wpmath0000007_4

Empirical_process.html

  1. F n ( x ) = 1 n i = 1 n I ( - , x ] ( X i ) , F_{n}(x)=\frac{1}{n}\sum_{i=1}^{n}I_{(-\infty,x]}(X_{i}),
  2. G n ( A ) = n ( P n ( A ) - P ( A ) ) G_{n}(A)=\sqrt{n}(P_{n}(A)-P(A))
  3. f G n f = n ( P n - P ) f = n ( 1 n i = 1 n f ( X i ) - 𝔼 f ) f\mapsto G_{n}f=\sqrt{n}(P_{n}-P)f=\sqrt{n}\left(\frac{1}{n}\sum_{i=1}^{n}f(X_% {i})-\mathbb{E}f\right)
  4. G n ( A ) G_{n}(A)
  5. G n f G_{n}f
  6. N ( 0 , 𝔼 ( f - 𝔼 f ) 2 ) N(0,\mathbb{E}(f-\mathbb{E}f)^{2})
  7. 𝔼 f \mathbb{E}f
  8. 𝔼 f 2 \mathbb{E}f^{2}
  9. ( G n ( c ) ) c 𝒞 \bigl(G_{n}(c)\bigr)_{c\in\mathcal{C}}
  10. 𝒞 \mathcal{C}
  11. ( G n f ) f \bigl(G_{n}f\bigr)_{f\in\mathcal{F}}
  12. \mathcal{F}
  13. \mathbb{R}
  14. F n ( x ) = P n ( ( - , x ] ) = P n I ( - , x ] . F_{n}(x)=P_{n}((-\infty,x])=P_{n}I_{(-\infty,x]}.
  15. 𝒞 = { ( - , x ] : x } . \mathcal{C}=\{(-\infty,x]:x\in\mathbb{R}\}.
  16. 𝒞 \mathcal{C}
  17. n ( F n ( x ) - F ( x ) ) \sqrt{n}(F_{n}(x)-F(x))
  18. ( ) \ell^{\infty}(\mathbb{R})

EN_207.html

  1. T < 10 - n T<10^{-n}
  2. l o g ( 500 ) - 1 = 1.69 log(500)-1=1.69
  3. n = 0 n=0

Encyclopedia_of_the_Brethren_of_Purity.html

  1. 1 + 2 + 3 + 4 = 10 1+2+3+4=10

End_(category_theory).html

  1. S : 𝐂 op × 𝐂 𝐗 S:\mathbf{C}^{\mathrm{op}}\times\mathbf{C}\to\mathbf{X}
  2. ( e , ω ) (e,\omega)
  3. ω : e ¨ S \omega:e\ddot{\to}S
  4. β : x ¨ S \beta:x\ddot{\to}S
  5. h : x e h:x\to e
  6. β a = ω a h \beta_{a}=\omega_{a}\circ h
  7. ω \omega
  8. e = c S ( c , c ) or just 𝐂 S . e=\int_{c}S(c,c)\,\text{ or just }\int_{\mathbf{C}}S.
  9. c S ( c , c ) c C S ( c , c ) c c S ( c , c ) , \int_{c}S(c,c)\to\prod_{c\in C}S(c,c)\rightrightarrows\prod_{c\to c^{\prime}}S% (c,c^{\prime}),
  10. S ( c , c ) S ( c , c ) S(c,c)\to S(c,c^{\prime})
  11. S ( c , c ) S ( c , c ) S(c^{\prime},c^{\prime})\to S(c,c^{\prime})
  12. S : 𝐂 op × 𝐂 𝐗 S:\mathbf{C}^{\mathrm{op}}\times\mathbf{C}\to\mathbf{X}
  13. ( d , ζ ) (d,\zeta)
  14. ζ : S ¨ d \zeta:S\ddot{\to}d
  15. γ : S ¨ x \gamma:S\ddot{\to}x
  16. g : d x g:d\to x
  17. γ a = g ζ a \gamma_{a}=g\circ\zeta_{a}
  18. d = c S ( c , c ) or 𝐂 S . d=\int^{c}S(c,c)\,\text{ or }\int^{\mathbf{C}}S.
  19. c S ( c , c ) c C S ( c , c ) c c S ( c , c ) . \int^{c}S(c,c)\leftarrow\coprod_{c\in C}S(c,c)\leftleftarrows\coprod_{c\to c^{% \prime}}S(c^{\prime},c).
  20. F , G : 𝐂 𝐗 F,G:\mathbf{C}\to\mathbf{X}
  21. Hom 𝐗 ( F ( - ) , G ( - ) ) : 𝐂 o p × 𝐂 𝐒𝐞𝐭 \mathrm{Hom}_{\mathbf{X}}(F(-),G(-)):\mathbf{C}^{op}\times\mathbf{C}\to\mathbf% {Set}
  22. c Hom 𝐗 ( F ( c ) , G ( c ) ) = Nat ( F , G ) \int_{c}\mathrm{Hom}_{\mathbf{X}}(F(c),G(c))=\mathrm{Nat}(F,G)
  23. F F
  24. G G
  25. F F
  26. G G
  27. F ( c ) F(c)
  28. G ( c ) G(c)
  29. c c
  30. T T
  31. T T
  32. Δ op 𝐒𝐞𝐭 \Delta^{\mathrm{op}}\to\mathbf{Set}
  33. 𝐒𝐞𝐭 𝐓𝐨𝐩 \mathbf{Set}\to\mathbf{Top}
  34. 𝐓𝐨𝐩 \mathbf{Top}
  35. γ : Δ 𝐓𝐨𝐩 \gamma:\Delta\to\mathbf{Top}
  36. [ n ] [n]
  37. Δ \Delta
  38. n n
  39. n + 1 \mathbb{R}^{n+1}
  40. 𝐓𝐨𝐩 × 𝐓𝐨𝐩 𝐓𝐨𝐩 \mathbf{Top}\times\mathbf{Top}\to\mathbf{Top}
  41. S S
  42. T × γ T\times\gamma
  43. S S
  44. T T

Endurance_(aeronautics).html

  1. E = t 1 t 2 d t = - W 1 W 2 d W F = W 2 W 1 d W F E=\int_{t_{1}}^{t_{2}}dt=-\int_{W_{1}}^{W_{2}}\frac{dW}{F}=\int_{W_{2}}^{W_{1}% }\frac{dW}{F}

Energy_landscape.html

  1. f : X f:X\to\mathbb{R}
  2. X X
  3. X = n X=\mathbb{R}^{n}
  4. n n
  5. n + 1 \mathbb{R}^{n+1}
  6. f f
  7. f : f:\mathbb{R}\to\mathbb{R}

Englert–Greenberger–Yasin_duality_relation.html

  1. V V
  2. D D
  3. D 2 + V 2 1 D^{2}+V^{2}\leq 1\,
  4. D D
  5. P P
  6. D D
  7. Ψ Total ( x ) = Ψ A ( x ) + Ψ B ( x ) . \Psi_{\,\text{Total}}(x)=\Psi_{A}(x)+\Psi_{B}(x).
  8. Ψ A ( x ) = C A Ψ 0 ( x - x A ) \Psi_{A}(x)=C_{A}\Psi_{0}(x-x_{A})
  9. x A x_{A}
  10. x x
  11. C A C_{A}
  12. C B C_{B}
  13. Ψ 0 ( x ) \Psi_{0}(x)
  14. p 0 = h / λ p_{0}=h/\lambda
  15. Ψ 0 ( x ) e i p 0 | x | / | x | \Psi_{0}(x)\propto\frac{e^{ip_{0}\cdot|x|/\hbar}}{|x|}
  16. | x | |x|
  17. D = | P A - P B | , D=|P_{A}-P_{B}|,\,
  18. P A P_{A}
  19. P B P_{B}
  20. P A = | C A | 2 | C A | 2 + | C B | 2 P_{A}=\frac{|C_{A}|^{2}}{|C_{A}|^{2}+|C_{B}|^{2}}
  21. P B = | C B | 2 | C A | 2 + | C B | 2 P_{B}=\frac{|C_{B}|^{2}}{|C_{A}|^{2}+|C_{B}|^{2}}
  22. D = | | C A | 2 - | C B | 2 | C A | 2 + | C B | 2 | D=\left|\;\frac{|C_{A}|^{2}-|C_{B}|^{2}}{|C_{A}|^{2}+|C_{B}|^{2}}\,\right|
  23. D = 0 D=0
  24. D = 1 D=1
  25. I ( y ) 1 + V cos ( p y d / + ϕ ) I(y)\propto 1+V\cos{(p_{y}d/\hbar+\phi)}
  26. p y = h / λ sin ( α ) p_{y}=h/\lambda\cdot\sin(\alpha)
  27. ϕ = Arg ( C A ) - Arg ( C B ) \phi=\,\text{Arg}(C_{A})-\,\text{Arg}(C_{B})
  28. d d
  29. sin ( α ) tan ( α ) = y / L \sin(\alpha)\simeq\tan(\alpha)=y/L
  30. L L
  31. sin ( α ) tan ( α ) = y / f \sin(\alpha)\simeq\tan(\alpha)=y/f
  32. f f
  33. V = I max - I min I max + I min V=\frac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}}
  34. I max I_{\mathrm{max}}
  35. I min I_{\mathrm{min}}
  36. I max | | C A | + | C B | | 2 I_{\mathrm{max}}\propto||C_{A}|+|C_{B}||^{2}
  37. I min | | C A | - | C B | | 2 I_{\mathrm{min}}\propto||C_{A}|-|C_{B}||^{2}
  38. V = 2 | C A C B * | | C A | 2 + | C B | 2 . V=2\frac{|C_{A}\cdot C_{B}^{*}|}{|C_{A}|^{2}+|C_{B}|^{2}}.
  39. V 2 + D 2 = 1 \begin{matrix}V^{2}+D^{2}=1\end{matrix}
  40. V = 0 V=0
  41. D = 1 D=1
  42. D = 0 D=0
  43. I min = 0 I_{\mathrm{min}}=0
  44. V = 1 V=1
  45. V 2 + D 2 = 1 V^{2}+D^{2}=1
  46. V 2 + D 2 1. V^{2}+D^{2}\leq 1.\,
  47. V 2 + D 2 = 1 V^{2}+D^{2}=1
  48. D D
  49. D = 1 D=1
  50. D = 0 D=0
  51. D = 0 D=0
  52. D = 0 D=0
  53. V = 1 V=1
  54. D = 1 D=1
  55. V = 0 V=0
  56. D D
  57. P P

Enneper_surface.html

  1. x = u ( 1 - u 2 / 3 + v 2 ) / 3 , x=u(1-u^{2}/3+v^{2})/3,
  2. y = - v ( 1 - v 2 / 3 + u 2 ) / 3 , y=-v(1-v^{2}/3+u^{2})/3,
  3. z = ( u 2 - v 2 ) / 3. z=(u^{2}-v^{2})/3.
  4. f ( z ) = 1 , g ( z ) = z f(z)=1,g(z)=z
  5. 64 z 9 - 128 z 7 + 64 z 5 - 702 x 2 y 2 z 3 - 18 x 2 y 2 z + 144 ( y 2 z 6 - x 2 z 6 ) 64z^{9}-128z^{7}+64z^{5}-702x^{2}y^{2}z^{3}-18x^{2}y^{2}z+144(y^{2}z^{6}-x^{2}% z^{6})
  6. + 162 ( y 4 z 2 - x 4 z 2 ) + 27 ( y 6 - x 6 ) + 9 ( x 4 z + y 4 z ) + 48 ( x 2 z 3 + y 2 z 3 ) {}+162(y^{4}z^{2}-x^{4}z^{2})+27(y^{6}-x^{6})+9(x^{4}z+y^{4}z)+48(x^{2}z^{3}+y% ^{2}z^{3})
  7. - 432 ( x 2 z 5 + y 2 z 5 ) + 81 ( x 4 y 2 - x 2 y 4 ) + 240 ( y 2 z 4 - x 2 z 4 ) - 135 ( x 4 z 3 + y 4 z 3 ) = 0. {}-432(x^{2}z^{5}+y^{2}z^{5})+81(x^{4}y^{2}-x^{2}y^{4})+240(y^{2}z^{4}-x^{2}z^% {4})-135(x^{4}z^{3}+y^{4}z^{3})=0.
  8. a + b x + c y + d z = 0 , a+bx+cy+dz=0,
  9. a = - ( u 2 - v 2 ) ( 1 + u 2 / 3 + v 2 / 3 ) , a=-(u^{2}-v^{2})(1+u^{2}/3+v^{2}/3),
  10. b = 6 u , b=6u,
  11. c = 6 v , c=6v,
  12. d = - 3 ( 1 - u 2 - v 2 ) . d=-3(1-u^{2}-v^{2}).
  13. 162 a 2 b 2 c 2 + 6 b 2 c 2 d 2 - 4 ( b 6 + c 6 ) + 54 ( a b 4 d - a c 4 d ) + 81 ( a 2 b 4 + a 2 c 4 ) 162a^{2}b^{2}c^{2}+6b^{2}c^{2}d^{2}-4(b^{6}+c^{6})+54(ab^{4}d-ac^{4}d)+81(a^{2% }b^{4}+a^{2}c^{4})
  14. + 4 ( b 4 c 2 + b 2 c 4 ) - 3 ( b 4 d 2 + c 4 d 2 ) + 36 ( a b 2 d 3 - a c 2 d 3 ) = 0. {}+4(b^{4}c^{2}+b^{2}c^{4})-3(b^{4}d^{2}+c^{4}d^{2})+36(ab^{2}d^{3}-ac^{2}d^{3% })=0.
  15. J = ( 1 + u 2 + v 2 ) 4 / 81 , J=(1+u^{2}+v^{2})^{4}/81,
  16. K = - ( 4 / 9 ) / J , K=-(4/9)/J,
  17. H = 0. H=0.
  18. - 4 π -4\pi
  19. \R 3 \R^{3}
  20. - 4 π -4\pi
  21. f ( z ) = 1 , g ( z ) = z k f(z)=1,g(z)=z^{k}
  22. \R n \R^{n}

Enolase.html

  1. \rightleftharpoons

Entanglement_witness.html

  1. H A H B H_{A}\otimes H_{B}
  2. ξ = i = 1 k p i ρ i A ρ i B , \xi=\sum_{i=1}^{k}p_{i}\,\rho_{i}^{A}\otimes\rho_{i}^{B},
  3. ρ i A \rho_{i}^{A}
  4. ρ i B \rho_{i}^{B}
  5. S 1 S_{1}
  6. S 2 S_{2}
  7. Tr ( A ρ ) < 0 \operatorname{Tr}(A\,\rho)<0
  8. Tr ( A σ ) 0 \operatorname{Tr}(A\,\sigma)\geq 0
  9. H A H_{A}
  10. H B H_{B}
  11. Tr ( A σ ) 0 \operatorname{Tr}(A\,\sigma)\geq 0
  12. Tr ( A P Q ) 0 \operatorname{Tr}(A\cdot P\otimes Q)\geq 0
  13. P Q P\otimes Q
  14. σ L ( H A ) L ( H B ) \sigma\in L(H_{A})\otimes L(H_{B})
  15. H B H_{B}
  16. H A H_{A}
  17. ( I A Λ ) ( σ ) (I_{A}\otimes\Lambda)(\sigma)
  18. I A I_{A}
  19. L ( H A ) \;L(H_{A})
  20. H A H_{A}

Entropy_(arrow_of_time).html

  1. S = Q T S=\frac{Q}{T}
  2. Δ S = S 𝑓𝑖𝑛𝑎𝑙 - S 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 \Delta S=S_{\mathit{final}}-S_{\mathit{initial}}\,
  3. Δ S = ( Q T 2 - Q T 1 ) \Delta S=\left(\frac{Q}{T_{2}}-\frac{Q}{T_{1}}\right)
  4. Δ S = Q ( 1 T 2 - 1 T 1 ) \Delta S=Q\left(\frac{1}{T_{2}}-\frac{1}{T_{1}}\right)
  5. τ e S \tau e^{S}
  6. τ \tau

Entropy_(classical_thermodynamics).html

  1. δ Q T = 0. \oint\frac{\delta Q}{T}=0.
  2. L δ Q T \int_{L}\frac{\delta Q}{T}
  3. d S = δ Q T . \mathrm{d}S=\frac{\delta Q}{T}.
  4. d S = ( S T ) P d T + ( S P ) T d P . \mathrm{d}S=\left(\frac{\partial S}{\partial T}\right)_{P}\mathrm{d}T+\left(% \frac{\partial S}{\partial P}\right)_{T}\mathrm{d}P.
  5. ( S T ) P = C P T . \left(\frac{\partial S}{\partial T}\right)_{P}=\frac{C_{P}}{T}.
  6. ( S P ) T = - ( V T ) P \left(\frac{\partial S}{\partial P}\right)_{T}=-\left(\frac{\partial V}{% \partial T}\right)_{P}
  7. α V = 1 V ( V T ) P \alpha_{V}=\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{P}
  8. d S = C P T d T - α V V d P . \mathrm{d}S=\frac{C_{P}}{T}\mathrm{d}T-\alpha_{V}V\mathrm{d}P.
  9. S ( P , T ) = S ( P 0 , T 0 ) + T 0 T C P ( P 0 , T ) T d T - P 0 P α V ( P , T ) V ( P , T ) d P . S(P,T)=S(P_{0},T_{0})+\int_{T_{0}}^{T}\frac{C_{P}(P_{0},T^{\prime})}{T^{\prime% }}\mathrm{d}T^{\prime}-\int_{P_{0}}^{P}\alpha_{V}(P^{\prime},T)V(P^{\prime},T)% \mathrm{d}P^{\prime}.
  10. S m ( P , T ) = S m ( P 0 , T 0 ) + C P ln T T 0 - R ln P P 0 . S_{m}(P,T)=S_{m}(P_{0},T_{0})+C_{P}\ln\frac{T}{T_{0}}-R\ln\frac{P}{P_{0}}.
  11. W = Q H - Q a . W=Q_{H}-Q_{a}.
  12. S i = - Q H T H + Q a T a . S_{i}=-\frac{Q_{H}}{T_{H}}+\frac{Q_{a}}{T_{a}}.
  13. W = ( 1 - T a T H ) Q H - T a S i . W=\left(1-\frac{T_{a}}{T_{H}}\right)Q_{H}-T_{a}S_{i}.
  14. W = W max - T a S i . W=W_{\mathrm{max}}-T_{a}S_{i}.
  15. Q a = T a T H Q H + T a S i = Q a , min + T a S i . Q_{a}=\frac{T_{a}}{T_{H}}Q_{H}+T_{a}S_{i}=Q_{a,\mathrm{min}}+T_{a}S_{i}.
  16. S i = Q a T a - Q L T L S_{i}=\frac{Q_{a}}{T_{a}}-\frac{Q_{L}}{T_{L}}
  17. W = Q L ( T a / T L - 1 ) + T a S i . W={Q_{L}}({T_{a}/T_{L}-1})+T_{a}S_{i}.
  18. W = W min + T a S i W=W_{\mathrm{min}}+T_{a}S_{i}

Entropy_(statistical_thermodynamics).html

  1. E i E_{i}
  2. p i p_{i}
  3. S = - k B i p i ln p i S=-k\text{B}\,\sum_{i}p_{i}\ln\,p_{i}
  4. d S = - k B i d p i ln p i dS=-k\text{B}\,\sum_{i}dp_{i}\ln p_{i}
  5. = - k B i d p i ( - E i / k B T - ln Z ) \,\,\,=-k\text{B}\,\sum_{i}dp_{i}(-E_{i}/k\text{B}T-\ln Z)
  6. = i E i d p i / T \,\,\,=\sum_{i}E_{i}dp_{i}/T
  7. = i [ d ( E i p i ) - ( d E i ) p i ] / T \,\,\,=\sum_{i}[d(E_{i}p_{i})-(dE_{i})p_{i}]/T
  8. d S = δ q rev T dS=\frac{\delta\langle q\text{rev}\rangle}{T}
  9. k B k\text{B}
  10. d S = δ Q T dS=\frac{\delta Q}{T}\!
  11. S = k B ln Ω S=k\text{B}\ln\Omega
  12. Ω \Omega
  13. S = k B ln Ω mic = k B ( ln Z can + β E ¯ ) = k B ( ln 𝒵 gr + β ( E ¯ - μ N ¯ ) ) S=k\text{B}\ln\Omega_{\rm mic}=k\text{B}(\ln Z_{\rm can}+\beta\bar{E})=k\text{% B}(\ln\mathcal{Z}_{\rm gr}+\beta(\bar{E}-\mu\bar{N}))
  14. Ω mic \Omega_{\rm mic}
  15. Z can Z_{\rm can}
  16. 𝒵 gr \mathcal{Z}_{\rm gr}
  17. E ν = h ν 0 ( n + 1 2 ) E_{\nu}=h\nu_{0}(n+\begin{matrix}\frac{1}{2}\end{matrix})
  18. h h
  19. ν 0 \nu_{0}
  20. n n
  21. n = 0 n=0
  22. E n E_{n}

Entropy_in_thermodynamics_and_information_theory.html

  1. S = - k B i p i ln p i , S=-k_{B}\sum_{i}p_{i}\ln p_{i},\,
  2. p i p_{i}
  3. H = - i p i log b p i , H=-\sum_{i}p_{i}\log_{b}p_{i},\,
  4. p i p_{i}
  5. m i m_{i}
  6. e e
  7. e e
  8. S = k B ln W S=k_{B}\ln W\,
  9. H = log b | M | H=\log_{b}|M|\,
  10. | M | |M|
  11. d E = - p d V + T d S dE=-pdV+TdS\,
  12. d E = - p d V + k B T d σ dE=-pdV+k_{B}Td\sigma\,
  13. k B T ln 2 k_{B}T\ln 2

Enumerated_type.html

  1. 1 + 1 + + 1 1+1+\cdots+1

Enumerative_combinatorics.html

  1. g ( n ) g(n)
  2. f ( n ) f(n)
  3. f ( n ) / g ( n ) 1 f(n)/g(n)\rightarrow 1
  4. n n\rightarrow\infty
  5. f ( n ) g ( n ) . f(n)\sim g(n).\,
  6. \mathcal{F}
  7. F ( x ) = n = 0 f n x n F(x)=\sum^{\infty}_{n=0}f_{n}x^{n}
  8. f n f_{n}
  9. x n x^{n}
  10. F ( x ) = n = 0 f n n ! x n F(x)=\sum^{\infty}_{n=0}\frac{f_{n}}{n!}x^{n}
  11. \mathcal{F}
  12. 𝒢 \mathcal{G}
  13. 𝒢 \mathcal{F}\cup\mathcal{G}
  14. × 𝒢 \mathcal{F}\times\mathcal{G}
  15. Seq ( ) = ϵ × × × \mbox{Seq}~{}(\mathcal{F})=\epsilon\ \cup\ \mathcal{F}\ \cup\ \mathcal{F}% \times\mathcal{F}\ \cup\ \mathcal{F}\times\mathcal{F}\times\mathcal{F}\ \cup\cdots
  16. 1 + F ( x ) + [ F ( x ) ] 2 + [ F ( x ) ] 3 + = 1 1 - F ( x ) 1+F(x)+[F(x)]^{2}+[F(x)]^{3}+\cdots=\frac{1}{1-F(x)}
  17. 𝒫 \mathcal{P}
  18. 𝒫 = { } × Seq ( 𝒫 ) \mathcal{P}=\{\bullet\}\times\mbox{Seq}~{}(\mathcal{P})
  19. { } \{\bullet\}
  20. 𝒫 \mathcal{P}
  21. P ( x ) = x 1 1 - P ( x ) P(x)=x\frac{1}{1-P(x)}
  22. P ( x ) = 1 - 1 - 4 x 2 P(x)=\frac{1-\sqrt{1-4x}}{2}
  23. p n \displaystyle p_{n}

Envelope_theorem.html

  1. X X
  2. t [ 0 , 1 ] t\in[0,1]
  3. f : X × [ 0 , 1 ] R f:X\times[0,1]\rightarrow R
  4. V V
  5. X X^{\ast}
  6. V ( t ) = sup x X f ( x , t ) V(t)=\sup_{x\in X}f(x,t)
  7. X ( t ) = { x X : f ( x , t ) = V ( t ) } X^{\ast}(t)=\{x\in X:f(x,t)=V(t)\}
  8. V V
  9. t t
  10. V ( t ) = f t ( x , t ) for each x X ( t ) . ( f t denotes partial derivative of f w.r.t to t ) V^{\prime}\left(t\right)=f_{t}\left(x,t\right)\,\text{ for each }x\in X^{\ast}% \left(t\right).(f_{t}\,\text{ denotes partial derivative of }f\,\text{ w.r.t % to }t)
  11. t t
  12. V V
  13. { f ( x , ) } x X \left\{f\left(x,\cdot\right)\right\}_{x\in X}
  14. X X
  15. f f
  16. x x
  17. t ( 0 , 1 ) t\in\left(0,1\right)
  18. x X ( t ) x\in X^{\ast}\left(t\right)
  19. V ( t ) V^{\prime}\left(t\right)
  20. f t ( x , t ) f_{t}\left(x,t\right)
  21. x X ( t ) x\in X^{\ast}\left(t\right)
  22. max s [ 0 , 1 ] [ f ( x , s ) - V ( s ) ] = f ( x , t ) - V ( t ) = 0. \max_{s\in\left[0,1\right]}\left[f\left(x,s\right)-V\left(s\right)\right]=f% \left(x,t\right)-V\left(t\right)=0.
  23. s = t s=t
  24. V V
  25. f ( x , ) f(x,\cdot)
  26. x X x\in X
  27. b : [ 0 , 1 ] b:[0,1]
  28. \rightarrow
  29. + \mathbb{R}_{+}
  30. | f t ( x , t ) | b ( t ) |f_{t}(x,t)|\leq b(t)
  31. x X x\in X
  32. t [ 0 , 1 ] t\in[0,1]
  33. V V
  34. f ( x , ) f(x,\cdot)
  35. x X x\in X
  36. X ( t ) X^{\ast}(t)\neq\varnothing
  37. [ 0 , 1 ] [0,1]
  38. x ( t ) X ( t ) x^{\ast}(t)\in X^{\ast}(t)
  39. V ( t ) = V ( 0 ) + 0 t f t ( x ( s ) , s ) d s . V(t)=V(0)+\int_{0}^{t}f_{t}(x^{\ast}(s),s)ds.
  40. t , t ′′ [ 0 , 1 ] t^{\prime},t^{\prime\prime}\in[0,1]
  41. t < t ′′ t^{\prime}<t^{\prime\prime}
  42. | V ( t ′′ ) - V ( t ) | sup x X | f ( x , t ′′ ) - f ( x , t ) | = sup x X | t t ′′ f t ( x , t ) d t | t t ′′ sup x X | f t ( x , t ) | d t t t ′′ b ( t ) d t . |V(t^{\prime\prime})-V(t^{\prime})|\leq\sup_{x\in X}|f(x,t^{\prime\prime})-f(x% ,t^{\prime})|=\sup_{x\in X}\left|\int_{t^{\prime}}^{t^{\prime\prime}}f_{t}(x,t% )dt\right|\leq\int_{t^{\prime}}^{t^{\prime\prime}}\sup_{x\in X}|f_{t}(x,t)|dt% \leq\int_{t^{\prime}}^{t^{\prime\prime}}b(t)dt.
  43. V V
  44. V V
  45. t = t 0 t=t_{0}
  46. { f ( x , ) } x X \left\{f\left(x,\cdot\right)\right\}_{x\in X}
  47. t 0 ( 0 , 1 ) t_{0}\in\left(0,1\right)
  48. f t ( X ( t ) , t 0 ) f_{t}\left(X^{\ast}\left(t\right),t_{0}\right)
  49. t = t 0 t=t_{0}
  50. t 0 t_{0}
  51. X X
  52. t 0 t_{0}
  53. π ( p ) \pi\left(p\right)
  54. X L X\subseteq\mathbb{R}^{L}
  55. p L p\in\mathbb{R}^{L}
  56. x ( p ) x^{\ast}\left(p\right)
  57. π ( p ) = max x X p x = p x ( p ) . \pi(p)=\max_{x\in X}p\cdot x=p\cdot x^{\ast}\left(p\right)\,\text{.}
  58. t = p i t=p_{i}
  59. i i
  60. p - i L - 1 p_{-i}\in\mathbb{R}^{L-1}
  61. f ( x , t ) = t x i + p - i x - i f(x,t)=tx_{i}+p_{-i}\cdot x_{-i}
  62. π ( p ) p i = x i ( p ) \frac{\partial\pi(p)}{\partial p_{i}}=x_{i}^{\ast}(p)
  63. i i
  64. p i p_{i}
  65. π ( t , p - i ) - π ( 0 , p - i ) = 0 p i x i ( s , p - i ) d s , \pi(t,p_{-i})-\pi(0,p_{-i})=\int_{0}^{p_{i}}x_{i}^{\ast}(s,p_{-i})ds,
  66. π ( t , p - i ) - π ( 0 , p - i ) \pi(t,p_{-i})-\pi(0,p_{-i})
  67. i i
  68. f ( x , t ) f(x,t)
  69. x X ¯ x\in\bar{X}
  70. t [ 0 , 1 ] t\in[0,1]
  71. X X ¯ X\subseteq\bar{X}
  72. V ( t ) V(t)
  73. X ( t ) X^{\ast}(t)
  74. x ( t ) X ( t ) x^{\ast}(t)\in X^{\ast}(t)
  75. f ( x , t ) f(x,t)
  76. t t
  77. x Y x\in Y
  78. sup x X ¯ | f t ( x , t ) | \sup_{x\in\bar{X}}|f_{t}(x,t)|
  79. [ 0 , 1 ] [0,1]
  80. V V
  81. x x^{\ast}
  82. x = ( y , z ) x=\left(y,z\right)
  83. y y
  84. z z
  85. f ( ( y , z ) , t ) = t y - z f\left(\left(y,z\right),t\right)=ty-z
  86. t ¯ \underline{t}
  87. V V
  88. V ( t ) - V ( t ¯ ) = 0 t y ( s ) d s . V(t)-V(\underline{t})=\int_{0}^{t}y^{\ast}(s)ds.
  89. z z
  90. y y
  91. t t
  92. y ( t ) y^{\ast}\left(t\right)
  93. t t
  94. V ( t ¯ ) V(\underline{t})
  95. T K T\subseteq\mathbb{R}^{K}
  96. f f
  97. V V
  98. t t
  99. V ( t ) = t f ( x , t ) \nabla V\left(t\right)=\nabla_{t}f\left(x,t\right)
  100. x X ( t ) x\in X^{\ast}\left(t\right)
  101. t 0 t_{0}
  102. t t
  103. f ( x , ) f(x,\cdot)
  104. x X x\in X
  105. | t f ( x , t ) | B |\nabla_{t}f(x,t)|\leq B
  106. x X , x\in X,
  107. t T t\in T
  108. t 0 t_{0}
  109. t t
  110. γ : [ 0 , 1 ] T \gamma:\left[0,1\right]\rightarrow T
  111. γ ( 0 ) = t 0 \gamma\left(0\right)=t_{0}
  112. γ ( 1 ) = t \gamma\left(1\right)=t
  113. t f ( x ( t ) , t ) \nabla_{t}f(x^{\ast}(t),t)
  114. V ( t ) - V ( t 0 ) = γ t f ( x ( s ) , s ) d s . V(t)-V(t_{0})=\int_{\gamma}\nabla_{t}f(x^{\ast}(s),s)\cdot ds.
  115. t = t 0 t=t_{0}
  116. γ \gamma
  117. t f ( x ( s ) , s ) d s = 0. \int\nabla_{t}f(x^{\ast}(s),s)\cdot ds=0.
  118. x x^{\ast}
  119. X X ¯ X\subseteq\bar{X}
  120. x X L x\in X\subseteq\mathbb{R}^{L}
  121. t L t\in\mathbb{R}^{L}
  122. f ( x , t ) = t x f\left(x,t\right)=t\cdot x
  123. x x^{\ast}
  124. x ( s ) d s = 0. \int x^{\ast}(s)\cdot ds=0.
  125. x x^{\ast}
  126. ( x i ( t ) / t j ) i , j = 1 L \left(\partial x_{i}^{\ast}\left(t\right)/\partial t_{j}\right)_{i,j=1}^{L}
  127. X ( t ) X\left(t\right)
  128. V ( t ) = sup x X ( t ) f ( x , t ) V(t)=\sup_{x\in X\left(t\right)}f(x,t)
  129. X ( t ) = { x X ( t ) : f ( x , t ) = V ( t ) } , X^{\ast}(t)=\{x\in X\left(t\right):f(x,t)=V(t)\}\,\text{, }
  130. X ( t ) = { x X : g ( x , t ) 0 } X\left(t\right)=\left\{x\in X:g\left(x,t\right)\geq 0\right\}
  131. g : X × [ 0 , 1 ] K . g:X\times\left[0,1\right]\rightarrow\mathbb{R}^{K}.
  132. X X
  133. f f
  134. g g
  135. x x
  136. x ^ X \hat{x}\in X
  137. g ( x ^ , t ) > 0 g\left(\hat{x},t\right)>0
  138. t [ 0 , 1 ] t\in\left[0,1\right]
  139. L ( x , y , t ) = f ( x , t ) + y g ( x , t ) L\left(x,y,t\right)=f(x,t)+y\cdot g\left(x,t\right)
  140. y + K y\in\mathbb{R}_{+}^{K}
  141. X X
  142. f f
  143. g g
  144. x x
  145. f t f_{t}
  146. g t g_{t}
  147. ( x , t ) \left(x,t\right)
  148. ( x ( t ) , y ( t ) ) \left(x^{\ast}(t),y^{\ast}\left(t\right)\right)
  149. t t
  150. V V
  151. V ( t ) = V ( 0 ) + 0 t L t ( x ( s ) , y ( s ) , s ) d s . V(t)=V(0)+\int_{0}^{t}L_{t}(x^{\ast}(s),y^{\ast}\left(s\right),s)ds.
  152. f ( x , t ) f\left(x,t\right)
  153. t t
  154. K = 1 K=1
  155. g ( x , t ) = h ( x ) + t g\left(x,t\right)=h\left(x\right)+t
  156. V ( t ) = L t ( x ( t ) , y ( t ) , t ) = y ( t ) V^{\prime}(t)=L_{t}(x^{\ast}(t),y^{\ast}\left(t\right),t)=y^{\ast}\left(t\right)
  157. t t
  158. y ( t ) y^{\ast}\left(t\right)

Eötvös_effect.html

  1. a r = 2 Ω u cos ϕ + u 2 + v 2 R . a_{r}=2\Omega u\cos\phi+\frac{u^{2}+v^{2}}{R}.
  2. Ω \Omega
  3. u u
  4. ϕ \phi
  5. v v
  6. R R
  7. a u a_{u}
  8. a s a_{s}
  9. Ω \Omega
  10. ω r \omega_{r}
  11. ( Ω + ω r ) (\Omega+\omega_{r})
  12. ω r * R = u \omega_{r}*R=u
  13. R R
  14. a r = a u - a s = ( Ω + ω r ) 2 R - Ω 2 R = Ω 2 R + 2 Ω ω r R + ω r 2 R - Ω 2 R = 2 Ω ω r R + ω r 2 R = 2 Ω u + u 2 / R \begin{aligned}\displaystyle a_{r}&\displaystyle=a_{u}-a_{s}\\ &\displaystyle=(\Omega+\omega_{r})^{2}R-\Omega^{2}R\\ &\displaystyle=\Omega^{2}R+2\Omega\omega_{r}R+\omega_{r}^{2}R-\Omega^{2}R\\ &\displaystyle=2\Omega\omega_{r}R+\omega_{r}^{2}R\\ &\displaystyle=2\Omega u+u^{2}/R\\ \end{aligned}
  15. a r = 2 Ω u cos ϕ + u 2 + v 2 R a_{r}=2\Omega u\cos\phi+\frac{u^{2}+v^{2}}{R}

Eötvös_number.html

  1. Eo = Bo = Δ ρ g L 2 σ \mathrm{Eo}=\mathrm{Bo}=\frac{\Delta\rho\,g\,L^{2}}{\sigma}
  2. Δ ρ \Delta\rho
  3. σ \sigma
  4. Bo = Eo = 2 Go 2 = 2 De 2 \mathrm{Bo}=\mathrm{Eo}=2\,\mathrm{Go}^{2}=2\,\mathrm{De}^{2}\,

Epispiral.html

  1. r = a sec n θ \ r=a\sec{n\theta}

Epistemic_modal_logic.html

  1. K 1 \mathit{K}_{1}
  2. K 2 \mathit{K}_{2}
  3. K a φ \mathit{K}_{a}\varphi
  4. a a
  5. φ \varphi
  6. \Diamond
  7. \Box
  8. ¬ K a ¬ φ \neg K_{a}\neg\varphi
  9. a a
  10. φ \varphi
  11. a a
  12. φ \varphi
  13. a a
  14. φ \varphi
  15. ¬ K a φ ¬ K a ¬ φ \neg K_{a}\varphi\land\neg K_{a}\neg\varphi
  16. E G \mathit{E}_{\mathit{G}}
  17. C G \mathit{C}_{\mathit{G}}
  18. D G \mathit{D}_{\mathit{G}}
  19. φ \varphi
  20. E G φ \mathit{E}_{G}\varphi
  21. C G φ \mathit{C}_{G}\varphi
  22. D G φ \mathit{D}_{G}\varphi
  23. K \mathit{K}
  24. E \mathit{E}
  25. C \mathit{C}
  26. D \mathit{D}
  27. Φ \Phi
  28. ( S , π , 𝒦 1 , , 𝒦 n ) (S,\pi,\mathcal{K}_{1},...,\mathcal{K}_{n})
  29. π \pi
  30. Φ \Phi
  31. 𝒦 1 , , 𝒦 n \mathcal{K}_{1},...,\mathcal{K}_{n}
  32. K i K_{i}
  33. 𝒦 i \mathcal{K}_{i}
  34. π ( s ) ( p ) \pi(s)(p)
  35. \mathcal{M}
  36. φ \varphi
  37. ( M , s ) φ (M,s)\models\varphi
  38. φ \varphi
  39. φ \varphi
  40. 𝒦 i \mathcal{K}_{i}
  41. 𝒦 i \mathcal{K}_{i}
  42. 𝒦 i \mathcal{K}_{i}
  43. φ \varphi
  44. φ ψ \varphi\implies\psi
  45. ψ \,\psi
  46. ( K i φ K i ( φ ψ ) ) K i ψ (K_{i}\varphi\land K_{i}(\varphi\implies\psi))\implies K_{i}\psi
  47. ϕ \phi
  48. K i ϕ K_{i}\phi
  49. ϕ \phi
  50. ϕ \phi
  51. ϕ \phi
  52. ϕ \phi
  53. if M φ then M K i φ . \,\text{if }M\models\varphi\,\text{ then }M\models K_{i}\varphi.\,
  54. K i φ φ K_{i}\varphi\implies\varphi
  55. K i φ K i K i φ K_{i}\varphi\implies K_{i}K_{i}\varphi
  56. ¬ K i φ K i ¬ K i φ \neg K_{i}\varphi\implies K_{i}\neg K_{i}\varphi
  57. ¬ B i \neg B_{i}\bot
  58. 𝒦 i \mathcal{K}_{i}
  59. Q Q
  60. P P
  61. P P
  62. Q Q
  63. P P
  64. P P
  65. Q Q
  66. Q Q
  67. P P
  68. Q Q
  69. P P
  70. 𝒦 a ( P Q ) \mathcal{K}_{a}(P\rightarrow Q)
  71. 𝒦 a P 𝒦 a Q \mathcal{K}_{a}P\rightarrow\mathcal{K}_{a}Q
  72. Q Q
  73. P P
  74. P P
  75. Q Q

Epsilon_numbers_(mathematics).html

  1. ε = ω ε , \varepsilon=\omega^{\varepsilon},\,
  2. ε = ω ε \varepsilon=\omega^{\varepsilon}
  3. ε 0 = ω ω ω = sup { ω , ω ω , ω ω ω , ω ω ω ω , } \varepsilon_{0}=\omega^{\omega^{\omega^{\cdot^{\cdot^{\cdot}}}}}=\sup\{\omega,% \omega^{\omega},\omega^{\omega^{\omega}},\omega^{\omega^{\omega^{\omega}}},\dots\}
  4. ε 1 , ε 2 , , ε ω , ε ω + 1 , , ε ε 0 , , ε ε 1 , , ε ε ε , \varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{\omega},\varepsilon_{% \omega+1},\ldots,\varepsilon_{\varepsilon_{0}},\ldots,\varepsilon_{\varepsilon% _{1}},\ldots,\varepsilon_{\varepsilon_{\varepsilon_{\cdot_{\cdot_{\cdot}}}}},\ldots
  5. α 0 = 1 , \alpha^{0}=1\,,
  6. α β + 1 = α β α , \alpha^{\beta+1}=\alpha^{\beta}\cdot\alpha\,,
  7. α κ = lim sup λ < κ α λ \alpha^{\kappa}=\limsup_{\lambda<\kappa}\,\alpha^{\lambda}
  8. κ \kappa
  9. β α β \beta\mapsto\alpha^{\beta}
  10. α = ω \alpha=\omega
  11. 0 , ω 0 = 1 , ω 1 = ω , ω ω , ω ω ω , , ω k , 0,\omega^{0}=1,\omega^{1}=\omega,\omega^{\omega},\omega^{\omega^{\omega}},% \ldots,\omega\uparrow\uparrow k,\ldots
  12. β ω β \beta\mapsto\omega^{\beta}
  13. \uparrow\uparrow
  14. ω ω \omega\uparrow\uparrow\omega
  15. ε 0 \varepsilon_{0}
  16. ε 1 = sup { ε 0 + 1 , ω ε 0 + 1 , ω ω ε 0 + 1 , ω ω ω ε 0 + 1 , } , \varepsilon_{1}=\sup\{\varepsilon_{0}+1,\omega^{\varepsilon_{0}+1},\omega^{% \omega^{\varepsilon_{0}+1}},\omega^{\omega^{\omega^{\varepsilon_{0}+1}}},\dots\},
  17. ε 0 + 1 \varepsilon_{0}+1
  18. ω ε 0 + 1 = ω ε 0 ω 1 = ε 0 ω , \omega^{\varepsilon_{0}+1}=\omega^{\varepsilon_{0}}\cdot\omega^{1}=\varepsilon% _{0}\cdot\omega\,,
  19. ω ω ε 0 + 1 = ω ( ε 0 ω ) = ( ω ε 0 ) ω = ε 0 ω , \omega^{\omega^{\varepsilon_{0}+1}}=\omega^{(\varepsilon_{0}\cdot\omega)}={(% \omega^{\varepsilon_{0}})}^{\omega}=\varepsilon_{0}^{\omega}\,,
  20. ω ω ω ε 0 + 1 = ω ε 0 ω = ω ε 0 1 + ω = ω ( ε 0 ε 0 ω ) = ( ω ε 0 ) ε 0 ω = ε 0 ε 0 ω . \omega^{\omega^{\omega^{\varepsilon_{0}+1}}}=\omega^{{\varepsilon_{0}}^{\omega% }}=\omega^{{\varepsilon_{0}}^{1+\omega}}=\omega^{(\varepsilon_{0}\cdot{% \varepsilon_{0}}^{\omega})}={(\omega^{\varepsilon_{0}})}^{{\varepsilon_{0}}^{% \omega}}={\varepsilon_{0}}^{{\varepsilon_{0}}^{\omega}}\,.
  21. ε 1 \varepsilon_{1}
  22. ε 1 = sup { 0 , 1 , ε 0 , ε 0 ε 0 , ε 0 ε 0 ε 0 , } , \varepsilon_{1}=\sup\{0,1,\varepsilon_{0},{\varepsilon_{0}}^{\varepsilon_{0}},% {\varepsilon_{0}}^{{\varepsilon_{0}}^{\varepsilon_{0}}},\ldots\},
  23. ε α + 1 \varepsilon_{\alpha+1}
  24. ε α + 1 \varepsilon_{\alpha}+1
  25. ε α \varepsilon_{\alpha}
  26. ε α + 1 = sup { ε α + 1 , ω ε α + 1 , ω ω ε α + 1 , } = sup { 0 , 1 , ε α , ε α ε α , ε α ε α ε α , } \varepsilon_{\alpha+1}=\sup\{\varepsilon_{\alpha}+1,\omega^{\varepsilon_{% \alpha}+1},\omega^{\omega^{\varepsilon_{\alpha}+1}},\dots\}=\sup\{0,1,% \varepsilon_{\alpha},\varepsilon_{\alpha}^{\varepsilon_{\alpha}},\varepsilon_{% \alpha}^{\varepsilon_{\alpha}^{\varepsilon_{\alpha}}},\dots\}
  27. ε α \varepsilon_{\alpha}
  28. { ε β , β < α } \{\varepsilon_{\beta},\beta<\alpha\}
  29. ε ω \varepsilon_{\omega}
  30. ε α \varepsilon_{\alpha}
  31. 1 < γ < ε α 1<\gamma<\varepsilon_{\alpha}
  32. α \alpha
  33. ε α \varepsilon_{\alpha}
  34. { ε β , β < α } \{\varepsilon_{\beta},\beta<\alpha\}
  35. ε 0 \varepsilon_{0}
  36. ε α \varepsilon_{\alpha}
  37. α \alpha
  38. ε ω = sup { ε 0 , ε 1 , ε 2 , } \varepsilon_{\omega}=\sup\{\varepsilon_{0},\varepsilon_{1},\varepsilon_{2},\ldots\}
  39. n ε n n\mapsto\varepsilon_{n}
  40. 1 α ε ω α = ω α . 1\leq\alpha\rightarrow\varepsilon_{\omega_{\alpha}}=\omega_{\alpha}\,.
  41. x ε x x\mapsto\varepsilon_{x}
  42. α ϕ α ( 0 ) \alpha\,\rightarrow\,\phi_{\alpha}(0)
  43. α ϕ α ( 0 ) \alpha\,\rightarrow\,\phi_{\alpha}(0)
  44. ω \omega
  45. n ω n n\mapsto\omega^{n}
  46. ε - 1 = { 0 , 1 , ω , ω ω , ε 0 - 1 , ω ε 0 - 1 , } \varepsilon_{-1}=\{0,1,\omega,\omega^{\omega},\ldots\mid\varepsilon_{0}-1,% \omega^{\varepsilon_{0}-1},\ldots\}
  47. ε 1 2 = { ε 0 + 1 , ω ε 0 + 1 , ε 1 - 1 , ω ε 1 - 1 , } . \varepsilon_{\frac{1}{2}}=\{\varepsilon_{0}+1,\omega^{\varepsilon_{0}+1},% \ldots\mid\varepsilon_{1}-1,\omega^{\varepsilon_{1}-1},\ldots\}.
  48. ε n \varepsilon_{n}

Equation_of_exchange.html

  1. M V = P Q M\cdot V=P\cdot Q
  2. M M\,
  3. V V\,
  4. P P\,
  5. Q Q\,
  6. M V T = P T M\cdot V_{T}=P\cdot T
  7. V T V_{T}\,
  8. T T\,
  9. M V T = i ( p i q i ) = 𝐩 T 𝐪 M\cdot V_{T}=\sum_{i}(p_{i}\cdot q_{i})=\mathbf{p}^{\mathrm{T}}\cdot\mathbf{q}
  10. p i p_{i}\,
  11. q i q_{i}\,
  12. 𝐩 𝐓 \mathbf{p^{T}}
  13. p i p_{i}\,
  14. 𝐪 \mathbf{q}
  15. q i q_{i}\,
  16. M V T = P T M\cdot V_{T}=P\cdot T
  17. M V T = P ( 𝐩 r e a l T 𝐪 ) = P T M\cdot V_{T}=P\cdot(\mathbf{p}_{real}^{\mathrm{T}}\cdot\mathbf{q})=P\cdot T
  18. 𝐩 r e a l T \mathbf{p}_{real}^{\mathrm{T}}
  19. M V = P Q . M\cdot V=P\cdot Q.
  20. P = M V Q P=\frac{M\cdot V}{Q}
  21. V V
  22. Q Q
  23. d P P = d M M \frac{dP}{P}=\frac{dM}{M}
  24. d P / P d t = d M / M d t \frac{dP/P}{dt}=\frac{dM/M}{dt}
  25. t t\,
  26. V V
  27. Q Q
  28. Q Q
  29. V V
  30. d M / M d t \frac{dM/M}{dt}
  31. k k
  32. n Y nY
  33. M D = k n Y M_{D}=k\cdot nY
  34. Q Q
  35. M D = k P Q M_{D}=k\cdot P\cdot Q
  36. M D = M M_{D}=M
  37. M 1 k = P Q M\cdot\frac{1}{k}=P\cdot Q
  38. L ( r , Y ) L(r,Y)
  39. M D = P L ( r , Y ) M_{D}=P\cdot L(r,Y)
  40. Y Y
  41. r r
  42. V V
  43. r r
  44. L ( r , Q ) = Q V ( r ) L(r,Q)=\frac{Q}{V(r)}

Equations_for_a_falling_body.html

  1. d \ d
  2. t \ t
  3. d = 1 2 g t 2 \ d=\frac{1}{2}gt^{2}
  4. t \ t
  5. d \ d
  6. t = 2 d g \ t=\ \sqrt{\frac{2d}{g}}
  7. v i \ v_{i}
  8. t \ t
  9. v i = g t \ v_{i}=gt
  10. v i \ v_{i}
  11. d \ d
  12. v i = 2 g d \ v_{i}=\sqrt{2gd}
  13. v a \ v_{a}
  14. t \ t
  15. v a = 1 2 g t \ v_{a}=\frac{1}{2}gt
  16. v a \ v_{a}
  17. d \ d
  18. v a = 2 g d 2 \ v_{a}=\frac{\sqrt{2gd}}{2}
  19. v i \ v_{i}
  20. d \ d
  21. M \ M
  22. r \ r
  23. g \ g
  24. g \ g
  25. g \ g
  26. v i = 2 G M d r 2 \ v_{i}=\sqrt{\frac{2GMd}{r^{2}}}
  27. v i \ v_{i}
  28. d \ d
  29. M \ M
  30. r \ r
  31. g \ g
  32. v i = 2 G M ( 1 r - 1 r + d ) \ v_{i}=\sqrt{2GM\Big(\frac{1}{r}-\frac{1}{r+d}\Big)}
  33. 2 h / g \sqrt{2h/g}
  34. t = arccos ( x r ) + x r ( 1 - x r ) 2 μ r 3 / 2 t=\frac{\arccos\Big(\sqrt{\frac{x}{r}}\Big)+\sqrt{\frac{x}{r}\ (1-\frac{x}{r})% }}{\sqrt{2\mu}}\,r^{3/2}
  35. μ = G ( m 1 + m 2 ) \mu=G(m_{1}+m_{2})

Equiconsistency.html

  1. ω 2 \omega_{2}
  2. ω 2 \omega_{2}

Erdős_conjecture_on_arithmetic_progressions.html

  1. n A 1 n = \sum_{n\in A}\frac{1}{n}=\infty

Erdős–Graham_problem.html

  1. n S 1 n = 1. \sum_{n\in S}\frac{1}{n}=1.

Erick_Weinberg.html

  1. V = λ 4 ! ϕ 4 + λ 2 ϕ 4 256 π 2 ( ln ϕ 2 M 2 - 25 6 ) V=\frac{\lambda}{4!}\phi^{4}+\frac{\lambda^{2}\phi^{4}}{256\pi^{2}}(\ln{\frac{% \phi^{2}}{M^{2}}}-\frac{25}{6})
  2. U ( 1 ) k U(1)^{k}
  3. U ( 1 ) U(1)
  4. 𝒢 \mathcal{G}

Error_vector_magnitude.html

  1. EVM ( dB ) = 10 log 10 ( P error P reference ) \mathrm{EVM(dB)}=10\log_{10}\left({P_{\mathrm{error}}\over P_{\mathrm{% reference}}}\right)
  2. EVM ( % ) = P error P reference * 100 % \mathrm{EVM(\%)}=\sqrt{{P_{\mathrm{error}}\over P_{\mathrm{reference}}}}*100\%

ESP_game.html

  1. X N X^{N}

Essential_supremum_and_essential_infimum.html

  1. f - 1 ( a , ) = { x X : f ( x ) > a } f^{-1}(a,\infty)=\{x\in X:f(x)>a\}
  2. U f = { a : f - 1 ( a , ) = } U_{f}=\{a\in\mathbb{R}:f^{-1}(a,\infty)=\emptyset\}\,
  3. sup f = inf U f \sup f=\inf U_{f}\,
  4. U f U_{f}
  5. U f ess = { a : μ ( f - 1 ( a , ) ) = 0 } U^{\mathrm{ess}}_{f}=\{a\in\mathbb{R}:\mu(f^{-1}(a,\infty))=0\}\,
  6. ess sup f = inf U f ess \mathrm{ess}\sup f=\inf U^{\mathrm{ess}}_{f}\,
  7. U f ess U^{\mathrm{ess}}_{f}\neq\emptyset
  8. ess inf f = sup { b : μ ( { x : f ( x ) < b } ) = 0 } \mathrm{ess}\inf f=\sup\{b\in\mathbb{R}:\mu(\{x:f(x)<b\})=0\}\,
  9. f ( x ) = { 5 , if x = 1 - 4 , if x = - 1 2 , otherwise. f(x)=\begin{cases}5,&\,\text{if }x=1\\ -4,&\,\text{if }x=-1\\ 2,&\,\text{ otherwise. }\end{cases}
  10. f ( x ) = { x 3 , if x arctan x , if x \ f(x)=\begin{cases}x^{3},&\,\text{if }x\in\mathbb{Q}\\ \arctan{x},&\,\text{if }x\in\mathbb{R}\backslash\mathbb{Q}\\ \end{cases}
  11. f ( x ) = { 1 / x , if x 0 0 , if x = 0. f(x)=\begin{cases}1/x,&\,\text{if }x\neq 0\\ 0,&\,\text{if }x=0.\\ \end{cases}
  12. a \textstyle a\in\mathbb{R}
  13. μ ( { x : 1 / x > a } ) 1 | a | \textstyle\mu(\{x\in\mathbb{R}:1/x>a\})\geq\tfrac{1}{|a|}
  14. U f = \textstyle U_{f}=\emptyset
  15. μ ( X ) > 0 \mu(X)>0
  16. inf f ess inf f ess sup f sup f \inf f\leq\mathrm{ess}\inf f\leq\mathrm{ess}\sup f\leq\sup f
  17. X X
  18. ess sup f = - \mathrm{ess}\sup f=-\infty
  19. ess inf f = + \mathrm{ess}\inf f=+\infty
  20. ess sup ( f g ) ( ess sup f ) ( ess sup g ) \mathrm{ess}\sup(fg)\leq(\mathrm{ess}\sup f)(\mathrm{ess}\sup g)

Essentially_surjective_functor.html

  1. F : C D F:C\to D
  2. d d
  3. D D
  4. F c Fc
  5. c c
  6. C C

Ethanol_precipitation.html

  1. q 1 q_{1}
  2. q 2 q_{2}
  3. r r
  4. ε r \varepsilon_{r}
  5. ε 0 \varepsilon_{0}
  6. F = 1 4 π ε r ε 0 q 1 q 2 r 2 F=\frac{1}{4\pi\varepsilon_{r}\varepsilon_{0}}\frac{q_{1}q_{2}}{r^{2}}

Euler's_theorem_in_geometry.html

  1. d 2 = R ( R - 2 r ) d^{2}=R(R-2r)\,
  2. 1 R - d + 1 R + d = 1 r , \frac{1}{R-d}+\frac{1}{R+d}=\frac{1}{r},
  3. R 2 r , R\geq 2r,
  4. R r a b c + a 3 + b 3 + c 3 2 a b c a b + b c + c a - 1 2 3 ( a b + b c + c a ) 2. \frac{R}{r}\geq\frac{abc+a^{3}+b^{3}+c^{3}}{2abc}\geq\frac{a}{b}+\frac{b}{c}+% \frac{c}{a}-1\geq\frac{2}{3}\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)% \geq 2.

Euler_number_(physics).html

  1. Eu = p upstream - p downstream ρ V 2 \mathrm{Eu}=\frac{p_{\mathrm{upstream}}-p_{\mathrm{downstream}}}{\rho V^{2}}
  2. ρ \rho
  3. p upstream p_{\mathrm{upstream}}
  4. p downstream p_{\mathrm{downstream}}
  5. V V
  6. Ca = p - p v 1 2 ρ V 2 \mathrm{Ca}=\frac{p-p_{\mathrm{v}}}{\frac{1}{2}\rho V^{2}}
  7. ρ \rho
  8. p p
  9. p v p_{\mathrm{v}}
  10. V V

Eulerian_number.html

  1. A n ( t ) = m = 0 n A ( n , m ) t m . A_{n}(t)=\sum_{m=0}^{n}A(n,m)\ t^{m}.
  2. n = 0 A n ( t ) x n n ! = t - 1 t - e ( t - 1 ) x . \sum_{n=0}^{\infty}A_{n}(t)\,\frac{x^{n}}{n!}=\frac{t-1}{t-\mathrm{e}^{(t-1)\,% x}}.
  3. A 0 ( t ) = 1 , A_{0}(t)=1,\,
  4. A n ( t ) = t ( 1 - t ) A n - 1 ( t ) + A n - 1 ( t ) ( 1 + ( n - 1 ) t ) , n 1. A_{n}(t)=t\,(1-t)\,A_{n-1}^{{}^{\prime}}(t)+A_{n-1}(t)\,(1+(n-1)\,t),\quad n% \geq 1.
  5. A 0 ( t ) = 1 , A_{0}(t)=1,\,
  6. A n ( t ) = k = 0 n - 1 ( n k ) A k ( t ) ( t - 1 ) n - 1 - k , n 1. A_{n}(t)=\sum_{k=0}^{n-1}{\left({{n}\atop{k}}\right)}\,A_{k}(t)\,(t-1)^{n-1-k}% ,\quad n\geq 1.
  7. n m \scriptstyle\left\langle{n\atop m}\right\rangle
  8. A ( n , m ) = ( n - m ) A ( n - 1 , m - 1 ) + ( m + 1 ) A ( n - 1 , m ) . A(n,m)=(n-m)A(n-1,m-1)+(m+1)A(n-1,m).
  9. A ( 4 , 1 ) = ( 4 - 1 ) A ( 3 , 0 ) + ( 1 + 1 ) A ( 3 , 1 ) = 3 × 1 + 2 × 4 = 11. A(4,1)=(4-1)A(3,0)+(1+1)A(3,1)=3\times 1+2\times 4=11.
  10. A ( n , m ) = k = 0 m ( - 1 ) k ( n + 1 k ) ( m + 1 - k ) n . A(n,m)=\sum_{k=0}^{m}(-1)^{k}{\left({{n+1}\atop{k}}\right)}(m+1-k)^{n}.
  11. m = 0 n - 1 A ( n , m ) = n ! for n 1. \sum_{m=0}^{n-1}A(n,m)=n!\,\text{ for }n\geq 1.
  12. m = 0 n - 1 ( - 1 ) m A ( n , m ) = 2 n + 1 ( 2 n + 1 - 1 ) B n + 1 n + 1 for n 1. \sum_{m=0}^{n-1}(-1)^{m}A(n,m)=\frac{2^{n+1}(2^{n+1}-1)B_{n+1}}{n+1}\,\text{ % for }n\geq 1.
  13. m = 0 n - 1 ( - 1 ) m A ( n , m ) ( n - 1 m ) = 0 for n 2 , \sum_{m=0}^{n-1}(-1)^{m}\frac{A(n,m)}{{\left({{n-1}\atop{m}}\right)}}=0\,\text% { for }n\geq 2,
  14. m = 0 n - 1 ( - 1 ) m A ( n , m ) ( n m ) = ( n + 1 ) B n for n 2 , \sum_{m=0}^{n-1}(-1)^{m}\frac{A(n,m)}{{\left({{n}\atop{m}}\right)}}=(n+1)B_{n}% \,\text{ for }n\geq 2,
  15. k = 0 k n x k = m = 0 n - 1 A ( n , m ) x m + 1 ( 1 - x ) n + 1 \sum_{k=0}^{\infty}k^{n}x^{k}=\frac{\sum_{m=0}^{n-1}A(n,m)x^{m+1}}{(1-x)^{n+1}}
  16. n 0 n\geq 0
  17. x n = m = 0 n - 1 A ( n , m ) ( x + m n ) . x^{n}=\sum_{m=0}^{n-1}A(n,m){\left({{x+m}\atop{n}}\right)}.
  18. k = 1 N k n = m = 0 n - 1 A ( n , m ) ( N + 1 + m n + 1 ) . \sum_{k=1}^{N}k^{n}=\sum_{m=0}^{n-1}A(n,m){\left({{N+1+m}\atop{n+1}}\right)}.
  19. e 1 - e x = n = 0 A n ( x ) n ! ( 1 - x ) n + 1 . \frac{e}{1-ex}=\sum_{n=0}^{\infty}\frac{A_{n}(x)}{n!(1-x)^{n+1}}.
  20. n m \scriptstyle\left\langle\!\!\left\langle{n\atop m}\right\rangle\!\!\right\rangle
  21. 332211 , 332211,\;
  22. 221133 , 221331 , 223311 , 233211 , 113322 , 133221 , 331122 , 331221 , 221133,\;221331,\;223311,\;233211,\;113322,\;133221,\;331122,\;331221,
  23. 112233 , 122133 , 112332 , 123321 , 133122 , 122331. 112233,\;122133,\;112332,\;123321,\;133122,\;122331.
  24. n m = ( 2 n - m - 1 ) n - 1 m - 1 + ( m + 1 ) n - 1 m , \left\langle\!\!\left\langle{n\atop m}\right\rangle\!\!\right\rangle=(2n-m-1)% \left\langle\!\!\left\langle{n-1\atop m-1}\right\rangle\!\!\right\rangle+(m+1)% \left\langle\!\!\left\langle{n-1\atop m}\right\rangle\!\!\right\rangle,
  25. 0 m = [ m = 0 ] . \left\langle\!\!\left\langle{0\atop m}\right\rangle\!\!\right\rangle=[m=0].
  26. P n ( x ) := m = 0 n n m x m P_{n}(x):=\sum_{m=0}^{n}\left\langle\!\!\left\langle{n\atop m}\right\rangle\!% \!\right\rangle x^{m}
  27. P n + 1 ( x ) = ( 2 n x + 1 ) P n ( x ) - x ( x - 1 ) P n ( x ) P_{n+1}(x)=(2nx+1)P_{n}(x)-x(x-1)P_{n}^{\prime}(x)
  28. P 0 ( x ) = 1. P_{0}(x)=1.
  29. ( x - 1 ) - 2 n - 2 P n + 1 ( x ) = ( x ( 1 - x ) - 2 n - 1 P n ( x ) ) (x-1)^{-2n-2}P_{n+1}(x)=\left(x(1-x)^{-2n-1}P_{n}(x)\right)^{\prime}
  30. u n ( x ) := ( x - 1 ) - 2 n P n ( x ) u_{n}(x):=(x-1)^{-2n}P_{n}(x)
  31. u n + 1 = ( x 1 - x u n ) , u 0 = 1 , u_{n+1}=\left(\frac{x}{1-x}u_{n}\right)^{\prime},\quad u_{0}=1,

Ewald_summation.html

  1. φ ( 𝐫 ) = def φ s r ( 𝐫 ) + φ r ( 𝐫 ) \varphi(\mathbf{r})\ \stackrel{\mathrm{def}}{=}\ \varphi_{sr}(\mathbf{r})+% \varphi_{\ell r}(\mathbf{r})
  2. φ s r ( 𝐫 ) \varphi_{sr}(\mathbf{r})
  3. φ r ( 𝐫 ) \varphi_{\ell r}(\mathbf{r})
  4. E r = d 𝐫 d 𝐫 ρ TOT ( 𝐫 ) ρ u c ( 𝐫 ) φ r ( 𝐫 - 𝐫 ) E_{\ell r}=\iint d\mathbf{r}\,d\mathbf{r}^{\prime}\,\rho\text{TOT}(\mathbf{r})% \rho_{uc}(\mathbf{r}^{\prime})\ \varphi_{\ell r}(\mathbf{r}-\mathbf{r}^{\prime})
  5. ρ u c ( 𝐫 ) \rho_{uc}(\mathbf{r})
  6. 𝐫 k \mathbf{r}_{k}
  7. q k q_{k}
  8. ρ u c ( 𝐫 ) = def charges k q k δ ( 𝐫 - 𝐫 k ) \rho_{uc}(\mathbf{r})\ \stackrel{\mathrm{def}}{=}\ \sum_{\mathrm{charges}\ k}q% _{k}\delta(\mathbf{r}-\mathbf{r}_{k})
  9. ρ TOT ( 𝐫 ) \rho\text{TOT}(\mathbf{r})
  10. q k q_{k}
  11. ρ TOT ( 𝐫 ) = def n 1 , n 2 , n 3 charges k q k δ ( 𝐫 - 𝐫 k - n 1 𝐚 1 - n 2 𝐚 2 - n 3 𝐚 3 ) \rho\text{TOT}(\mathbf{r})\ \stackrel{\mathrm{def}}{=}\ \sum_{n_{1},n_{2},n_{3% }}\sum_{\mathrm{charges}\ k}q_{k}\delta(\mathbf{r}-\mathbf{r}_{k}-n_{1}\mathbf% {a}_{1}-n_{2}\mathbf{a}_{2}-n_{3}\mathbf{a}_{3})
  12. δ ( 𝐱 ) \delta(\mathbf{x})
  13. 𝐚 1 \mathbf{a}_{1}
  14. 𝐚 2 \mathbf{a}_{2}
  15. 𝐚 3 \mathbf{a}_{3}
  16. n 1 n_{1}
  17. n 2 n_{2}
  18. n 3 n_{3}
  19. ρ TOT ( 𝐫 ) \rho\text{TOT}(\mathbf{r})
  20. ρ u c ( 𝐫 ) \rho_{uc}(\mathbf{r})
  21. L ( 𝐫 ) L(\mathbf{r})
  22. L ( 𝐫 ) = def n 1 , n 2 , n 3 δ ( 𝐫 - n 1 𝐚 1 - n 2 𝐚 2 - n 3 𝐚 3 ) L(\mathbf{r})\ \stackrel{\mathrm{def}}{=}\ \sum_{n_{1},n_{2},n_{3}}\delta(% \mathbf{r}-n_{1}\mathbf{a}_{1}-n_{2}\mathbf{a}_{2}-n_{3}\mathbf{a}_{3})
  23. ρ TOT ( 𝐫 ) \rho\text{TOT}(\mathbf{r})
  24. ρ ~ TOT ( 𝐤 ) = L ~ ( 𝐤 ) ρ ~ u c ( 𝐤 ) \tilde{\rho}\text{TOT}(\mathbf{k})=\tilde{L}(\mathbf{k})\tilde{\rho}_{uc}(% \mathbf{k})
  25. L ~ ( 𝐤 ) = ( 2 π ) 3 Ω m 1 , m 2 , m 3 δ ( 𝐤 - m 1 𝐛 1 - m 2 𝐛 2 - m 3 𝐛 3 ) \tilde{L}(\mathbf{k})=\frac{\left(2\pi\right)^{3}}{\Omega}\sum_{m_{1},m_{2},m_% {3}}\delta(\mathbf{k}-m_{1}\mathbf{b}_{1}-m_{2}\mathbf{b}_{2}-m_{3}\mathbf{b}_% {3})
  26. 𝐛 1 = def 𝐚 2 × 𝐚 3 Ω \mathbf{b}_{1}\ \stackrel{\mathrm{def}}{=}\ \frac{\mathbf{a}_{2}\times\mathbf{% a}_{3}}{\Omega}
  27. Ω = def 𝐚 1 ( 𝐚 2 × 𝐚 3 ) \Omega\ \stackrel{\mathrm{def}}{=}\ \mathbf{a}_{1}\cdot\left(\mathbf{a}_{2}% \times\mathbf{a}_{3}\right)
  28. L ( 𝐫 ) L(\mathbf{r})
  29. L ~ ( 𝐤 ) \tilde{L}(\mathbf{k})
  30. v ( 𝐫 ) = def d 𝐫 ρ u c ( 𝐫 ) φ r ( 𝐫 - 𝐫 ) v(\mathbf{r})\ \stackrel{\mathrm{def}}{=}\ \int d\mathbf{r}^{\prime}\,\rho_{uc% }(\mathbf{r}^{\prime})\ \varphi_{\ell r}(\mathbf{r}-\mathbf{r}^{\prime})
  31. V ~ ( 𝐤 ) = def ρ ~ u c ( 𝐤 ) Φ ~ ( 𝐤 ) \tilde{V}(\mathbf{k})\ \stackrel{\mathrm{def}}{=}\ \tilde{\rho}_{uc}(\mathbf{k% })\tilde{\Phi}(\mathbf{k})
  32. V ~ ( 𝐤 ) = d 𝐫 v ( 𝐫 ) e - i 𝐤 𝐫 \tilde{V}(\mathbf{k})=\int d\mathbf{r}\ v(\mathbf{r})\ e^{-i\mathbf{k}\cdot% \mathbf{r}}
  33. E r = d 𝐫 ρ TOT ( 𝐫 ) v ( 𝐫 ) E_{\ell r}=\int d\mathbf{r}\ \rho\text{TOT}(\mathbf{r})\ v(\mathbf{r})
  34. E r = d 𝐤 ( 2 π ) 3 ρ ~ TOT * ( 𝐤 ) V ~ ( 𝐤 ) = d 𝐤 ( 2 π ) 3 L ~ * ( 𝐤 ) | ρ ~ u c ( 𝐤 ) | 2 Φ ~ ( 𝐤 ) = 1 Ω m 1 , m 2 , m 3 | ρ ~ u c ( 𝐤 ) | 2 Φ ~ ( 𝐤 ) E_{\ell r}=\int\frac{d\mathbf{k}}{\left(2\pi\right)^{3}}\ \tilde{\rho}\text{% TOT}^{*}(\mathbf{k})\tilde{V}(\mathbf{k})=\int\frac{d\mathbf{k}}{\left(2\pi% \right)^{3}}\tilde{L}^{*}(\mathbf{k})\left|\tilde{\rho}_{uc}(\mathbf{k})\right% |^{2}\tilde{\Phi}(\mathbf{k})=\frac{1}{\Omega}\sum_{m_{1},m_{2},m_{3}}\left|% \tilde{\rho}_{uc}(\mathbf{k})\right|^{2}\tilde{\Phi}(\mathbf{k})
  35. 𝐤 = m 1 𝐛 1 + m 2 𝐛 2 + m 3 𝐛 3 \mathbf{k}=m_{1}\mathbf{b}_{1}+m_{2}\mathbf{b}_{2}+m_{3}\mathbf{b}_{3}
  36. ρ ~ u c ( 𝐤 ) \tilde{\rho}_{uc}(\mathbf{k})
  37. 𝐤 \mathbf{k}
  38. φ ( 𝐫 ) = def φ s r ( 𝐫 ) + φ r ( 𝐫 ) \varphi(\mathbf{r})\ \stackrel{\mathrm{def}}{=}\ \varphi_{sr}(\mathbf{r})+% \varphi_{\ell r}(\mathbf{r})
  39. E TOT = i , j φ ( 𝐫 j - 𝐫 i ) = E s r + E r E\text{TOT}=\sum_{i,j}\varphi(\mathbf{r}_{j}-\mathbf{r}_{i})=E_{sr}+E_{\ell r}
  40. E s r E_{sr}
  41. E s r = i , j φ s r ( 𝐫 j - 𝐫 i ) E_{sr}=\sum_{i,j}\varphi_{sr}(\mathbf{r}_{j}-\mathbf{r}_{i})
  42. E r = 𝐤 Φ ~ r ( 𝐤 ) | ρ ~ ( 𝐤 ) | 2 E_{\ell r}=\sum_{\mathbf{k}}\tilde{\Phi}_{\ell r}(\mathbf{k})\left|\tilde{\rho% }(\mathbf{k})\right|^{2}
  43. Φ ~ r \tilde{\Phi}_{\ell r}
  44. ρ ~ ( 𝐤 ) \tilde{\rho}(\mathbf{k})
  45. ρ ~ ( 𝐤 ) \tilde{\rho}(\mathbf{k})
  46. 𝐩 u c \mathbf{p}_{uc}
  47. R R
  48. R 2 R^{2}
  49. 1 R 3 \frac{1}{R^{3}}
  50. n = 1 1 n \sum_{n=1}^{\infty}\frac{1}{n}
  51. σ = 𝐏 𝐧 \sigma=\mathbf{P}\cdot\mathbf{n}
  52. 𝐧 \mathbf{n}
  53. 𝐏 \mathbf{P}
  54. U U
  55. U = 1 2 V u c ( 𝐩 u c 𝐫 ) ( 𝐩 u c 𝐧 ) d S r 3 U=\frac{1}{2V_{uc}}\int\frac{\left(\mathbf{p}_{uc}\cdot\mathbf{r}\right)\left(% \mathbf{p}_{uc}\cdot\mathbf{n}\right)dS}{r^{3}}
  56. 𝐩 u c \mathbf{p}_{uc}
  57. V u c V_{uc}
  58. d S dS
  59. 𝐫 \mathbf{r}
  60. d U = - 𝐩 u c 𝐝𝐄 dU=-\mathbf{p}_{uc}\cdot\mathbf{dE}
  61. d 𝐄 d\mathbf{E}
  62. d q = def σ d S dq\ \stackrel{\mathrm{def}}{=}\ \sigma dS
  63. d 𝐄 = def ( - 1 4 π ϵ ) d q 𝐫 r 3 = ( - 1 4 π ϵ ) σ d S 𝐫 r 3 d\mathbf{E}\ \stackrel{\mathrm{def}}{=}\ \left(\frac{-1}{4\pi\epsilon}\right)% \frac{dq\ \mathbf{r}}{r^{3}}=\left(\frac{-1}{4\pi\epsilon}\right)\frac{\sigma% \,dS\ \mathbf{r}}{r^{3}}
  64. 𝐫 \mathbf{r}
  65. O ( N 2 ) O(N^{2})
  66. N N
  67. O ( N log N ) O(N\,\log N)

Exchange_interaction.html

  1. Φ a ( r 1 ) \Phi_{a}(r_{1})
  2. Φ b ( r 2 ) \Phi_{b}(r_{2})
  3. Φ a \Phi_{a}
  4. Φ b \Phi_{b}
  5. Ψ A ( r 1 , r 2 ) = 1 2 [ Φ a ( r 1 ) Φ b ( r 2 ) - Φ b ( r 1 ) Φ a ( r 2 ) ] \Psi_{A}(\vec{r}_{1},\vec{r}_{2})=\frac{1}{\sqrt{2}}[\Phi_{a}(\vec{r}_{1})\Phi% _{b}(\vec{r}_{2})-\Phi_{b}(\vec{r}_{1})\Phi_{a}(\vec{r}_{2})]
  6. Ψ S ( r 1 , r 2 ) = 1 2 [ Φ a ( r 1 ) Φ b ( r 2 ) + Φ b ( r 1 ) Φ a ( r 2 ) ] \Psi_{S}(\vec{r}_{1},\vec{r}_{2})=\frac{1}{\sqrt{2}}[\Phi_{a}(\vec{r}_{1})\Phi% _{b}(\vec{r}_{2})+\Phi_{b}(\vec{r}_{1})\Phi_{a}(\vec{r}_{2})]
  7. \mathcal{H}
  8. ( 0 ) \mathcal{H}^{(0)}
  9. ( 1 ) \mathcal{H}^{(1)}
  10. ( 0 ) = - 2 2 m ( 1 2 + 2 2 ) - e 2 r 1 - e 2 r 2 \mathcal{H}^{(0)}=-\frac{\hbar^{2}}{2m}\left(\nabla^{2}_{1}+\nabla^{2}_{2}% \right)-\frac{e^{2}}{r_{1}}-\frac{e^{2}}{r_{2}}
  11. ( 1 ) = ( e 2 R a b + e 2 r 12 - e 2 r a 1 - e 2 r b 2 ) \mathcal{H}^{(1)}=\left(\frac{e^{2}}{R_{ab}}+\frac{e^{2}}{r_{12}}-\frac{e^{2}}% {r_{a1}}-\frac{e^{2}}{r_{b2}}\right)
  12. E + / - = E ( 0 ) + C ± J e x 1 ± B 2 \ E_{+/-}=E_{(0)}+\frac{C\pm J_{ex}}{1\pm B^{2}}
  13. \mathcal{H}
  14. C = Φ a ( r 1 ) 2 ( 1 R a b + 1 r 12 - 1 r a 1 - 1 r b 2 ) Φ b ( r 2 ) 2 d 3 r 1 d 3 r 2 C=\int\Phi_{a}(\vec{r}_{1})^{2}\left(\frac{1}{R_{ab}}+\frac{1}{r_{12}}-\frac{1% }{r_{a1}}-\frac{1}{r_{b2}}\right)\Phi_{b}(\vec{r}_{2})^{2}\,d^{3}r_{1}\,d^{3}r% _{2}
  15. J e x = Φ a ( r 1 ) Φ b ( r 2 ) ( 1 R a b + 1 r 12 - 1 r a 1 - 1 r b 2 ) Φ b ( r 1 ) Φ a ( r 2 ) d 3 r 1 d 3 r 2 J_{ex}=\int\Phi_{a}(\vec{r}_{1})\Phi_{b}(\vec{r}_{2})\left(\frac{1}{R_{ab}}+% \frac{1}{r_{12}}-\frac{1}{r_{a1}}-\frac{1}{r_{b2}}\right)\Phi_{b}(\vec{r}_{1})% \Phi_{a}(\vec{r}_{2})\,d^{3}r_{1}\,d^{3}r_{2}
  16. β ( 2 ) \beta(2)
  17. α ( 2 ) \alpha(2)
  18. β ( 1 ) \beta(1)
  19. ( s a + s b ) 2 (\vec{s}_{a}+\vec{s}_{b})^{2}
  20. S ( S + 1 ) S(S+1)
  21. s a 2 \vec{s}_{a}^{\;2}
  22. s b 2 \vec{s}_{b}^{\;2}
  23. 1 2 ( 1 2 + 1 ) = 3 4 \tfrac{1}{2}(\tfrac{1}{2}+1)=\tfrac{3}{4}
  24. ( s a + s b ) 2 = s a 2 + s b 2 + 2 s a s b (\vec{s}_{a}+\vec{s}_{b})^{2}=\vec{s}_{a}^{\;2}+\vec{s}_{b}^{\;2}+2\vec{s}_{a}% \cdot\vec{s}_{b}
  25. s a s b \vec{s}_{a}\cdot\vec{s}_{b}
  26. 1 2 ( 0 - 6 4 ) = - 3 4 \tfrac{1}{2}(0-\tfrac{6}{4})=-\tfrac{3}{4}
  27. 1 2 ( 2 - 6 4 ) = 1 4 \tfrac{1}{2}(2-\tfrac{6}{4})=\tfrac{1}{4}
  28. s a s b \vec{s}_{a}\cdot\vec{s}_{b}
  29. s a s b \vec{s}_{a}\cdot\vec{s}_{b}
  30. E e x - C + 1 2 J e x + 2 J e x s a s b = 0 E_{ex}-C+\frac{1}{2}J_{ex}+2J_{ex}\vec{s}_{a}\cdot\vec{s}_{b}=0
  31. E e x = C - 1 2 J e x - 2 J e x s a s b E_{ex}=C-\frac{1}{2}J_{ex}-2J_{ex}\vec{s}_{a}\cdot\vec{s}_{b}
  32. s a \vec{s}_{a}
  33. s b \vec{s}_{b}
  34. - 2 J a b s a s b \ -2J_{ab}\vec{s}_{a}\cdot\vec{s}_{b}
  35. s a \vec{s}_{a}
  36. s b \vec{s}_{b}
  37. H e i s = - 2 J a b s a s b \mathcal{H}_{Heis}=-2J_{ab}\vec{s}_{a}\cdot\vec{s}_{b}
  38. J a b = 1 2 ( E + - E - ) = J e x - C B 2 1 - B 4 \ J_{ab}=\frac{1}{2}(E_{+}-E_{-})=\frac{J_{ex}-CB^{2}}{1-B^{4}}
  39. E - - E + = 2 ( C B 2 - J e x ) 1 - B 4 \ E_{-}-E_{+}=\frac{2(CB^{2}-J_{ex})}{1-B^{4}}
  40. H e i s = 1 2 ( - 2 J i , j S i S j ) = - i , j J S i S j \mathcal{H}_{Heis}=\frac{1}{2}(-2J\sum_{i,j}\vec{S}_{i}\cdot\vec{S}_{j}\quad)=% -\sum_{i,j}J\vec{S}_{i}\cdot\vec{S}_{j}
  41. a a
  42. A s c = J e x S 2 a A_{sc}=\frac{J_{ex}S^{2}}{a}
  43. A b c c = 2 J e x S 2 a A_{bcc}=\frac{2J_{ex}S^{2}}{a}
  44. A f c c = 4 J e x S 2 a A_{fcc}=\frac{4J_{ex}S^{2}}{a}
  45. E = - i j J i j S i z S j z E=-\sum_{i\neq j}J_{ij}S_{i}^{z}S_{j}^{z}\,
  46. μ S = - g μ B [ S ( S + 1 ) ] 1 / 2 \mu_{S}=-g\mu_{B}[S(S+1)]^{1/2}
  47. S = 1 \vec{S}=1
  48. μ S = 2.83 μ B \mu_{S}=2.83\mu_{B}
  49. μ S = g μ B S = 2 μ B \vec{\mu}_{S}=g\mu_{B}\vec{S}=2\mu_{B}

Exchange_operator.html

  1. P ^ \hat{P}
  2. | x 1 , x 2 \left|x_{1},x_{2}\right\rangle
  3. P ^ 2 | x 1 , x 2 = P ^ | x 2 , x 1 = | x 1 , x 2 \hat{P}^{2}\left|x_{1},x_{2}\right\rangle=\hat{P}\left|x_{2},x_{1}\right% \rangle=\left|x_{1},x_{2}\right\rangle
  4. P ^ \hat{P}
  5. P ^ | x 1 , x 2 = ± | x 1 , x 2 . \hat{P}\left|x_{1},x_{2}\right\rangle=\pm\left|x_{1},x_{2}\right\rangle\,.
  6. P ^ \hat{P}
  7. K ^ j ( x 1 ) f ( x 1 ) = ϕ j ( x 1 ) ϕ j * ( x 2 ) f ( x 2 ) r 12 d x 2 \hat{K}_{j}(x_{1})f(x_{1})=\phi_{j}(x_{1})\int{\frac{\phi_{j}^{*}(x_{2})f(x_{2% })}{r_{12}}dx_{2}}
  8. K ^ j ( x 1 ) \hat{K}_{j}(x_{1})
  9. f ( x 1 ) f(x_{1})
  10. f ( x 2 ) f(x_{2})
  11. ϕ j ( x 1 ) \phi_{j}(x_{1})
  12. ϕ j ( x 2 ) \phi_{j}(x_{2})

Exergy_efficiency.html

  1. B i n = B o u t + B l o s t + B d e s t r o y e d (1) B_{in}=B_{out}+B_{lost}+B_{destroyed}\qquad\mbox{(1)}~{}
  2. η B = B o u t B i n = 1 - ( B l o s t + B d e s t r o y e d ) B i n (2) \eta_{B}=\frac{B_{out}}{B_{in}}=1-\frac{(B_{lost}+B_{destroyed})}{B_{in}}% \qquad\mbox{(2)}~{}
  3. η B = W ˙ n e t m ˙ f u e l Δ G T 0 (3) \eta_{B}=\frac{\dot{W}_{net}}{\dot{m}_{fuel}\Delta G^{0}_{T}}\qquad\mbox{(3)}~{}
  4. Δ G T 0 \Delta G^{0}_{T}
  5. T \mathrm{T}
  6. p 0 = 1 b a r p_{0}=1\mathrm{bar}
  7. W ˙ n e t \dot{W}_{net}
  8. m ˙ f u e l \dot{m}_{fuel}
  9. η E = W ˙ n e t m ˙ f u e l Δ H T 0 (4) \eta_{E}=\frac{\dot{W}_{net}}{\dot{m}_{fuel}\Delta H^{0}_{T}}\qquad\mbox{(4)}~{}
  10. Δ H T 0 \Delta H^{0}_{T}
  11. T \mathrm{T}
  12. p 0 = 1 b a r p_{0}=1\mathrm{bar}
  13. Δ G T 0 < Δ H T 0 \Delta G^{0}_{T}<\Delta H^{0}_{T}

Exhaustion_by_compact_sets.html

  1. K j K_{j}
  2. K j K_{j}
  3. K j + 1 K_{j+1}
  4. K j K_{j}
  5. K j + 1 K_{j+1}
  6. j j
  7. E = { z ; | z | < 1 } E=\{z;|z|<1\}
  8. K j = { z ; | z | ( 1 - 1 / j ) } K_{j}=\{z;|z|\leq(1-1/j)\}
  9. K j K_{j}

Exner_equation.html

  1. η \eta
  2. t t
  3. ε o \varepsilon_{o}
  4. q s q_{s}
  5. η t = - 1 ε o 𝐪 𝐬 \frac{\partial\eta}{\partial t}=-\frac{1}{\varepsilon_{o}}\nabla\cdot\mathbf{q% _{s}}
  6. ε o \varepsilon_{o}
  7. ( 1 - λ p ) (1-\lambda_{p})
  8. λ p \lambda_{p}
  9. ε o \varepsilon_{o}
  10. x x
  11. η t = - 1 ε o 𝐪 𝐬 x \frac{\partial\eta}{\partial t}=-\frac{1}{\varepsilon_{o}}\frac{\partial% \mathbf{q_{s}}}{\partial x}
  12. σ \sigma
  13. η \eta
  14. σ \sigma
  15. η t = - 1 ε o 𝐪 𝐬 + σ \frac{\partial\eta}{\partial t}=-\frac{1}{\varepsilon_{o}}\nabla\cdot\mathbf{q% _{s}}+\sigma

Experience_modifier.html

  1. I + ( C * ( 1 - A ) + G ) + ( A * F ) E + ( C * ( 1 - A ) + G ) + ( A * C ) \frac{I+(C*(1-A)+G)+(A*F)}{E+(C*(1-A)+G)+(A*C)}
  2. I I
  3. E E
  4. ( C * ( 1 - A ) + G ) (C*(1-A)+G)
  5. ( A * F ) (A*F)
  6. ( A * C ) (A*C)
  7. A c t u a l P r i m a r y L o s s e s + S t a b i l i z i n g V a l u e + A c t u a l R a t a b l e E x c e s s E x p e c t e d P r i m a r y L o s s e s + S t a b i l i z i n g V a l u e + E x p e c t e d R a t a b l e E x c e s s \frac{ActualPrimaryLosses+StabilizingValue+ActualRatableExcess}{% ExpectedPrimaryLosses+StabilizingValue+ExpectedRatableExcess}
  8. ( D - E ) (D-E)
  9. ( H - I ) (H-I)

Explicit_and_implicit_methods.html

  1. Y ( t ) Y(t)
  2. Y ( t + Δ t ) Y(t+\Delta t)
  3. Δ t \Delta t
  4. Y ( t + Δ t ) = F ( Y ( t ) ) Y(t+\Delta t)=F(Y(t))\,
  5. G ( Y ( t ) , Y ( t + Δ t ) ) = 0 ( 1 ) G(Y(t),Y(t+\Delta t))=0\quad\quad(1)\,
  6. Y ( t + Δ t ) . Y(t+\Delta t).
  7. Δ t \Delta t
  8. d y d t = - y 2 , t [ 0 , a ] ( 2 ) \frac{dy}{dt}=-y^{2},\ t\in[0,a]\quad\quad(2)
  9. y ( 0 ) = 1. y(0)=1.
  10. t k = a k n t_{k}=a\frac{k}{n}
  11. Δ t = a / n , \Delta t=a/n,
  12. y k = y ( t k ) y_{k}=y(t_{k})
  13. k k
  14. ( d y d t ) k = y k + 1 - y k Δ t = - y k 2 \left(\frac{dy}{dt}\right)_{k}=\frac{y_{k+1}-y_{k}}{\Delta t}=-y_{k}^{2}
  15. y k + 1 = y k - Δ t y k 2 ( 3 ) y_{k+1}=y_{k}-\Delta ty_{k}^{2}\quad\quad\quad(3)\,
  16. k = 0 , 1 , , n . k=0,1,\dots,n.
  17. y k + 1 y_{k+1}
  18. y k + 1 - y k Δ t = - y k + 1 2 \frac{y_{k+1}-y_{k}}{\Delta t}=-y_{k+1}^{2}
  19. y k + 1 + Δ t y k + 1 2 = y k y_{k+1}+\Delta ty_{k+1}^{2}=y_{k}
  20. y k + 1 y_{k+1}
  21. y k + 1 y_{k+1}
  22. y y
  23. y k + 1 = - 1 + 1 + 4 Δ t y k 2 Δ t . ( 4 ) y_{k+1}=\frac{-1+\sqrt{1+4\Delta ty_{k}}}{2\Delta t}.\quad\quad(4)

Exponential_discounting.html

  1. U ( { c t } t = t 1 t 2 ) = t = t 1 t 2 δ t - t 1 ( u ( c t ) ) , U(\{c_{t}\}_{t=t_{1}}^{t_{2}})=\sum_{t=t_{1}}^{t_{2}}\delta^{t-t_{1}}(u(c_{t})),
  2. δ \delta
  3. U ( { c ( t ) } t = t 1 t 2 ) = t 1 t 2 e - ρ ( t - t 1 ) u ( c ( t ) ) d t , U(\{c(t)\}_{t=t_{1}}^{t_{2}})=\int_{t_{1}}^{t_{2}}e^{-\rho(t-t_{1})}u(c(t))\,dt,

Exponential_factorial.html

  1. n ( n - 1 ) ( n - 2 ) . n^{(n-1)^{(n-2)\cdots}}.\,
  2. a 0 = 1 , a n = n a n - 1 . a_{0}=1,\quad a_{n}=n^{a_{n-1}}.\,
  3. 262144 = 4 3 2 1 . 262144=4^{3^{2^{1}}}.\,

Exponential_type.html

  1. e - π z 2 e^{-\pi z^{2}}
  2. 2 π 2\pi
  3. | f ( r e i θ ) | M e τ r |f(re^{i\theta})|\leq Me^{\tau r}
  4. r r\to\infty
  5. z = r e i θ z=re^{i\theta}
  6. f ( z ) = sin ( π z ) f(z)=\sin(\pi z)
  7. sin ( π z ) \sin(\pi z)
  8. sin ( π z ) \sin(\pi z)
  9. F ( z ) F(z)
  10. σ > 0 \sigma>0
  11. ε > 0 \varepsilon>0
  12. A ε A_{\varepsilon}
  13. | F ( z ) | A ε e ( σ + ε ) | z | |F(z)|\leq A_{\varepsilon}e^{(\sigma+\varepsilon)|z|}
  14. | z | |z|\to\infty
  15. z z\in\mathbb{C}
  16. F ( z ) F(z)
  17. F ( z ) F(z)
  18. σ \sigma
  19. σ > 0 \sigma>0
  20. τ ( F ) = σ = lim sup | z | | z | - 1 log | F ( z ) | \tau(F)=\sigma=\displaystyle\limsup_{|z|\rightarrow\infty}|z|^{-1}\log|F(z)|
  21. F ( z ) F(z)
  22. K K
  23. n \mathbb{R}^{n}
  24. K K
  25. K \|\cdot\|_{K}
  26. K = { x n : x K 1 } K=\{x\in\mathbb{R}^{n}:\|x\|_{K}\leq 1\}
  27. K K
  28. n \mathbb{R}^{n}
  29. K \|\cdot\|_{K}
  30. K * = { y n : x y 1 for all x K } K^{*}=\{y\in\mathbb{R}^{n}:x\cdot y\leq 1\,\text{ for all }x\in{K}\}
  31. n \mathbb{R}^{n}
  32. x K = sup y K * | x y | \|x\|_{K}=\displaystyle\sup_{y\in K^{*}}|x\cdot y|
  33. K \|\cdot\|_{K}
  34. n \mathbb{R}^{n}
  35. n \mathbb{C}^{n}
  36. z K = sup y K * | z y | . \|z\|_{K}=\displaystyle\sup_{y\in K^{*}}|z\cdot y|.
  37. F ( z ) F(z)
  38. n n
  39. K K
  40. ε > 0 \varepsilon>0
  41. A ε A_{\varepsilon}
  42. | F ( z ) | < A ε e 2 π ( 1 + ε ) z K |F(z)|<A_{\varepsilon}e^{2\pi(1+\varepsilon)\|z\|_{K}}
  43. z n z\in\mathbb{C}^{n}
  44. τ \tau
  45. f n = sup z exp [ - ( τ + 1 n ) | z | ] | f ( z ) | \|f\|_{n}=\sup_{z\in\mathbb{C}}\exp\left[-\left(\tau+\frac{1}{n}\right)|z|% \right]|f(z)|

Extension_(predicate_logic).html

  1. Φ \Phi
  2. { ( x 1 , , x n ) Φ ( x 1 , , x n ) } . \{(x_{1},...,x_{n})\mid\Phi(x_{1},...,x_{n})\}\,.
  3. Φ \Phi
  4. Φ \Phi
  5. Φ \Phi

Extent_of_reaction.html

  1. d ξ = d n i ν i d\xi=\frac{dn_{i}}{\nu_{i}}
  2. n i n_{i}
  3. ν i \nu_{i}
  4. Δ ξ = Δ n i ν i \Delta\xi=\frac{\Delta n_{i}}{\nu_{i}}
  5. ξ = Δ n i ν i = n e q u i l i b r i u m - n i n i t i a l ν i \xi=\frac{\Delta n_{i}}{\nu_{i}}=\frac{n_{equilibrium}-n_{initial}}{\nu_{i}}
  6. Δ r G = ( G ξ ) p , T \Delta_{r}G=\left(\frac{\partial G}{\partial\xi}\right)_{p,T}
  7. Δ r H = ( H ξ ) p , T \Delta_{r}H=\left(\frac{\partial H}{\partial\xi}\right)_{p,T}
  8. n A = 2 m o l , n B = 1 m o l , n C = 0 m o l n_{A}=2mol,n_{B}=1mol,n_{C}=0mol
  9. ξ = Δ n A ν A = 0.5 - 2 - 2 = 0.75 \xi=\frac{\Delta n_{A}}{\nu_{A}}=\frac{0.5-2}{-2}=0.75
  10. n e q u i l i b r i u m = ξ ν i + n i n i t i a l n_{equilibrium}=\xi\nu_{i}+n_{initial}
  11. n B = 0.75 * 1 + 1 = 1.75 m o l n_{B}=0.75*1+1=1.75mol
  12. n C = 0.75 * 3 + 0 = 2.25 m o l n_{C}=0.75*3+0=2.25mol

External_ray.html

  1. K K\,
  2. K K\,
  3. K K\,
  4. K K\,
  5. Ψ c \Psi_{c}\,
  6. 𝔻 ¯ \overline{\mathbb{D}}
  7. K c \ Kc
  8. Ψ c : ^ 𝔻 ¯ ^ K c \Psi_{c}:\mathbb{\hat{C}}\setminus\overline{\mathbb{D}}\to\mathbb{\hat{C}}% \setminus Kc
  9. Φ c \Phi_{c}\,
  10. K c \ Kc
  11. Φ c : ^ K c ^ 𝔻 ¯ \Phi_{c}:\mathbb{\hat{C}}\setminus Kc\to\mathbb{\hat{C}}\setminus\overline{% \mathbb{D}}
  12. ^ \mathbb{\hat{C}}
  13. Φ c \Phi_{c}\,
  14. Ψ c = Φ c - 1 \Psi_{c}=\Phi_{c}^{-1}\,
  15. w = Φ c ( z ) = lim n ( f c n ( z ) ) 2 - n w=\Phi_{c}(z)=\lim_{n\rightarrow\infty}(f_{c}^{n}(z))^{2^{-n}}
  16. z ^ K c z\in\mathbb{\hat{C}}\setminus K_{c}
  17. w ^ 𝔻 ¯ w\in\mathbb{\hat{C}}\setminus\overline{\mathbb{D}}
  18. w w\,
  19. θ \theta\,
  20. θ K \mathcal{R}^{K}_{\theta}
  21. Ψ c \Psi_{c}\,
  22. θ = { ( r * e 2 π i θ ) : r > 1 } \mathcal{R}_{\theta}=\{\left(r*e^{2\pi i\theta}\right):\ r>1\}
  23. θ K = Ψ c ( θ ) \mathcal{R}^{K}_{\theta}=\Psi_{c}(\mathcal{R}_{\theta})
  24. θ \theta
  25. θ K = { z ^ K c : arg ( Φ c ( z ) ) = θ } \mathcal{R}^{K}_{\theta}=\{z\in\mathbb{\hat{C}}\setminus Kc:\arg(\Phi_{c}(z))=\theta\}
  26. θ \theta\,
  27. f ( θ K ) = 2 θ K f(\mathcal{R}^{K}_{\theta})=\mathcal{R}^{K}_{2\theta}
  28. γ f ( θ ) ) \gamma_{f}(\theta))
  29. f ( γ f ( θ ) ) = γ f ( 2 θ ) f(\gamma_{f}(\theta))=\gamma_{f}(2\theta)
  30. Ψ M \Psi_{M}\,
  31. 𝔻 ¯ \overline{\mathbb{D}}
  32. M \ M
  33. Ψ M : ^ 𝔻 ¯ ^ M \Psi_{M}:\mathbb{\hat{C}}\setminus\overline{\mathbb{D}}\to\mathbb{\hat{C}}\setminus M
  34. Φ M \Phi_{M}\,
  35. M \ M
  36. Φ M : ^ M ^ 𝔻 ¯ \Phi_{M}:\mathbb{\hat{C}}\setminus M\to\mathbb{\hat{C}}\setminus\overline{% \mathbb{D}}
  37. Φ M ( c ) c 1 a s c \frac{\Phi_{M}(c)}{c}\to 1\ as\ c\to\infty\,
  38. ^ \mathbb{\hat{C}}
  39. Ψ M \Psi_{M}\,
  40. Ψ M = Φ M - 1 \Psi_{M}=\Phi_{M}^{-1}\,
  41. c = Ψ M ( w ) = w + m = 0 b m w - m = w - 1 2 + 1 8 w - 1 4 w 2 + 15 128 w 3 + c=\Psi_{M}(w)=w+\sum_{m=0}^{\infty}b_{m}w^{-m}=w-\frac{1}{2}+\frac{1}{8w}-% \frac{1}{4w^{2}}+\frac{15}{128w^{3}}+...\,
  42. c ^ M c\in\mathbb{\hat{C}}\setminus M
  43. w ^ 𝔻 ¯ w\in\mathbb{\hat{C}}\setminus\overline{\mathbb{D}}
  44. θ \theta\,
  45. Ψ c \Psi_{c}\,
  46. θ = { ( r * e 2 π i θ ) : r > 1 } \mathcal{R}_{\theta}=\{\left(r*e^{2\pi i\theta}\right):\ r>1\}
  47. θ M = Ψ M ( θ ) \mathcal{R}^{M}_{\theta}=\Psi_{M}(\mathcal{R}_{\theta})
  48. θ \theta
  49. θ M = { c ^ M : arg ( Φ M ( c ) ) = θ } \mathcal{R}^{M}_{\theta}=\{c\in\mathbb{\hat{C}}\setminus M:\arg(\Phi_{M}(c))=\theta\}
  50. Φ M \Phi_{M}\,
  51. Φ M ( c ) = def Φ c ( z = c ) \Phi_{M}(c)\ \overset{\underset{\mathrm{def}}{}}{=}\ \Phi_{c}(z=c)\,
  52. c c\,
  53. z = c z=c\,
  54. θ \theta\,
  55. a r g ( Φ M ( c ) ) arg(\Phi_{M}(c))\,
  56. a r g ( ρ n ( c ) ) arg(\rho_{n}(c))\,
  57. a r g ( c ) arg(c)\,
  58. a r g ( Φ c ( z ) ) arg(\Phi_{c}(z))\,
  59. a r g ( z ) arg(z)\,
  60. a r g M ( c ) = a r g ( Φ M ( c ) ) arg_{M}(c)=arg(\Phi_{M}(c))\,
  61. a r g c ( z ) = a r g ( Φ c ( z ) ) arg_{c}(z)=arg(\Phi_{c}(z))\,
  62. f c ( z ) = z 2 - 1 f_{c}(z)=z^{2}-1
  63. α c \alpha_{c}\,
  64. α c \alpha_{c}\,

Eyring_equation.html

  1. k = k B T h e - Δ G \Dagger R T \ k=\frac{k_{\mathrm{B}}T}{h}\mathrm{e}^{-\frac{\Delta G^{\Dagger}}{RT}}
  2. k = k B T h e Δ S R e - Δ H R T k=\frac{k_{\mathrm{B}}T}{h}\mathrm{e}^{\frac{\Delta S^{\ddagger}}{R}}\mathrm{e% }^{-\frac{\Delta H^{\ddagger}}{RT}}
  3. ln k T = - Δ H R 1 T + ln k B h + Δ S R \ln\frac{k}{T}=\frac{-\Delta H^{\ddagger}}{R}\cdot\frac{1}{T}+\ln\frac{k_{% \mathrm{B}}}{h}+\frac{\Delta S^{\ddagger}}{R}
  4. k \ k
  5. T \ T
  6. Δ H \ \Delta H^{\ddagger}
  7. R \ R
  8. k B \ k_{\mathrm{B}}
  9. h \ h
  10. Δ S \ \Delta S^{\ddagger}
  11. ln ( k / T ) \ \ln(k/T)
  12. 1 / T \ 1/T
  13. - Δ H / R \ -\Delta H^{\ddagger}/R
  14. ln ( k B / h ) + Δ S / R \ \ln(k_{\mathrm{B}}/h)+\Delta S^{\ddagger}/R
  15. κ \ \kappa
  16. A B \ AB^{\ddagger}
  17. A B \ AB
  18. A \ A
  19. B \ B
  20. κ \ \kappa
  21. k ( T ) / k ( T R e f ) \ k(T)/k(T_{Ref})
  22. κ \ \kappa

F._and_M._Riesz_theorem.html

  1. μ \mu
  2. μ ^ n = 0 2 π e - i n θ d μ ( θ ) 2 π = 0 , \hat{\mu}_{n}=\int_{0}^{2\pi}{\rm e}^{-in\theta}\frac{d\mu(\theta)}{2\pi}=0,
  3. n < 0 n<0

F1_score.html

  1. F 1 = 2 precision recall precision + recall F_{1}=2\cdot\frac{\mathrm{precision}\cdot\mathrm{recall}}{\mathrm{precision}+% \mathrm{recall}}
  2. F β = ( 1 + β 2 ) precision recall ( β 2 precision ) + recall F_{\beta}=(1+\beta^{2})\cdot\frac{\mathrm{precision}\cdot\mathrm{recall}}{(% \beta^{2}\cdot\mathrm{precision})+\mathrm{recall}}
  3. F β = ( 1 + β 2 ) true positive ( 1 + β 2 ) true positive + β 2 false negative + false positive F_{\beta}=\frac{(1+\beta^{2})\cdot\mathrm{true\ positive}}{(1+\beta^{2})\cdot% \mathrm{true\ positive}+\beta^{2}\cdot\mathrm{false\ negative}+\mathrm{false\ % positive}}\,
  4. F 2 F_{2}
  5. F 0.5 F_{0.5}
  6. F β F_{\beta}
  7. E = 1 - ( α P + 1 - α R ) - 1 E=1-\left(\frac{\alpha}{P}+\frac{1-\alpha}{R}\right)^{-1}
  8. F β = 1 - E F_{\beta}=1-E
  9. α = 1 1 + β 2 \alpha=\frac{1}{1+\beta^{2}}
  10. F β F_{\beta}
  11. G = precision recall G=\sqrt{\mathrm{precision}\cdot\mathrm{recall}}

Factorial_moment_generating_function.html

  1. M X ( t ) = E [ t X ] M_{X}(t)=\operatorname{E}\bigl[t^{X}\bigr]
  2. | t | = 1 |t|=1
  3. M X M_{X}
  4. M X ( t ) M_{X}(t)
  5. | t | 1 |t|\leq 1
  6. M X M_{X}
  7. E [ ( X ) n ] = M X ( n ) ( 1 ) = d n d t n | t = 1 M X ( t ) , \operatorname{E}[(X)_{n}]=M_{X}^{(n)}(1)=\left.\frac{\mathrm{d}^{n}}{\mathrm{d% }t^{n}}\right|_{t=1}M_{X}(t),
  8. ( x ) n = x ( x - 1 ) ( x - 2 ) ( x - n + 1 ) . (x)_{n}=x(x-1)(x-2)\cdots(x-n+1).\,
  9. M X ( t ) = k = 0 t k P ( X = k ) = λ k e - λ / k ! = e - λ k = 0 ( t λ ) k k ! = e λ ( t - 1 ) , t , M_{X}(t)=\sum_{k=0}^{\infty}t^{k}\underbrace{\operatorname{P}(X=k)}_{=\,% \lambda^{k}e^{-\lambda}/k!}=e^{-\lambda}\sum_{k=0}^{\infty}\frac{(t\lambda)^{k% }}{k!}=e^{\lambda(t-1)},\qquad t\in\mathbb{C},
  10. E [ ( X ) n ] = λ n . \operatorname{E}[(X)_{n}]=\lambda^{n}.

Factorization_lemma.html

  1. T : Ω Ω T:\Omega\rightarrow\Omega^{\prime}
  2. Ω \Omega
  3. ( Ω , 𝒜 ) (\Omega^{\prime},\mathcal{A}^{\prime})
  4. f : Ω ¯ f:\Omega\rightarrow\overline{\mathbb{R}}
  5. Ω \Omega
  6. f f
  7. σ ( T ) = T - 1 ( 𝒜 ) \sigma(T)=T^{-1}(\mathcal{A}^{\prime})
  8. T T
  9. Ω \Omega
  10. g : ( Ω , 𝒜 ) ( ¯ , ( ¯ ) ) g:(\Omega^{\prime},\mathcal{A}^{\prime})\rightarrow(\overline{\mathbb{R}},% \mathcal{B}(\overline{\mathbb{R}}))
  11. f = g T f=g\circ T
  12. ( ¯ ) \mathcal{B}(\overline{\mathbb{R}})
  13. f f
  14. g g
  15. f = g T f=g\circ T
  16. σ ( T ) - ( ¯ ) \sigma(T)-\mathcal{B}(\overline{\mathbb{R}})
  17. σ ( T ) - 𝒜 \sigma(T)-\mathcal{A}^{\prime}
  18. 𝒜 - ( ¯ ) \mathcal{A}^{\prime}-\mathcal{B}(\overline{\mathbb{R}})
  19. f = i = 1 n α i 1 A i f=\sum_{i=1}^{n}\alpha_{i}1_{A_{i}}
  20. n * , i [ [ 1 , n ] ] , A i σ ( T ) n\in\mathbb{N}^{*},\forall i\in[\![1,n]\!],A_{i}\in\sigma(T)
  21. α i + \alpha_{i}\in\mathbb{R}^{+}
  22. A i 𝒜 A_{i}^{\prime}\in\mathcal{A}^{\prime}
  23. A i = T - 1 ( A i ) A_{i}=T^{-1}(A_{i}^{\prime})
  24. g = i = 1 n α i 1 A i g=\sum_{i=1}^{n}\alpha_{i}1_{A_{i}^{\prime}}
  25. ( u n ) n (u_{n})_{n\in\mathbb{N}}
  26. g n g_{n}
  27. u n = g n T u_{n}=g_{n}\circ T
  28. lim n + g n \lim_{n\rightarrow+\infty}g_{n}
  29. f + f^{+}
  30. f - f^{-}
  31. g 0 + g_{0}^{+}
  32. g 0 - g_{0}^{-}
  33. f + = g 0 + T f^{+}=g_{0}^{+}\circ T
  34. f - = g 0 - T f^{-}=g_{0}^{-}\circ T
  35. g := g + - g - g:=g^{+}-g^{-}
  36. U = { x : g 0 + ( x ) = + } { x : g 0 - ( x ) = + } U=\{x:g_{0}^{+}(x)=+\infty\}\cap\{x:g_{0}^{-}(x)=+\infty\}
  37. T ( Ω ) U = T(\Omega)\cap U=\varnothing
  38. g 0 + ( T ( ω ) ) = f + ( ω ) = + g_{0}^{+}(T(\omega))=f^{+}(\omega)=+\infty
  39. g 0 - ( T ( ω ) ) = f - ( ω ) = 0 g_{0}^{-}(T(\omega))=f^{-}(\omega)=0
  40. g + = 1 Ω \ U g 0 + g^{+}=1_{\Omega^{\prime}\backslash U}g_{0}^{+}
  41. g - = 1 Ω \ U g 0 - g^{-}=1_{\Omega^{\prime}\backslash U}g_{0}^{-}
  42. g = g + - g - g=g^{+}-g^{-}
  43. U = { ω : | g ( ω ) | = + } U^{\prime}=\{\omega:|g(\omega)|=+\infty\}
  44. g 0 = 1 Ω \ U g g_{0}=1_{\Omega^{\prime}\backslash U^{\prime}}g
  45. U T ( Ω ) = U^{\prime}\cap T(\Omega)=\varnothing
  46. f f
  47. \mathbb{R}
  48. ( ) \mathcal{B}(\mathbb{R})
  49. f f

Factorization_of_polynomials.html

  1. - 10 x 2 + 5 x + 5 = ( - 5 ) ( 2 x 2 - x - 1 ) -10x^{2}+5x+5=(-5)\cdot(2x^{2}-x-1)\,
  2. q = p c , q=\frac{p}{c},
  3. cont ( q ) = cont ( p ) c , \,\text{cont}(q)=\frac{\,\text{cont}(p)}{c},
  4. 1 3 x 5 + 7 2 x 2 + 2 x + 1 = 1 6 ( 2 x 5 + 21 x 2 + 12 x + 6 ) \frac{1}{3}x^{5}+\frac{7}{2}x^{2}+2x+1=\frac{1}{6}(2x^{5}+21x^{2}+12x+6)
  5. a n x n + a n - 1 x n - 1 + + a 1 x + a 0 a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}
  6. b 1 x - b 0 b_{1}x-b_{0}
  7. b 1 b_{1}
  8. a n a_{n}
  9. b 0 b_{0}
  10. a 0 a_{0}
  11. f ( x ) = x 5 + x 4 + x 2 + x + 2 f(x)=x^{5}+x^{4}+x^{2}+x+2
  12. f ( 0 ) = 2 f(0)=2
  13. f ( 1 ) = 6 f(1)=6
  14. f ( - 1 ) = 2 f(-1)=2
  15. x = 0 x=0
  16. x = - 1 x=-1
  17. f ( x ) f(x)
  18. p ( x ) = x 2 + x + 1 p(x)=x^{2}+x+1
  19. p ( 0 ) = 1 p(0)=1
  20. p ( 1 ) = 3 p(1)=3
  21. p ( - 1 ) = 1 p(-1)=1
  22. f ( x ) f(x)
  23. f f
  24. p p
  25. q ( x ) = x 3 - x + 2 q(x)=x^{3}-x+2
  26. f = p q f=pq
  27. p p
  28. q q
  29. f f
  30. f ( x ) = p ( x ) q ( x ) = ( x 2 + x + 1 ) ( x 3 - x + 2 ) f(x)=p(x)q(x)=(x^{2}+x+1)(x^{3}-x+2)
  31. f ( x ) f(x)
  32. B B
  33. g ( x ) g(x)
  34. B B
  35. m m
  36. 2 B 2B
  37. g ( x ) g(x)
  38. m m
  39. g ( x ) g(x)
  40. m m
  41. p p
  42. f ( x ) f(x)
  43. p p
  44. f ( x ) f(x)
  45. f ( x ) f(x)
  46. p p
  47. f 1 ( x ) , , f r ( x ) f_{1}(x),...,f_{r}(x)
  48. f ( x ) f(x)
  49. p p
  50. f i ( x ) f_{i}(x)
  51. f ( x ) f(x)
  52. p a p^{a}
  53. a a
  54. p a p^{a}
  55. 2 B 2B
  56. p a p^{a}
  57. f ( x ) f(x)
  58. 2 r 2^{r}
  59. f 1 ( x ) , , f r ( x ) {f_{1}(x),...,f_{r}(x)}
  60. f ( x ) f(x)
  61. p a p^{a}
  62. p a p^{a}
  63. f ( x ) f(x)
  64. Z [ x ] Z[x]
  65. p a p^{a}
  66. p a p^{a}
  67. 2 B 2B
  68. 2 r 2^{r}
  69. 2 r - 1 2^{r-1}
  70. f ( x ) f(x)
  71. f i ( x ) f_{i}(x)
  72. f ( x ) f(x)
  73. f ( x ) f(x)
  74. f 1 ( x ) , , f r ( x ) f_{1}(x),...,f_{r}(x)
  75. r r
  76. f 1 ( x ) , , f r ( x ) f_{1}(x),...,f_{r}(x)
  77. p ( x ) K [ x ] p(x)\in K[x]
  78. K K
  79. \mathbb{Q}
  80. L = K [ x ] / p ( x ) L=K[x]/p(x)
  81. \mathbb{Q}
  82. α L \alpha\in L
  83. α \alpha
  84. L L
  85. \mathbb{Q}
  86. q ( y ) [ y ] q(y)\in\mathbb{Q}[y]
  87. α \alpha
  88. \mathbb{Q}
  89. q ( y ) = i = 1 n q i ( y ) q(y)=\prod_{i=1}^{n}q_{i}(y)
  90. [ y ] \mathbb{Q}[y]
  91. L = [ α ] = [ y ] / q ( y ) = i = 1 n [ y ] / q i ( y ) L=\mathbb{Q}[\alpha]=\mathbb{Q}[y]/q(y)=\prod_{i=1}^{n}\mathbb{Q}[y]/q_{i}(y)
  92. L L
  93. p ( x ) p(x)
  94. α \alpha
  95. ( y , y , , y ) (y,y,\ldots,y)
  96. L L
  97. i = 1 m K [ x ] / p i ( x ) \prod_{i=1}^{m}K[x]/p_{i}(x)
  98. p ( x ) = i = 1 m p i ( x ) p(x)=\prod_{i=1}^{m}p_{i}(x)
  99. p ( x ) p(x)
  100. K [ x ] K[x]
  101. x L x\in L
  102. K K
  103. α \alpha
  104. x x
  105. K K
  106. [ y ] / q i ( y ) = K [ x ] / p i ( x ) \mathbb{Q}[y]/q_{i}(y)=K[x]/p_{i}(x)
  107. x x
  108. p i ( x ) p_{i}(x)
  109. p ( x ) p(x)
  110. K . K.

Failure_mode,_effects,_and_criticality_analysis.html

  1. C m C_{m}
  2. C r C_{r}
  3. λ p \lambda_{p}
  4. α \alpha
  5. β \beta
  6. t t
  7. C m = λ p α β t C_{m}=\lambda_{p}\alpha\beta t
  8. C r = n = 1 N ( C m ) n C_{r}=\sum_{n=1}^{N}(C_{m})_{n}
  9. λ p \lambda_{p}
  10. β \beta
  11. C m C_{m}
  12. C r C_{r}

Fair_queuing.html

  1. \infty

Fairness_measure.html

  1. 𝒥 ( x 1 , x 2 , , x n ) = ( i = 1 n x i ) 2 n i = 1 n x i 2 \mathcal{J}(x_{1},x_{2},\dots,x_{n})=\frac{(\sum_{i=1}^{n}x_{i})^{2}}{n\cdot% \sum_{i=1}^{n}{x_{i}}^{2}}
  2. n n
  3. x i x_{i}
  4. i i
  5. 1 n \tfrac{1}{n}
  6. k n \tfrac{k}{n}
  7. k k
  8. n - k n-k

False_discovery_rate.html

  1. m 0 m_{0}
  2. m 0 m_{0}
  3. ( E [ V ] R ) \left(\frac{E[V]}{R}\right)
  4. m m
  5. m m
  6. V V
  7. S S
  8. R R
  9. U U
  10. T T
  11. m - R m-R
  12. m 0 m_{0}
  13. m - m 0 m-m_{0}
  14. m m
  15. m m
  16. m 0 m_{0}
  17. m - m 0 m-m_{0}
  18. V V
  19. S S
  20. T T
  21. U U
  22. R R
  23. m m
  24. m 0 m_{0}
  25. R R
  26. S S
  27. T T
  28. U U
  29. V V
  30. Q Q
  31. ( Q = V R ) \left(Q=\frac{V}{R}\right)
  32. F D R = Q e = E [ Q ] = E [ V V + S ] = E [ V R ] , FDR=Q_{e}=\mathrm{E}\!\left[Q\right]=\mathrm{E}\!\left[\frac{V}{V+S}\right]=% \mathrm{E}\!\left[\frac{V}{R}\right],
  33. V R \frac{V}{R}
  34. R = 0 R=0
  35. q = 5 % q=5\%
  36. q = 5 % q=5\%
  37. m m
  38. F D R m 0 m q FDR\leq\frac{m_{0}}{m}q
  39. F D R = m 0 m q FDR=\frac{m_{0}}{m}q
  40. F D R m 0 m q FDR\leq\frac{m_{0}}{m}q
  41. F D R m 0 m q ( 1 + 1 2 + 1 3 + + 1 m ) m 0 m q log ( m ) FDR\leq\frac{m_{0}}{m}q\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{m}% \right)\approx\frac{m_{0}}{m}q\log(m)
  42. m 0 = m m_{0}=m
  43. F W E R = P ( V 1 ) = E ( V R ) = F D R q FWER=P\left(V\geq 1\right)=E\left(\frac{V}{R}\right)=FDR\leq q
  44. m 0 < m m_{0}<m
  45. F W E R F D R FWER\geq FDR
  46. H 1 H m H_{1}\ldots H_{m}
  47. P 1 P m P_{1}\ldots P_{m}
  48. P ( 1 ) P ( m ) P_{(1)}\ldots P_{(m)}
  49. α \alpha
  50. α \alpha
  51. k k
  52. P ( k ) k m α . P_{(k)}\leq\frac{k}{m}\alpha.
  53. H ( i ) H_{(i)}
  54. i = 1 , , k i=1,\ldots,k
  55. m m
  56. E ( Q ) m 0 m α α E(Q)\leq\frac{m_{0}}{m}\alpha\leq\alpha
  57. m 0 m_{0}
  58. α \alpha
  59. m m
  60. α ( m + 1 ) 2 m \frac{\alpha(m+1)}{2m}
  61. α \alpha
  62. α \alpha
  63. m m
  64. k k
  65. P ( k ) k m c ( m ) α P_{(k)}\leq\frac{k}{m\cdot c(m)}\alpha
  66. c ( m ) = 1 c(m)=1
  67. c ( m ) = i = 1 m 1 i c(m)=\sum_{i=1}^{m}\frac{1}{i}
  68. c ( m ) c(m)
  69. i = 1 m 1 i ln ( m ) + γ . \sum_{i=1}^{m}\frac{1}{i}\approx\ln(m)+\gamma.
  70. α \alpha
  71. m m
  72. / c ( m ) /c(m)
  73. π 0 \pi_{0}
  74. π 1 = 1 - π 0 \pi_{1}=1-\pi_{0}
  75. N π 0 N\pi_{0}
  76. 1 - α 1-\alpha
  77. P C E R PCER
  78. P C E R = E [ V m ] PCER=E\left[\frac{V}{m}\right]
  79. α \alpha
  80. P C E R α PCER\leq\alpha
  81. F W E R FWER
  82. F W E R = P all ( V 1 ) FWER=P\text{all}(V\geq 1)
  83. F W E R FWER
  84. F W E R = P any ( V 1 ) FWER=P\text{any}(V\geq 1)
  85. k -FWER k\,\text{-FWER}
  86. k - F W E R = P ( V k ) q k-FWER=P(V\geq k)\leq q
  87. k -FDR k\,\text{-FDR}
  88. k -FDR = E ( V R I ( V > k ) ) q k\,\text{-FDR}=E\left(\frac{V}{R}I_{(V>k)}\right)\leq q
  89. Q Q^{\prime}
  90. Q = E [ V ] R Q^{\prime}=\frac{E[V]}{R}
  91. m 0 = m m_{0}=m
  92. F D R - 1 FDR_{-1}
  93. F D R - 1 = F d r = E [ V ] E [ R ] FDR_{-1}=Fdr=\frac{E[V]}{E[R]}
  94. F D R + 1 FDR_{+1}
  95. F D R + 1 = p F D R = E [ V R | R > 0 ] FDR_{+1}=pFDR=E\left[\left.{\frac{V}{R}}\right|R>0\right]
  96. P ( V R > q ) P\left(\frac{V}{R}>q\right)
  97. W -FDR W\,\text{-FDR}
  98. w i 0 w_{i}\geq 0
  99. W -FDR = E ( w i V i w i R i ) W\,\text{-FDR}=E\left(\frac{\sum w_{i}V_{i}}{\sum w_{i}R_{i}}\right)
  100. F D C R FDCR
  101. c i c_{i}
  102. H 00 H_{00}
  103. c 0 c_{0}
  104. F D C R = E ( c 0 V 0 + c i V i c 0 R 0 + c i R i ) FDCR=E\left(c_{0}V_{0}+\frac{\sum c_{i}V_{i}}{c_{0}R_{0}+\sum c_{i}R_{i}}\right)
  105. P F E R PFER
  106. α \alpha
  107. P F E R = E ( V ) α PFER=E(V)\leq\alpha
  108. F N R FNR
  109. F N R = E ( T m - R ) = E ( m - m 0 - ( R - V ) m - R ) FNR=E\left(\frac{T}{m-R}\right)=E\left(\frac{m-m_{0}-(R-V)}{m-R}\right)
  110. F d r ( z ) Fdr(z)
  111. F d r ( z ) = p 0 F 0 ( z ) F ( z ) Fdr(z)=\frac{p_{0}F_{0}(z)}{F(z)}
  112. f d r fdr
  113. f d r = p 0 f 0 ( z ) f ( z ) fdr=\frac{p_{0}f_{0}(z)}{f(z)}

Familywise_error_rate.html

  1. V V
  2. S S
  3. R R
  4. U U
  5. T T
  6. m - R m-R
  7. m 0 m_{0}
  8. m - m 0 m-m_{0}
  9. m m
  10. m 0 m_{0}
  11. m - m 0 m-m_{0}
  12. V V
  13. S S
  14. T T
  15. U U
  16. R R
  17. R R
  18. S S
  19. T T
  20. U U
  21. V V
  22. FWER = Pr ( V 1 ) , \mathrm{FWER}=\Pr(V\geq 1),\,
  23. FWER = 1 - Pr ( V = 0 ) . \mathrm{FWER}=1-\Pr(V=0).
  24. FWER α \mathrm{FWER}\leq\alpha\,\!\,
  25. α \alpha\,\!
  26. α \alpha\,\!
  27. m 0 m_{0}
  28. m m
  29. α \alpha\,\!
  30. α \alpha
  31. p i p_{i}
  32. H i H_{i}
  33. H i H_{i}
  34. p i α m p_{i}\leq\frac{\alpha}{m}
  35. α S I D = 1 - ( 1 - α ) 1 m \alpha_{SID}=1-(1-\alpha)^{\frac{1}{m}}
  36. Y A - Y B S E \frac{Y_{A}-Y_{B}}{SE}
  37. Y A Y_{A}
  38. Y B Y_{B}
  39. S E SE
  40. α \alpha
  41. P ( 1 ) P ( m ) P_{(1)}\ldots P_{(m)}
  42. H ( 1 ) H ( m ) H_{(1)}\ldots H_{(m)}
  43. R R
  44. k k
  45. P ( k ) > α m + 1 - k P_{(k)}>\frac{\alpha}{m+1-k}
  46. H ( 1 ) H ( R - 1 ) H_{(1)}\ldots H_{(R-1)}
  47. R = 1 R=1
  48. P ( 1 ) P ( m ) P_{(1)}\ldots P_{(m)}
  49. H ( 1 ) H ( m ) H_{(1)}\ldots H_{(m)}
  50. α \alpha
  51. R R
  52. k k
  53. P ( k ) α m + 1 - k P_{(k)}\leq\frac{\alpha}{m+1-k}
  54. H ( 1 ) H ( R ) H_{(1)}\ldots H_{(R)}
  55. α \alpha
  56. α / m \alpha/m
  57. α \alpha

Fano's_inequality.html

  1. P ( x , y ) P(x,y)
  2. X X ~ X\neq\tilde{X}
  3. X ~ = f ( Y ) \tilde{X}=f(Y)
  4. X X
  5. H ( X | Y ) H ( e ) + P ( e ) log ( | 𝒳 | - 1 ) , H(X|Y)\leq H(e)+P(e)\log(|\mathcal{X}|-1),
  6. 𝒳 \mathcal{X}
  7. H ( X | Y ) = - i , j P ( x i , y j ) log P ( x i | y j ) H\left(X|Y\right)=-\sum_{i,j}P(x_{i},y_{j})\log P\left(x_{i}|y_{j}\right)
  8. P ( e ) = P ( X X ~ ) P(e)=P(X\neq\tilde{X})
  9. H ( e ) = - P ( e ) log P ( e ) - ( 1 - P ( e ) ) log ( 1 - P ( e ) ) H(e)=-P(e)\log P(e)-(1-P(e))\log(1-P(e))
  10. r + 1 r+1
  11. f 1 , , f r + 1 f_{1},\ldots,f_{r+1}
  12. D K L ( f i f j ) β D_{KL}(f_{i}\|f_{j})\leq\beta
  13. i j . i\not=j.
  14. ψ ( X ) { 1 , , r + 1 } \psi(X)\in\{1,\ldots,r+1\}
  15. sup i P i ( ψ ( X ) i ) 1 - β + log 2 log r \sup_{i}P_{i}(\psi(X)\not=i)\geq 1-\frac{\beta+\log 2}{\log r}
  16. P i P_{i}
  17. f i f_{i}
  18. f θ - f θ L 1 α , \|f_{\theta}-f_{\theta^{\prime}}\|_{L_{1}}\geq\alpha,\,
  19. D K L ( f θ f θ ) β . D_{KL}(f_{\theta}\|f_{\theta^{\prime}})\leq\beta.\,
  20. sup f 𝐅 E f n - f L 1 α 2 ( 1 - n β + log 2 log r ) \sup_{f\in\mathbf{F}}E\|f_{n}-f\|_{L_{1}}\geq\frac{\alpha}{2}\left(1-\frac{n% \beta+\log 2}{\log r}\right)

Fano_factor.html

  1. F = σ W 2 μ W , F=\frac{\sigma_{W}^{2}}{\mu_{W}},
  2. σ W 2 \sigma_{W}^{2}
  3. μ W \mu_{W}
  4. R = F W H M μ = 2.35 F w E , R=\frac{FWHM}{\mu}=2.35\sqrt{\frac{Fw}{E}},

Fat-tailed_distribution.html

  1. Pr [ X > x ] x - α as x , α > 0. \Pr[X>x]\sim x^{-\alpha}\,\text{ as }x\to\infty,\qquad\alpha>0.\,
  2. f X ( x ) f_{X}(x)
  3. f X ( x ) x - ( 1 + α ) as x , α > 0. f_{X}(x)\sim x^{-(1+\alpha)}\,\text{ as }x\to\infty,\qquad\alpha>0.\,
  4. \sim

Fatou–Bieberbach_domain.html

  1. n \mathbb{C}^{n}
  2. n \mathbb{C}^{n}
  3. Ω n \Omega\subsetneq\mathbb{C}^{n}
  4. f : Ω n f:\Omega\rightarrow\mathbb{C}^{n}
  5. f - 1 : n Ω f^{-1}:\mathbb{C}^{n}\rightarrow\Omega
  6. 4 \mathcal{R}_{4}
  7. n \mathbb{C}^{n}
  8. n \mathbb{C}^{n}

Fatou–Lebesgue_theorem.html

  1. S lim inf n f n d μ lim inf n S f n d μ lim sup n S f n d μ S lim sup n f n d μ . \int_{S}\liminf_{n\to\infty}f_{n}\,d\mu\leq\liminf_{n\to\infty}\int_{S}f_{n}\,% d\mu\leq\limsup_{n\to\infty}\int_{S}f_{n}\,d\mu\leq\int_{S}\limsup_{n\to\infty% }f_{n}\,d\mu\,.
  2. 0 | S lim inf n f n d μ | S | lim inf n f n | d μ S lim sup n | f n | d μ S g d μ 0\leq\biggl|\int_{S}\liminf_{n\to\infty}f_{n}\,d\mu\biggr|\leq\int_{S}\Bigl|% \liminf_{n\to\infty}f_{n}\Bigr|\,d\mu\leq\int_{S}\limsup_{n\to\infty}|f_{n}|\,% d\mu\leq\int_{S}g\,d\mu

Feasibility_condition.html

  1. x p x + y p y I xp_{x}+yp_{y}\leq I

Fekete_polynomial.html

  1. f p ( t ) := a = 0 p - 1 ( a p ) t a f_{p}(t):=\sum_{a=0}^{p-1}\left(\frac{a}{p}\right)t^{a}\,
  2. ( p ) \left(\frac{\cdot}{p}\right)\,

Fermat's_theorem_(stationary_points).html

  1. f \displaystyle f
  2. f \displaystyle f^{\prime}
  3. f \displaystyle f^{\prime}
  4. f : ( a , b ) f\colon(a,b)\rightarrow\mathbb{R}
  5. x 0 ( a , b ) \displaystyle x_{0}\in(a,b)
  6. f \displaystyle f
  7. f \displaystyle f
  8. x 0 \displaystyle x_{0}
  9. f ( x 0 ) = 0 \displaystyle f^{\prime}(x_{0})=0
  10. f \displaystyle f
  11. x 0 ( a , b ) \displaystyle x_{0}\in(a,b)
  12. f ( x 0 ) 0 \displaystyle f^{\prime}(x_{0})\neq 0
  13. x 0 x_{0}
  14. x 0 x_{0}
  15. x 0 x_{0}
  16. x 0 x_{0}
  17. x 0 x_{0}
  18. a + b x , a+bx,
  19. f ( x 0 ) + f ( x 0 ) ( x - x 0 ) . f(x_{0})+f^{\prime}(x_{0})\cdot(x-x_{0}).
  20. x 0 , x_{0},
  21. x 0 , x_{0},
  22. x 0 x_{0}
  23. x 0 , x_{0},
  24. x 0 . x_{0}.
  25. x 0 x_{0}
  26. x 0 x_{0}
  27. x 0 . x_{0}.
  28. x 0 ( a , b ) \displaystyle x_{0}\in(a,b)
  29. x 0 \displaystyle x_{0}
  30. x 0 \displaystyle x_{0}
  31. f \displaystyle f^{\prime}
  32. x 0 \displaystyle x_{0}
  33. f \displaystyle f^{\prime}
  34. f ( x ) = 0 \displaystyle f^{\prime}(x)=0
  35. x 0 \displaystyle x_{0}
  36. x 0 \displaystyle x_{0}
  37. x 0 ( a , b ) , x_{0}\in(a,b),
  38. K > 0 , K>0,
  39. x 0 x_{0}
  40. x 0 x_{0}
  41. x 0 x_{0}
  42. x 0 , x_{0},
  43. x 0 , x_{0},
  44. x 0 x_{0}
  45. x 0 , x_{0},
  46. x 0 . x_{0}.
  47. f ( x 0 ) = K f^{\prime}(x_{0})=K
  48. lim ϵ 0 f ( x 0 + ϵ ) - f ( x 0 ) ϵ = K . \lim_{\epsilon\to 0}\frac{f(x_{0}+\epsilon)-f(x_{0})}{\epsilon}=K.
  49. ϵ \epsilon
  50. ϵ 0 \epsilon_{0}
  51. K / 2 , K/2,
  52. ( x 0 - ϵ 0 , x 0 + ϵ 0 ) (x_{0}-\epsilon_{0},x_{0}+\epsilon_{0})
  53. f ( x 0 + ϵ ) - f ( x 0 ) ϵ > K / 2 ; \frac{f(x_{0}+\epsilon)-f(x_{0})}{\epsilon}>K/2;
  54. ϵ > 0 , \epsilon>0,
  55. f ( x 0 + ϵ ) > f ( x 0 ) + ( K / 2 ) ϵ > f ( x 0 ) , f(x_{0}+\epsilon)>f(x_{0})+(K/2)\epsilon>f(x_{0}),\,
  56. f ( x 0 ) , f(x_{0}),
  57. ϵ < 0 , \epsilon<0,
  58. f ( x 0 + ϵ ) < f ( x 0 ) + ( K / 2 ) ϵ < f ( x 0 ) , f(x_{0}+\epsilon)<f(x_{0})+(K/2)\epsilon<f(x_{0}),\,
  59. f ( x 0 ) . f(x_{0}).
  60. x 0 x_{0}
  61. x 0 \displaystyle x_{0}
  62. x 0 \displaystyle x_{0}
  63. x 0 \displaystyle x_{0}
  64. δ > 0 \exists\,\delta>0
  65. ( x 0 - δ , x 0 + δ ) ( a , b ) (x_{0}-\delta,x_{0}+\delta)\subset(a,b)
  66. f ( x 0 ) f ( x ) x f(x_{0})\geq f(x)\,\forall x
  67. | x - x 0 | < δ \displaystyle|x-x_{0}|<\delta
  68. h ( 0 , δ ) h\in(0,\delta)
  69. f ( x 0 + h ) - f ( x 0 ) h 0. \frac{f(x_{0}+h)-f(x_{0})}{h}\leq 0.
  70. h \displaystyle h
  71. f ( x 0 ) \displaystyle f^{\prime}(x_{0})
  72. f ( x 0 ) 0 f^{\prime}(x_{0})\leq 0
  73. h ( - δ , 0 ) h\in(-\delta,0)
  74. f ( x 0 + h ) - f ( x 0 ) h 0 \frac{f(x_{0}+h)-f(x_{0})}{h}\geq 0
  75. h \displaystyle h
  76. f ( x 0 ) \displaystyle f^{\prime}(x_{0})
  77. f ( x 0 ) 0 f^{\prime}(x_{0})\geq 0
  78. f ( x 0 ) = 0. \displaystyle f^{\prime}(x_{0})=0.
  79. C 1 C^{1}
  80. x 0 , x_{0},
  81. f ( x 0 ) > 0 f^{\prime}(x_{0})>0
  82. x 0 , x_{0},
  83. f ( x 0 ) = K > 0 f^{\prime}(x_{0})=K>0
  84. f C 1 , f\in C^{1},
  85. ( x 0 - ϵ 0 , x 0 + ϵ 0 ) (x_{0}-\epsilon_{0},x_{0}+\epsilon_{0})
  86. x 0 x_{0}
  87. f ( x 0 ) > K / 2. f^{\prime}(x_{0})>K/2.
  88. K / 2 , K/2,
  89. x 0 x_{0}
  90. x 0 x_{0}
  91. x 0 x_{0}
  92. x 0 x_{0}
  93. x 3 x^{3}
  94. x 4 x^{4}
  95. - x 4 -x^{4}
  96. x 2 ( sin ( 1 / x ) ) x^{2}(\sin(1/x))
  97. x 0 x_{0}
  98. f ( k ) ( x 0 ) 0 f^{(k)}(x_{0})\neq 0
  99. f ( k ) f^{(k)}
  100. f C k f\in C^{k}
  101. f ( k ) ( x 0 ) ( x - x 0 ) k , f^{(k)}(x_{0})(x-x_{0})^{k},
  102. sin ( 1 / x ) \sin(1/x)
  103. - 1 -1
  104. 1 1
  105. f ( x ) = ( 1 + sin ( 1 / x ) ) x 2 f(x)=(1+\sin(1/x))x^{2}
  106. 2 x 2 2x^{2}
  107. f ( 0 ) := 0 , f(0):=0,
  108. 2 x 2 2x^{2}
  109. f ( x ) = ( 2 + sin ( 1 / x ) ) x 2 f(x)=(2+\sin(1/x))x^{2}
  110. x 2 x^{2}
  111. 3 x 2 , 3x^{2},
  112. x = 0 x=0
  113. f ( x ) f^{\prime}(x)
  114. x 0 x\to 0
  115. C 1 C^{1}

Fermat_(computer_algebra_system).html

  1. F F
  2. \mathbb{Z}
  3. / n \mathbb{Z}/n
  4. t 1 , t 2 , , t n , t_{1},t_{2},\dots,t_{n},
  5. F [ t 1 , t 2 , , t n ] F[t_{1},t_{2},\dots,t_{n}]
  6. p , q , p,q,\dots
  7. t i t_{i}
  8. F ( t 1 , t 2 , ) / ( p , q , ) . F(t_{1},t_{2},\dots)/(p,q,\dots).
  9. p ( t 1 ) , p(t_{1}),
  10. q ( t 2 , t 1 ) , q(t_{2},t_{1}),
  11. G F ( 2 8 ) GF(2^{8})
  12. G F ( 2 16 ) , GF(2^{16}),

Fermat_cubic.html

  1. x 3 + y 3 + z 3 = 1. x^{3}+y^{3}+z^{3}=1.
  2. x ( s , t ) = 3 t - 1 3 ( s 2 + s t + t 2 ) 2 t ( s 2 + s t + t 2 ) - 3 x(s,t)={3t-{1\over 3}(s^{2}+st+t^{2})^{2}\over t(s^{2}+st+t^{2})-3}
  3. y ( s , t ) = 3 s + 3 t + 1 3 ( s 2 + s t + t 2 ) 2 t ( s 2 + s t + t 2 ) - 3 y(s,t)={3s+3t+{1\over 3}(s^{2}+st+t^{2})^{2}\over t(s^{2}+st+t^{2})-3}
  4. z ( s , t ) = - 3 - ( s 2 + s t + t 2 ) ( s + t ) t ( s 2 + s t + t 2 ) - 3 . z(s,t)={-3-(s^{2}+st+t^{2})(s+t)\over t(s^{2}+st+t^{2})-3}.
  5. w 3 + x 3 + y 3 + z 3 = 0. w^{3}+x^{3}+y^{3}+z^{3}=0.

Fermi_heap_and_Fermi_hole.html

  1. Ψ = Ψ 0 Ψ s \Psi=\Psi_{0}\Psi_{s}
  2. Ψ \Psi
  3. χ + ( 1 ) χ + ( 2 ) \chi_{+}(1)\chi_{+}(2)
  4. χ - ( 1 ) χ - ( 2 ) \chi_{-}(1)\chi_{-}(2)
  5. χ + ( 1 ) χ - ( 2 ) \chi_{+}(1)\chi_{-}(2)
  6. χ - ( 1 ) χ + ( 2 ) \chi_{-}(1)\chi_{+}(2)
  7. χ 1 , 1 = χ + ( 1 ) χ + ( 2 ) \chi_{1,1}=\chi_{+}(1)\chi_{+}(2)
  8. χ 1 , - 1 = χ - ( 1 ) χ - ( 2 ) \chi_{1,-1}=\chi_{-}(1)\chi_{-}(2)
  9. χ 1 , 0 = 1 2 ( χ - ( 1 ) χ + ( 2 ) + χ + ( 1 ) χ - ( 2 ) ) \chi_{1,0}=\frac{1}{\sqrt{2}}(\chi_{-}(1)\chi_{+}(2)+\chi_{+}(1)\chi_{-}(2))
  10. χ 0 , 0 = 1 2 ( χ - ( 1 ) χ + ( 2 ) - χ + ( 1 ) χ - ( 2 ) ) \chi_{0,0}=\frac{1}{\sqrt{2}}(\chi_{-}(1)\chi_{+}(2)-\chi_{+}(1)\chi_{-}(2))
  11. Ψ 0 = ψ 1 s ( r 1 ) ψ 2 s ( r 2 ) - ψ 2 s ( r 1 ) ψ 1 s ( r 2 ) \Psi_{0}=\psi_{1s}(\vec{r}_{1})\psi_{2s}(\vec{r}_{2})-\psi_{2s}(\vec{r}_{1})% \psi_{1s}(\vec{r}_{2})
  12. Ψ 0 = ψ 1 s ( r 1 ) ψ 2 s ( r 2 ) + ψ 2 s ( r 1 ) ψ 1 s ( r 2 ) \Psi_{0}=\psi_{1s}(\vec{r}_{1})\psi_{2s}(\vec{r}_{2})+\psi_{2s}(\vec{r}_{1})% \psi_{1s}(\vec{r}_{2})
  13. r 1 r 2 \vec{r}_{1}\approx\vec{r}_{2}
  14. χ 0 , 0 \chi_{0,0}

Fermi–Walker_transport.html

  1. γ ( s ) \gamma(s)
  2. D F X d s = D X d s - ( X , D V d s ) V + ( X , V ) D V d s , \frac{D_{F}X}{ds}=\frac{DX}{ds}-(X,\frac{DV}{ds})V+(X,V)\frac{DV}{ds},
  3. D F X d s = 0 , \frac{D_{F}X}{ds}=0,
  4. D F a τ d s = 2 μ ( F τ λ - u τ u σ F σ λ ) a λ , \frac{D_{F}a^{\tau}}{ds}=2\mu(F^{\tau\lambda}-u^{\tau}u_{\sigma}F^{\sigma% \lambda})a_{\lambda},
  5. a τ a^{\tau}
  6. μ \mu
  7. u τ u^{\tau}
  8. a τ a τ = - u τ u τ = - 1 a^{\tau}a_{\tau}=-u^{\tau}u_{\tau}=-1
  9. u τ a τ = 0 u^{\tau}a_{\tau}=0
  10. F τ σ F^{\tau\sigma}
  11. v μ v^{\mu}

Ferroics.html

  1. T c T_{c}
  2. T c T_{c}
  3. T c T_{c}
  4. 10 24 10^{24}
  5. ( x , y , z ) \left(x,y,z\right)
  6. ( - x , - y , - z ) \left(-x,-y,-z\right)
  7. T c T_{c}

Ferromagnetic_resonance.html

  1. M \scriptstyle\vec{M}
  2. H \scriptstyle\vec{H}
  3. B B
  4. f = γ 2 π B ( B + μ 0 M ) f=\frac{\gamma}{2\pi}\sqrt{B(B+\mu_{0}M)}
  5. M M
  6. γ \gamma

Fibered_manifold.html

  1. π : E B \pi\colon E\to B\,
  2. y E y∈E
  3. T y π : T y E T π ( y ) B T_{y}\pi\colon T_{y}E\to T_{\pi(y)}B
  4. B B
  5. ( E , π , B ) (E,π,B)
  6. E E
  7. B B
  8. π : E B π:E→B
  9. π : E B π:E→B
  10. V E V⊂E
  11. π ( V ) B π(V)⊂B
  12. B B
  13. E E
  14. d i m E d i m B dimE−dimB
  15. y E y∈E
  16. U U
  17. π ( y ) π(y)
  18. B B
  19. s : U E s:U→E
  20. s ( π ( y ) ) = y s(π(y))=y
  21. B B
  22. E E
  23. n n
  24. p p
  25. ( E , π , B ) (E,π,B)
  26. ( V , ψ ) (V,ψ)
  27. E E
  28. π : E B π:E→B
  29. ( U , φ ) (U,φ)
  30. B B
  31. U = π ( V ) U=π(V)
  32. u 1 = x 1 π , u 2 = x 2 π , , u n = x n π , u^{1}=x^{1}\circ\pi,\,u^{2}=x^{2}\circ\pi,\,\dots,\,u^{n}=x^{n}\circ\pi\,,
  33. ψ = ( u 1 , , u n , y 1 , , y p - n ) . y 0 V , φ = ( x 1 , , x n ) , π ( y 0 ) U . \begin{aligned}\displaystyle\psi&\displaystyle=(u^{1},\dots,u^{n},y^{1},\dots,% y^{p-n}).\quad y_{0}\in V,\\ \displaystyle\varphi&\displaystyle=(x^{1},\dots,x^{n}),\quad\pi(y_{0})\in U.% \end{aligned}
  34. φ π = π 1 ψ , \varphi\circ\pi={\mathrm{\pi}_{1}}\circ\psi,
  35. pr 1 : n × p - n n {\mathrm{pr}_{1}}\colon{\mathbb{R}^{n}}\times{\mathbb{R}^{p-n}}\to{\mathbb{R}^% {n}}\,
  36. ( U , φ ) (U,φ)
  37. ( V , ψ ) (V,ψ)
  38. ψ = ( x < s u p > i , y σ ) ψ=(x<sup>i,y^{σ})

Field_line.html

  1. 1 / r 2 1/r^{2}
  2. 1 / r 1/r

Fierz_identity.html

  1. ( χ ¯ γ μ ψ ) ( ψ ¯ γ μ χ ) = ( χ ¯ χ ) ( ψ ¯ ψ ) - 1 2 ( χ ¯ γ μ χ ) ( ψ ¯ γ μ ψ ) - 1 2 ( χ ¯ γ μ γ 5 χ ) ( ψ ¯ γ μ γ 5 ψ ) - ( χ ¯ γ 5 χ ) ( ψ ¯ γ 5 ψ ) . \left(\bar{\chi}\gamma^{\mu}\psi\right)\left(\bar{\psi}\gamma_{\mu}\chi\right)% =\left(\bar{\chi}\chi\right)\left(\bar{\psi}\psi\right)-\frac{1}{2}\left(\bar{% \chi}\gamma^{\mu}\chi\right)\left(\bar{\psi}\gamma_{\mu}\psi\right)-\frac{1}{2% }\left(\bar{\chi}\gamma^{\mu}\gamma_{5}\chi\right)\left(\bar{\psi}\gamma_{\mu}% \gamma_{5}\psi\right)-\left(\bar{\chi}\gamma_{5}\chi\right)\left(\bar{\psi}% \gamma_{5}\psi\right).

Filament_propagation.html

  1. n = n 0 + n ¯ 2 I n={n_{0}+\bar{n}_{2}I}
  2. n 0 n_{0}
  3. n ¯ 2 \bar{n}_{2}
  4. I I
  5. Δ z \Delta z
  6. ϕ d i f f r a c t i o n = k Δ z 2 ρ 0 2 r 2 \phi_{diffraction}={k\Delta z\over 2\rho_{0}^{2}}r^{2}
  7. ϕ K e r r = 2 π n ¯ 2 I 0 Δ z λ e x p ( - 2 r 2 w 0 2 ) 2 π n ¯ 2 I 0 Δ z λ ( 1 - 2 r 2 w 0 2 ) \phi_{Kerr}={2\pi\bar{n}_{2}I_{0}\Delta z\over\lambda}exp({-2r^{2}\over w_{0}^% {2}})\approx{2\pi\bar{n}_{2}I_{0}\Delta z\over\lambda}(1-{2r^{2}\over w_{0}^{2% }})
  8. k = 2 π n 0 λ k={2\pi n_{0}\over\lambda}
  9. ρ 0 = π w 0 2 n 0 λ \rho_{0}={\pi w_{0}^{2}n_{0}\over\lambda}
  10. w 0 w_{0}
  11. r 2 r^{2}
  12. I 0 = w 0 2 4 ρ 0 2 n ¯ 2 I_{0}={w_{0}^{2}\over 4\rho_{0}^{2}\bar{n}_{2}}
  13. π w 0 2 2 \pi w_{0}^{2}\over 2
  14. P c = λ 2 8 π n 0 n ¯ 2 P_{c}={\lambda^{2}\over 8\pi n_{0}\bar{n}_{2}}
  15. n ¯ 2 ( c m 2 W ) = n 2 n 0 ϵ 0 c \bar{n}_{2}({cm^{2}\over W})=n_{2}n_{0}\epsilon_{0}c
  16. P c P_{c}

File:Fractal_sine.jpg.html

  1. k = 0 40 - 1 2 k + 1 sin ( 2 k + 1 ) x | sin ( 2 k + 1 ) x | \sum^{40}_{k=0}-\frac{1}{2k+1}\frac{\sin(2k+1)x}{\left|\sin(2k+1)x\right|}

File:Illuminated-arimass.png.html

  1. γ ρ ( h γ ) d s \int_{\gamma}\rho(h\circ\gamma)\,ds

File:Phase_Plots.svg.html

  1. h [ n ] = δ [ n - 3 ] + δ [ n - 2 ] + δ [ n - 1 ] + δ [ n ] h[n]=\delta[n-3]+\delta[n-2]+\delta[n-1]+\delta[n]
  2. h [ n ] = δ [ n - 3 ] + δ [ n - 2 ] - δ [ n - 1 ] - δ [ n ] h[n]=\delta[n-3]+\delta[n-2]-\delta[n-1]-\delta[n]
  3. h [ n ] = δ [ n - 1 ] + 2 δ [ n ] - 3 h [ n - 1 ] 4 h[n]={{\delta[n-1]+2\delta[n]-3h[n-1]}\over{4}}
  4. h [ n ] = 4 δ [ n - 3 ] + 3 δ [ n - 2 ] + 2 δ [ n - 1 ] + δ [ n ] h[n]=4\delta[n-3]+3\delta[n-2]+2\delta[n-1]+\delta[n]
  5. H ( e j ω ) H(e^{j\omega})
  6. ω \omega
  7. p i pi

File:Time_dilation_spacetime_diagram03.gif.html

  1. ( t 2 - x 2 ) = 3 \sqrt{(}t^{2}-x^{2})=3
  2. ( t 2 - x 2 ) = 3 \sqrt{(}t^{2}-x^{2})=3

File:Trigloop.JPG.html

  1. ( x ( θ ) , y ( θ ) ) = ( sin ( 5 θ ) , 2 sin ( 4 θ ) ) . (x(\theta)\,,y(\theta))=(\sin(5\theta)\,,2\sin(4\theta)).

File:Xml_text_editor.png.html

  1. a + b a+b

Filtered_category.html

  1. J J
  2. j j
  3. j j^{\prime}
  4. J J
  5. k k
  6. f : j k f:j\to k
  7. f : j k f^{\prime}:j^{\prime}\to k
  8. J J
  9. u , v : i j u,v:i\to j
  10. J J
  11. k k
  12. w : j k w:j\to k
  13. w u = w v wu=wv
  14. κ \kappa
  15. κ \kappa
  16. J J
  17. d : D J d:D\to J
  18. κ \kappa
  19. J J
  20. κ \kappa
  21. d d
  22. J J
  23. κ \kappa
  24. d d
  25. F : J C F:J\to C
  26. J J
  27. κ \kappa
  28. C C
  29. C o p S e t C^{op}\to Set
  30. C C
  31. I n d ( C ) Ind(C)
  32. C o p S e t C^{op}\to Set
  33. P r o ( C ) = I n d ( C o p ) o p Pro(C)=Ind(C^{op})^{op}
  34. C C
  35. C o p C^{op}
  36. J J
  37. J op J^{\mathrm{op}}
  38. j j
  39. j j^{\prime}
  40. J J
  41. k k
  42. f : k j f:k\to j
  43. f : k j f^{\prime}:k\to j^{\prime}
  44. J J
  45. u , v : j i u,v:j\to i
  46. J J
  47. k k
  48. w : k j w:k\to j
  49. u w = v w uw=vw
  50. F : J C F:J\to C
  51. J J

Finite_element_method_in_structural_mechanics.html

  1. External virtual work = V \deltasymbol ϵ T s y m b o l σ d V ( 1 ) \mbox{External virtual work}~{}=\int_{V}\deltasymbol{\epsilon}^{T}symbol{% \sigma}\,dV\qquad\mathrm{(1)}
  2. 𝐑 = 𝐊𝐫 + 𝐑 o ( 2 ) \mathbf{R}=\mathbf{Kr}+\mathbf{R}^{o}\qquad\qquad\qquad\mathrm{(2)}
  3. 𝐑 \mathbf{R}
  4. 𝐫 \mathbf{r}
  5. 𝐑 o \mathbf{R}^{o}
  6. 𝐊 \mathbf{K}
  7. 𝐤 e \mathbf{k}^{e}
  8. 𝐫 = 𝐊 - 1 ( 𝐑 - 𝐑 o ) ( 3 ) \mathbf{r}=\mathbf{K}^{-1}(\mathbf{R}-\mathbf{R}^{o})\qquad\qquad\qquad\mathrm% {(3)}
  9. ϵ = 𝐁𝐪 ( 4 ) \mathbf{\epsilon}=\mathbf{Bq}\qquad\qquad\qquad\qquad\mathrm{(4)}
  10. σ = 𝐄 ( ϵ - ϵ o ) + σ o = 𝐄 ( 𝐁𝐪 - ϵ o ) + σ o ( 5 ) \mathbf{\sigma}=\mathbf{E}(\mathbf{\epsilon}-\mathbf{\epsilon}^{o})+\mathbf{% \sigma}^{o}=\mathbf{E}(\mathbf{Bq}-\mathbf{\epsilon}^{o})+\mathbf{\sigma}^{o}% \qquad\qquad\qquad\mathrm{(5)}
  11. 𝐪 \mathbf{q}
  12. 𝐁 \mathbf{B}
  13. 𝐄 \mathbf{E}
  14. ϵ o \mathbf{\epsilon}^{o}
  15. σ o \mathbf{\sigma}^{o}
  16. 𝐁 \mathbf{B}
  17. 𝐤 e \mathbf{k}^{e}
  18. 𝐑 o \mathbf{R}^{o}
  19. 𝐊 \mathbf{K}
  20. ϵ o \mathbf{\epsilon}^{o}
  21. σ o \mathbf{\sigma}^{o}
  22. 𝐑 \mathbf{R}
  23. 𝐄 \mathbf{E}
  24. 𝐪 \mathbf{q}
  25. 𝐮 = 𝐍𝐪 ( 6 ) \mathbf{u}=\mathbf{N}\mathbf{q}\qquad\qquad\qquad\mathrm{(6)}
  26. 𝐮 \mathbf{u}
  27. 𝐍 \mathbf{N}
  28. δ 𝐮 = 𝐍 δ 𝐪 ( 6 b ) \delta\mathbf{u}=\mathbf{N}\delta\mathbf{q}\qquad\qquad\qquad\mathrm{(6b)}
  29. ϵ = 𝐃𝐮 = 𝐃𝐍𝐪 ( 7 ) \mathbf{\epsilon}=\mathbf{Du}=\mathbf{DNq}\qquad\qquad\qquad\qquad\mathrm{(7)}
  30. 𝐃 \mathbf{D}
  31. 𝐁 = 𝐃𝐍 ( 8 ) \mathbf{B}=\mathbf{DN}\qquad\qquad\qquad\qquad\mathrm{(8)}
  32. δ s y m b o l ϵ = 𝐁 δ 𝐪 ( 9 ) \delta symbol{\epsilon}=\mathbf{B}\delta\mathbf{q}\qquad\qquad\qquad\qquad% \mathrm{(9)}
  33. V e V^{e}
  34. Internal virtual work = V e \deltasymbol ϵ T s y m b o l σ d V e = δ 𝐪 T V e 𝐁 T { 𝐄 ( 𝐁𝐪 - ϵ o ) + σ o } d V e ( 10 ) \mbox{Internal virtual work}~{}=\int_{V^{e}}\deltasymbol{\epsilon}^{T}symbol{% \sigma}\,dV^{e}=\delta\ \mathbf{q}^{T}\int_{V^{e}}\mathbf{B}^{T}\big\{\mathbf{% E}(\mathbf{Bq}-\mathbf{\epsilon}^{o})+\mathbf{\sigma}^{o}\big\}\,dV^{e}\qquad% \mathrm{(10)}
  35. 𝐤 e = V e 𝐁 T 𝐄𝐁 d V e ( 11 ) \mathbf{k}^{e}=\int_{V^{e}}\mathbf{B}^{T}\mathbf{E}\mathbf{B}\,dV^{e}\qquad% \mathrm{(11)}
  36. 𝐐 o e = V e - 𝐁 T ( 𝐄 ϵ o - σ o ) d V e ( 12 ) \mathbf{Q}^{oe}=\int_{V^{e}}-\mathbf{B}^{T}\big(\mathbf{E}\mathbf{\epsilon}^{o% }-\mathbf{\sigma}^{o}\big)\,dV^{e}\qquad\mathrm{(12)}
  37. Internal virtual work = δ 𝐪 T ( 𝐤 e 𝐪 + 𝐐 o e ) ( 13 ) \mbox{Internal virtual work}~{}=\delta\ \mathbf{q}^{T}\big(\mathbf{k}^{e}% \mathbf{q}+\mathbf{Q}^{oe}\big)\qquad\mathrm{(13)}
  38. Internal virtual work = δ 𝐫 T ( 𝐤 e 𝐫 + 𝐐 o e ) ( 14 ) \mbox{Internal virtual work}~{}=\delta\ \mathbf{r}^{T}\big(\mathbf{k}^{e}% \mathbf{r}+\mathbf{Q}^{oe}\big)\qquad\mathrm{(14)}
  39. System internal virtual work = e δ 𝐫 T ( 𝐤 e 𝐫 + 𝐐 o e ) = δ 𝐫 T ( e 𝐤 e ) 𝐫 + δ 𝐫 T e 𝐐 o e ( 15 ) \mbox{System internal virtual work}~{}=\sum_{e}\delta\ \mathbf{r}^{T}\big(% \mathbf{k}^{e}\mathbf{r}+\mathbf{Q}^{oe}\big)=\delta\ \mathbf{r}^{T}\big(\sum_% {e}\mathbf{k}^{e}\big)\mathbf{r}+\delta\ \mathbf{r}^{T}\sum_{e}\mathbf{Q}^{oe}% \qquad\mathrm{(15)}
  40. δ 𝐫 T 𝐑 ( 16 ) \delta\ \mathbf{r}^{T}\mathbf{R}\qquad\mathrm{(16)}
  41. 𝐓 e \mathbf{T}^{e}
  42. 𝐒 e \mathbf{S}^{e}
  43. 𝐟 e \mathbf{f}^{e}
  44. e S e δ 𝐮 T 𝐓 e d S e + e V e δ 𝐮 T 𝐟 e d V e \sum_{e}\int_{S^{e}}\delta\ \mathbf{u}^{T}\mathbf{T}^{e}\,dS^{e}+\sum_{e}\int_% {V^{e}}\delta\ \mathbf{u}^{T}\mathbf{f}^{e}\,dV^{e}
  45. δ 𝐪 T e S e 𝐍 T 𝐓 e d S e + δ 𝐪 T e V e 𝐍 T 𝐟 e d V e \delta\ \mathbf{q}^{T}\sum_{e}\int_{S^{e}}\mathbf{N}^{T}\mathbf{T}^{e}\,dS^{e}% +\delta\ \mathbf{q}^{T}\sum_{e}\int_{V^{e}}\mathbf{N}^{T}\mathbf{f}^{e}\,dV^{e}
  46. - δ 𝐪 T e ( 𝐐 t e + 𝐐 f e ) ( 17 a ) -\delta\ \mathbf{q}^{T}\sum_{e}(\mathbf{Q}^{te}+\mathbf{Q}^{fe})\qquad\mathrm{% (17a)}
  47. 𝐐 t e = - S e 𝐍 T 𝐓 e d S e ( 18 a ) \mathbf{Q}^{te}=-\int_{S^{e}}\mathbf{N}^{T}\mathbf{T}^{e}\,dS^{e}\qquad\mathrm% {(18a)}
  48. 𝐐 f e = - V e 𝐍 T 𝐟 e d V e ( 18 b ) \mathbf{Q}^{fe}=-\int_{V^{e}}\mathbf{N}^{T}\mathbf{f}^{e}\,dV^{e}\qquad\mathrm% {(18b)}
  49. 𝐐 t e , 𝐐 f e \mathbf{Q}^{te},\mathbf{Q}^{fe}
  50. - δ 𝐫 T e ( 𝐐 t e + 𝐐 f e ) ( 17 b ) -\delta\ \mathbf{r}^{T}\sum_{e}(\mathbf{Q}^{te}+\mathbf{Q}^{fe})\qquad\mathrm{% (17b)}
  51. δ 𝐫 T 𝐑 - δ 𝐫 T e ( 𝐐 t e + 𝐐 f e ) = δ 𝐫 T ( e 𝐤 e ) 𝐫 + δ 𝐫 T e 𝐐 o e \delta\ \mathbf{r}^{T}\mathbf{R}-\delta\ \mathbf{r}^{T}\sum_{e}(\mathbf{Q}^{te% }+\mathbf{Q}^{fe})=\delta\ \mathbf{r}^{T}\big(\sum_{e}\mathbf{k}^{e}\big)% \mathbf{r}+\delta\ \mathbf{r}^{T}\sum_{e}\mathbf{Q}^{oe}
  52. δ 𝐫 \delta\ \mathbf{r}
  53. 𝐑 = ( e 𝐤 e ) 𝐫 + e ( 𝐐 o e + 𝐐 t e + 𝐐 f e ) \mathbf{R}=\big(\sum_{e}\mathbf{k}^{e}\big)\mathbf{r}+\sum_{e}\big(\mathbf{Q}^% {oe}+\mathbf{Q}^{te}+\mathbf{Q}^{fe}\big)
  54. 𝐊 = e 𝐤 e \mathbf{K}=\sum_{e}\mathbf{k}^{e}
  55. 𝐑 o = e ( 𝐐 o e + 𝐐 t e + 𝐐 f e ) \mathbf{R}^{o}=\sum_{e}\big(\mathbf{Q}^{oe}+\mathbf{Q}^{te}+\mathbf{Q}^{fe}\big)
  56. 𝐊 \mathbf{K}
  57. k i j e {k}_{ij}^{e}
  58. K k l {K}_{kl}
  59. q i e , q j e {q}_{i}^{e},{q}_{j}^{e}
  60. r k , r l {r}_{k},{r}_{l}
  61. 𝐑 o \mathbf{R}^{o}
  62. Q i e {Q}_{i}^{e}
  63. R k o {R}^{o}_{k}
  64. q i e {q}_{i}^{e}
  65. r k {r}_{k}
  66. k i j e {k}_{ij}^{e}
  67. K k l {K}_{kl}

First-order_partial_differential_equation.html

  1. F ( x 1 , , x n , u , u x 1 , u x n ) = 0. F(x_{1},\ldots,x_{n},u,u_{x_{1}},\ldots u_{x_{n}})=0.\,
  2. u t 2 = c 2 ( u x 2 + u y 2 + u z 2 ) . u_{t}^{2}=c^{2}\left(u_{x}^{2}+u_{y}^{2}+u_{z}^{2}\right).\,
  3. u t = 1 u_{t}=1
  4. u x 2 + u y 2 + u z 2 = 1 c 2 . u_{x}^{2}+u_{y}^{2}+u_{z}^{2}=\frac{1}{c^{2}}.\,
  5. x = ( x , y , z ) and p = ( u x , u y , u z ) . \vec{x}=(x,y,z)\quad\hbox{and}\quad\vec{p}=(u_{x},u_{y},u_{z}).\,
  6. u ( x ) = p ( x - x 0 ) , u(\vec{x})=\vec{p}\cdot(\vec{x}-\vec{x_{0}}),\,
  7. | p | = 1 c , and x 0 is arbitrary . |\vec{p}\,|=\frac{1}{c},\quad\,\text{and}\quad\vec{x_{0}}\quad\,\text{is % arbitrary}.\,
  8. p \vec{p}
  9. x - x 0 \vec{x}-\vec{x_{0}}
  10. u ( x ) = ± 1 c | x - x 0 | . u(\vec{x})=\pm\frac{1}{c}|\vec{x}-\vec{x_{0}}\,|.
  11. 1 c | x - x 0 | is stationary for x 0 S . \frac{1}{c}|\vec{x}-\vec{x_{0}}\,|\quad\hbox{is stationary for}\quad\vec{x_{0}% }\in S.\,
  12. | x - x 0 | |\vec{x}-\vec{x_{0}}\,|
  13. F ( x , y , u , p , q ) = 0 , F(x,y,u,p,q)=0,\,
  14. p = u x , q = u y . p=u_{x},\quad q=u_{y}.\,
  15. d φ d a = φ a ( x , y , u , A , w ( A ) ) + w ( A ) φ b ( x , y , u , A , w ( A ) ) = 0. \frac{d\varphi}{da}=\varphi_{a}(x,y,u,A,w(A))+w^{\prime}(A)\varphi_{b}(x,y,u,A% ,w(A))=0.\,
  16. u w u_{w}
  17. u w = ϕ ( x , y , u , A , w ( A ) ) u_{w}=\phi(x,y,u,A,w(A))\,
  18. u - u 0 = p ( x - x 0 ) + q ( y - y 0 ) , u-u_{0}=p(x-x_{0})+q(y-y_{0}),\,
  19. F ( x 0 , y 0 , u 0 , p , q ) = 0. F(x_{0},y_{0},u_{0},p,q)=0.\,
  20. F p d p + F q d q = 0 , F_{p}\,dp+F_{q}\,dq=0,\,
  21. ( x 0 , y 0 , u 0 , p , q ) (x_{0},y_{0},u_{0},p,q)
  22. d x : d y : d u = F p : F q : ( p F p + q F q ) . dx:dy:du=F_{p}:F_{q}:(pF_{p}+qF_{q}).\,
  23. F x + F u p + F p p x + F q p y = 0 , F_{x}+F_{u}p+F_{p}p_{x}+F_{q}p_{y}=0,\,
  24. F y + F u q + F p q x + F q q y = 0 , F_{y}+F_{u}q+F_{p}q_{x}+F_{q}q_{y}=0,\,
  25. ( x , y , u , p , q ) (x,y,u,p,q)
  26. d x : d y : d u : d p : d q = F p : F q : ( p F p + q F q ) : ( - F x - F u p ) : ( - F y - F u q ) . dx:dy:du:dp:dq=F_{p}:F_{q}:(pF_{p}+qF_{q}):(-F_{x}-F_{u}p):(-F_{y}-F_{u}q).\,
  27. ( x 0 , y 0 , u 0 ) (x_{0},y_{0},u_{0})

First_moment_of_area.html

  1. S x = A y ¯ = i = 1 n y i d A i = A y d A S_{x}=A\bar{y}=\sum_{i=1}^{n}{y_{i}\,dA_{i}}=\int_{A}ydA
  2. S y = i = 1 n x i d A i = A x d A S_{y}=\sum_{i=1}^{n}{x_{i}\,dA_{i}}=\int_{A}xdA
  3. Q j , x = y i d A Q_{j,x}=\int y_{i}dA
  4. q = V y S x I x q=\frac{V_{y}S_{x}}{I_{x}}
  5. τ = q t {\tau}=\frac{q}{t}
  6. τ {\tau}

First_quantization.html

  1. A ^ \hat{A}
  2. | ψ α |\psi_{\alpha}\rangle
  3. α \alpha
  4. A ^ | ψ α = α | ψ α \hat{A}|\psi_{\alpha}\rangle=\alpha|\psi_{\alpha}\rangle
  5. | ψ α |\psi_{\alpha}\rangle
  6. | ψ β |\psi_{\beta}\rangle
  7. A ^ \hat{A}
  8. B ^ \hat{B}
  9. B ^ \hat{B}
  10. | ψ β |\psi_{\beta}\rangle
  11. | ψ α |\psi_{\alpha}\rangle
  12. | ψ β = α | ψ α C α β |\psi_{\beta}\rangle=\sum_{\alpha}|\psi_{\alpha}\rangle C_{\alpha\beta}
  13. A ^ \hat{A}
  14. | ψ α |\psi_{\alpha}\rangle
  15. | C α β | 2 |C_{\alpha\beta}|^{2}
  16. | ψ ( t ) |\psi(t)\rangle
  17. H ^ \hat{H}
  18. i t | ψ ( t ) = H ^ | ψ ( t ) i\hbar\frac{\partial}{\partial t}|\psi(t)\rangle=\hat{H}|\psi(t)\rangle
  19. | ψ |\psi\rangle
  20. ( | ψ ) = ψ | (|\psi\rangle)^{\dagger}=\langle\psi|
  21. ψ | ϕ \langle\psi|\phi\rangle
  22. ψ | \langle\psi|
  23. | ϕ |\phi\rangle
  24. | ψ |\psi\rangle
  25. ψ | ψ \langle\psi|\psi\rangle
  26. | ψ |\psi\rangle
  27. | ϕ |\phi\rangle
  28. ψ | ϕ = 0 \langle\psi|\phi\rangle=0
  29. | ψ α {|\psi_{\alpha}\rangle}
  30. C α β C_{\alpha\beta}
  31. C α β = ψ α | ψ β C_{\alpha\beta}=\langle\psi_{\alpha}|\psi_{\beta}\rangle
  32. ψ | ϕ = ϕ | ψ * \langle\psi|\phi\rangle=\langle\phi|\psi\rangle^{*}
  33. A ^ \hat{A}
  34. A ^ \hat{A}^{\dagger}
  35. ψ | A ^ | ϕ = ϕ | A ^ | ψ * \langle\psi|\hat{A}|\phi\rangle=\langle\phi|\hat{A}^{\dagger}|\psi\rangle^{*}
  36. | ψ |\psi\rangle
  37. | ϕ |\phi\rangle
  38. ν \nu
  39. n , l , m n,l,m
  40. | ν |\nu\rangle
  41. ψ ν ( r ) = r | ν \psi_{\nu}({r})=\langle{r}|\nu\rangle
  42. | ψ = ν | ν ν | ψ |\psi\rangle=\sum_{\nu}|\nu\rangle\langle\nu|\psi\rangle
  43. ν | ν ν | = 1 ^ \sum_{\nu}|\nu\rangle\langle\nu|={\hat{1}}
  44. ψ ( r ) \psi({r})
  45. ψ ( r 1 , r 2 , , r N ) \psi({r}_{1},{r}_{2},...,{r}_{N})
  46. ψ ( r 1 , , r j , , r k , , r N ) = + ψ ( r 1 , , r k , , r j , , r N ) \psi({r}_{1},...,{r}_{j},...,{r}_{k},...,{r_{N}})=+\psi({r}_{1},...,{r}_{k},..% .,{r}_{j},...,{r}_{N})
  47. ψ ( r 1 , , r j , , r k , , r N ) = - ψ ( r 1 , , r k , , r j , , r N ) \psi({r}_{1},...,{r}_{j},...,{r}_{k},...,{r_{N}})=-\psi({r}_{1},...,{r}_{k},..% .,{r}_{j},...,{r}_{N})
  48. ( r j , r k ) ({r}_{j},{r}_{k})

First_variation.html

  1. δ J ( y ) \delta J(y)
  2. δ J ( y , h ) = lim ε 0 J ( y + ε h ) - J ( y ) ε = d d ε J ( y + ε h ) | ε = 0 , \delta J(y,h)=\lim_{\varepsilon\to 0}\frac{J(y+\varepsilon h)-J(y)}{% \varepsilon}=\left.\frac{d}{d\varepsilon}J(y+\varepsilon h)\right|_{% \varepsilon=0},
  3. J ( y ) = a b y y d x . J(y)=\int_{a}^{b}yy^{\prime}dx.
  4. δ J ( y , h ) = d d ε J ( y + ε h ) | ε = 0 = d d ε a b ( y + ε h ) ( y + ε h ) d x | ε = 0 = d d ε a b ( y y + y ε h + y ε h + ε 2 h h ) d x | ε = 0 = a b d d ε ( y y + y ε h + y ε h + ε 2 h h ) d x | ε = 0 = a b ( y h + y h + 2 ε h h ) d x | ε = 0 = a b ( y h + y h ) d x \begin{aligned}\displaystyle\delta J(y,h)&\displaystyle=\left.\frac{d}{d% \varepsilon}J(y+\varepsilon h)\right|_{\varepsilon=0}\\ &\displaystyle=\left.\frac{d}{d\varepsilon}\int_{a}^{b}(y+\varepsilon h)(y^{% \prime}+\varepsilon h^{\prime})\ dx\right|_{\varepsilon=0}\\ &\displaystyle=\left.\frac{d}{d\varepsilon}\int_{a}^{b}(yy^{\prime}+y% \varepsilon h^{\prime}+y^{\prime}\varepsilon h+\varepsilon^{2}hh^{\prime})\ dx% \right|_{\varepsilon=0}\\ &\displaystyle=\left.\int_{a}^{b}\frac{d}{d\varepsilon}(yy^{\prime}+y% \varepsilon h^{\prime}+y^{\prime}\varepsilon h+\varepsilon^{2}hh^{\prime})\ dx% \right|_{\varepsilon=0}\\ &\displaystyle=\left.\int_{a}^{b}(yh^{\prime}+y^{\prime}h+2\varepsilon hh^{% \prime})\ dx\right|_{\varepsilon=0}\\ &\displaystyle=\int_{a}^{b}(yh^{\prime}+y^{\prime}h)\ dx\end{aligned}

Fish_curve.html

  1. e 2 = 1 2 e^{2}=\frac{1}{2}
  2. y = f ( x ) y=f(x)
  3. x = g ( t ) x=g(t)
  4. y = h ( t ) y=h(t)
  5. x = a cos ( t ) - a sin 2 ( t ) 2 , y = a cos ( t ) sin ( t ) \textstyle{x=a\cos(t)-\frac{a\sin^{2}(t)}{\sqrt{2}},\qquad y=a\cos(t)\sin(t)}
  6. x = a cos ( t ) , y = a sin ( t ) 2 \textstyle{x=a\cos(t),\qquad y=\frac{a\sin(t)}{\sqrt{2}}}
  7. - 2 a 4 2 a 3 x - 2 a 2 ( x 2 - 5 y 2 ) + ( 2 x 2 + y 2 ) 2 + 2 2 a x ( 2 x 2 - 3 y 2 ) + 2 a 2 ( y 2 - x 2 ) = 0 -2a^{4}\sqrt{2}a^{3}x-2a^{2}\left(x^{2}-5y^{2}\right)+\left(2x^{2}+y^{2}\right% )^{2}+2\sqrt{2}ax\left(2x^{2}-3y^{2}\right)+2a^{2}\left(y^{2}-x^{2}\right)=0
  8. ( 2 x 2 + y 2 ) 2 + 2 2 a x ( 2 x 2 - 3 y 2 ) + 2 a 2 ( y 2 - x 2 ) = 0 \left(2x^{2}+y^{2}\right)^{2}+2\sqrt{2}ax\left(2x^{2}-3y^{2}\right)+2a^{2}% \left(y^{2}-x^{2}\right)=0
  9. A = 1 2 | ( x y - y x ) d t | A=\frac{1}{2}\left|\int{\left(xy^{\prime}-yx^{\prime}\right)dt}\right|
  10. = 1 8 a 2 | [ 3 cos ( t ) + cos ( 3 t ) + 2 2 sin 2 ( t ) ] d t | =\frac{1}{8}a^{2}\left|\int{\left[3\cos(t)+\cos(3t)+2\sqrt{2}\sin^{2}(t)\right% ]dt}\right|
  11. A Tail = ( 2 3 - π 4 2 ) a 2 A_{\mathrm{Tail}}=\left(\frac{2}{3}-\frac{\pi}{4\sqrt{2}}\right)a^{2}
  12. A Head = ( 2 3 + π 4 2 ) a 2 A_{\mathrm{Head}}=\left(\frac{2}{3}+\frac{\pi}{4\sqrt{2}}\right)a^{2}
  13. A = 4 3 a 2 A=\frac{4}{3}a^{2}
  14. a 2 ( 1 2 π + 3 ) a\sqrt{2}\left(\frac{1}{2}\pi+3\right)
  15. K ( t ) = 2 2 + 3 cos ( t ) - cos ( 3 t ) 2 a [ cos 4 t + sin 2 t + sin 4 t + 2 sin ( t ) sin ( 2 t ) ] 3 2 K(t)=\frac{2\sqrt{2}+3\cos(t)-\cos(3t)}{2a\left[\cos^{4}t+\sin^{2}t+\sin^{4}t+% \sqrt{2}\sin(t)\sin(2t)\right]^{\frac{3}{2}}}
  16. ϕ ( t ) = π - arg ( 2 - 1 - 2 ( 1 + 2 ) e i t - 1 ) \phi(t)=\pi-\arg\left(\sqrt{2}-1-\frac{2}{\left(1+\sqrt{2}\right)e^{it}-1}\right)
  17. arg ( z ) \arg(z)
  18. x = x ( t ) , y = y ( t ) x=x(t),y=y(t)

Fisher's_z-distribution.html

  1. 0
  2. z = 1 2 log F z=\frac{1}{2}\log F
  3. x = e 2 x x^{\prime}=e^{2x}\,
  4. f ( x ; d 1 , d 2 ) = 2 d 1 d 1 / 2 d 2 d 2 / 2 B ( d 1 / 2 , d 2 / 2 ) e d 1 x ( d 1 e 2 x + d 2 ) ( d 1 + d 2 ) / 2 , f(x;d_{1},d_{2})=\frac{2d_{1}^{d_{1}/2}d_{2}^{d_{2}/2}}{B(d_{1}/2,d_{2}/2)}% \frac{e^{d_{1}x}}{\left(d_{1}e^{2x}+d_{2}\right)^{(d_{1}+d_{2})/2}},
  5. d 1 , d 2 d_{1},d_{2}\rightarrow\infty
  6. x ¯ = ( 1 / d 2 - 1 / d 1 ) / 2 \bar{x}=(1/d_{2}-1/d_{1})/2
  7. σ x 2 = ( 1 / d 1 + 1 / d 2 ) / 2. \sigma^{2}_{x}=(1/d_{1}+1/d_{2})/2.
  8. X FisherZ ( n , m ) X\sim\operatorname{FisherZ}(n,m)
  9. e 2 X F ( n , m ) e^{2X}\sim\operatorname{F}(n,m)\,
  10. X F ( n , m ) X\sim\operatorname{F}(n,m)
  11. log X 2 FisherZ ( n , m ) \tfrac{\log{X}}{2}\sim\operatorname{FisherZ}(n,m)

Five-point_stencil.html

  1. { x - 2 h , x - h , x , x + h , x + 2 h } . \ \{x-2h,x-h,x,x+h,x+2h\}.
  2. f ( x ) - f ( x + 2 h ) + 8 f ( x + h ) - 8 f ( x - h ) + f ( x - 2 h ) 12 h f^{\prime}(x)\approx\frac{-f(x+2h)+8f(x+h)-8f(x-h)+f(x-2h)}{12h}
  3. f ( x ± h ) = f ( x ) ± h f ( x ) + h 2 2 f ′′ ( x ) ± h 3 6 f ( 3 ) ( x ) + O 1 ± ( h 4 ) . ( E 1 ± ) . f(x\pm h)=f(x)\pm hf^{\prime}(x)+\frac{h^{2}}{2}f^{\prime\prime}(x)\pm\frac{h^% {3}}{6}f^{(3)}(x)+O_{1\pm}(h^{4}).\qquad(E_{1\pm}).
  4. f ( x + h ) - f ( x - h ) = 2 h f ( x ) + h 3 3 f ( 3 ) ( x ) + O 1 ( h 4 ) . ( E 1 ) . f(x+h)-f(x-h)=2hf^{\prime}(x)+\frac{h^{3}}{3}f^{(3)}(x)+O_{1}(h^{4}).\qquad(E_% {1}).
  5. f ( x ± 2 h ) = f ( x ) ± 2 h f ( x ) + 2 h 2 f ′′ ( x ) ± 4 h 3 3 f ( 3 ) ( x ) + O 2 ± ( h 4 ) . ( E 2 ± ) f(x\pm 2h)=f(x)\pm 2hf^{\prime}(x)+2h^{2}f^{\prime\prime}(x)\pm\frac{4h^{3}}{3% }f^{(3)}(x)+O_{2\pm}(h^{4}).\qquad(E_{2\pm})
  6. ( E 2 + ) - ( E 2 - ) (E_{2+})-(E_{2-})
  7. f ( x + 2 h ) - f ( x - 2 h ) = 4 h f ( x ) + 8 h 3 3 f ( 3 ) ( x ) + O 2 ( h 4 ) . ( E 2 ) . f(x+2h)-f(x-2h)=4hf^{\prime}(x)+\frac{8h^{3}}{3}f^{(3)}(x)+O_{2}(h^{4}).\qquad% (E_{2}).
  8. 8 f ( x + h ) - 8 f ( x - h ) - f ( x + 2 h ) + f ( x - 2 h ) = 12 h f ( x ) + O ( h 4 ) 8f(x+h)-8f(x-h)-f(x+2h)+f(x-2h)=12hf^{\prime}(x)+O(h^{4})\,
  9. - f ( x + 2 h ) + 8 f ( x + h ) - 8 f ( x - h ) + f ( x - 2 h ) 12 h = f ( x ) - 1 30 f ( 5 ) ( x ) h 4 + O ( h 5 ) \frac{-f(x+2h)+8f(x+h)-8f(x-h)+f(x-2h)}{12h}=f^{\prime}(x)-\frac{1}{30}f^{(5)}% (x)h^{4}+O(h^{5})
  10. f ( x ) f^{\prime}(x)
  11. f ′′ ( x ) - f ( x + 2 h ) + 16 f ( x + h ) - 30 f ( x ) + 16 f ( x - h ) - f ( x - 2 h ) 12 h 2 f^{\prime\prime}(x)\approx\frac{-f(x+2h)+16f(x+h)-30f(x)+16f(x-h)-f(x-2h)}{12h% ^{2}}
  12. f ( 3 ) ( x ) f ( x + 2 h ) - 2 f ( x + h ) + 2 f ( x - h ) - f ( x - 2 h ) 2 h 3 f^{(3)}(x)\approx\frac{f(x+2h)-2f(x+h)+2f(x-h)-f(x-2h)}{2h^{3}}
  13. f ( 4 ) ( x ) f ( x + 2 h ) - 4 f ( x + h ) + 6 f ( x ) - 4 f ( x - h ) + f ( x - 2 h ) h 4 f^{(4)}(x)\approx\frac{f(x+2h)-4f(x+h)+6f(x)-4f(x-h)+f(x-2h)}{h^{4}}
  14. j ( ξ ) = i = 0 , i j k ξ - x i x j - x i , \ell_{j}(\xi)=\prod_{i=0,\,i\neq j}^{k}\frac{\xi-x_{i}}{x_{j}-x_{i}},
  15. x 0 = x - 2 h , x 1 = x - h , x 2 = x , x 3 = x + h , x 4 = x + 2 h . x_{0}=x-2h,\quad x_{1}=x-h,\quad x_{2}=x,\quad x_{3}=x+h,\quad x_{4}=x+2h.
  16. p 4 ( x ) p_{4}(x)
  17. p 4 ( x ) = j = 0 4 f ( x j ) j ( x ) p_{4}(x)=\sum\limits_{j=0}^{4}f(x_{j})\ell_{j}(x)
  18. p 4 ( x ) = j = 0 4 f ( x j ) j ( x ) . p_{4}^{\prime}(x)=\sum\limits_{j=0}^{4}f(x_{j})\ell^{\prime}_{j}(x).
  19. f ( x 2 ) = 0 ( x 2 ) f ( x 0 ) + 1 ( x 2 ) f ( x 1 ) + 2 ( x 2 ) f ( x 2 ) + 3 ( x 2 ) f ( x 3 ) + 4 ( x 2 ) f ( x 4 ) + O ( h 4 ) f^{\prime}(x_{2})=\ell_{0}^{\prime}(x_{2})f(x_{0})+\ell_{1}^{\prime}(x_{2})f(x% _{1})+\ell_{2}^{\prime}(x_{2})f(x_{2})+\ell_{3}^{\prime}(x_{2})f(x_{3})+\ell_{% 4}^{\prime}(x_{2})f(x_{4})+O(h^{4})
  20. { ( x - h , y ) , ( x , y ) , ( x + h , y ) , ( x , y - h ) , ( x , y + h ) } , \{(x-h,y),(x,y),(x+h,y),(x,y-h),(x,y+h)\},\,
  21. Δ f ( x , y ) f ( x - h , y ) + f ( x + h , y ) + f ( x , y - h ) + f ( x , y + h ) - 4 f ( x , y ) h 2 . \Delta f(x,y)\approx\frac{f(x-h,y)+f(x+h,y)+f(x,y-h)+f(x,y+h)-4f(x,y)}{h^{2}}.
  22. 2 f x 2 = f ( x + Δ x , y ) + f ( x - Δ x , y ) - 2 f ( x , y ) Δ x 2 - 2 f ( 4 ) ( x , y ) 4 ! Δ x 2 + \begin{array}[]{l}\frac{\partial^{2}f}{\partial x^{2}}=\frac{f\left(x+\Delta x% ,y\right)+f\left(x-\Delta x,y\right)-2f(x,y)}{\Delta x^{2}}-2\frac{f^{(4)}(x,y% )}{4!}\Delta x^{2}+\cdots\end{array}
  23. 2 f y 2 = f ( x , y + Δ y ) + f ( x , y - Δ y ) - 2 f ( x , y ) Δ y 2 - 2 f ( 4 ) ( x , y ) 4 ! Δ y 2 + \begin{array}[]{l}\frac{\partial^{2}f}{\partial y^{2}}=\frac{f\left(x,y+\Delta y% \right)+f\left(x,y-\Delta y\right)-2f(x,y)}{\Delta y^{2}}-2\frac{f^{(4)}(x,y)}% {4!}\Delta y^{2}+\cdots\end{array}
  24. Δ x = Δ y = h \Delta x=\Delta y=h
  25. 2 f = 2 f x 2 + 2 f y 2 = f ( x + h , y ) + f ( x - h , y ) + f ( x , y + h ) + f ( x , y - h ) - 4 f ( x , y ) h 2 - 4 f ( 4 ) ( x , y ) 4 ! h 2 + = f ( x + h , y ) + f ( x - h , y ) + f ( x , y + h ) + f ( x , y - h ) - 4 f ( x , y ) h 2 + O ( h 2 ) \begin{array}[]{ll}\nabla^{2}f&=\frac{\partial^{2}f}{\partial x^{2}}+\frac{% \partial^{2}f}{\partial y^{2}}\\ &=\frac{f\left(x+h,y\right)+f\left(x-h,y\right)+f\left(x,y+h\right)+f\left(x,y% -h\right)-4f(x,y)}{h^{2}}-4\frac{f^{(4)}(x,y)}{4!}h^{2}+\cdots\\ &=\frac{f\left(x+h,y\right)+f\left(x-h,y\right)+f\left(x,y+h\right)+f\left(x,y% -h\right)-4f(x,y)}{h^{2}}+O\left(h^{2}\right)\\ \end{array}

Fixation_index.html

  1. p ¯ \bar{p}
  2. σ S 2 \sigma^{2}_{S}
  3. σ T 2 \sigma^{2}_{T}
  4. F S T = σ S 2 σ T 2 = σ S 2 p ¯ ( 1 - p ¯ ) F_{ST}=\frac{\sigma^{2}_{S}}{\sigma^{2}_{T}}=\frac{\sigma^{2}_{S}}{\bar{p}(1-% \bar{p})}
  5. 2 p ( 1 - p ) 2p(1-p)
  6. i i
  7. p i p_{i}
  8. i i
  9. c i c_{i}
  10. F S T = p ¯ ( 1 - p ¯ ) - c i p i ( 1 - p i ) p ¯ ( 1 - p ¯ ) = p ¯ ( 1 - p ¯ ) - p ( 1 - p ) ¯ p ¯ ( 1 - p ¯ ) F_{ST}=\frac{\bar{p}(1-\bar{p})-\sum c_{i}p_{i}(1-p_{i})}{\bar{p}(1-\bar{p})}=% \frac{\bar{p}(1-\bar{p})-\overline{p(1-p)}}{\bar{p}(1-\bar{p})}
  11. F S T = f 0 - f ¯ 1 - f ¯ F_{ST}=\frac{f_{0}-\bar{f}}{1-\bar{f}}
  12. f 0 f_{0}
  13. f ¯ \bar{f}
  14. F S T 1 - T 0 T F_{ST}\approx 1-\frac{T_{0}}{T}
  15. F S T = π Between - π Within π Between F_{ST}=\frac{\pi\text{Between}-\pi\text{Within}}{\pi\text{Between}}
  16. π Between \pi\text{Between}
  17. π Within \pi\text{Within}
  18. π Between \pi\text{Between}
  19. π Within \pi\text{Within}
  20. M ^ 1 2 ( 1 F S T - 1 ) \hat{M}\approx\frac{1}{2}\left(\frac{1}{F_{ST}}-1\right)

Fixed_effects_model.html

  1. N N
  2. T T
  3. y i t = X i t β + α i + u i t y_{it}=X_{it}\mathbf{\beta}+\alpha_{i}+u_{it}
  4. t = 1 , . . , T t=1,..,T
  5. i = 1 , , N i=1,...,N
  6. y i t y_{it}
  7. i i
  8. t , t,
  9. X i t X_{it}
  10. 1 × k 1\times k
  11. α i \alpha_{i}
  12. u i t u_{it}
  13. X i t X_{it}
  14. α i \alpha_{i}
  15. α i \alpha_{i}
  16. α i \alpha_{i}
  17. x i t x_{it}
  18. t = 1 , , T t=1,...,T
  19. α i \alpha_{i}
  20. x i t x_{it}
  21. α i \alpha_{i}
  22. α i \alpha_{i}
  23. y i t - y i ¯ = ( X i t - X i ¯ ) β + ( α i - α i ¯ ) + ( u i t - u i ¯ ) y i t ¨ = X i t ¨ β + u i t ¨ y_{it}-\overline{y_{i}}=\left(X_{it}-\overline{X_{i}}\right)\beta+\left(\alpha% _{i}-\overline{\alpha_{i}}\right)+\left(u_{it}-\overline{u_{i}}\right)\implies% \ddot{y_{it}}=\ddot{X_{it}}\beta+\ddot{u_{it}}
  24. X i ¯ = 1 T t = 1 T X i t \overline{X_{i}}=\frac{1}{T}\sum\limits_{t=1}^{T}X_{it}
  25. u i ¯ = 1 T t = 1 T u i t \overline{u_{i}}=\frac{1}{T}\sum\limits_{t=1}^{T}u_{it}
  26. α i \alpha_{i}
  27. α i ¯ = α i \overline{\alpha_{i}}=\alpha_{i}
  28. β ^ F E \hat{\beta}_{FE}
  29. y ¨ \ddot{y}
  30. X ¨ \ddot{X}
  31. i i
  32. T , T,
  33. T = 2 T=2
  34. F E T = 2 = [ ( x i 1 - x ¯ i ) ( x i 1 - x ¯ i ) + ( x i 2 - x ¯ i ) ( x i 2 - x ¯ i ) ] - 1 [ ( x i 1 - x ¯ i ) ( y i 1 - y ¯ i ) + ( x i 2 - x ¯ i ) ( y i 2 - y ¯ i ) ] {FE}_{T=2}=\left[(x_{i1}-\bar{x}_{i})(x_{i1}-\bar{x}_{i})^{\prime}+(x_{i2}-% \bar{x}_{i})(x_{i2}-\bar{x}_{i})^{\prime}\right]^{-1}\left[(x_{i1}-\bar{x}_{i}% )(y_{i1}-\bar{y}_{i})+(x_{i2}-\bar{x}_{i})(y_{i2}-\bar{y}_{i})\right]
  35. ( x i 1 - x ¯ i ) (x_{i1}-\bar{x}_{i})
  36. ( x i 1 - x i 1 + x i 2 2 ) = x i 1 - x i 2 2 (x_{i1}-\dfrac{x_{i1}+x_{i2}}{2})=\dfrac{x_{i1}-x_{i2}}{2}
  37. F E T = 2 = [ i = 1 N x i 1 - x i 2 2 x i 1 - x i 2 2 + x i 2 - x i 1 2 x i 2 - x i 1 2 ] - 1 [ i = 1 N x i 1 - x i 2 2 y i 1 - y i 2 2 + x i 2 - x i 1 2 y i 2 - y i 1 2 ] {FE}_{T=2}=\left[\sum_{i=1}^{N}\dfrac{x_{i1}-x_{i2}}{2}\dfrac{x_{i1}-x_{i2}}{2% }^{\prime}+\dfrac{x_{i2}-x_{i1}}{2}\dfrac{x_{i2}-x_{i1}}{2}^{\prime}\right]^{-% 1}\left[\sum_{i=1}^{N}\dfrac{x_{i1}-x_{i2}}{2}\dfrac{y_{i1}-y_{i2}}{2}+\dfrac{% x_{i2}-x_{i1}}{2}\dfrac{y_{i2}-y_{i1}}{2}\right]
  38. = [ i = 1 N 2 x i 2 - x i 1 2 x i 2 - x i 1 2 ] - 1 [ i = 1 N 2 x i 2 - x i 1 2 y i 2 - y i 1 2 ] =\left[\sum_{i=1}^{N}2\dfrac{x_{i2}-x_{i1}}{2}\dfrac{x_{i2}-x_{i1}}{2}^{\prime% }\right]^{-1}\left[\sum_{i=1}^{N}2\dfrac{x_{i2}-x_{i1}}{2}\dfrac{y_{i2}-y_{i1}% }{2}\right]
  39. = 2 [ i = 1 N ( x i 2 - x i 1 ) ( x i 2 - x i 1 ) ] - 1 [ i = 1 N 1 2 ( x i 2 - x i 1 ) ( y i 2 - y i 1 ) ] =2\left[\sum_{i=1}^{N}(x_{i2}-x_{i1})(x_{i2}-x_{i1})^{\prime}\right]^{-1}\left% [\sum_{i=1}^{N}\frac{1}{2}(x_{i2}-x_{i1})(y_{i2}-y_{i1})\right]
  40. = [ i = 1 N ( x i 2 - x i 1 ) ( x i 2 - x i 1 ) ] - 1 i = 1 N ( x i 2 - x i 1 ) ( y i 2 - y i 1 ) = F D T = 2 =\left[\sum_{i=1}^{N}(x_{i2}-x_{i1})(x_{i2}-x_{i1})^{\prime}\right]^{-1}\sum_{% i=1}^{N}(x_{i2}-x_{i1})(y_{i2}-y_{i1})={FD}_{T=2}
  41. X X
  42. Z Z
  43. X X
  44. Z Z
  45. α i \alpha_{i}
  46. X X
  47. Z Z
  48. X = [ X 1 i t T N × K 1 X 2 i t T N × K 2 ] Z = [ Z 1 i t T N × G 1 Z 2 i t T N × G 2 ] \begin{array}[c]{c}X=[\underset{TN\times K1}{X_{1it}}\vdots\underset{TN\times K% 2}{X_{2it}}]\\ Z=[\underset{TN\times G1}{Z_{1it}}\vdots\underset{TN\times G2}{Z_{2it}}]\end{array}
  49. X 1 X_{1}
  50. Z 1 Z_{1}
  51. α i \alpha_{i}
  52. K 1 > G 2 K1>G2
  53. γ \gamma
  54. d i ^ = Z i γ + φ i t \widehat{di}=Z_{i}\gamma+\varphi_{it}
  55. X 1 X_{1}
  56. Z 1 Z_{1}
  57. H 0 H_{0}
  58. α i X i t , Z i \alpha_{i}\perp X_{it},Z_{i}
  59. H a H_{a}
  60. α i ⟂̸ X i t , Z i \alpha_{i}\not\perp X_{it},Z_{i}
  61. H 0 H_{0}
  62. β ^ R E \widehat{\beta}_{RE}
  63. β ^ F E \widehat{\beta}_{FE}
  64. β ^ R E \widehat{\beta}_{RE}
  65. H a H_{a}
  66. β ^ F E \widehat{\beta}_{FE}
  67. β ^ R E \widehat{\beta}_{RE}
  68. Q ^ = \widehat{Q}=
  69. β ^ R E - β ^ F E \widehat{\beta}_{RE}-\widehat{\beta}_{FE}
  70. H T ^ = T Q ^ [ V a r ( β ^ F E ) - V a r ( β ^ R E ) ] - 1 Q ^ χ K 2 \widehat{HT}=T\widehat{Q}^{\prime}[Var(\widehat{\beta}_{FE})-Var(\widehat{% \beta}_{RE})]^{-1}\widehat{Q}\sim\chi_{K}^{2}
  71. K = dim ( Q ) K=\dim(Q)
  72. β ^ L D β ^ F D β ^ F E \widehat{\beta}_{LD}\approx\widehat{\beta}_{FD}\approx\widehat{\beta}_{FE}
  73. | β ^ L D | > | β ^ F E | > | β ^ F D | \left|\widehat{\beta}_{LD}\right|>\left|\widehat{\beta}_{FE}\right|>\left|% \widehat{\beta}_{FD}\right|

Flammability_limit.html

  1. L E L m i x = 1 x i L E L i LEL_{mix}=\frac{1}{\sum\frac{x_{i}}{LEL_{i}}}

Flash_ADC.html

  1. 2 n - 1 \scriptstyle 2^{n}-1
  2. 2 n - 1 \scriptstyle 2^{n}-1
  3. 2 n m \scriptstyle\frac{2^{n}}{m}

Flexural_strength.html

  1. σ \sigma
  2. σ = ϝ b d \sigma=\frac{\digamma}{bd}
  3. ϝ \digamma
  4. σ = 3 F L 2 b d 2 \sigma=\frac{3FL}{2bd^{2}}
  5. 3 L 2 d \frac{3L}{2d}
  6. σ = 3 F L 2 b d 2 \sigma=\frac{3FL}{2bd^{2}}
  7. σ = F L b d 2 \sigma=\frac{FL}{bd^{2}}
  8. σ = 3 F L 4 b d 2 \sigma=\frac{3FL}{4bd^{2}}
  9. σ = 3 F ( L - L i ) 2 b d 2 \sigma=\frac{3F(L-L_{i})}{2bd^{2}}

Flora_family.html

  1. ν 6 \nu_{6}\,\!

Flow_separation.html

  1. u u s = - 1 ρ d p d s + ν 2 u y 2 u{\partial u\over\partial s}=-{1\over\rho}{dp\over ds}+{\nu}{\partial^{2}u% \over\partial y^{2}}
  2. s , y s,y
  3. d p / d s > 0 dp/ds>0
  4. u u
  5. s s
  6. d u o / d s ( s ) < 0 du_{o}/ds(s)<0
  7. ρ u o d u o d s = - d p d s \rho u_{o}{du_{o}\over ds}=-{dp\over ds}
  8. d u o / d s du_{o}/ds
  9. d u o / d s du_{o}/ds

Fluid_coupling.html

  1. r ( n 2 ) ( d 5 ) r(n^{2})(d^{5})
  2. r r
  3. n n
  4. d d
  5. r ( n 2 ) ( d 5 ) r(n^{2})(d^{5})

Fluoroantimonic_acid.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Flux_balance_analysis.html

  1. Input = Output + Accumulation \mathrm{Input}=\mathrm{Output}+\mathrm{Accumulation}
  2. Input = Output \mathrm{Input}=\mathrm{Output}
  3. Input - Output = 0 \mathrm{Input}-\mathrm{Output}=\mathrm{0}
  4. A 𝐱 = 𝟎 \qquad A\cdot\mathbf{x}=\mathbf{0}
  5. S S
  6. 𝐯 \mathbf{v}
  7. S 𝐯 = 𝟎 \qquad S\cdot\mathbf{v}=\mathbf{0}
  8. maximize \displaystyle\,\text{maximize}
  9. ( ) T (\cdot)^{\mathrm{T}}
  10. maximize \displaystyle\,\text{maximize}
  11. 𝐯 \mathbf{v}
  12. S S
  13. 𝐜 T 𝐯 \mathbf{c}^{\mathrm{T}}\mathbf{v}
  14. 𝐥𝐨𝐰𝐞𝐫𝐛𝐨𝐮𝐧𝐝 𝐯 \mathbf{lowerbound}\leq\mathbf{v}
  15. 𝐯 𝐮𝐩𝐩𝐞𝐫𝐛𝐨𝐮𝐧𝐝 \mathbf{v}\leq\mathbf{upperbound}
  16. S S
  17. - < v i < -\infty<v_{i}<\infty\,
  18. 0 < v i < 0<v_{i}<\infty\,
  19. 0 < v i < v max 0<v_{i}<v_{\max}\,
  20. v i , m v_{i,m}\,
  21. ε \varepsilon\,
  22. v i , m - ε < v i < v i , m + ε v_{i,m}-\varepsilon<v_{i}<v_{i,m}+\varepsilon\,
  23. ( S v = 0 ) (S\vec{v}=0)
  24. v \vec{v}
  25. v biomass v_{\textrm{biomass}}
  26. v b v_{b}
  27. max v v b s. t. S v = 0 \max_{\vec{v}}\ v_{b}\qquad\textrm{s. t.}\qquad{S}\,\vec{v}=0
  28. c \vec{c}
  29. max v v c s. t. S v = 0 \max_{\vec{v}}\ \vec{v}\cdot\vec{c}\qquad\textrm{s. t.}\qquad{S}\,\vec{v}=0
  30. c \vec{c}
  31. c \vec{c}
  32. S \scriptstyle{S}
  33. v 1 v_{1}
  34. v 2 v_{2}
  35. S v = 0 \scriptstyle{S}\,\vec{v}=0
  36. min | | 𝐯 𝐰 - 𝐯 𝐝 | | 2 s . t . 𝐒 𝐯 𝐝 = 0 \min\ ||\mathbf{v_{w}}-\mathbf{v_{d}}||^{2}\qquad s.t.\quad\mathbf{S}\cdot% \mathbf{v_{d}}=0
  37. 𝐯 𝐰 \mathbf{v_{w}}
  38. 𝐯 𝐝 \mathbf{v_{d}}
  39. min 1 2 𝐯 𝐝 T 𝐈 𝐯 𝐝 + ( - 𝐯 𝐰 ) 𝐯 𝐝 s . t . 𝐒 𝐯 𝐝 = 0 \min\ \frac{1}{2}\,{\mathbf{v_{d}}}^{T}\,\mathbf{I}\,\mathbf{v_{d}}+(\mathbf{-% v_{w}})\cdot\mathbf{v_{d}}\qquad s.t.\quad\mathbf{S}\cdot\mathbf{v_{d}}=0

Flux_linkage.html

  1. λ = d t \lambda=\int\mathcal{E}\,dt
  2. \mathcal{E}
  3. = d λ d t \mathcal{E}=\frac{d\lambda}{dt}
  4. λ = S B d S \lambda=\int\limits_{S}\,\vec{B}\cdot dS
  5. B \vec{B}
  6. λ = | B | A = B A \lambda=|\vec{B}|\,A=BA
  7. λ = N B A \lambda=NBA
  8. N N

Fokker_periodicity_block.html

  1. 𝐮 = ( u x , u y ) \mathbf{u}=(u_{x},u_{y})
  2. 𝐯 = ( v x , v y ) \mathbf{v}=(v_{x},v_{y})
  3. | u x u y v x v y | = u x v y - u y v x . \left|\begin{matrix}u_{x}&u_{y}\\ v_{x}&v_{y}\end{matrix}\right|=u_{x}v_{y}-u_{y}v_{x}.
  4. ϕ B ( x , y ) := ( x 0 , y 0 ) + ( x , y ) ( u x u y v x v y ) \phi_{B}(x,y):=(x_{0},y_{0})+(x,y)\begin{pmatrix}u_{x}&u_{y}\\ v_{x}&v_{y}\end{pmatrix}
  5. ϕ B ( x , y ) := ( x 0 , y 0 ) + x 𝐮 + y 𝐯 \phi_{B}(x,y):=(x_{0},y_{0})+x\mathbf{u}+y\mathbf{v}
  6. ϕ B - 1 ( x , y ) := ( ( x , y ) - ( x 0 , y 0 ) ) ( u x u y v x v y ) - 1 \phi_{B}^{-1}(x,y):=\left((x,y)-(x_{0},y_{0})\right)\begin{pmatrix}u_{x}&u_{y}% \\ v_{x}&v_{y}\end{pmatrix}^{-1}
  7. = ( ( x , y ) - ( x 0 , y 0 ) ) u x v y - u y v x ( v y - u y - v x u x ) ={\left((x,y)-(x_{0},y_{0})\right)\over u_{x}v_{y}-u_{y}v_{x}}\begin{pmatrix}v% _{y}&-u_{y}\\ -v_{x}&u_{x}\end{pmatrix}
  8. ν B ( x , y ) := ( x , y ) , \nu_{B}(x,y):=(\lfloor x\rfloor,\lfloor y\rfloor),
  9. μ B ( x , y ) := ν B ( ϕ B - 1 ( x , y ) ) , \mu_{B}(x,y):=\nu_{B}(\phi_{B}^{-1}(x,y)),
  10. μ B ( x , y ) = μ B ( 0 , 0 ) , \mu_{B}(x,y)=\mu_{B}(0,0),
  11. M B = { B ( x , y ) : μ B ( x , y ) = μ B ( 0 , 0 ) } . M_{B}=\{B(x,y):\mu_{B}(x,y)=\mu_{B}(0,0)\}.
  12. ϕ A ( x ) := x 0 + L x \phi_{A}(x):=x_{0}+Lx
  13. ϕ A - 1 ( x ) = x - x 0 L \phi_{A}^{-1}(x)={x-x_{0}\over L}
  14. μ A ( x ) := x - x 0 L , \mu_{A}(x):=\left\lfloor{x-x_{0}\over L}\right\rfloor,
  15. M A = { A ( x ) : μ A ( x ) = μ A ( 0 ) } . M_{A}=\{A(x):\mu_{A}(x)=\mu_{A}(0)\}.
  16. ϕ C ( x , y , z ) := ( x 0 , y 0 , z 0 ) + ( x , y , z ) ( u x u y u z v x v y v z w x w y w z ) \phi_{C}(x,y,z):=(x_{0},y_{0},z_{0})+(x,y,z)\begin{pmatrix}u_{x}&u_{y}&u_{z}\\ v_{x}&v_{y}&v_{z}\\ w_{x}&w_{y}&w_{z}\end{pmatrix}
  17. ϕ C - 1 ( x , y , z ) = ( ( x , y , z ) - ( x 0 , y 0 , z 0 ) ) Δ ( v y w z - v z w y u z w y - u y w z u y v z - u z v y v z w x - v x w z u x w z - u z w x u z v x - u x v z v x w y - v y w x u y w x - u x w y u x v y - u y v x ) \phi_{C}^{-1}(x,y,z)={((x,y,z)-(x_{0},y_{0},z_{0}))\over\Delta}\begin{pmatrix}% v_{y}w_{z}-v_{z}w_{y}&u_{z}w_{y}-u_{y}w_{z}&u_{y}v_{z}-u_{z}v_{y}\\ v_{z}w_{x}-v_{x}w_{z}&u_{x}w_{z}-u_{z}w_{x}&u_{z}v_{x}-u_{x}v_{z}\\ v_{x}w_{y}-v_{y}w_{x}&u_{y}w_{x}-u_{x}w_{y}&u_{x}v_{y}-u_{y}v_{x}\end{pmatrix}
  18. Δ = u x v y w z + u y v z w x + u z v x w y - u x v z w y - u y v x w z - u z v y w x \Delta=u_{x}v_{y}w_{z}+u_{y}v_{z}w_{x}+u_{z}v_{x}w_{y}-u_{x}v_{z}w_{y}-u_{y}v_% {x}w_{z}-u_{z}v_{y}w_{x}
  19. ν C ( x , y , z ) := ( x , y , z ) \nu_{C}(x,y,z):=(\lfloor x\rfloor,\lfloor y\rfloor,\lfloor z\rfloor)
  20. μ C ( x , y , z ) := ν C ( ϕ C - 1 ( x , y , z ) ) \mu_{C}(x,y,z):=\nu_{C}(\phi_{C}^{-1}(x,y,z))
  21. M C = { C ( x , y , z ) : μ C ( x , y , z ) = μ C ( 0 , 0 , 0 ) } . M_{C}=\{C(x,y,z):\mu_{C}(x,y,z)=\mu_{C}(0,0,0)\}.

Foot–pound–second_system.html

  1. 1 pdl = 1 lb m 1 ft s 2 1\,\,\text{pdl}=1\,\,\text{lb}_{m}\cdot 1\,\frac{\,\text{ft}}{\,\text{s}^{2}}
  2. 1 slug = 1 lb F 1 s 2 ft 1\,\,\text{slug}=1\,\,\text{lb}_{F}\cdot 1\,\frac{\,\text{s}^{2}}{\,\text{ft}}
  3. 1 lb F = 1 lb m g 1\,\,\text{lb}_{F}=1\,\,\text{lb}_{m}\cdot g

Forbidden_mechanism.html

  1. \hbar
  2. L L
  3. L > 0 L>0
  4. J J
  5. L L
  6. Δ J = L - 1 , L , L + 1 ; Δ π = ( - 1 ) L , \Delta J=L-1,L,L+1;\Delta\pi=(-1)^{L},
  7. Δ π = 1 Δπ=1
  8. 1 −1
  9. J J
  10. L L
  11. J J

Force_of_mortality.html

  1. P x ( Δ x ) = P ( x < X < x + Δ x X > x ) = F X ( x + Δ x ) - F X ( x ) ( 1 - F X ( x ) ) P_{x}(\Delta x)=P(x<X<x+\Delta\;x\mid\;X>x)=\frac{F_{X}(x+\Delta\;x)-F_{X}(x)}% {(1-F_{X}(x))}
  2. μ ( x ) \mu(x)
  3. μ ( x ) = F X ( x ) 1 - F X ( x ) \mu\,(x)=\frac{F^{\prime}_{X}(x)}{1-F_{X}(x)}
  4. μ ( x ) = f X ( x ) 1 - F X ( x ) = - S ( x ) S ( x ) = - d d x ln [ S ( x ) ] . \mu\,(x)=\frac{f_{X}(x)}{1-F_{X}(x)}=-\frac{S^{\prime}(x)}{S(x)}=-{\frac{d}{dx% }}\ln[S(x)].
  5. μ ( x ) \mu(x)
  6. μ ( x ) S ( x ) = f X ( x ) \,\mu(x)S(x)=f_{X}(x)
  7. μ ( x ) = f X ( x ) S ( x ) . \mu(x)=\frac{f_{X}(x)}{S(x)}.
  8. x x + t - d d y ln [ S ( y ) ] d y \int_{x}^{x+t}-\frac{d}{dy}\ln[S(y)]\,dy
  9. ln [ S ( x + t ) ] - ln [ S ( x ) ] , \ln[S(x+t)]-\ln[S(x)],
  10. S ( x + t ) S ( x ) = S x ( t ) . \frac{S(x+t)}{S(x)}=S_{x}(t).
  11. S x ( t ) = e - x x + t μ ( y ) d y . S_{x}(t)=e^{-\int_{x}^{x+t}\mu(y)\,dy\,}.
  12. μ ( y ) = A + B c y for y 0. \mu(y)=A+Bc^{y}\quad\,\text{for }y\geqslant 0.
  13. x x + t A + B c y d y = A t + B ( c x + t - c x ) / ln [ c ] . \int_{x}^{x+t}A+Bc^{y}dy=At+B(c^{x+t}-c^{x})/\ln[c].
  14. S x ( t ) = e - ( A t + B ( c x + t - c x ) / ln [ c ] ) = e - A t g c x ( c t - 1 ) S_{x}(t)=e^{-(At+B(c^{x+t}-c^{x})/\ln[c])}=e^{-At}g^{c^{x}(c^{t}-1)}
  15. g = e - B / ln [ c ] . g=e^{-B/\ln[c]}.

Forced_convection.html

  1. P e = U L α Pe=\frac{UL}{\alpha}

FORK-256.html

  1. 2 63 2^{63}
  2. 2 80 2^{80}
  3. 2 126.6 2^{126.6}
  4. 2 108 , 2^{108},
  5. 2 112.9 2^{112.9}
  6. 2 128 2^{128}

Formal_calculation.html

  1. n = 0 q n = 1 1 - q \sum_{n=0}^{\infty}q^{n}=\frac{1}{1-q}
  2. n = 0 2 n = - 1. \sum_{n=0}^{\infty}2^{n}=-1.
  3. d y d x = y 2 \frac{dy}{dx}=y^{2}
  4. d x d y = 1 y 2 \frac{dx}{dy}=\frac{1}{y^{2}}
  5. x = - 1 y + C x=\frac{-1}{y}+C
  6. y = 1 C - x y=\frac{1}{C-x}
  7. C = C=\infty
  8. y = 1 - x = 1 = 0 y=\frac{1}{\infty-x}=\frac{1}{\infty}=0

Formal_derivative.html

  1. f ( x ) = a n x n + + a 1 x + a 0 f(x)\,=\,a_{n}x^{n}+\cdots+a_{1}x+a_{0}
  2. f ( x ) = D f ( x ) = n a n x n - 1 + + 2 a 2 x + a 1 f^{\prime}(x)\,=\,Df(x)=na_{n}x^{n-1}+\cdots+2a_{2}x+a_{1}
  3. n a n na_{n}
  4. k = 1 n a n , \sum_{k=1}^{n}a_{n},
  5. k k
  6. ( f ( x ) b ) = f ( x ) b (f(x)\cdot b)^{\prime}=f^{\prime}(x)\cdot b
  7. r R r\in R
  8. r = 0 , r^{\prime}=0,
  9. x = 1. x^{\prime}=1.
  10. ( a + b ) = a + b , (a+b)^{\prime}=a^{\prime}+b^{\prime},
  11. ( a b ) = a b + a b . (a\cdot b)^{\prime}=a^{\prime}\cdot b+a\cdot b^{\prime}.
  12. ( a + b ) = a + b = b + a = ( b + a ) , (a+b)^{\prime}=a^{\prime}+b^{\prime}=b^{\prime}+a^{\prime}=(b+a)^{\prime},
  13. ( ( a + b ) + c ) = ( a + b ) + c = ( a + b ) + c = a + ( b + c ) = a + ( b + c ) = ( a + ( b + c ) ) , ((a+b)+c)^{\prime}=(a+b)^{\prime}+c^{\prime}=(a^{\prime}+b^{\prime})+c^{\prime% }=a^{\prime}+(b^{\prime}+c^{\prime})=a^{\prime}+(b+c)^{\prime}=(a+(b+c))^{% \prime},
  14. ( a ( b c ) ) = a ( b c ) + a ( b c ) = a b c + a ( b c + b c ) = a b c + a b c + a b c = (a(bc))^{\prime}=a^{\prime}(bc)+a(bc)^{\prime}=a^{\prime}bc+a(b^{\prime}c+bc^{% \prime})=a^{\prime}bc+ab^{\prime}c+abc^{\prime}=
  15. = ( a b + a b ) c + ( a b ) c = ( a b ) c + ( a b ) c = ( ( a b ) c ) , =(a^{\prime}b+ab^{\prime})c+(ab)c^{\prime}=(ab)^{\prime}c+(ab)c^{\prime}=((ab)% c)^{\prime},
  16. ( ( a + b ) c ) = ( a + b ) c + ( a + b ) c = ( a + b ) c + ( a + b ) c = ( a c + b c ) + ( a c + b c ) = ((a+b)c)^{\prime}=(a+b)^{\prime}c+(a+b)c^{\prime}=(a^{\prime}+b^{\prime})c+(a+% b)c^{\prime}=(a^{\prime}c+b^{\prime}c)+(ac^{\prime}+bc^{\prime})=
  17. = = ( a c + ( b c + a c ) ) + b c = ( a c + ( a c + b c ) ) + b c = = ( a c + a c ) + ( b c + b c ) = =\cdots=(a^{\prime}c+(b^{\prime}c+ac^{\prime}))+bc^{\prime}=(a^{\prime}c+(ac^{% \prime}+b^{\prime}c))+bc^{\prime}=\cdots=(a^{\prime}c+ac^{\prime})+(b^{\prime}% c+bc^{\prime})=
  18. = ( a c ) + ( b c ) = ( a c + b c ) , =(ac)^{\prime}+(bc)^{\prime}=(ac+bc)^{\prime},
  19. ( i a i x i ) = i ( a i x i ) = i ( ( a i ) x i + a i ( x i ) ) = i ( 0 x i + a i ( j = 1 i x j - 1 ( x ) x i - j ) ) = i j = 1 i a i x i - 1 . (\sum_{i}a_{i}x^{i})^{\prime}=\sum_{i}(a_{i}x^{i})^{\prime}=\sum_{i}((a_{i})^{% \prime}x^{i}+a_{i}(x^{i})^{\prime})=\sum_{i}(0x^{i}+a_{i}(\sum_{j=1}^{i}x^{j-1% }(x^{\prime})x^{i-j}))=\sum_{i}\sum_{j=1}^{i}a_{i}x^{i-1}.
  20. ( r f + s g ) ( x ) = r f ( x ) + s g ( x ) . (r\cdot f+s\cdot g)^{\prime}(x)=r\cdot f^{\prime}(x)+s\cdot g^{\prime}(x).
  21. ( f g ) ( x ) = f ( x ) g ( x ) + f ( x ) g ( x ) . (f\cdot g)^{\prime}(x)=f^{\prime}(x)\cdot g(x)+f(x)\cdot g^{\prime}(x).
  22. f ( x ) = ( x - r ) m r g ( x ) f(x)=(x-r)^{m_{r}}g(x)
  23. f ( x ) = x 6 + 1 f(x)\,=\,x^{6}+1
  24. g ( X , Y ) = f ( Y ) - f ( X ) Y - X g(X,Y)=\frac{f(Y)-f(X)}{Y-X}
  25. Y Y
  26. X X
  27. X X
  28. Y Y
  29. f f

Forward_exchange_rate.html

  1. ( 1 + i d ) = S F ( 1 + i f ) (1+i_{d})=\frac{S}{F}(1+i_{f})
  2. F = S ( 1 + i f ) ( 1 + i d ) F=S\frac{(1+i_{f})}{(1+i_{d})}
  3. F = S ( 1 + P ) F=S(1+P)
  4. P = F S - 1 P=\frac{F}{S}-1
  5. P N = ( F S - 1 ) 360 d P_{N}=(\frac{F}{S}-1)\frac{360}{d}
  6. P 6 = ( 1.2260 1.2238 - 1 ) 360 30 = 0.021572 = 2.16 % P_{6}=(\frac{1.2260}{1.2238}-1)\frac{360}{30}=0.021572=2.16\%
  7. F t = E t ( S t + k ) F_{t}=E_{t}(S_{t+k})
  8. F t F_{t}
  9. E t ( S t + k ) E_{t}(S_{t+k})
  10. F t = E t ( S t + 1 ) + P t F_{t}=E_{t}(S_{t+1})+P_{t}
  11. F t - S t = E t ( S t + 1 - S t ) + P t F_{t}-S_{t}=E_{t}(S_{t+1}-S_{t})+P_{t}
  12. F t - S t F_{t}-S_{t}
  13. F t - S t + 1 F_{t}-S_{t+1}
  14. E t ( S t + 1 - S t ) E_{t}(S_{t+1}-S_{t})
  15. F t - S t F_{t}-S_{t}

Foster's_reactance_theorem.html

  1. Z = i X Z=iX\,
  2. X \scriptstyle X
  3. i \scriptstyle i
  4. Y = 1 i X = - i 1 X = i B Y=\frac{1}{iX}=-i\frac{1}{X}=iB
  5. B \scriptstyle B
  6. Z = i ω L Z=i\omega L\,
  7. L \scriptstyle L
  8. ω \scriptstyle\omega
  9. X = ω L X=\omega L\,
  10. Z = 1 i ω C Z=\frac{1}{i\omega C}
  11. C \scriptstyle C
  12. X = - 1 ω C X=-\frac{1}{\omega C}
  13. Z = i ω L + 1 i ω C = i ( ω L - 1 ω C ) Z=i\omega L+\frac{1}{i\omega C}=i\left(\omega L-\frac{1}{\omega C}\right)
  14. Y = i ω C + 1 i ω L Y=i\omega C+\frac{1}{i\omega L}
  15. Z = i ( ω L 1 - ω 2 L C ) Z=i\left(\frac{\omega L}{1-\omega^{2}LC}\right)
  16. Z ( s ) = P ( s ) Q ( s ) Z(s)=\frac{P(s)}{Q(s)}
  17. Z ( s ) \scriptstyle Z(s)
  18. P ( s ) , Q ( s ) \scriptstyle P(s),\ Q(s)
  19. s \scriptstyle s
  20. i ω \scriptstyle i\omega

Foucault_pendulum_vector_diagrams.html

  1. v e l o c i t y v e c t o r a t a g i v e n l a t i t u d e v e l o c i t y v e c t o r a t e q u a t o r \begin{matrix}\frac{velocity\;vector\;at\;a\;given\;latitude}{velocity\;vector% \;at\;equator}\end{matrix}
  2. 1 E V U × c o s i n e o f l a t i t u d e 1 E V U × c o s i n e o f z e r o \begin{matrix}\frac{1\;EVU\;\times\;cosine\;of\;latitude}{1\;EVU\;\times\;% cosine\;of\;zero}\end{matrix}
  3. c o s i n e o f l a t i t u d e 1 \begin{matrix}\frac{cosine\;of\;latitude}{1}\end{matrix}
  4. v e l o c i t y v e c t o r a t l a t i t u d e A v e l o c i t y v e c t o r a t l a t i t u d e B \begin{matrix}\frac{velocity\;vector\;at\;latitude\;A}{velocity\;vector\;at\;% latitude\;B}\end{matrix}
  5. 1 E V U × c o s i n e o f l a t i t u d e A 1 E V U × c o s i n e o f l a t i t u d e B \begin{matrix}\frac{1\;EVU\;\times\;cosine\;of\;latitude\;A}{1\;EVU\;\times\;% cosine\;of\;latitude\;B}\end{matrix}
  6. c o s i n e o f l a t i t u d e A c o s i n e o f l a t i t u d e B \begin{matrix}\frac{cosine\;of\;latitude\;A}{cosine\;of\;latitude\;B}\end{matrix}
  7. O R T R P a t a g i v e n l a t i t u d e O R T R P a t t h e N o r t h P o l e \begin{matrix}\frac{ORTRP\;at\;a\;given\;latitude}{ORTRP\;at\;the\;North\;Pole% }\end{matrix}
  8. 1 d a y × s i n e o f 90 1 d a y × s i n e o f g i v e n l a t i t u d e \begin{matrix}\frac{1\;day\;\times\;sine\;of\;90}{1\;day\;\times\;sine\;of\;% given\;latitude}\end{matrix}
  9. 1 s i n e o f g i v e n l a t i t u d e \begin{matrix}\frac{1}{sine\;of\;given\;latitude}\end{matrix}
  10. O R T R P a t l a t i t u d e A O R T R P a t l a t i t u d e B \begin{matrix}\frac{ORTRP\;at\;latitude\;A}{ORTRP\;at\;latitude\;B}\end{matrix}
  11. 1 d a y × s i n e o f l a t i t u d e B 1 d a y × s i n e o f l a t i t u d e A \begin{matrix}\frac{1\;day\;\times\;sine\;of\;latitude\;B}{1\;day\;\times\;% sine\;of\;latitude\;A}\end{matrix}
  12. s i n e o f l a t i t u d e B s i n e o f l a t i t u d e A \begin{matrix}\frac{sine\;of\;latitude\;B}{sine\;of\;latitude\;A}\end{matrix}
  13. r o t a t i o n a l v e l o c i t y v e c t o r a t a g i v e n l a t i t u d e r o t a t i o n a l v e l o c i t y v e c t o r a t N o r t h P o l e \begin{matrix}\frac{rotational\;velocity\;vector\;at\;a\;given\;latitude}{% rotational\;velocity\;vector\;at\;North\;Pole}\end{matrix}
  14. 1 R V U × c o s i n e o f m i s a l i g n m e n t 1 R V U × c o s i n e o f z e r o \begin{matrix}\frac{1\;RVU\;\times\;cosine\;of\;misalignment}{1\;RVU\;\times\;% cosine\;of\;zero}\end{matrix}
  15. c o s i n e o f m i s a l i g n m e n t 1 \begin{matrix}\frac{cosine\;of\;misalignment}{1}\end{matrix}

Fox_n-coloring.html

  1. ρ \rho
  2. D 2 n D_{2n}
  3. ρ \rho
  4. S 3 S^{3}
  5. t s i ts^{i}
  6. 2 π / n 2\pi/n
  7. t s i D 2 p ts^{i}\in D_{2p}
  8. i / p i\in\mathbb{Z}/p\mathbb{Z}
  9. ρ \rho
  10. ρ \rho
  11. ρ \rho
  12. ρ \rho
  13. col n ( L ) , \mathrm{col}_{n}(L),
  14. C n ( K ) C_{n}(K)\,
  15. C n ( K ) n C n 0 ( K ) C_{n}(K)\cong\mathbb{Z}_{n}\oplus C_{n}^{0}(K)\,
  16. C n 0 ( K ) C_{n}^{0}(K)
  17. # \#
  18. L 1 L_{1}
  19. L 2 L_{2}
  20. col n ( L 1 ) col n ( L 2 ) = n col n ( L 1 # L 2 ) . \mathrm{col}_{n}(L_{1})\mathrm{col}_{n}(L_{2})=n\mathrm{col}_{n}(L_{1}\#L_{2}).
  21. ρ \rho

Fractional_supersymmetry.html

  1. 1 / 2 1/2
  2. 1 / N 1/N

Fracture_toughness.html

  1. K K
  2. Pa m \,\text{Pa}\sqrt{\rm{m}}
  3. psi in \,\text{psi}\sqrt{\rm{in}}
  4. γ w o f \gamma_{wof}
  5. K I c 2 / E K_{Ic}^{2}/E
  6. E E
  7. γ w o f \gamma_{wof}
  8. M N / m 3 / 2 MN/m^{3/2}
  9. M N / m 3 / 2 MN/m^{3/2}
  10. K I K_{I}
  11. G I G_{I}
  12. G I = K I 2 E G_{I}=\cfrac{K_{I}^{2}}{E^{\prime}}
  13. E E
  14. E = E E^{\prime}=E
  15. E = E / ( 1 - ν 2 ) E^{\prime}=E/(1-\nu^{2})

Fragmentation_(computing).html

  1. External Memory Fragmentation = 1 - Largest Block Of Free Memory Total Free Memory {\,\text{External Memory Fragmentation}=1-}\frac{\,\text{Largest Block Of Free% Memory}}{\,\text{Total Free Memory}}

Frame_of_a_vector_space.html

  1. { 𝐞 k } \{\mathbf{e}_{k}\}
  2. 𝐯 V \mathbf{v}\in V
  3. { 𝐞 k } \{\mathbf{e}_{k}\}
  4. c k c_{k}
  5. 𝐯 = k c k 𝐞 k \mathbf{v}=\sum_{k}c_{k}\mathbf{e}_{k}
  6. { 𝐞 k } \{\mathbf{e}_{k}\}
  7. V V
  8. 𝐯 \mathbf{v}
  9. { 𝐞 k } \{\mathbf{e}_{k}\}
  10. V V
  11. V V
  12. c k c_{k}
  13. 𝐯 \mathbf{v}
  14. { 𝐞 k } \{\mathbf{e}_{k}\}
  15. V V
  16. V V
  17. { 𝐞 k } \{\mathbf{e}_{k}\}
  18. V V
  19. V V
  20. 𝐯 \mathbf{v}
  21. c k c_{k}
  22. { 𝐞 k } \{\mathbf{e}_{k}\}
  23. c k c_{k}
  24. 𝐯 \mathbf{v}
  25. 𝐯 \mathbf{v}
  26. { 𝐞 k } \{\mathbf{e}_{k}\}
  27. { 𝐞 k } k \{\mathbf{e}_{k}\}_{k\in\mathbb{N}}
  28. V V
  29. 𝐯 \mathbf{v}
  30. A 𝐯 2 k | 𝐯 , 𝐞 k | 2 B 𝐯 2 . A\left\|\mathbf{v}\right\|^{2}\leq\sum_{k\in\mathbb{N}}\left|\langle\mathbf{v}% ,\mathbf{e}_{k}\rangle\right|^{2}\leq B\left\|\mathbf{v}\right\|^{2}.
  31. { 𝐞 k } \{\mathbf{e}_{k}\}
  32. 𝐯 V \mathbf{v}\in V
  33. 𝐞 k \mathbf{e}_{k}
  34. 𝐯 \mathbf{v}
  35. A 𝐯 2 0 B 𝐯 2 ; A\left\|\mathbf{v}\right\|^{2}\leq 0\leq B\left\|\mathbf{v}\right\|^{2};
  36. A 0 A\leq 0
  37. V = 2 V=\mathbb{R}^{2}
  38. { 𝐞 k } \{\mathbf{e}_{k}\}
  39. { ( 1 , 0 ) , ( 0 , 1 ) , ( 0 , 1 2 ) , ( 0 , 1 3 ) , } . \left\{(1,0),\,(0,1),\,\left(0,\frac{1}{\sqrt{2}}\right),\,\left(0,\frac{1}{% \sqrt{3}}\right),\ldots\right\}.
  40. k | 𝐞 k , ( 0 , 1 ) | 2 = 0 + 1 + 1 2 + 1 3 + = \sum_{k}\left|\langle\mathbf{e}_{k},(0,1)\rangle\right|^{2}=0+1+\frac{1}{2}+% \frac{1}{3}+\cdots=\infty
  41. { 𝐞 k } \{\mathbf{e}_{k}\}
  42. e i = c \|e_{i}\|=c
  43. | e i , e j | = c |\langle e_{i},e_{j}\rangle|=c
  44. { 𝐞 ~ k } \{\mathbf{\tilde{e}}_{k}\}
  45. 𝐯 = k 𝐯 , 𝐞 ~ k 𝐞 k = k 𝐯 , 𝐞 k 𝐞 ~ k \mathbf{v}=\sum_{k}\langle\mathbf{v},\mathbf{\tilde{e}}_{k}\rangle\mathbf{e}_{% k}=\sum_{k}\langle\mathbf{v},\mathbf{e}_{k}\rangle\mathbf{\tilde{e}}_{k}
  46. 𝐯 V \mathbf{v}\in V
  47. 𝐒 : V V \mathbf{S}:V\rightarrow V
  48. 𝐒𝐯 = k 𝐯 , 𝐞 k 𝐞 k \mathbf{S}\mathbf{v}=\sum_{k}\langle\mathbf{v},\mathbf{e}_{k}\rangle\mathbf{e}% _{k}
  49. 𝐒 \mathbf{S}
  50. 𝐒𝐯 , 𝐯 = k | 𝐯 , 𝐞 k | 2 , \langle\mathbf{S}\mathbf{v},\mathbf{v}\rangle=\sum_{k}\left|\langle\mathbf{v},% \mathbf{e}_{k}\rangle\right|^{2},
  51. A 𝐯 2 𝐒𝐯 , 𝐯 B 𝐯 2 , A\left\|\mathbf{v}\right\|^{2}\leq\langle\mathbf{S}\mathbf{v},\mathbf{v}% \rangle\leq B\left\|\mathbf{v}\right\|^{2},
  52. 𝐯 V \mathbf{v}\in V
  53. 𝐒 \mathbf{S}
  54. 𝐒 - 1 \mathbf{S}^{-1}
  55. 𝐒 \mathbf{S}
  56. 𝐒 - 1 \mathbf{S}^{-1}
  57. 𝐞 ~ k = 𝐒 - 1 𝐞 k \tilde{\mathbf{e}}_{k}=\mathbf{S}^{-1}\mathbf{e}_{k}
  58. 𝐯 \mathbf{v}
  59. V V
  60. 𝐮 = k 𝐯 , 𝐞 k 𝐞 ~ k \mathbf{u}=\sum_{k}\langle\mathbf{v},\mathbf{e}_{k}\rangle\tilde{\mathbf{e}}_{k}
  61. 𝐮 = k 𝐯 , 𝐞 k ( 𝐒 - 1 𝐞 k ) = 𝐒 - 1 ( k 𝐯 , 𝐞 k 𝐞 k ) = 𝐒 - 1 𝐒𝐯 = 𝐯 \mathbf{u}=\sum_{k}\langle\mathbf{v},\mathbf{e}_{k}\rangle(\mathbf{S}^{-1}% \mathbf{e}_{k})=\mathbf{S}^{-1}\left(\sum_{k}\langle\mathbf{v},\mathbf{e}_{k}% \rangle\mathbf{e}_{k}\right)=\mathbf{S}^{-1}\mathbf{S}\mathbf{v}=\mathbf{v}
  62. 𝐯 = k 𝐯 , 𝐞 k 𝐞 ~ k \mathbf{v}=\sum_{k}\langle\mathbf{v},\mathbf{e}_{k}\rangle\tilde{\mathbf{e}}_{k}
  63. 𝐮 = k 𝐯 , 𝐞 ~ k 𝐞 k \mathbf{u}=\sum_{k}\langle\mathbf{v},\tilde{\mathbf{e}}_{k}\rangle\mathbf{e}_{k}
  64. 𝐞 ~ k \tilde{\mathbf{e}}_{k}
  65. 𝐒 \mathbf{S}
  66. 𝐮 = k 𝐯 , 𝐒 - 1 𝐞 k 𝐞 k = k 𝐒 - 1 𝐯 , 𝐞 k 𝐞 k = 𝐒 ( 𝐒 - 1 𝐯 ) = 𝐯 \mathbf{u}=\sum_{k}\langle\mathbf{v},\mathbf{S}^{-1}\mathbf{e}_{k}\rangle% \mathbf{e}_{k}=\sum_{k}\langle\mathbf{S}^{-1}\mathbf{v},\mathbf{e}_{k}\rangle% \mathbf{e}_{k}=\mathbf{S}(\mathbf{S}^{-1}\mathbf{v})=\mathbf{v}
  67. 𝐯 = k 𝐯 , 𝐞 ~ k 𝐞 k \mathbf{v}=\sum_{k}\langle\mathbf{v},\tilde{\mathbf{e}}_{k}\rangle\mathbf{e}_{k}
  68. 𝐯 , 𝐞 ~ k \langle\mathbf{v},\tilde{\mathbf{e}}_{k}\rangle
  69. { 𝐞 ~ k } \{\tilde{\mathbf{e}}_{k}\}
  70. { 𝐞 k } \{\mathbf{e}_{k}\}
  71. { 𝐞 k } \{\mathbf{e}_{k}\}
  72. 𝐯 \mathbf{v}
  73. { 𝐞 k } \{\mathbf{e}_{k}\}
  74. { c k } \{c_{k}\}
  75. 𝐯 = k c k 𝐞 k \mathbf{v}=\sum_{k}c_{k}\mathbf{e}_{k}
  76. { c k } \{c_{k}\}
  77. 𝐯 , 𝐞 ~ k \langle\mathbf{v},\tilde{\mathbf{e}}_{k}\rangle
  78. { 𝐞 k } \{\mathbf{e}_{k}\}
  79. { c k } \{c_{k}\}
  80. { 𝐠 k } { 𝐞 ~ k } \{\mathbf{g}_{k}\}\neq\{\tilde{\mathbf{e}}_{k}\}
  81. 𝐯 = k 𝐯 , 𝐠 k 𝐞 k \mathbf{v}=\sum_{k}\langle\mathbf{v},\mathbf{g}_{k}\rangle\mathbf{e}_{k}
  82. 𝐯 V \mathbf{v}\in V
  83. { 𝐠 k } \{\mathbf{g}_{k}\}
  84. { 𝐞 k } \{\mathbf{e}_{k}\}

Fransén–Robinson_constant.html

  1. F = 0 1 Γ ( x ) d x . F=\int_{0}^{\infty}\frac{1}{\Gamma(x)}\,\mathrm{d}x.
  2. F n = 1 1 Γ ( n ) = n = 0 1 n ! , F\approx\sum_{n=1}^{\infty}\frac{1}{\Gamma(n)}=\sum_{n=0}^{\infty}\frac{1}{n!},
  3. F = e + 0 e - x π 2 + ln 2 x d x F=e+\int_{0}^{\infty}\frac{e^{-x}}{\pi^{2}+\ln^{2}x}\,\mathrm{d}x
  4. F = e + 1 π - π / 2 π / 2 e π tan θ e - e π tan θ d θ . F=e+\frac{1}{\pi}\int_{-\pi/2}^{\pi/2}e^{\pi\tan\theta}e^{-e^{\pi\tan\theta}}% \,\mathrm{d}\theta.
  5. F = lim α 0 α E α , 0 ( 1 ) . F=\lim_{\alpha\to 0}\alpha E_{\alpha,0}(1).

Franz_Rellich.html

  1. E ε ( λ ) E_{\varepsilon}(\lambda)
  2. A ε A_{\varepsilon}
  3. ε \varepsilon

Franz–Keldysh_effect.html

  1. α = 2 ω k 0 c = ω κ 2 n 0 c . \alpha=\frac{2\omega k_{0}}{c}={{\omega\kappa_{2}}\over{n_{0}c}}.
  2. H = 1 2 m ( p + e A ) 2 + V ( r ) H={1\over 2m}(p+eA)^{2}+V(r)
  3. A = 1 2 A 0 e [ e i ( k p r - ω t ) + e - i ( k p r - ω t ) ] A={1\over 2}A_{0}e[e^{i(k_{p}\cdot r-\omega t)}+e^{-i(k_{p}\cdot r-\omega t)}]
  4. A 2 A^{2}
  5. \cdot
  6. \cdot
  7. H p 2 2 m + V ( r ) + e m A p H\sim{p^{2}\over 2m}+V(r)+{e\over m}A\cdot p
  8. | j k e i k r u j k ( r ) |jk>=e^{ik\cdot r}u_{jk}(r)
  9. w c v = 2 π 2 | < c k | e m A p | v k > | 2 δ [ \Epsilon c ( k ) - \Epsilon v ( k ) - ω ] w_{cv}={2\pi\over\hbar}^{2}|<ck^{\prime}|{e\over m}A\cdot p|vk>|^{2}\delta[% \Epsilon_{c}(k^{\prime})-\Epsilon_{v}(k)-\hbar\omega]
  10. = π e 2 2 m 2 A 0 2 | < c k | e x p ( i k p r ) e p | v k > | 2 δ [ \Epsilon c ( k ) - \Epsilon v ( k ) - ω ] ={\pi e^{2}\over 2\hbar m^{2}}A_{0}^{2}|<ck^{\prime}|exp(ik_{p}\cdot r)e\cdot p% |vk>|^{2}\delta[\Epsilon_{c}(k^{\prime})-\Epsilon_{v}(k)-\hbar\omega]
  11. k p k_{p}
  12. e p c v = 1 V v e i ( k p + k - k ) r u * c k ( r ) e ( p + k ) u v k ( r ) d 3 r e\cdot p_{cv}={1\over V}\int_{v}e^{i(k_{p}+k-k^{\prime})\cdot r}{u^{*}}_{ck^{% \prime}}(r)e\cdot(p+\hbar k)u_{vk}(r)d^{3}r
  13. ω w c v = 1 2 ω κ 2 ϵ 0 E 0 2 \hbar\omega w_{cv}={1\over 2}\omega\kappa_{2}\epsilon_{0}{E_{0}}^{2}
  14. E = - A t {E}=-{{\partial A}\over{\partial t}}
  15. E 0 = ω A 0 E_{0}=\omega A_{0}
  16. κ = π e 2 ϵ 0 m 2 ω 2 k , k | e p c v | 2 δ [ \Epsilon c ( k ) - \Epsilon v ( k ) - ω ] δ k k \kappa={{\pi e^{2}}\over{\epsilon_{0}m^{2}\omega^{2}}}\sum_{k,k^{\prime}}|e% \cdot p_{cv}|^{2}\delta[\Epsilon_{c}(k^{\prime})-\Epsilon_{v}(k)-\hbar\omega]% \delta_{kk^{\prime}}
  17. k = k e k^{\prime}=k_{e}
  18. k h = - k k_{h}=-k
  19. V ( r e - r h ) V(r_{e}-r_{h})
  20. | k e | , | k h | |k_{e}|,|k_{h}|
  21. Ψ i j ( r e , r h ) = ψ i k e ( r e ) ψ j k h ( r h ) \Psi_{ij}(r_{e},r_{h})=\psi_{ik_{e}}(r_{e})\psi_{jk_{h}}(r_{h})
  22. K = k e + k h . K=k_{e}+k_{h}.
  23. H = H e + H h + V ( r e - r h ) H=H_{e}+H_{h}+V(r_{e}-r_{h})
  24. A c v n , K A^{n,K}_{cv}
  25. Ψ n , K ( r e , r h ) = c , k e , v , k h A c v n , K ( k e , k h ) ψ c k e ( r e ) ψ v k h ( r h ) \Psi^{n,K}(r_{e},r_{h})=\sum_{c,k_{e},v,k_{h}}A^{n,K}_{cv}(k_{e},k_{h})\psi_{% ck_{e}}(r_{e})\psi_{vk_{h}}(r_{h})
  26. [ \Epsilon c ( k e ) + \Epsilon h ( k h ) + V ( r e - r h ) - ϵ ] A c , V n , K ( k e , k h ) = 0 ( * ) [\Epsilon_{c}(k_{e})+\Epsilon_{h}(k_{h})+V(r_{e}-r_{h})-\epsilon]A^{n,K}_{c,V}% (k_{e},k_{h})=0(*)
  27. \Epsilon v = 0 \Epsilon_{v}=0
  28. \Epsilon c ( k e ) = 2 k e 2 2 m e + \Epsilon G , \Epsilon h ( k h ) = 2 k h 2 2 m h \Epsilon_{c}(k_{e})={{\hbar^{2}k_{e}^{2}}\over{2m_{e}}}+\Epsilon_{G},\Epsilon_% {h}(k_{h})={{\hbar^{2}k_{h}^{2}}\over{2m_{h}}}
  29. \Epsilon G \Epsilon_{G}
  30. A c v n , K ( k e , k h ) A^{n,K}_{cv}(k_{e},k_{h})
  31. [ ( - 2 2 M 2 ) + ( - 2 2 μ 2 - e 2 4 π ϵ r ) ] Φ n , k ( r , R ) = [ \Epsilon - \Epsilon G ] Φ n , K ( r , R ) [(-{\hbar^{2}\over 2M}\nabla^{2})+(-{\hbar^{2}\over 2\mu}\nabla^{2}-{e^{2}% \over 4\pi\epsilon r})]\Phi^{n,k}(r,R)=[\Epsilon-\Epsilon_{G}]\cdot\Phi^{n,K}(% r,R)
  32. r = r e - r h , R = m e r e + m h r h m e + m h , 1 μ = 1 m e + 1 m h , M = m e + m h r=r_{e}-r_{h},R={{m_{e}r_{e}+m_{h}r_{h}}\over{m_{e}+m_{h}}},{1\over\mu}={1% \over m_{e}}+{1\over m_{h}},M=m_{e}+m_{h}
  33. Ψ n , K ( r , R ) = Ψ K ( R ) ψ n ( r ) \Psi^{n,K}(r,R)=\Psi_{K}(R)\psi_{n}(r)
  34. Ψ n , K ( r , R ) = 1 V e x p ( i K R ) ϕ n ( r ) \Psi^{n,K}(r,R)={1\over\sqrt{V}}exp(iK\cdot R)\phi_{n}(r)
  35. ϕ n ( r ) \phi_{n}(r)
  36. κ 2 ( ω ) = π e 2 ϵ 0 m 2 ω 2 | e p c v | 2 λ | ϕ λ ( 0 ) | 2 δ ( \Epsilon G + \Epsilon λ - ω ) ( * * ) \kappa_{2}(\omega)={\pi e^{2}\over{\epsilon_{0}m^{2}\omega^{2}}}|e\cdot p_{cv}% |^{2}\sum_{\lambda}|\phi_{\lambda}(0)|^{2}\delta(\Epsilon_{G}+\Epsilon_{% \lambda}-\hbar\omega)(**)
  37. [ - 2 2 μ 2 - e E r ] ψ ( r ) = ϵ ψ ( r ) [-{\hbar^{2}\over 2\mu}\nabla^{2}-eE\cdot r]\psi(r)=\epsilon\psi(r)
  38. [ - 2 2 μ d 2 d r i 2 - e E i r i - ϵ i ] ψ ( r i ) = 0 [-{\hbar^{2}\over 2\mu}{d^{2}\over{dr_{i}^{2}}}-eE_{i}r_{i}-\epsilon_{i}]\psi(% r_{i})=0
  39. μ i \mu_{i}
  40. θ i = ( e 2 E i 2 2 2 μ i ) 1 / 3 , ξ i = ϵ i + e E i r i θ i \hbar\theta_{i}=({{e^{2}E_{i}^{2}\hbar^{2}}\over{2\mu_{i}}})^{1/3},\xi_{i}={{% \epsilon_{i}+eE_{i}r_{i}}\over{\hbar\theta_{i}}}
  41. ψ ( ξ x , ξ y , ξ z ) = C x C y C z A i ( - ξ x ) A i ( - ξ y ) A i ( - ξ z ) \psi(\xi_{x},\xi_{y},\xi_{z})=C_{x}C_{y}C_{z}Ai(-\xi_{x})Ai(-\xi_{y})Ai(-\xi_{% z})
  42. C i = e | E i | θ C_{i}={{\sqrt{e|E_{i}|}}\over{\hbar\theta}}
  43. E y = E z = 0 , E x E_{y}=E_{z}=0,E_{x}
  44. ψ ( x , y , z ) = C A i ( - e E x - ϵ + 2 k y 2 / 2 μ y + 2 k z 2 / 2 μ z θ x ) \psi(x,y,z)=C\cdot Ai({{-eEx-\epsilon+\hbar^{2}k_{y}^{2}/2\mu_{y}+\hbar^{2}k_{% z}^{2}/2\mu_{z}}\over{\hbar\theta_{x}}})
  45. - d ϵ x d ϵ y d ϵ z \int_{-\infty}^{\infty}d\epsilon_{x}d\epsilon_{y}d\epsilon_{z}
  46. d ϵ x d ϵ y d\epsilon_{x}d\epsilon_{y}
  47. κ ( ω , E ) = π 2 ϵ 0 m 2 ω 2 | e p c v | 2 e | E x | θ x 2 - J 2 D c v ( ω - ϵ G - ϵ x ) | A i ( - ϵ x θ ) 2 | d ϵ x . {\kappa(\omega,E)}={{\pi^{2}}\over{\epsilon_{0}m^{2}\omega^{2}}}|e\cdot p_{c}v% |^{2}{{e|E_{x}|}\over{\hbar\theta_{x}^{2}}}\int_{-\infty}^{\infty}{J^{2D}}_{cv% }(\hbar\omega-\epsilon_{G}-\epsilon_{x})\cdot|Ai(-{{\epsilon_{x}}\over{\hbar% \theta}})^{2}|d\epsilon_{x}.
  48. J c v 2 D ( ω ) = ( μ y μ z ) 1 / 2 π 2 , ω > ϵ G . J^{2D}_{cv}(\hbar\omega)={(\mu_{y}\mu_{z})^{1/2}\over\pi\hbar^{2}},\hbar\omega% >\epsilon_{G}.
  49. = 0 , ω < ϵ G . =0,\hbar\omega<\epsilon_{G}.
  50. η = ω - ϵ G θ x \eta={{\hbar\omega-\epsilon_{G}}\over{\hbar\theta_{x}}}
  51. η 0 \eta<<0
  52. ω ϵ G \hbar\omega<<\epsilon_{G}
  53. κ 2 ( ω , E x ) = 1 / 2 κ 2 ( ω ) e x p [ - 4 3 ( ϵ G - ω θ x ) ] \kappa_{2}(\omega,E_{x})={1/2}\kappa_{2}(\omega)exp[{-4\over 3}({{\epsilon_{G}% -\hbar\omega}\over{\hbar\theta_{x}}})]

Fredholm_theory.html

  1. g ( x ) = a b K ( x , y ) f ( y ) d y . g(x)=\int_{a}^{b}K(x,y)f(y)\,dy.
  2. L g ( x ) = f ( x ) Lg(x)=f(x)
  3. L = d 2 d x 2 L=\frac{d^{2}}{dx^{2}}\,
  4. L K ( x , y ) = δ ( x - y ) LK(x,y)=\delta(x-y)
  5. δ ( x ) \delta(x)
  6. g ( x ) = K ( x , y ) f ( y ) d y . g(x)=\int K(x,y)f(y)\,dy.
  7. K ( x , y ) K(x,y)
  8. L ψ n ( x ) = ω n ψ n ( x ) L\psi_{n}(x)=\omega_{n}\psi_{n}(x)
  9. ω n \omega_{n}
  10. ψ n ( x ) \psi_{n}(x)
  11. K ( x , y ) = n ψ n * ( x ) ψ n ( y ) ω n K(x,y)=\sum_{n}\frac{\psi_{n}^{*}(x)\psi_{n}(y)}{\omega_{n}}
  12. ψ n * \psi_{n}^{*}
  13. ψ n \psi_{n}
  14. K ( x , y ) K(x,y)
  15. δ ( x - y ) = n ψ n * ( x ) ψ n ( y ) . \delta(x-y)=\sum_{n}\psi_{n}^{*}(x)\psi_{n}(y).
  16. ω n \omega_{n}
  17. K ( x , y ) K(x,y)
  18. f ( x ) = - ω ϕ ( x ) + K ( x , y ) ϕ ( y ) d y f(x)=-\omega\phi(x)+\int K(x,y)\phi(y)\,dy
  19. f = ( K - ω ) ϕ f=(K-\omega)\phi
  20. ϕ = 1 K - ω f . \phi=\frac{1}{K-\omega}f.
  21. R ( ω ) = 1 K - ω I . R(\omega)=\frac{1}{K-\omega I}.
  22. R ( ω ; x , y ) = n ψ n * ( y ) ψ n ( x ) ω n - ω R(\omega;x,y)=\sum_{n}\frac{\psi_{n}^{*}(y)\psi_{n}(x)}{\omega_{n}-\omega}
  23. ϕ ( x ) = R ( ω ; x , y ) f ( y ) d y . \phi(x)=\int R(\omega;x,y)f(y)\,dy.
  24. λ = 1 / ω \lambda=1/\omega
  25. g ( x ) = ϕ ( x ) - λ K ( x , y ) ϕ ( y ) d y g(x)=\phi(x)-\lambda\int K(x,y)\phi(y)\,dy
  26. R ( λ ) = 1 I - λ K . R(\lambda)=\frac{1}{I-\lambda K}.
  27. det ( I - λ K ) = exp [ - n λ n n Tr K n ] \det(I-\lambda K)=\exp\left[-\sum_{n}\frac{\lambda^{n}}{n}\operatorname{Tr}\,K% ^{n}\right]
  28. Tr K = K ( x , x ) d x \operatorname{Tr}\,K=\int K(x,x)\,dx
  29. Tr K 2 = K ( x , y ) K ( y , x ) d x d y \operatorname{Tr}\,K^{2}=\iint K(x,y)K(y,x)\,dx\,dy
  30. ζ ( s ) = 1 det ( I - s K ) . \zeta(s)=\frac{1}{\det(I-sK)}.