n}^{n}c_{k}e^{ikx},
  2. c k = 1 2 π - π π f ( t ) e - i k t d t , c_{k}=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(t)e^{-ikt}dt,
  3. σ n ( x ) = 1 n k = 0 n - 1 s k ( x ) = 1 2 π - π π f ( x - t ) F n ( t ) d t , \sigma_{n}(x)=\frac{1}{n}\sum_{k=0}^{n-1}s_{k}(x)=\frac{1}{2\pi}\int_{-\pi}^{% \pi}f(x-t)F_{n}(t)dt,
  4. σ n ( x 0 ) 1 2 ( f ( x 0 + 0 ) + f ( x 0 - 0 ) ) . \sigma_{n}(x_{0})\to\frac{1}{2}\left(f(x_{0}+0)+f(x_{0}-0)\right).

FELICS.html

  1. Δ = H - L \Delta=H-L
  2. H = m a x ( P 1 , P 2 ) H=max(P1,P2)
  3. L = m i n ( P 1 , P 2 ) L=min(P1,P2)
  4. P 1 , P 2 P1,P2
  5. P P
  6. ( L + H ) / 2 (L+H)/2

Feller_process.html

  1. A f = lim t 0 T t f - f t , Af=\lim_{t\rightarrow 0}\frac{T_{t}f-f}{t},
  2. R λ f = 0 e - λ t T t f d t . R_{\lambda}f=\int_{0}^{\infty}e^{-\lambda t}T_{t}f\,dt.
  3. R λ R μ = R μ R λ = ( R μ - R λ ) / ( λ - μ ) . R_{\lambda}R_{\mu}=R_{\mu}R_{\lambda}=(R_{\mu}-R_{\lambda})/(\lambda-\mu).
  4. R λ = ( λ - A ) - 1 , A = λ - R λ - 1 . \begin{aligned}&\displaystyle R_{\lambda}=(\lambda-A)^{-1},\\ &\displaystyle A=\lambda-R_{\lambda}^{-1}.\end{aligned}

Femtosecond_pulse_shaping.html

  1. E ( t ) E(t)
  2. E ( t ) = - 1 { { E ( t ) } ( ω ) f ( ω ) } ( t ) . E^{\prime}(t)=\mathcal{F}^{-1}\{\mathcal{F}\{E(t)\}(\omega)f(\omega)\}(t).
  3. f ( ω ) f(\omega)
  4. | f ( ω ) | 1 |f(\omega)|\leq 1

Fenske_equation.html

  1. N = log [ ( X d 1 - X d ) ( 1 - X b X b ) ] log α a v g \ N=\frac{\log\,\bigg[\Big(\frac{X_{d}}{1-X_{d}}\Big)\Big(\frac{1-X_{b}}{X_{b}% }\Big)\bigg]}{\log\,\alpha_{avg}}
  2. N N
  3. X d X_{d}
  4. X b X_{b}
  5. α a v g \alpha_{avg}
  6. N = log [ ( L K d H K d ) ( H K b L K b ) ] log α a v g \ N=\frac{\log\,\bigg[\Big(\frac{LK_{d}}{HK_{d}}\Big)\Big(\frac{HK_{b}}{LK_{b}% }\Big)\bigg]}{\log\,\alpha_{avg}}
  7. N = log [ ( L K d 1 - L K d ) ( 1 - L K b L K b ) ] log α a v g \ N=\frac{\log\,\bigg[\Big(\frac{LK_{d}}{1-LK_{d}}\Big)\Big(\frac{1-LK_{b}}{LK% _{b}}\Big)\bigg]}{\log\,\alpha_{avg}}
  8. α a v g . \alpha_{avg.}
  9. α \alpha
  10. α a v g . = ( α t ) ( α b ) \alpha_{avg.}=\sqrt{(\alpha_{t})(\alpha_{b})}
  11. α t \alpha_{t}
  12. α b \alpha_{b}
  13. Z a Z b = X a X b ( P a 0 P b 0 ) N \ \frac{Z_{a}}{Z_{b}}=\frac{X_{a}}{X_{b}}\left(\frac{P^{0}_{a}}{P^{0}_{b}}% \right)^{N}
  14. N N
  15. Z n Z_{n}
  16. X n X_{n}
  17. P n 0 {P^{0}_{n}}

Fiber_(mathematics).html

  1. { y } \{y\}
  2. y Y y\in Y
  3. f - 1 ( y ) f^{-1}(y)
  4. f - 1 ( { y } ) = { x X | f ( x ) = y } . f^{-1}(\{y\})=\{x\in X\,|\,f(x)=y\}.
  5. { y } \{y\}
  6. { y } \{y\}
  7. X × Y Spec k ( p ) X\times_{Y}\mathrm{Spec}\,k(p)
  8. { y } \{y\}
  9. { y } \{y\}

Fiber_bundle_construction_theorem.html

  1. t i j : U i U j G t_{ij}:U_{i}\cap U_{j}\to G\,
  2. t i k ( x ) = t i j ( x ) t j k ( x ) x U i U j U k t_{ik}(x)=t_{ij}(x)t_{jk}(x)\qquad\forall x\in U_{i}\cap U_{j}\cap U_{k}
  3. t i : U i G t_{i}:U_{i}\to G\,
  4. t i j ( x ) = t i ( x ) - 1 t i j ( x ) t j ( x ) x U i U j . t^{\prime}_{ij}(x)=t_{i}(x)^{-1}t_{ij}(x)t_{j}(x)\qquad\forall x\in U_{i}\cap U% _{j}.
  5. T = i I U i × F = { ( i , x , y ) : i I , x U i , y F } T=\coprod_{i\in I}U_{i}\times F=\{(i,x,y):i\in I,x\in U_{i},y\in F\}
  6. ( j , x , y ) ( i , x , t i j ( x ) y ) x U i U j , y F . (j,x,y)\sim(i,x,t_{ij}(x)\cdot y)\qquad\forall x\in U_{i}\cap U_{j},y\in F.
  7. ϕ i : π - 1 ( U i ) U i × F \phi_{i}:\pi^{-1}(U_{i})\to U_{i}\times F\,
  8. ϕ i - 1 ( x , y ) = [ ( i , x , y ) ] . \phi_{i}^{-1}(x,y)=[(i,x,y)].

Fiber_diffraction.html

  1. h k l hkl
  2. | h | + | k | 0 |h|+|k|\neq 0
  3. l 0 l\neq 0
  4. ϕ ~{}\phi
  5. β \beta
  6. β \beta
  7. 2 β 2\beta

Fidelity_of_quantum_states.html

  1. F ( X , Y ) = i p i q i F(X,Y)=\sum_{i}\sqrt{p_{i}q_{i}}
  2. ( p 1 , , p n ) (\sqrt{p_{1}},\cdots,\sqrt{p_{n}})
  3. ( q 1 , , q n ) (\sqrt{q_{1}},\cdots,\sqrt{q_{n}})
  4. 0 F ( X , Y ) 1 0\leq F(X,Y)\leq 1
  5. ρ \rho
  6. σ \sigma
  7. { F i } \{F_{i}\}
  8. ρ \rho
  9. i i
  10. p i = Tr [ ρ F i ] p_{i}=\mathrm{Tr}[\rho F_{i}]
  11. q i = Tr [ σ F i ] q_{i}=\mathrm{Tr}[\sigma F_{i}]
  12. σ \sigma
  13. ρ \rho
  14. σ \sigma
  15. p p
  16. q q
  17. { F i } \{F_{i}\}
  18. F ( ρ , σ ) = min { F i } F ( X , Y ) F(\rho,\sigma)=\min_{\{F_{i}\}}F(X,Y)
  19. = min { F i } i Tr [ ρ F i ] Tr [ σ F i ] =\min_{\{F_{i}\}}\sum_{i}\sqrt{\mathrm{Tr}[\rho F_{i}]\mathrm{Tr}[\sigma F_{i}]}
  20. F ( ρ , σ ) = Tr [ ρ σ ρ ] . F(\rho,\sigma)=\operatorname{Tr}\left[\sqrt{\sqrt{\rho}\sigma\sqrt{\rho}}% \right].
  21. F ( ρ , σ ) = ρ σ tr , F(\rho,\sigma)=\lVert\sqrt{\rho}\sqrt{\sigma}\rVert_{\mathrm{tr}},
  22. F = F 2 F\;^{\prime}=F^{2}
  23. ρ = | ϕ ϕ | \rho=|\phi\rangle\langle\phi|
  24. ρ = ρ = | ϕ ϕ | \sqrt{\rho}=\rho=|\phi\rangle\langle\phi|
  25. F ( ρ , σ ) = Tr [ | ϕ ϕ | σ | ϕ ϕ | ] = ϕ | σ | ϕ Tr [ | ϕ ϕ | ] = ϕ | σ | ϕ . F(\rho,\sigma)=\operatorname{Tr}\left[\sqrt{|\phi\rangle\langle\phi|\sigma|% \phi\rangle\langle\phi|}\right]=\sqrt{\langle\phi|\sigma|\phi\rangle}% \operatorname{Tr}\left[\sqrt{|\phi\rangle\langle\phi|}\right]=\sqrt{\langle% \phi|\sigma|\phi\rangle}.
  26. σ = | ψ ψ | \sigma=|\psi\rangle\langle\psi|
  27. F ( ρ , σ ) = ϕ | ψ ψ | ϕ = | ϕ | ψ | . F(\rho,\sigma)=\sqrt{\langle\phi|\psi\rangle\langle\psi|\phi\rangle}=|\langle% \phi|\psi\rangle|.
  28. | ϕ |\phi\rangle
  29. | ψ |\psi\rangle
  30. | ϕ |\phi\rangle
  31. ρ = i p i | i i | \rho=\sum_{i}p_{i}|i\rangle\langle i|
  32. σ = i q i | i i | \sigma=\sum_{i}q_{i}|i\rangle\langle i|
  33. { | i } \{|i\rangle\}
  34. F ( ρ , σ ) = i p i q i . F(\rho,\sigma)=\sum_{i}\sqrt{p_{i}q_{i}}.
  35. F ( ρ , σ ) = F ( U ρ U * , U σ U * ) \;F(\rho,\sigma)=F(U\rho\;U^{*},U\sigma U^{*})
  36. | ψ ρ = i = 1 n ( ρ 1 2 | e i ) | e i n n |\psi_{\rho}\rangle=\sum_{i=1}^{n}(\rho^{\frac{1}{2}}|e_{i}\rangle)\otimes|e_{% i}\rangle\in\mathbb{C}^{n}\otimes\mathbb{C}^{n}
  37. { | e i } \textstyle\{|e_{i}\rangle\}
  38. F ( ρ , σ ) = max | ψ σ | ψ ρ | ψ σ | F(\rho,\sigma)=\max_{|\psi_{\sigma}\rangle}|\langle\psi_{\rho}|\psi_{\sigma}\rangle|
  39. | ψ σ |\psi_{\sigma}\rangle
  40. | Ω \textstyle|\Omega\rangle
  41. | Ω = i = 1 n | e i | e i |\Omega\rangle=\sum_{i=1}^{n}|e_{i}\rangle\otimes|e_{i}\rangle
  42. | ψ σ = ( σ 1 2 V 1 V 2 ) | Ω |\psi_{\sigma}\rangle=(\sigma^{\frac{1}{2}}V_{1}\otimes V_{2})|\Omega\rangle
  43. | ψ ρ | ψ σ | = | Ω | ( ρ 1 2 I ) ( σ 1 2 V 1 V 2 ) | Ω | = | Tr ( ρ 1 2 σ 1 2 V 1 V 2 T ) | . |\langle\psi_{\rho}|\psi_{\sigma}\rangle|=|\langle\Omega|(\rho^{\frac{1}{2}}% \otimes I)(\sigma^{\frac{1}{2}}V_{1}\otimes V_{2})|\Omega\rangle|=|% \operatorname{Tr}(\rho^{\frac{1}{2}}\sigma^{\frac{1}{2}}V_{1}V_{2}^{T})|.
  44. cos θ ρ σ = F ( ρ , σ ) \cos\theta_{\rho\sigma}=F(\rho,\sigma)\,
  45. ρ \rho
  46. σ \sigma
  47. θ ρ σ \theta_{\rho\sigma}
  48. ρ = σ \rho=\sigma
  49. D ( A , B ) = 1 2 A - B tr . D(A,B)=\frac{1}{2}\|A-B\|_{\rm tr}\,.
  50. 1 - F ( ρ , σ ) D ( ρ , σ ) 1 - F ( ρ , σ ) 2 . 1-F(\rho,\sigma)\leq D(\rho,\sigma)\leq\sqrt{1-F(\rho,\sigma)^{2}}\,.
  51. 1 - F ( ψ , ρ ) 2 D ( ψ , ρ ) . 1-F(\psi,\rho)^{2}\leq D(\psi,\rho)\,.

Fiducial_inference.html

  1. [ 0 , ω ] [0,\omega]
  2. [ 0 , X ] [0,X]
  3. F ( x ) = P ( X x ) = P ( all observations x ) = ( x ω ) n . F(x)=P(X\leq x)=P\left(\mathrm{all\ observations}\leq x\right)=\left(\frac{x}{% \omega}\right)^{n}.
  4. P ( a < X ω ) = 1 - a n = α . P\left(a<\frac{X}{\omega}\right)=1-a^{n}=\alpha.
  5. a = ( 1 - α ) 1 n . a=(1-\alpha)^{\frac{1}{n}}.
  6. P ( ω < X a ) = α . P\left(\omega<\frac{X}{a}\right)=\alpha.

FIFA_World_Ranking_system_(1999–2006).html

  1. ( w + g + a - c ) s r = m (w+g+a-c)\ s\ r=m\,

Figure_of_merit.html

  1. ( SNR ) O , AM ( SNR ) C , AM = k a 2 P 1 + k a 2 P \mathrm{\frac{(SNR)_{O,AM}}{(SNR)_{C,AM}}}=\frac{k_{a}^{2}P}{1+k_{a}^{2}P}
  2. ( SNR ) O , FM ( SNR ) C , FM = 3 k f 2 P W 2 \mathrm{\frac{(SNR)_{O,FM}}{(SNR)_{C,FM}}}=\frac{3k_{f}^{2}P}{W^{2}}

File:Bell_observers_expansion.png.html

  1. k t k\,t
  2. θ [ X ] 11 = k 2 T 1 + k 2 T 2 \theta[\vec{X}]_{11}=\frac{k^{2}\,T}{\sqrt{1+k^{2}\,T^{2}}}
  3. k t k\,t

File:Bell_observers_experiment2.png.html

  1. k k
  2. σ \sigma

File:BikeLeanForces2.svg.html

  1. N N
  2. F F
  3. m m

File:Bjt_equilibrium_bands_v2.svg.html

  1. E c E_{c}
  2. E f E_{f}
  3. E i E_{i}
  4. E v E_{v}