wpmath0000012_7

Lambert_summation.html

  1. a n \sum a_{n}
  2. a n = A ( L ) \sum a_{n}=A\,(\mathrm{L})
  3. lim r 1 - ( 1 - r ) n = 1 n a n r n 1 - r n = A . \lim_{r\rightarrow 1-}(1-r)\sum_{n=1}^{\infty}\frac{na_{n}r^{n}}{1-r^{n}}=A.\,
  4. n = 1 μ ( n ) n = 0 ( L ) \sum_{n=1}^{\infty}\frac{\mu(n)}{n}=0(\mathrm{L})

Lamé_function.html

  1. d 2 y d x 2 = ( A + B \weierp ( x ) ) y \frac{d^{2}y}{dx^{2}}=(A+B\weierp(x))y
  2. \wp
  3. d 2 y d t 2 + 1 2 ( 1 t - e 1 + 1 t - e 2 + 1 t - e 3 ) d y d t = A + B t 4 ( t - e 1 ) ( t - e 2 ) ( t - e 3 ) y \frac{d^{2}y}{dt^{2}}+\frac{1}{2}\left(\frac{1}{t-e_{1}}+\frac{1}{t-e_{2}}+% \frac{1}{t-e_{3}}\right)\frac{dy}{dt}=\frac{A+Bt}{4(t-e_{1})(t-e_{2})(t-e_{3})}y

Landau–Lifshitz_model.html

  1. J = diag ( J 1 , J 2 , J 3 ) J=\operatorname{diag}(J_{1},J_{2},J_{3})
  2. H = 1 2 [ i ( 𝐒 x i ) 2 - J ( 𝐒 ) ] d x ( 1 ) H=\frac{1}{2}\int\left[\sum_{i}\left(\frac{\partial\mathbf{S}}{\partial x_{i}}% \right)^{2}-J(\mathbf{S})\right]\,dx\qquad(1)
  3. 𝐒 t = 𝐒 i 2 𝐒 x i 2 + 𝐒 J 𝐒 . ( 2 ) \frac{\partial\mathbf{S}}{\partial t}=\mathbf{S}\wedge\sum_{i}\frac{\partial^{% 2}\mathbf{S}}{\partial x_{i}^{2}}+\mathbf{S}\wedge J\mathbf{S}.\qquad(2)
  4. 𝐒 t = 𝐒 2 𝐒 x 2 + 𝐒 J 𝐒 . ( 3 ) \frac{\partial\mathbf{S}}{\partial t}=\mathbf{S}\wedge\frac{\partial^{2}% \mathbf{S}}{\partial x^{2}}+\mathbf{S}\wedge J\mathbf{S}.\qquad(3)
  5. 𝐒 t = 𝐒 ( 2 𝐒 x 2 + 2 𝐒 y 2 ) + 𝐒 J 𝐒 ( 4 ) \frac{\partial\mathbf{S}}{\partial t}=\mathbf{S}\wedge\left(\frac{\partial^{2}% \mathbf{S}}{\partial x^{2}}+\frac{\partial^{2}\mathbf{S}}{\partial y^{2}}% \right)+\mathbf{S}\wedge J\mathbf{S}\qquad(4)
  6. 𝐒 t = 𝐒 ( 2 𝐒 x 2 + 2 𝐒 y 2 + 2 𝐒 z 2 ) + 𝐒 J 𝐒 . ( 5 ) \frac{\partial\mathbf{S}}{\partial t}=\mathbf{S}\wedge\left(\frac{\partial^{2}% \mathbf{S}}{\partial x^{2}}+\frac{\partial^{2}\mathbf{S}}{\partial y^{2}}+% \frac{\partial^{2}\mathbf{S}}{\partial z^{2}}\right)+\mathbf{S}\wedge J\mathbf% {S}.\qquad(5)
  7. J = 0 J=0

Landau–Lifshitz–Gilbert_equation.html

  1. 𝐌 \mathbf{M}
  2. 𝐌 \mathbf{M}
  3. d 𝐌 d t = - γ 𝐌 × 𝐇 eff - λ 𝐌 × ( 𝐌 × 𝐇 eff ) \frac{d\mathbf{M}}{dt}=-\gamma\mathbf{M}\times\mathbf{H_{\mathrm{eff}}}-% \lambda\mathbf{M}\times\left(\mathbf{M}\times\mathbf{H_{\mathrm{eff}}}\right)
  4. γ γ
  5. λ λ
  6. λ = α γ M s , \lambda=\alpha\frac{\gamma}{M_{\mathrm{s}}},
  7. α α
  8. d 𝐌 d t = - γ ( 𝐌 × 𝐇 eff - η 𝐌 × d 𝐌 d t ) \frac{d\mathbf{M}}{dt}=-\gamma\left(\mathbf{M}\times\mathbf{H}_{\mathrm{eff}}-% \eta\mathbf{M}\times\frac{d\mathbf{M}}{dt}\right)
  9. η η
  10. d 𝐌 d t = - γ 𝐌 × 𝐇 eff - λ 𝐌 × ( 𝐌 × 𝐇 eff ) \frac{d\mathbf{M}}{dt}=-\gamma^{\prime}\mathbf{M}\times\mathbf{H}_{\mathrm{eff% }}-\lambda\mathbf{M}\times(\mathbf{M}\times\mathbf{H}_{\mathrm{eff}})
  11. γ = γ 1 + γ 2 η 2 M s 2 and λ = γ 2 η 1 + γ 2 η 2 M s 2 . \gamma^{\prime}=\frac{\gamma}{1+\gamma^{2}\eta^{2}M_{s}^{2}}\qquad\,\text{and}% \qquad\lambda=\frac{\gamma^{2}\eta}{1+\gamma^{2}\eta^{2}M_{s}^{2}}.
  12. $\mathbf{ }$
  13. m ^ = M / M S \hat{m}=\vec{M}/M_{S}
  14. m ^ ˙ = - γ m ^ × H 𝑒𝑓𝑓 + α m ^ × m ^ ˙ + τ m ^ × ( x ^ × m ^ ) | x ^ × m ^ | + τ x ^ × m ^ | x ^ × m ^ | \dot{\hat{m}}=-\gamma\hat{m}\times\vec{H}_{\mathit{eff}}+\alpha\hat{m}\times% \dot{\hat{m}}+\tau_{\parallel}\frac{\hat{m}\times(\hat{x}^{\prime}\times\hat{m% })}{\left|\hat{x}^{\prime}\times\hat{m}\right|}+\tau_{\perp}\frac{\hat{x}^{% \prime}\times\hat{m}}{\left|\hat{x}^{\prime}\times\hat{m}\right|}
  15. α \alpha
  16. τ \tau_{\perp}
  17. τ \tau_{\parallel}
  18. x ^ \hat{x}^{\prime}
  19. γ γ
  20. λ = α γ λ=αγ
  21. γ < s u b > G = γ / ( 1 + α 2 ) γ<sub>G=γ/(1+α^{2})

Landen's_transformation.html

  1. F F
  2. F ( φ , k ) = F ( φ | k 2 ) = F ( sin φ ; k ) = 0 φ d θ 1 - k 2 sin 2 θ . F(\varphi,k)=F(\varphi\,|\,k^{2})=F(\sin\varphi;k)=\int_{0}^{\varphi}\frac{d% \theta}{\sqrt{1-k^{2}\sin^{2}\theta}}.
  3. F ( φ , k ) = 0 φ d θ 1 - k 2 sin 2 θ . F(\varphi^{\prime},k^{\prime})=\int_{0}^{\varphi^{\prime}}\frac{d\theta}{\sqrt% {1-k^{\prime 2}\sin^{2}\theta}}.
  4. K ( k ) = 2 1 + k K ( 1 - k 1 + k ) K(k)=\frac{2}{1+k^{\prime}}K(\frac{1-k^{\prime}}{1+k^{\prime}})
  5. k = 1 - k 2 k^{\prime}={\sqrt{1-k^{2}}}
  6. K ( k ) = 0 π 2 d θ 1 - k 2 sin 2 θ . K(k)=\int_{0}^{\frac{\pi}{2}}\frac{d\theta}{\sqrt{1-k^{2}\sin^{2}\theta}}.
  7. I = 0 π 2 1 a 2 cos 2 ( θ ) + b 2 sin 2 ( θ ) d θ I=\int_{0}^{\frac{\pi}{2}}\frac{1}{\sqrt{a^{2}\cos^{2}(\theta)+b^{2}\sin^{2}(% \theta)}}\,d\theta
  8. a \scriptstyle{a}
  9. b \scriptstyle{b}
  10. a 1 = a + b 2 , b 1 = a b . a_{1}=\frac{a+b}{2},\qquad b_{1}=\sqrt{ab}.\,
  11. I 1 = 0 π 2 1 a 1 2 cos 2 ( θ ) + b 1 2 sin 2 ( θ ) d θ I_{1}=\int_{0}^{\frac{\pi}{2}}\frac{1}{\sqrt{a_{1}^{2}\cos^{2}(\theta)+b_{1}^{% 2}\sin^{2}(\theta)}}\,d\theta
  12. I = 1 a K ( ( a 2 - b 2 ) a ) I=\frac{1}{a}K(\frac{\sqrt{(a^{2}-b^{2})}}{a})
  13. I 1 = 2 a + b K ( a - b a + b ) I_{1}=\frac{2}{a+b}K(\frac{a-b}{a+b})
  14. K ( ( a 2 - b 2 ) a ) = 2 a a + b K ( a - b a + b ) K(\frac{\sqrt{(a^{2}-b^{2})}}{a})=\frac{2a}{a+b}K(\frac{a-b}{a+b})
  15. I 1 = I I_{1}=I
  16. θ = arctan ( x b ) \scriptstyle{\theta=\arctan\left(\frac{x}{b}\right)}
  17. d θ = ( 1 b cos 2 ( θ ) ) d x \scriptstyle{d\theta=\left(\frac{1}{b}\cos^{2}(\theta)\right)dx}
  18. I = 0 π 2 1 a 2 cos 2 ( θ ) + b 2 sin 2 ( θ ) d θ = 0 1 ( x 2 + a 2 ) ( x 2 + b 2 ) d x I=\int_{0}^{\frac{\pi}{2}}\frac{1}{\sqrt{a^{2}\cos^{2}(\theta)+b^{2}\sin^{2}(% \theta)}}\,d\theta=\int_{0}^{\infty}\frac{1}{\sqrt{(x^{2}+a^{2})(x^{2}+b^{2})}% }\,dx
  19. x = t + t 2 + a b \scriptstyle{x=t+\sqrt{t^{2}+ab}}
  20. I = 0 1 ( x 2 + a 2 ) ( x 2 + b 2 ) d x = - 1 2 ( t 2 + ( a + b 2 ) 2 ) ( t 2 + a b ) d t = 0 1 ( t 2 + ( a + b 2 ) 2 ) ( t 2 + ( a b ) 2 ) d t \begin{aligned}\displaystyle I&\displaystyle=\int_{0}^{\infty}\frac{1}{\sqrt{(% x^{2}+a^{2})(x^{2}+b^{2})}}\,dx\\ &\displaystyle=\int_{-\infty}^{\infty}\frac{1}{2\sqrt{\left(t^{2}+\left(\frac{% a+b}{2}\right)^{2}\right)(t^{2}+ab)}}\,dt\\ &\displaystyle=\int_{0}^{\infty}\frac{1}{\sqrt{\left(t^{2}+\left(\frac{a+b}{2}% \right)^{2}\right)\left(t^{2}+\left(\sqrt{ab}\right)^{2}\right)}}\,dt\end{aligned}
  21. ( x 2 + a 2 ) ( x 2 + b 2 ) = 2 x t 2 + ( a + b 2 ) 2 \sqrt{(x^{2}+a^{2})(x^{2}+b^{2})}=2x\sqrt{t^{2}+\left(\frac{a+b}{2}\right)^{2}}
  22. d x = x t 2 + a b d t dx=\frac{x}{\sqrt{t^{2}+ab}}\,dt
  23. x \scriptstyle{x}
  24. a \scriptstyle{a}
  25. b \scriptstyle{b}
  26. a \scriptstyle{a}
  27. b \scriptstyle{b}
  28. AGM ( a , b ) \scriptstyle{\operatorname{AGM}(a,b)}
  29. I = 0 π 2 1 a 2 cos 2 ( θ ) + b 2 sin 2 ( θ ) d θ = 0 π 2 1 AGM ( a , b ) d θ = π 2 AGM ( a , b ) I=\int_{0}^{\frac{\pi}{2}}\frac{1}{\sqrt{a^{2}\cos^{2}(\theta)+b^{2}\sin^{2}(% \theta)}}\,d\theta=\int_{0}^{\frac{\pi}{2}}\frac{1}{\operatorname{AGM}(a,b)}\,% d\theta=\frac{\pi}{2\,\operatorname{AGM}(a,b)}
  30. b 2 = a 2 ( 1 - k 2 ) \scriptstyle{b^{2}=a^{2}(1-k^{2})}
  31. I = 1 a 0 π 2 1 1 - k 2 sin 2 ( θ ) d θ = 1 a F ( π 2 , k ) = 1 a K ( k ) I=\frac{1}{a}\int_{0}^{\frac{\pi}{2}}\frac{1}{\sqrt{1-k^{2}\sin^{2}(\theta)}}% \,d\theta=\frac{1}{a}F\left(\frac{\pi}{2},k\right)=\frac{1}{a}K(k)
  32. a \scriptstyle{a}
  33. K ( k ) = π a 2 AGM ( a , a 1 - k 2 ) K(k)=\frac{\pi a}{2\,\operatorname{AGM}(a,a\sqrt{1-k^{2}})}
  34. a - 1 = a + a 2 - b 2 a_{-1}=a+\sqrt{a^{2}-b^{2}}\,
  35. b - 1 = a - a 2 - b 2 b_{-1}=a-\sqrt{a^{2}-b^{2}}\,
  36. AGM ( a , b ) = AGM ( a + a 2 - b 2 , a - a 2 - b 2 ) \operatorname{AGM}(a,b)=\operatorname{AGM}(a+\sqrt{a^{2}-b^{2}},a-\sqrt{a^{2}-% b^{2}})\,
  37. K ( k ) = π a 2 AGM ( a ( 1 + k ) , a ( 1 - k ) ) K(k)=\frac{\pi a}{2\,\operatorname{AGM}(a(1+k),a(1-k))}\,
  38. AGM ( u , v ) = π ( u + v ) 4 K ( u - v v + u ) . \operatorname{AGM}(u,v)=\frac{\pi(u+v)}{4K\left(\frac{u-v}{v+u}\right)}.
  39. K ( k ) \scriptstyle{K(k)}
  40. K ( k ) \scriptstyle{K(k)}
  41. K ( m 2 ) \scriptstyle{K(m^{2})}

Landsberg–Schaar_relation.html

  1. 1 p n = 0 p - 1 exp ( 2 π i n 2 q p ) = e π i / 4 2 q n = 0 2 q - 1 exp ( - π i n 2 p 2 q ) . \frac{1}{\sqrt{p}}\sum_{n=0}^{p-1}\exp\left(\frac{2\pi in^{2}q}{p}\right)=% \frac{e^{\pi i/4}}{\sqrt{2q}}\sum_{n=0}^{2q-1}\exp\left(-\frac{\pi in^{2}p}{2q% }\right).
  2. τ = 2 i q / p + ε \tau=2iq/p+\varepsilon
  3. ε > 0 \varepsilon>0
  4. n = - + e - π n 2 τ = 1 τ n = - + e - π n 2 / τ \sum_{n=-\infty}^{+\infty}e^{-\pi n^{2}\tau}=\frac{1}{\sqrt{\tau}}\sum_{n=-% \infty}^{+\infty}e^{-\pi n^{2}/\tau}
  5. ε 0. \varepsilon\to 0.
  6. 1 p n = 0 p - 1 exp ( π i n 2 q p ) = e π i / 4 q n = 0 q - 1 exp ( - π i n 2 p q ) \frac{1}{\sqrt{p}}\sum_{n=0}^{p-1}\exp\left(\frac{\pi in^{2}q}{p}\right)=\frac% {e^{\pi i/4}}{\sqrt{q}}\sum_{n=0}^{q-1}\exp\left(-\frac{\pi in^{2}p}{q}\right)

Langmuir_adsorption_model.html

  1. p A p_{A}
  2. V V
  3. A g A_{g}
  4. S S
  5. A a d A_{ad}
  6. K e q K_{eq}
  7. A g + S A a d A_{g}\,\,\,+\,\,\,S\,\,\,\rightleftharpoons\,\,\,A_{ad}
  8. θ A = V V m = K e q A p A 1 + K e q A p A \theta_{A}=\frac{V}{V_{m}}=\frac{K_{eq}^{A}\,p_{A}}{1+K_{eq}^{A}\,p_{A}}
  9. θ A \theta_{A}
  10. V m V_{m}
  11. r a d = k a d p A [ S ] r_{ad}=k_{ad}\,p_{A}\,[S]
  12. r d = k d [ A a d ] r_{d}=k_{d}\,[A_{ad}]
  13. [ A a d ] p A [ S ] = k a d k d = K e q A \frac{[A_{ad}]}{p_{A}[S]}=\frac{k_{ad}}{k_{d}}=K_{eq}^{A}
  14. [ S 0 ] = [ S ] + [ A a d ] [S_{0}]=[S]+[A_{ad}]\,
  15. [ S 0 ] = [ A a d ] K e q A p A + [ A a d ] = 1 + K e q A p A K e q A p A [ A a d ] [S_{0}]=\frac{[A_{ad}]}{K_{eq}^{A}\,p_{A}}+[A_{ad}]=\frac{1+K_{eq}^{A}\,p_{A}}% {K_{eq}^{A}\,p_{A}}\,[A_{ad}]
  16. θ A = [ A a d ] [ S 0 ] \theta_{A}=\frac{[A_{ad}]}{[S_{0}]}
  17. θ A = K e q A p A 1 + K e q A p A \theta_{A}=\frac{K_{eq}^{A}\,p_{A}}{1+K_{eq}^{A}\,p_{A}}
  18. Z ( N ) = ζ L N N ! N S ! ( N S - N ) ! Z(N)\,=\,\frac{\zeta^{N}_{L}}{N!}\,\frac{N_{S}!}{(N_{S}-N)!}
  19. ζ L \zeta_{L}
  20. N S N_{S}
  21. 𝒵 ( μ ) = N = 0 N S exp ( N μ k B T ) ζ L N N ! N S ! ( N S - N ) ! \mathcal{Z}(\mu)\,=\,\sum_{N=0}^{N_{S}}\,\exp\left(\,\frac{N\mu}{k_{B}T}\right% )\frac{\zeta^{N}_{L}}{N!}\,\frac{N_{S}!}{(N_{S}-N)!}\,
  22. 𝒵 = ( 1 + x ) N S \mathcal{Z}\,=\,(1+x)^{N_{S}}
  23. x = ζ L exp ( μ k B T ) x\,=\,\zeta_{L}\exp\left(\frac{\mu}{k_{B}T}\right)
  24. L = - k B T ln ( 𝒵 ) = - k B T N S ln ( 1 + x ) L\,=\,-k_{B}T\ln(\mathcal{Z})\,=\,-k_{B}TN_{S}\ln(1+x)
  25. ( L μ ) T , V , A r e a = - N \left(\frac{\partial L}{\partial\mu}\right)_{T,V,Area}=\,-N
  26. θ A = N N S = x 1 + x \theta_{A}\,=\,\frac{N}{N_{S}}\,=\,\frac{x}{1+x}
  27. μ A = μ g \mu_{A}\,=\,\mu_{g}
  28. μ g = ( A g N i ) T , V , N j i = k B T ln N 3 D Z 3 D \mu_{g}=\left(\frac{\partial A_{g}}{\partial N_{i}}\right)_{T,V,N_{j\neq i}}\,% \,=k_{B}T\ln\frac{N^{3D}}{Z^{3D}}
  29. x = θ A 1 - θ A = ζ L N 3 D ζ 3 D = ζ L ( h 2 2 π m k B T ) 3 / 2 P k B T = P P 0 x\,=\,\frac{\theta_{A}}{1-\theta_{A}}\,=\,\zeta_{L}\frac{N^{3D}}{\zeta^{3D}}\,% =\,\zeta_{L}\left(\frac{h^{2}}{2\pi mk_{B}T}\right)^{3/2}\frac{P}{k_{B}T}\,=\,% \frac{P}{P_{0}}
  30. P 0 = k B T ζ L ( 2 π m k B T h 2 ) 3 / 2 P_{0}=\frac{k_{B}T}{\zeta_{L}}\left(\frac{2\pi mk_{B}T}{h^{2}}\right)^{3/2}
  31. θ A = P P + P 0 \theta_{A}=\frac{P}{P+P_{0}}
  32. [ A a d ] p A [ S ] = K e q A \frac{[A_{ad}]}{p_{A}\,[S]}=K^{A}_{eq}
  33. [ B a d ] p B [ S ] = K e q B \frac{[B_{ad}]}{p_{B}\,[S]}=K^{B}_{eq}
  34. [ S 0 ] = [ S ] + [ A a d ] + [ B a d ] [S_{0}]=[S]+[A_{ad}]+[B_{ad}]\,
  35. θ A = K e q A p A 1 + K e q A p A + K e q B p B \theta_{A}=\frac{K^{A}_{eq}\,p_{A}}{1+K^{A}_{eq}\,p_{A}+K^{B}_{eq}\,p_{B}}
  36. θ B = K e q B p B 1 + K e q A p A + K e q B p B \theta_{B}=\frac{K^{B}_{eq}\,p_{B}}{1+K^{A}_{eq}\,p_{A}+K^{B}_{eq}\,p_{B}}
  37. [ D a d ] p D 2 1 / 2 [ S ] = K e q D \frac{[D_{ad}]}{p^{1/2}_{D_{2}}[S]}=K^{D}_{eq}
  38. θ D = K e q D p D 2 1 / 2 1 + K e q D p D 2 1 / 2 \theta_{D}=\frac{K^{D}_{eq}\,p^{1/2}_{D_{2}}}{1+K^{D}_{eq}\,p^{1/2}_{D_{2}}}
  39. S = S c o n f i g u r a t i o n a l + S v i b r a t i o n a l S\,=\,S_{configurational}\,+\,S_{vibrational}
  40. S c o n f = k B l n Ω c o n f S_{conf}=k_{B}\,ln\Omega_{conf}
  41. Ω c o n f = N S ! N ! ( N S - N ) ! \Omega_{conf}\,=\,\frac{N_{S}!}{N!(N_{S}-N)!}
  42. l n N ! = N l n N - N lnN!\,=\,NlnN-N
  43. S c o n f / k B = - θ A l n ( θ A ) - ( 1 - θ A ) l n ( 1 - θ A ) S_{conf}/k_{B}=-\theta_{A}\,ln(\theta_{A})-(1-\theta_{A})\,ln(1-\theta_{A})
  44. S g a s N k B = l n ( k B T P λ 3 ) + 5 / 2 \frac{S_{gas}}{Nk_{B}}\,=\,ln\left(\frac{k_{B}T}{P\lambda^{3}}\right)+5/2
  45. λ \lambda
  46. θ A = α F p C F \theta_{A}=\alpha_{F}\,p^{C_{F}}
  47. θ A = p A 1 K e q A + p A \theta_{A}=\frac{p_{A}}{\frac{1}{K_{eq}^{A}}+p_{A}}
  48. θ C T 0 = α T 0 p A C T 0 1 K e q A + p A C T 0 \theta^{C_{T_{0}}}=\frac{\alpha_{T_{0}}\,p_{A}^{C_{T_{0}}}}{\frac{1}{K_{eq}^{A% }}+p_{A}^{C_{T_{0}}}}
  49. [ A a d ] p A [ S ] = K e q A e - Δ G a d / R T = e Δ S a d / R e - Δ H a d / R T \frac{[A_{ad}]}{p_{A}\,[S]}=K^{A}_{eq}\propto\mathrm{e}^{-\Delta G_{ad}/RT}=% \mathrm{e}^{\Delta S_{ad}/R}\,\mathrm{e}^{-\Delta H_{ad}/RT}
  50. Δ H a d = Δ H a d 0 ( 1 - α T θ ) \Delta H_{ad}=\Delta H^{0}_{ad}\,(1-\alpha_{T}\,\theta)
  51. K e q A K_{eq}^{A}
  52. K e q A = K e q A , 0 e ( Δ H a d 0 α T θ / k T ) K^{A}_{eq}=K^{A,0}_{eq}\mathrm{e}^{(\Delta H^{0}_{ad}\,\alpha_{T}\,\theta/k\,T)}
  53. K e q A p A = θ 1 - θ K^{A}_{eq}\,p_{A}=\frac{\theta}{1-\theta}
  54. ln ( K e q A , 0 p A ) = - Δ H a d 0 α T θ k T + ln ( θ 1 - θ ) \ln(K^{A,0}_{eq}\,p_{A})=\frac{-\Delta H^{0}_{ad}\,\alpha_{T}\,\theta}{k\,T}+% \ln\left(\frac{\theta}{1-\theta}\right)
  55. [ A ] S 0 = c B x B ( 1 - x B ) [ 1 + ( c B - 1 ) x B ] \frac{[A]}{S_{0}}=\frac{c_{B}\,x_{B}}{(1-x_{B})\,[1+(c_{B}-1)\,x_{B}]}
  56. x B = p A K m , c B = K 1 K m x_{B}=p_{A}\,K_{m},\qquad c_{B}=\frac{K_{1}}{K_{m}}
  57. [ A ] = i = 1 i [ A ] i = i = 1 i K 1 K m i - 1 p A i [ A ] 0 [A]=\sum^{\infty}_{i=1}i\,[A]_{i}=\sum^{\infty}_{i=1}i\,K_{1}\,K^{i-1}_{m}\,p^% {i}_{A}\,[A]_{0}
  58. K i = [ A ] i p A [ A ] i - 1 K_{i}=\frac{[A]_{i}}{p_{A}\,[A]_{i-1}}
  59. x 1 s = K x 1 l 1 + ( K - 1 ) x 1 l x_{1}^{s}\,=\,\frac{Kx_{1}^{l}}{1+(K-1)x_{1}^{l}}
  60. i = 1 k x i s = 1 i = 1 k x i l = 1 \sum_{i=1}^{k}x^{s}_{i}=1\,\,\sum_{i=1}^{k}x^{l}_{i}=1
  61. x 1 s = K [ x 1 l / ( 1 - x 1 l ) ] 1 + K [ x 1 l / ( 1 - x 1 l ) ] x_{1}^{s}\,=\,\frac{K[x_{1}^{l}/(1-x_{1}^{l})]}{1+K[x_{1}^{l}/(1-x_{1}^{l})]}

Lankford_coefficient.html

  1. x x
  2. y y
  3. z z
  4. R = ϵ xy p ϵ z p R=\cfrac{\epsilon^{p}_{\mathrm{xy}}}{\epsilon^{p}_{\mathrm{z}}}
  5. ϵ xy p \epsilon^{p}_{\mathrm{xy}}
  6. ϵ z p \epsilon^{p}_{\mathrm{z}}
  7. R R
  8. R R
  9. 0 , 45 , 90 0^{\circ},45^{\circ},90^{\circ}
  10. R = 1 4 ( R 0 + 2 R 45 + R 90 ) . R=\cfrac{1}{4}\left(R_{0}+2~{}R_{45}+R_{90}\right)~{}.
  11. R R
  12. R p = 1 2 ( R 0 - 2 R 45 + R 90 ) . R_{p}=\cfrac{1}{2}\left(R_{0}-2~{}R_{45}+R_{90}\right)~{}.
  13. R R
  14. R R
  15. R p R_{p}
  16. R 90 R_{90}
  17. R 45 R_{45}
  18. R 45 R_{45}
  19. R 0 R_{0}
  20. R 90 R_{90}
  21. R R
  22. R 45 R_{45}

Laplace_invariant.html

  1. x y + a x + b y + c , \partial_{x}\,\partial_{y}+a\,\partial_{x}+b\,\partial_{y}+c,\,
  2. a = a ( x , y ) , b = c ( x , y ) , c = c ( x , y ) , a=a(x,y),\ \ b=c(x,y),\ \ c=c(x,y),
  3. a ^ = c - a b - a x and b ^ = c - a b - b y . \hat{a}=c-ab-a_{x}\quad\,\text{and}\quad\hat{b}=c-ab-b_{y}.
  4. A and A ~ A\quad\,\text{and}\quad\tilde{A}
  5. A ~ g = e - φ A ( e φ g ) A φ g . \tilde{A}g=e^{-\varphi}A(e^{\varphi}g)\equiv A_{\varphi}g.
  6. x y + a x + b y + c = { ( x + b ) ( y + a ) - a b - a x + c , ( y + a ) ( x + b ) - a b - b y + c . \partial_{x}\,\partial_{y}+a\,\partial_{x}+b\,\partial_{y}+c=\left\{\begin{% array}[]{c}(\partial_{x}+b)(\partial_{y}+a)-ab-a_{x}+c,\\ (\partial_{y}+a)(\partial_{x}+b)-ab-b_{y}+c.\end{array}\right.
  7. c - a b - a x 0 and/or c - a b - b y 0 , c-ab-a_{x}\neq 0\quad\,\text{and/or}\quad c-ab-b_{y}\neq 0,
  8. c - a b - a x = 0 and c - a b - b y = 0 , c-ab-a_{x}=0\quad\,\text{and}\quad c-ab-b_{y}=0,

Laplace_pressure.html

  1. Δ P P inside - P outside = γ ( 1 R 1 + 1 R 2 ) , \Delta P\equiv P\text{inside}-P\text{outside}=\gamma\left(\frac{1}{R_{1}}+% \frac{1}{R_{2}}\right),
  2. R 1 R_{1}
  3. R 2 R_{2}
  4. γ \gamma
  5. R 1 R_{1}
  6. R 2 R_{2}
  7. Δ P = 2 γ R , \Delta P=\frac{2\gamma}{R},
  8. γ \gamma
  9. σ \sigma
  10. γ \gamma
  11. Δ P \Delta P
  12. Δ P \Delta P
  13. γ \gamma

Large_extra_dimension.html

  1. G F / G N = 10 - 32 G_{F}/G_{N}=10^{-32}
  2. d d
  3. 1 / r 4 1/r^{4}
  4. r d \scriptstyle r\ll d
  5. 1 / r 2 1/r^{2}
  6. r d \scriptstyle r\gg d
  7. d d
  8. ψ ¯ γ μ μ ψ + 1 4 F μ ν F μ ν + ψ ¯ e γ μ A μ ψ \int\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi+{1\over 4}F^{\mu\nu}F_{\mu\nu}+% \bar{\psi}e\gamma^{\mu}A_{\mu}\psi\,
  9. A ψ ¯ F μ ν σ μ ν ψ A\bar{\psi}F^{\mu\nu}\sigma_{\mu\nu}\psi\,
  10. Y μ ν σ μ ν Y^{\mu\nu}\sigma_{\mu\nu}
  11. L ¯ H H L \bar{L}HHL
  12. 10 - 2 10^{-2}
  13. H 2 M = 0.01 eV . \frac{H^{2}}{M}=0.01\,\,\text{eV}.\,
  14. H 1 H\simeq 1
  15. M 10 26 M\simeq 10^{26}
  16. 10 17 \simeq 10^{17}
  17. L ¯ L q ¯ q \bar{L}L\bar{q}q
  18. f π 3 / T e V 2 \scriptstyle{f_{\pi}}^{3}/TeV^{2}

Large_Veblen_ordinal.html

  1. ϕ < m t p l > Ω Ω ( 0 ) \phi_{<}mtpl>{{\Omega^{\Omega}}}(0)
  2. θ ( Ω Ω ) \theta(\Omega^{\Omega})
  3. ψ ( Ω Ω Ω ) \psi(\Omega^{\Omega^{\Omega}})

Larson-Miller_Parameter.html

  1. P ( L . M . ) = T [ log t r + C ] P(L.M.)=T[\log t_{r}+C]
  2. t r t_{r}

Larson–Miller_relation.html

  1. r = A e - Δ H / ( R T ) r=A\cdot e^{-\Delta H/(R\cdot T)}
  2. Δ H \Delta H
  3. l n ( r ) = l n ( A ) - Δ H / ( R T ) ln(r)=ln(A)-\Delta H/(R\cdot T)
  4. Δ H / R = T ( l n ( A ) - l n ( r ) ) \Delta H/R=T\cdot(ln(A)-ln(r))
  5. Δ l / Δ t = A e - Δ H / ( R T ) \Delta l/\Delta t=A^{\prime}\cdot e^{-\Delta H/(R\cdot T)}
  6. ln ( Δ l / Δ t ) = l n ( A ) - Δ H / ( R T ) \ln(\Delta l/\Delta t)=ln(A^{\prime})-\Delta H/(R\cdot T)
  7. Δ H / R = T ( B + l n ( Δ t ) ) \Delta H/R=T\cdot(B+ln(\Delta t))
  8. ln ( A / Δ l ) \ln(A^{\prime}/\Delta l)
  9. L M P = T ( C + l o g ( t ) ) LMP=T\cdot(C+log(t))
  10. Δ L M P = L M P T Δ T + L M P l o g ( t ) Δ l o g ( t ) + \Delta LMP=\frac{\partial LMP}{\partial T}\cdot\Delta T+\frac{\partial LMP}{% \partial log(t)}\cdot\Delta log(t)+...
  11. L M P T = 20 + l o g ( t ) L M P l o g ( t ) = T \frac{\partial LMP}{\partial T}=20+log(t)\qquad\qquad\frac{\partial LMP}{% \partial log(t)}=T
  12. Δ L M P = L M P l o g ( t ) Δ l o g ( t ) = T Δ l o g ( t ) = T 1 = T \Delta LMP=\frac{\partial LMP}{\partial log(t)}\cdot\Delta log(t)=T\cdot\Delta log% (t)=T\cdot 1=T
  13. Δ L M P = T = L M P T Δ T Δ T = Δ L M P L M P T = T ( 20 + l o g ( t ) ) T 20 = 5 % T \Delta LMP=T=\frac{\partial LMP}{\partial T}\cdot\Delta T\qquad\Delta T=\frac{% \Delta LMP}{\frac{\partial LMP}{\partial T}}=\frac{T}{(20+log(t))}\approx\frac% {T}{20}=5\%T

Laser-based_angle-resolved_photoemission_spectroscopy.html

  1. E = h f - ϕ E=hf-\phi
  2. h f hf
  3. ϕ \phi

Latent_semantic_indexing.html

  1. A A
  2. m m
  3. n n
  4. a i j a_{ij}
  5. tf ij \mathrm{tf_{ij}}
  6. a i j a_{ij}
  7. A A
  8. l i j l_{ij}
  9. g i g_{i}
  10. l i j = 1 l_{ij}=1
  11. 0
  12. l i j = tf i j l_{ij}=\mathrm{tf}_{ij}
  13. i i
  14. j j
  15. l i j = log ( tf i j + 1 ) l_{ij}=\log(\mathrm{tf}_{ij}+1)
  16. l i j = ( tf i j max i ( tf i j ) ) + 1 2 l_{ij}=\frac{\Big(\frac{\mathrm{tf}_{ij}}{\max_{i}(\mathrm{tf}_{ij})}\Big)+1}{2}
  17. g i = 1 g_{i}=1
  18. g i = 1 j tf i j 2 g_{i}=\frac{1}{\sqrt{\sum_{j}\mathrm{tf}_{ij}^{2}}}
  19. g i = gf i / df i g_{i}=\mathrm{gf}_{i}/\mathrm{df}_{i}
  20. gf i \mathrm{gf}_{i}
  21. i i
  22. df i \mathrm{df}_{i}
  23. i i
  24. g i = log 2 n 1 + df i g_{i}=\log_{2}\frac{n}{1+\mathrm{df}_{i}}
  25. g i = 1 + j p i j log p i j log n g_{i}=1+\sum_{j}\frac{p_{ij}\log p_{ij}}{\log n}
  26. p i j = tf i j gf i p_{ij}=\frac{\mathrm{tf}_{ij}}{\mathrm{gf}_{i}}
  27. a i j a_{ij}
  28. A A
  29. g i = 1 + j p i j log p i j log n g_{i}=1+\sum_{j}\frac{p_{ij}\log p_{ij}}{\log n}
  30. a i j = g i log ( tf i j + 1 ) a_{ij}=g_{i}\ \log(\mathrm{tf}_{ij}+1)
  31. A A
  32. T T
  33. S S
  34. D D
  35. A T S D T A\approx TSD^{T}
  36. T T T = I r D T D = I r T^{T}T=I_{r}\quad D^{T}D=I_{r}
  37. S 1 , 1 S 2 , 2 S r , r > 0 S i , j = 0 where i j S_{1,1}\geq S_{2,2}\geq\ldots\geq S_{r,r}>0\quad S_{i,j}=0\;\,\text{where}\;i\neq j

Lateral_surface.html

  1. a a

Lattice_(discrete_subgroup).html

  1. S L ( 2 , [ 1 p ] ) S L ( 2 , ) × S L ( 2 , p ) , S = { p , } . SL\left(2,\mathbb{Z}\left[\frac{1}{p}\right]\right)\subset SL(2,\mathbb{R})% \times SL(2,\mathbb{Q}_{p}),S=\{p,\infty\}.
  2. X / Γ X/\Gamma
  3. G \ X G\backslash X

Lattice_phase_equaliser.html

  1. Z o 2 = Z Z Z_{o}^{2}=ZZ^{\prime}
  2. H ( ω ) = Z o - Z Z o + Z H(\omega)=\frac{Z_{o}-Z}{Z_{o}+Z}
  3. Z R 0 = R 0 Z \frac{Z}{R_{0}}=\frac{R_{0}}{Z^{\prime}}
  4. tan φ 2 = - X R 0 \tan\frac{\varphi}{2}=-\frac{X}{R_{0}}
  5. tan φ 2 = - ω ω m \tan\frac{\varphi}{2}=-\frac{\omega}{\omega_{m}}
  6. ω m = 1 L C \omega_{m}=\frac{1}{\sqrt{LC}}

Lattice_problem.html

  1. n \mathbb{Q}^{n}
  2. n \mathbb{Z}^{n}
  3. λ ( L ) \lambda(L)
  4. λ ( L ) = min v L { 𝟎 } v N \lambda(L)=\min_{v\in L\setminus\{\mathbf{0}\}}\|v\|_{N}
  5. N ( v ) = λ ( L ) N(v)=\lambda(L)
  6. γ \gamma
  7. S V P γ SVP_{\gamma}
  8. γ λ ( L ) \gamma\lambda(L)
  9. n n 2 e + o ( n ) n^{\frac{n}{2e}+o(n)}
  10. G a p S V P β GapSVP_{\beta}
  11. β \beta
  12. β \beta
  13. n n
  14. λ ( L ) 1 \lambda(L)\leq 1
  15. λ ( L ) > β \lambda(L)>\beta
  16. G a p S V P ζ , γ GapSVP_{\zeta,\gamma}
  17. ζ , γ \zeta,\gamma
  18. B B
  19. d d
  20. λ ( L ( B ) ) ζ ( n ) \lambda(L(B))\leq\zeta(n)
  21. 1 d ζ ( n ) / γ ( n ) 1\leq d\leq\zeta(n)/\gamma(n)
  22. n n
  23. λ ( L ( B ) ) d \lambda(L(B))\leq d
  24. λ ( L ( B ) ) γ ( n ) . d \lambda(L(B))\geq\gamma(n).d
  25. ζ \zeta
  26. ζ ( n ) > 2 n / 2 \zeta(n)>2^{n/2}
  27. G a p S V P γ GapSVP_{\gamma}
  28. ζ \zeta
  29. γ \gamma
  30. C V P γ CVP_{\gamma}
  31. γ \gamma
  32. C V P γ CVP_{\gamma}
  33. S V P γ SVP_{\gamma}
  34. C V P γ CVP_{\gamma}
  35. S V P γ SVP_{\gamma}
  36. C V P γ CVP_{\gamma}
  37. S V P γ SVP_{\gamma}
  38. B = [ b 1 , b 2 , , b n ] B=[b_{1},b_{2},\ldots,b_{n}]
  39. B i = [ b 1 , , 2 b i , , b n ] B^{i}=[b_{1},\ldots,2b_{i},\ldots,b_{n}]
  40. x i x_{i}
  41. C V P γ ( B i , b i ) CVP_{\gamma}(B^{i},b_{i})
  42. { x i - b i } \{x_{i}-b_{i}\}
  43. 2 log 1 - ϵ ( n ) 2^{\log^{1-\epsilon}(n)}
  44. NP DTIME ( 2 p o l y ( log n ) ) \operatorname{NP}\subseteq\operatorname{DTIME}(2^{poly(\log n)})
  45. ϵ = ( log log n ) c \epsilon=(\log\log n)^{c}
  46. c < 1 / 2 c<1/2
  47. G a p C V P β GapCVP_{\beta}
  48. v v
  49. v v
  50. β \beta
  51. v v
  52. β = 2 O ( n ( log log n ) 2 / log n ) \beta=2^{O(n(\log\log n)^{2}/\log n)}
  53. β = 2 O ( n log log n / log n ) \beta=2^{O(n\log\log n/\log n)}
  54. G a p C V P n GapCVP_{n}
  55. N P c o N P NP\cap coNP
  56. β = n / log n \beta=\sqrt{n/\log n}
  57. c c
  58. β = c n \beta=c\sqrt{n}
  59. N P c o N P NP\cap coNP
  60. β = n o ( 1 / log log n ) \beta=n^{o(1/\log{\log{n}})}
  61. v 1 , v 2 , , v n v_{1},v_{2},\ldots,v_{n}
  62. max v i < max B b i \max\|v_{i}\|<\max_{B}\|b_{i}\|
  63. B = { b 1 , , b n } B=\{b_{1},\ldots,b_{n}\}
  64. γ \gamma
  65. v 1 , v 2 , , v n v_{1},v_{2},\ldots,v_{n}
  66. v i v_{i}
  67. γ λ n ( L ) \gamma\lambda_{n}(L)
  68. λ n ( L ) \lambda_{n}(L)
  69. n n
  70. L L
  71. λ ( L ) / 2 \lambda(L)/2
  72. B B
  73. B B^{\prime}
  74. B B^{\prime}
  75. S B P γ SBP_{\gamma}
  76. γ \gamma

Laue_equations.html

  1. 𝐤 i \mathbf{k}_{i}
  2. 𝐤 o \mathbf{k}_{o}
  3. 𝐤 o - 𝐤 i = 𝚫 𝐤 \mathbf{k}_{o}-\mathbf{k}_{i}=\mathbf{\Delta k}
  4. 𝐚 , 𝐛 , 𝐜 \mathbf{a}\,,\mathbf{b}\,,\mathbf{c}
  5. 𝐚 𝚫 𝐤 = 2 π h \mathbf{a}\cdot\mathbf{\Delta k}=2\pi h
  6. 𝐛 𝚫 𝐤 = 2 π k \mathbf{b}\cdot\mathbf{\Delta k}=2\pi k
  7. 𝐜 𝚫 𝐤 = 2 π l \mathbf{c}\cdot\mathbf{\Delta k}=2\pi l
  8. 𝐆 = h 𝐀 + k 𝐁 + l 𝐂 \mathbf{G}=h\mathbf{A}+k\mathbf{B}+l\mathbf{C}
  9. 𝐆 ( 𝐚 + 𝐛 + 𝐜 ) = 2 π ( h + k + l ) \mathbf{G}\cdot(\mathbf{a}+\mathbf{b}+\mathbf{c})=2\pi(h+k+l)
  10. 𝚫 𝐤 ( 𝐚 + 𝐛 + 𝐜 ) = 2 π ( h + k + l ) \mathbf{\Delta k}\cdot(\mathbf{a}+\mathbf{b}+\mathbf{c})=2\pi(h+k+l)
  11. 𝚫 𝐤 = 𝐆 \mathbf{\Delta k}=\mathbf{G}
  12. 𝐤 o - 𝐤 i = 𝐆 \mathbf{k}_{o}-\mathbf{k}_{i}=\mathbf{G}
  13. 𝐤 o - 𝐤 i \displaystyle\mathbf{k}_{o}-\mathbf{k}_{i}
  14. ( 𝐤 o ) 2 = ( 𝐤 i ) 2 (\mathbf{k}_{o})^{2}=(\mathbf{k}_{i})^{2}
  15. 𝐆 = - 𝐆 \mathbf{G}=-\mathbf{G}
  16. 2 𝐤 i 𝐆 = G 2 2\mathbf{k}_{i}\cdot\mathbf{G}=G^{2}
  17. 2 𝐤 i 𝐆 = G 2 \;2\mathbf{k}_{i}\cdot\mathbf{G}=G^{2}
  18. 2 d sin θ = n λ \;2d\sin\theta=n\lambda

Lauricella's_theorem.html

  1. { u k } \{u_{k}\}
  2. { v k } \{v_{k}\}
  3. { u k } \{u_{k}\}

Law_of_cosines.html

  1. c 2 = a 2 + b 2 - 2 a b cos γ c^{2}=a^{2}+b^{2}-2ab\cos\gamma\,
  2. γ \gamma\,
  3. γ \gamma\,
  4. c 2 = a 2 + b 2 . c^{2}=a^{2}+b^{2}.\,
  5. a 2 = b 2 + c 2 - 2 b c cos α a^{2}=b^{2}+c^{2}-2bc\cos\alpha\,
  6. b 2 = a 2 + c 2 - 2 a c cos β . b^{2}=a^{2}+c^{2}-2ac\cos\beta.\,
  7. A B 2 = C A 2 + C B 2 + 2 ( C A ) ( C H ) . AB^{2}=CA^{2}+CB^{2}+2(CA)(CH)\,.
  8. c = a 2 + b 2 - 2 a b cos γ ; \,c=\sqrt{a^{2}+b^{2}-2ab\cos\gamma}\,;
  9. γ = arccos ( a 2 + b 2 - c 2 2 a b ) ; \,\gamma=\arccos\left(\frac{a^{2}+b^{2}-c^{2}}{2ab}\right)\,;
  10. a = b cos γ ± c 2 - b 2 sin 2 γ . \,a=b\cos\gamma\pm\sqrt{c^{2}-b^{2}\sin^{2}\gamma}\,.
  11. A = ( b cos θ , b sin θ ) , B = ( a , 0 ) , and C = ( 0 , 0 ) . A=(b\cos\theta,\ b\sin\theta),\ B=(a,\ 0),\ \,\text{and}\ C=(0,\ 0)\,.
  12. c = ( a - b cos θ ) 2 + ( 0 - b sin θ ) 2 . c=\sqrt{(a-b\cos\theta)^{2}+(0-b\sin\theta)^{2}}\,.
  13. c 2 \displaystyle c^{2}
  14. c = a cos β + b cos α . c=a\cos\beta+b\cos\alpha\,.
  15. c 2 = a c cos β + b c cos α . c^{2}=ac\cos\beta+bc\cos\alpha.\,
  16. a 2 = a c cos β + a b cos γ , a^{2}=ac\cos\beta+ab\cos\gamma,\,
  17. b 2 = b c cos α + a b cos γ . b^{2}=bc\cos\alpha+ab\cos\gamma.\,
  18. a 2 + b 2 = a c cos β + b c cos α + 2 a b cos γ . a^{2}+b^{2}=ac\cos\beta+bc\cos\alpha+2ab\cos\gamma.\,
  19. a 2 + b 2 - c 2 = - a c cos β - b c cos α + a c cos β + b c cos α + 2 a b cos γ a^{2}+b^{2}-c^{2}=-ac\cos\beta-bc\cos\alpha+ac\cos\beta+bc\cos\alpha+2ab\cos\gamma\,
  20. c 2 = a 2 + b 2 - 2 a b cos γ . c^{2}=a^{2}+b^{2}-2ab\cos\gamma.\,
  21. c 2 = ( b + d ) 2 + h 2 , c^{2}=(b+d)^{2}+h^{2},\,
  22. d 2 + h 2 = a 2 . d^{2}+h^{2}=a^{2}.\,
  23. c 2 = b 2 + 2 b d + d 2 + h 2 . c^{2}=b^{2}+2bd+d^{2}+h^{2}.\,
  24. c 2 = a 2 + b 2 + 2 b d . c^{2}=a^{2}+b^{2}+2bd.\,
  25. d = a cos ( π - γ ) = - a cos γ . d=a\cos(\pi-\gamma)=-a\cos\gamma.\,
  26. c 2 \displaystyle c^{2}
  27. cos 2 γ + sin 2 γ = 1. \cos^{2}\gamma+\sin^{2}\gamma=1.\,
  28. tan α = a sin γ b - a cos γ . \tan\alpha=\frac{a\sin\gamma}{b-a\cos\gamma}.
  29. B F = A E = B C cos B ^ = a cos B ^ \displaystyle BF=AE=BC\cos\hat{B}=a\cos\hat{B}
  30. A D × B C + A B × D C = A C × B D \displaystyle AD\times BC+AB\times DC=AC\times BD
  31. a 2 + c 2 = b 2 . a^{2}+c^{2}=b^{2}.\quad
  32. γ = π / 2 - γ \scriptstyle\gamma^{\prime}\,=\,\pi/2-\gamma
  33. γ = γ - π / 2 \scriptstyle\gamma^{\prime}\,=\,\gamma-\pi/2
  34. a 2 + b 2 = c 2 + 2 a b cos γ . \,a^{2}+b^{2}=c^{2}+2ab\cos\gamma\,.
  35. a 2 + b 2 - 2 a b cos ( γ ) = c 2 . \,a^{2}+b^{2}-2ab\cos(\gamma)=c^{2}.
  36. c 2 = b 2 + h 2 . c^{2}=b^{2}+h^{2}.\,
  37. h 2 = a ( a - 2 b cos γ ) . h^{2}=a(a-2b\cos\gamma).\,
  38. c 2 = b 2 + a ( a - 2 b cos γ ) . c^{2}=b^{2}+a(a-2b\cos\gamma).\,
  39. b 2 = c 2 + h 2 . b^{2}=c^{2}+h^{2}.\,
  40. h 2 = a ( 2 b cos γ - a ) . h^{2}=a(2b\cos\gamma-a).\,
  41. b 2 = c 2 + a ( 2 b cos γ - a ) . b^{2}=c^{2}+a(2b\cos\gamma-a)\,.
  42. c 2 - a 2 = b ( b + 2 a cos ( π - γ ) ) = b ( b - 2 a cos γ ) , \begin{aligned}\displaystyle c^{2}-a^{2}&\displaystyle{}=b(b+2a\cos(\pi-\gamma% ))\\ &\displaystyle{}=b(b-2a\cos\gamma),\end{aligned}
  43. sin α a = sin β b = sin γ c . \frac{\sin\alpha}{a}=\frac{\sin\beta}{b}=\frac{\sin\gamma}{c}.
  44. c sin γ = b sin β , \frac{c}{\sin\gamma}=\frac{b}{\sin\beta},
  45. c sin γ = a sin α , \frac{c}{\sin\gamma}=\frac{a}{\sin\alpha},
  46. α + β + γ = 180. \alpha+\beta+\gamma=180.
  47. c sin γ = b sin ( α + γ ) , \frac{c}{\sin\gamma}=\frac{b}{\sin(\alpha+\gamma)},
  48. c sin γ = a sin α , \frac{c}{\sin\gamma}=\frac{a}{\sin\alpha},
  49. sin ( α + γ ) = sin α cos γ + sin γ cos α \sin(\alpha+\gamma)=\sin\alpha\cos\gamma+\sin\gamma\cos\alpha
  50. c ( sin α cos γ + sin γ cos α ) = b sin γ , c\cdot(\sin\alpha\cos\gamma+\sin\gamma\cos\alpha)=b\sin\gamma,
  51. c sin α = a sin γ . c\cdot\sin\alpha=a\sin\gamma.
  52. cos γ \cos\gamma
  53. c ( sin α + tan γ cos α ) = b tan γ , c\cdot(\sin\alpha+\tan\gamma\cos\alpha)=b\tan\gamma,
  54. c sin α cos γ = a tan γ . c\cdot\frac{\sin\alpha}{\cos\gamma}=a\tan\gamma.
  55. c sin α b - c cos α = tan γ \frac{c\sin\alpha}{b-c\cos\alpha}=\tan\gamma
  56. cos γ = ± 1 1 + tan 2 γ \cos\gamma=\pm\frac{1}{\sqrt{1+\tan^{2}\gamma}}
  57. c sin α ( ± 1 + c 2 sin 2 α ( b - c cos α ) 2 ) = a c sin α b - c cos α . c\sin\alpha\left(\pm\sqrt{1+\frac{c^{2}\sin^{2}\alpha}{(b-c\cos\alpha)^{2}}}% \right)=a\frac{c\sin\alpha}{b-c\cos\alpha}.
  58. ( b - c cos α ) 2 (b-c\cos\alpha)^{2}
  59. ( b - c cos α ) 2 + c 2 sin 2 α = a 2 . (b-c\cos\alpha)^{2}+c^{2}\sin^{2}\alpha=a^{2}.
  60. c 2 cos 2 α + b 2 - 2 b c cos α + c 2 sin 2 α = a 2 . c^{2}\cos^{2}\alpha+b^{2}-2bc\cos\alpha+c^{2}\sin^{2}\alpha=a^{2}.
  61. a 2 = b 2 + c 2 - 2 b c cos α . a^{2}=b^{2}+c^{2}-2bc\cos\alpha.
  62. b c = b c cos θ \vec{b}\cdot\vec{c}=\|\vec{b}\|\|\vec{c}\|\cos\theta
  63. a = b - c , \vec{a}=\vec{b}-\vec{c}\,,
  64. a 2 \displaystyle\|\vec{a}\|^{2}
  65. a 2 \displaystyle\|\vec{a}\|^{2}
  66. cos γ = 1 - c 2 2 a 2 \cos\gamma=1-\frac{c^{2}}{2a^{2}}
  67. c 2 = 2 a 2 ( 1 - cos γ ) . c^{2}=2a^{2}(1-\cos\gamma).\;
  68. β γ ^ , \scriptstyle{\widehat{\beta\gamma},}
  69. α 2 = β 2 + γ 2 + δ 2 - 2 ( β γ cos ( β γ ^ ) + γ δ cos ( γ δ ^ ) + δ β cos ( δ β ^ ) ) . \alpha^{2}=\beta^{2}+\gamma^{2}+\delta^{2}-2\left(\beta\gamma\cos\left(% \widehat{\beta\gamma}\right)+\gamma\delta\cos\left(\widehat{\gamma\delta}% \right)+\delta\beta\cos\left(\widehat{\delta\beta}\right)\right).\,
  70. cos a \displaystyle\cos a
  71. cosh a = cosh b cosh c - sinh b sinh c cos A \cosh a=\cosh b\cosh c-\sinh b\sinh c\cos A\,
  72. cos A = - cos B cos C + sin B sin C cosh a . \cos A=-\cos B\cos C+\sin B\sin C\cosh a.\,

Lawrence–Krammer_representation.html

  1. B n B_{n}
  2. P n P_{n}
  3. B n B_{n}
  4. C 2 P n C_{2}P_{n}
  5. H 1 C 2 P n n + 1 H_{1}C_{2}P_{n}\simeq\mathbb{Z}^{n+1}
  6. H 1 C 2 P n H_{1}C_{2}P_{n}
  7. B n B_{n}
  8. q , t q,t
  9. C 2 P n C_{2}P_{n}
  10. π 1 C 2 P n 2 q , t \pi_{1}C_{2}P_{n}\to\mathbb{Z}^{2}\langle q,t\rangle
  11. C 2 P n ¯ \overline{C_{2}P_{n}}
  12. P n P_{n}
  13. P n P_{n}
  14. C 2 P n C_{2}P_{n}
  15. C 2 P n ¯ \overline{C_{2}P_{n}}
  16. B n B_{n}
  17. H 2 C 2 P n ¯ , H_{2}\overline{C_{2}P_{n}},
  18. t ± , q ± \mathbb{Z}\langle t^{\pm},q^{\pm}\rangle
  19. H 2 C 2 P n ¯ H_{2}\overline{C_{2}P_{n}}
  20. t ± , q ± \mathbb{Z}\langle t^{\pm},q^{\pm}\rangle
  21. ( n 2 ) n\choose 2
  22. H 2 C 2 P n ¯ H_{2}\overline{C_{2}P_{n}}
  23. v j , k v_{j,k}
  24. 1 j < k n 1\leq j<k\leq n
  25. σ i \sigma_{i}
  26. σ i v j , k = { v j , k i { j - 1 , j , k - 1 , k } , q v i , k + ( q 2 - q ) v i , j + ( 1 - q ) v j , k i = j - 1 v j + 1 , k i = j k - 1 , q v j , i + ( 1 - q ) v j , k - ( q 2 - q ) t v i , k i = k - 1 j , v j , k + 1 i = k , - t q 2 v j , k i = j = k - 1. \sigma_{i}\cdot v_{j,k}=\left\{\begin{array}[]{lr}v_{j,k}&i\notin\{j-1,j,k-1,k% \},\\ qv_{i,k}+(q^{2}-q)v_{i,j}+(1-q)v_{j,k}&i=j-1\\ v_{j+1,k}&i=j\neq k-1,\\ qv_{j,i}+(1-q)v_{j,k}-(q^{2}-q)tv_{i,k}&i=k-1\neq j,\\ v_{j,k+1}&i=k,\\ -tq^{2}v_{j,k}&i=j=k-1.\end{array}\right.
  27. q , t q,t
  28. n ( n - 1 ) 2 \frac{n(n-1)}{2}
  29. v i , j , v k , l = - ( 1 - t ) ( 1 + q t ) ( q - 1 ) 2 t - 2 q - 3 { - q 2 t 2 ( q - 1 ) i = k < j < l or i < k < j = l - ( q - 1 ) k = i < l < j or k < i < j = l t ( q - 1 ) i < j = k < l q 2 t ( q - 1 ) k < l = i < j - t ( q - 1 ) 2 ( 1 + q t ) i < k < j < l ( q - 1 ) 2 ( 1 + q t ) k < i < l < j ( 1 - q t ) ( 1 + q 2 t ) k = i , j = l 0 otherwise \langle v_{i,j},v_{k,l}\rangle=-(1-t)(1+qt)(q-1)^{2}t^{-2}q^{-3}\left\{\begin{% array}[]{lr}-q^{2}t^{2}(q-1)&i=k<j<l\,\text{ or }i<k<j=l\\ -(q-1)&k=i<l<j\,\text{ or }k<i<j=l\\ t(q-1)&i<j=k<l\\ q^{2}t(q-1)&k<l=i<j\\ -t(q-1)^{2}(1+qt)&i<k<j<l\\ (q-1)^{2}(1+qt)&k<i<l<j\\ (1-qt)(1+q^{2}t)&k=i,j=l\\ 0&\,\text{otherwise}\\ \end{array}\right.

Lax–Friedrichs_method.html

  1. u ( x , t ) u(x,t)
  2. u t + a u x = 0 u_{t}+au_{x}=0\,
  3. b x c , 0 t d b\leq x\leq c,\;0\leq t\leq d
  4. u ( x , 0 ) = u 0 ( x ) u(x,0)=u_{0}(x)\,
  5. u ( b , t ) = u b ( t ) u(b,t)=u_{b}(t)\,
  6. u ( c , t ) = u c ( t ) . u(c,t)=u_{c}(t).\,
  7. ( b , c ) × ( 0 , d ) (b,c)\times(0,d)
  8. Δ x \Delta x
  9. x x
  10. Δ t \Delta t
  11. t t
  12. u i n = u ( x i , t n ) with x i = b + i Δ x , t n = n Δ t for i = 0 , , N , n = 0 , , M , u_{i}^{n}=u(x_{i},t^{n})\,\text{ with }x_{i}=b+i\,\Delta x,\,t^{n}=n\,\Delta t% \,\text{ for }i=0,\ldots,N,\,n=0,\ldots,M,
  13. N = c - b Δ x , M = d Δ t N=\frac{c-b}{\Delta x},\,M=\frac{d}{\Delta t}
  14. u i n + 1 - 1 2 ( u i + 1 n + u i - 1 n ) Δ t + a u i + 1 n - u i - 1 n 2 Δ x = 0 \frac{u_{i}^{n+1}-\frac{1}{2}(u_{i+1}^{n}+u_{i-1}^{n})}{\Delta t}+a\frac{u_{i+% 1}^{n}-u_{i-1}^{n}}{2\,\Delta x}=0
  15. u i n + 1 , u_{i}^{n+1},
  16. u i n + 1 = 1 2 ( u i + 1 n + u i - 1 n ) - a Δ t 2 Δ x ( u i + 1 n - u i - 1 n ) u_{i}^{n+1}=\frac{1}{2}(u_{i+1}^{n}+u_{i-1}^{n})-a\frac{\Delta t}{2\,\Delta x}% (u_{i+1}^{n}-u_{i-1}^{n})\,
  17. u i 0 = u 0 ( x i ) u_{i}^{0}=u_{0}(x_{i})
  18. u 0 n = u b ( t n ) u_{0}^{n}=u_{b}(t^{n})
  19. u N n = u c ( t n ) . u_{N}^{n}=u_{c}(t^{n}).
  20. f f
  21. u t + ( f ( u ) ) x = 0. u_{t}+(f(u))_{x}=0.
  22. f ( u ) = a u f(u)=au
  23. u u
  24. m m
  25. u i n + 1 = 1 2 ( u i + 1 n + u i - 1 n ) - Δ t 2 Δ x ( f ( u i + 1 n ) - f ( u i - 1 n ) ) . u_{i}^{n+1}=\frac{1}{2}(u_{i+1}^{n}+u_{i-1}^{n})-\frac{\Delta t}{2\,\Delta x}(% f(u_{i+1}^{n})-f(u_{i-1}^{n})).
  26. u i n + 1 = u i n - Δ t Δ x ( f ^ i + 1 / 2 n - f ^ i - 1 / 2 n ) , u_{i}^{n+1}=u^{n}_{i}-\frac{\Delta t}{\Delta x}\left(\hat{f}^{n}_{i+1/2}-\hat{% f}^{n}_{i-1/2}\right),
  27. f ^ i - 1 / 2 n = 1 2 ( f i - 1 + f i ) - Δ x 2 Δ t ( u i n - u i - 1 n ) . \hat{f}^{n}_{i-1/2}=\frac{1}{2}\left(f_{i-1}+f_{i}\right)-\frac{\Delta x}{2% \Delta t}\left(u^{n}_{i}-u^{n}_{i-1}\right).
  28. u i n u^{n}_{i}
  29. u i - 1 n u^{n}_{i-1}
  30. f ^ i - 1 / 2 n \hat{f}^{n}_{i-1/2}
  31. u 0 ( x ) , u b ( t ) , u c ( t ) u_{0}(x),\,u_{b}(t),\,u_{c}(t)
  32. | a Δ t Δ x | 1. \left|a\frac{\Delta t}{\Delta x}\right|\leq 1.

Leaching_(chemistry).html

  1. t = ρ B r s 2 6 D e b M B c A b [ 1 - 3 ( r c r s ) 2 + 2 ( r c r s ) 3 ] t=\frac{\rho_{B}r_{s}^{2}}{6D_{e}bM_{B}c_{A_{b}}}\left[1-3\left(\frac{r_{c}}{r% _{s}}\right)^{2}+2\left(\frac{r_{c}}{r_{s}}\right)^{3}\right]

Leak_detection.html

  1. M ˙ I \dot{M}_{I}
  2. M ˙ O \dot{M}_{O}
  3. M ˙ I - M ˙ O \dot{M}_{I}-\dot{M}_{O}
  4. M ˙ I \dot{M}_{I}
  5. M ˙ O \dot{M}_{O}
  6. γ \gamma
  7. Hypothesis H 0 : No leak \,\text{Hypothesis }H_{0}:\,\text{ No leak}
  8. Hypothesis H 1 : Leak \,\text{Hypothesis }H_{1}:\,\text{ Leak}
  9. M ˙ ^ I \hat{\dot{M}}_{I}
  10. M ˙ ^ O \hat{\dot{M}}_{O}
  11. p I p_{I}
  12. T I T_{I}
  13. p O p_{O}
  14. T O T_{O}
  15. M ˙ I \dot{M}_{I}
  16. M ˙ O \dot{M}_{O}
  17. x = M ˙ I - M ˙ ^ I x=\dot{M}_{I}-\hat{\dot{M}}_{I}
  18. y = M ˙ O - M ˙ ^ O y=\dot{M}_{O}-\hat{\dot{M}}_{O}

Leak_noise_correlator.html

  1. ( f g ) ( t ) = def - f * ( τ ) g ( t + τ ) d τ , (f\star g)(t)\ \stackrel{\mathrm{def}}{=}\int_{-\infty}^{\infty}f^{*}(\tau)\ g% (t+\tau)\,d\tau,

Least_absolute_deviations.html

  1. f ( x i ) y i . f(x_{i})\approx y_{i}.
  2. S = i = 1 n | y i - f ( x i ) | . S=\sum_{i=1}^{n}|y_{i}-f(x_{i})|.
  3. S ( β , b ) = i | 𝐱 i β + b - y i | S(\mathbf{\beta},b)=\sum_{i}|\mathbf{x}^{\prime}_{i}\mathbf{\beta}+b-y_{i}|
  4. 𝐱 1 β + b - y 1 k \mathbf{x}^{\prime}_{1}\mathbf{\beta}+b-y_{1}\leq k
  5. β \mathbf{\beta}
  6. Minimize i = 1 n | y i - a 0 - a 1 x i 1 - a 2 x i 2 - - a k x i k | \,\text{Minimize}\sum_{i=1}^{n}|y_{i}-a_{0}-a_{1}x_{i1}-a_{2}x_{i2}-\cdots-a_{% k}x_{ik}|
  7. a 0 , , a k a_{0},...,a_{k}
  8. Minimize i = 1 n u i \,\text{Minimize}\sum_{i=1}^{n}u_{i}
  9. a 0 , , a k a_{0},...,a_{k}
  10. u 1 , , u n u_{1},...,u_{n}
  11. u i y i - a 0 - a 1 x i 1 - a 2 x i 2 - - a k x i k for i = 1 , , n u_{i}\geq y_{i}-a_{0}-a_{1}x_{i1}-a_{2}x_{i2}-\cdots-a_{k}x_{ik}\,\ \,\ \,\ \,% \ \,\ \,\text{for}\,\ i=1,...,n
  12. u i - [ y i - a 0 - a 1 x i 1 - a 2 x i 2 - - a k x i k ] for i = 1 , , n . u_{i}\geq-[y_{i}-a_{0}-a_{1}x_{i1}-a_{2}x_{i2}-\cdots-a_{k}x_{ik}]\,\ \,\ \,% \text{for}\,\ i=1,...,n.
  13. u i u_{i}
  14. | y i - a 0 - a 1 x i 1 - a 2 x i 2 - - a k x i k | |y_{i}-a_{0}-a_{1}x_{i1}-a_{2}x_{i2}-\cdots-a_{k}x_{ik}|

Lebedev–Milin_inequality.html

  1. k 0 β k z k = exp ( k 1 α k z k ) \sum_{k\geq 0}\beta_{k}z^{k}=\exp\left(\sum_{k\geq 1}\alpha_{k}z^{k}\right)
  2. k = 0 | β k | 2 exp ( k = 1 k | α k | 2 ) , \sum_{k=0}^{\infty}|\beta_{k}|^{2}\leq\exp\left(\sum_{k=1}^{\infty}k|\alpha_{k% }|^{2}\right),
  3. k = 0 n | β k | 2 ( n + 1 ) exp ( 1 n + 1 m = 1 n k = 1 m ( k | α k | 2 - 1 / k ) ) , \sum_{k=0}^{n}|\beta_{k}|^{2}\leq(n+1)\exp\left(\frac{1}{n+1}\sum_{m=1}^{n}% \sum_{k=1}^{m}(k|\alpha_{k}|^{2}-1/k)\right),
  4. | β n | 2 exp ( k = 1 n ( k | α k | 2 - 1 / k ) ) . |\beta_{n}|^{2}\leq\exp\left(\sum_{k=1}^{n}(k|\alpha_{k}|^{2}-1/k)\right).

Leda_Atomica.html

  1. φ = 1 + 5 2 \varphi=\frac{1+\sqrt{5}}{2}

Lee-Kesler_method.html

  1. ln P r = f ( 0 ) + ω f ( 1 ) \ln P_{r}=f^{(0)}+\omega\cdot f^{(1)}
  2. f ( 0 ) = 5.92714 - 6.09648 T r - 1.28862 ln T r + 0.169347 T r 6 f^{(0)}=5.92714-\frac{6.09648}{T_{r}}-1.28862\cdot\ln T_{r}+0.169347\cdot T_{r% }^{6}
  3. f ( 1 ) = 15.2518 - 15.6875 T r - 13.4721 ln T r + 0.43577 T r 6 f^{(1)}=15.2518-\frac{15.6875}{T_{r}}-13.4721\cdot\ln T_{r}+0.43577\cdot T_{r}% ^{6}
  4. P r = P P c P_{r}=\frac{P}{P_{c}}
  5. T r = T T c T_{r}=\frac{T}{T_{c}}

Lee_distance.html

  1. x 1 x 2 x n x_{1}x_{2}\dots x_{n}
  2. y 1 y 2 y n y_{1}y_{2}\dots y_{n}
  3. i = 1 n min ( | x i - y i | , q - | x i - y i | ) . \sum_{i=1}^{n}\min(|x_{i}-y_{i}|,q-|x_{i}-y_{i}|).
  4. 4 \mathbb{Z}_{4}
  5. 2 2 \mathbb{Z}_{2}^{2}

Lee–Yang_theorem.html

  1. H = - J j k S j S k - z j S j H=-\sum J_{jk}S_{j}S_{k}-\sum z_{j}S_{j}
  2. Z = e - H d μ 1 ( S 1 ) d μ N ( S N ) Z=\int e^{-H}d\mu_{1}(S_{1})\cdots d\mu_{N}(S_{N})
  3. e b S 2 d | μ j ( S ) | < , b . \int e^{bS^{2}}d|\mu_{j}(S)|<\infty,\,\forall b\in\mathbb{R}.
  4. e h S d μ j ( S ) 0 , h + := { z | z > 0 } \int e^{hS}d\mu_{j}(S)\neq 0,\,\forall h\in\mathbb{H}_{+}:=\{z\in\mathbb{C}|% \Re{z}>0\}
  5. Z ( { z j } ) 0 , z j + Z(\{z_{j}\})\neq 0,\,\forall z_{j}\in\mathbb{H}_{+}
  6. exp ( - λ cosh ( S ) ) d S \exp(-\lambda\cosh(S))\,dS
  7. exp ( - λ S 4 - b S 2 ) d S \exp(-\lambda S^{4}-bS^{2})\,dS
  8. exp ( - λ S 6 - a S 4 - b S 2 ) d S \exp(-\lambda S^{6}-aS^{4}-bS^{2})\,dS

Lefschetz_duality.html

  1. D M : H p ( M , A , Z ) H n - p ( M , B , Z ) . D_{M}:H^{p}(M,A,Z)\to H_{n-p}(M,B,Z).

Legendre–Clebsch_condition.html

  1. a b L ( t , x , x ) d t . \int_{a}^{b}L(t,x,x^{\prime})\,dt.\,
  2. 0 L x x ( t , x ( t ) , x ( t ) ) , t [ a , b ] 0\geq L_{x^{\prime}x^{\prime}}(t,x(t),x^{\prime}(t)),\,\forall t\in[a,b]
  3. H u = 0 \frac{\partial H}{\partial u}=0
  4. 2 H u 2 > 0 \frac{\partial^{2}H}{\partial u^{2}}>0

Leggett–Garg_inequality.html

  1. Q = ± 1 Q=\pm 1
  2. t 1 < t 2 < t 3 t_{1}<t_{2}<t_{3}
  3. C 13 C_{13}
  4. t 1 t_{1}
  5. t 3 t_{3}
  6. C 13 = 1 N r = 1 N Q r ( t 1 ) Q r ( t 3 ) = 1. C_{13}=\frac{1}{N}\sum_{r=1}^{N}Q_{r}(t_{1})Q_{r}(t_{3})=1.
  7. t 2 t_{2}
  8. C 12 = C 23 = 1 C_{12}=C_{23}=1
  9. t 1 = ± 1 t_{1}=\pm 1
  10. ± 1 \pm 1
  11. t 2 t_{2}
  12. t 3 t_{3}
  13. C 12 = C 23 = - 1 C_{12}=C_{23}=-1
  14. t 1 t_{1}
  15. t 3 t_{3}
  16. t 1 t_{1}
  17. Q ( t 1 ) Q(t_{1})
  18. Q ( t 2 ) Q(t_{2})
  19. Q ( t 2 ) Q(t_{2})
  20. Q ( t 3 ) Q(t_{3})
  21. Q ( t 1 ) Q(t_{1})
  22. Q ( t 2 ) Q(t_{2})
  23. C 12 = C 23 = 0 C_{12}=C_{23}=0
  24. Q = ± 1 Q=\pm 1
  25. t 1 t_{1}
  26. ± 1 \pm 1
  27. t 3 t_{3}
  28. t 2 t_{2}
  29. K K
  30. K = C 12 + C 23 - C 13 K=C_{12}+C_{23}-C_{13}
  31. K = 1 , - 3 , K=1,-3,
  32. - 1 -1
  33. t 1 t_{1}
  34. t 3 t_{3}
  35. K = C 12 + C 23 - C 13 1 K=C_{12}+C_{23}-C_{13}\leq 1
  36. K = 1 N r = 0 N ( Q ( t 1 ) Q ( t 2 ) + Q ( t 2 ) Q ( t 3 ) - Q ( t 1 ) Q ( t 3 ) ) r . K=\frac{1}{N}\sum_{r=0}^{N}\left(Q(t_{1})Q(t_{2})+Q(t_{2})Q(t_{3})-Q(t_{1})Q(t% _{3})\right)_{r}.
  37. r r
  38. K = C 12 + C 23 + C 34 - C 14 2 K=C_{12}+C_{23}+C_{34}-C_{14}\leq 2
  39. Q ( start ) Q ( end ) \langle Q(\,\text{start})Q(\,\text{end})\rangle

Lehmer_random_number_generator.html

  1. X k + 1 = g X k mod n X_{k+1}=g\cdot X_{k}~{}~{}\bmod~{}~{}n
  2. 0 {}_{0}
  3. 31 {}^{31}
  4. 31 {}_{31}
  5. 5 {}^{5}
  6. 31 {}_{31}
  7. 16 {}^{16}
  8. 4 {}_{4}
  9. 4 {}_{4}
  10. 48 {}^{48}
  11. 0 {}_{0}
  12. k {}_{k}
  13. n \mathbb{Z}_{n}
  14. φ ( n ) \varphi(n)

Leonard–Merritt_mass_estimator.html

  1. M ( r ) = 16 3 π G R ( 2 V R 2 + V T 2 ) . \langle M(r)\rangle={16\over 3\pi G}\langle R\left(2V_{R}^{2}+V_{T}^{2}\right)\rangle.
  2. M ( r ) M(r)
  3. r r
  4. R R
  5. V R V_{R}
  6. V T V_{T}
  7. G G
  8. M T M_{T}
  9. M T = 32 3 π G R ( 2 V R 2 + V T 2 ) . M_{T}={32\over 3\pi G}\langle R\left(2V_{R}^{2}+V_{T}^{2}\right)\rangle.
  10. M 0 M_{0}
  11. M 0 = 16 3 π G R ( 2 V R 2 + V T 2 ) . M_{0}={16\over 3\pi G}\langle R\left(2V_{R}^{2}+V_{T}^{2}\right)\rangle.

LeRoy_radius.html

  1. V ( r ) = 𝔇 - C n / r n V(r)=\mathfrak{D}-C_{n}/r^{n}
  2. G ( v ) = 𝔇 - X n ( C n ) [ v 𝔇 - v ] 2 n n - 2 . G(v)=\mathfrak{D}-X_{n}(C_{n})[v_{\mathfrak{D}}-v]^{\frac{2n}{n-2}}.
  3. X n ( C n ) X_{n}(C_{n})
  4. v 𝔇 v_{\mathfrak{D}}
  5. R LR = 2 [ r A 2 1 / 2 + r B 2 1 / 2 ] R_{\mathrm{LR}}=2[\langle r_{A}^{2}\rangle^{1/2}+\langle r_{B}^{2}\rangle^{1/2}]
  6. r > R L R r>R_{LR}
  7. r < R L R r<R_{LR}

Leverage_(statistics).html

  1. i t h i^{th}
  2. h i i = ( H ) i i h_{ii}=(H)_{ii}
  3. i t h i^{th}
  4. H = X ( X X ) - 1 X H=X(X^{\prime}X)^{-1}X^{\prime}
  5. h i i = y ^ i y i , h_{ii}=\frac{\partial\hat{y}_{i}}{\partial y_{i}},
  6. y ^ i \hat{y}_{i}
  7. y i {y}_{i}
  8. 0 h i i 1 0\leq h_{ii}\leq 1
  9. H 2 = X ( X X ) - 1 X X ( X X ) - 1 X = X I ( X X ) - 1 X = H H^{2}=X(X^{\prime}X)^{-1}X^{\prime}X(X^{\prime}X)^{-1}X^{\prime}=XI(X^{\prime}% X)^{-1}X^{\prime}=H
  10. H H
  11. h i i = h i i 2 + i j h i j 2 0 h_{ii}=h_{ii}^{2}+\sum_{i\neq j}h_{ij}^{2}\geq 0
  12. h i i h i i 2 h i i 1 h_{ii}\geq h_{ii}^{2}\implies h_{ii}\leq 1
  13. Y = X β + ϵ Y=X\beta+\epsilon
  14. v a r ( ϵ ) = σ 2 I var(\epsilon)=\sigma^{2}I
  15. v a r ( e i ) = ( 1 - h i i ) σ 2 var(e_{i})=(1-h_{ii})\sigma^{2}
  16. e i = Y i - Y ^ i e_{i}=Y_{i}-\hat{Y}_{i}
  17. ϵ \epsilon
  18. I - H I-H
  19. v a r ( e ) = v a r ( ( I - H ) Y ) = ( I - H ) v a r ( Y ) ( I - H ) = σ 2 ( I - H ) 2 = σ 2 ( I - H ) var(e)=var((I-H)Y)=(I-H)var(Y)(I-H)^{\prime}=\sigma^{2}(I-H)^{2}=\sigma^{2}(I-H)
  20. v a r ( e i ) = ( 1 - h i i ) σ 2 var(e_{i})=(1-h_{ii})\sigma^{2}

Levi-Civita_parallelogramoid.html

  1. | A B | 2 = | A B | 2 + 8 3 R ( X , Y ) X , Y + higher order terms |A^{\prime}B^{\prime}|^{2}=|AB|^{2}+\frac{8}{3}\langle R(X,Y)X,Y\rangle+\,% \text{higher order terms}

Levich_equation.html

  1. I L = ( 0.620 ) n F A D 2 3 w 1 2 v - 1 6 C I_{L}=(0.620)nFAD^{\frac{2}{3}}w^{\frac{1}{2}}v^{\frac{-1}{6}}C

Lévy_family_of_graphs.html

  1. ε > 0 \varepsilon>0
  2. lim n α ( G n , ε ) = 0 \lim_{n\longrightarrow\infty}\alpha\left(G_{n},\varepsilon\right)=0
  3. α ( G , ε ) = max { 1 - | A ( ε D ) | | G | : A G , | A | > | G | / 2 } . \alpha(G,\varepsilon)=\max\left\{1-\frac{\left|A_{(\varepsilon D)}\right|}{|G|% }\,:\,A\subseteq G,|A|>|G|/2\right\}.
  4. ϵ \epsilon
  5. ϵ D \epsilon D

Lichnerowicz_formula.html

  1. D 2 ψ = * ψ + 1 4 Sc ψ D^{2}\psi=\nabla^{*}\nabla\psi+\frac{1}{4}\operatorname{Sc}\psi
  2. * \nabla^{*}\nabla
  3. ϕ \phi
  4. D A * D A ϕ = A * A ϕ + 1 4 R ϕ + 1 2 F A + , ϕ . D_{A}^{*}D_{A}\phi=\nabla_{A}^{*}\nabla_{A}\phi+\frac{1}{4}R\phi+\frac{1}{2}% \langle F_{A}^{+},\phi\rangle.
  5. D A D_{A}
  6. D A : Γ ( W + ) Γ ( W - ) , D_{A}:\Gamma(W^{+})\to\Gamma(W^{-}),
  7. A \nabla_{A}
  8. A : Γ ( W + ) Γ ( W + T M * ) \nabla_{A}:\Gamma(W^{+})\to\Gamma(W^{+}\otimes T_{M}^{*})
  9. R R
  10. F A + F_{A}^{+}
  11. , \langle,\rangle

Lie_product_formula.html

  1. e A + B = lim N ( e A / N e B / N ) N , e^{A+B}=\lim_{N\rightarrow\infty}(e^{A/N}e^{B/N})^{N},
  2. e x + y = e x e y e^{x+y}=e^{x}e^{y}\,

Lie_sphere_geometry.html

  1. ( x 0 , x 1 , x 2 , x 3 , x 4 ) ( y 0 , y 1 , y 2 , y 3 , y 4 ) = - x 0 y 0 + x 1 y 1 + x 2 y 2 + x 3 y 4 + x 4 y 3 . (x_{0},x_{1},x_{2},x_{3},x_{4})\cdot(y_{0},y_{1},y_{2},y_{3},y_{4})=-x_{0}y_{0% }+x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{4}+x_{4}y_{3}.
  2. ( x 0 , x 1 , x n , x n + 1 , x n + 2 ) ( y 0 , y 1 , y n , y n + 1 , y n + 2 ) (x_{0},x_{1},\ldots x_{n},x_{n+1},x_{n+2})\cdot(y_{0},y_{1},\ldots y_{n},y_{n+% 1},y_{n+2})
  3. = - x 0 y 0 + x 1 y 1 + + x n y n + x n + 1 y n + 2 + x n + 2 y n + 1 . =-x_{0}y_{0}+x_{1}y_{1}+\cdots+x_{n}y_{n}+x_{n+1}y_{n+2}+x_{n+2}y_{n+1}.
  4. i , j = 1 5 a i j x i x j = 0 \sum_{i,j=1}^{5}a_{ij}x_{i}x_{j}=0

Lieb's_square_ice_constant.html

  1. 1 + 1 1 + 1 1 + 1 5 + 1 1 + 1 4 + 1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{5+\cfrac{1}{1+\cfrac{1}{4+\ddots}}}}}
  2. 8 3 9 \frac{8\sqrt{3}}{9}
  3. lim n f ( n ) n 2 = ( 4 3 ) 3 2 = 8 3 9 = 1.5396007 \lim_{n\to\infty}\sqrt[n^{2}]{f(n)}=\left(\frac{4}{3}\right)^{\frac{3}{2}}=% \frac{8\sqrt{3}}{9}=1.5396007\dots

Light-cone_coordinates.html

  1. x x_{\perp}
  2. d s 2 = - d t 2 + δ i j d x i d x j ds^{2}=-dt^{2}+\delta_{ij}dx^{i}dx^{j}
  3. i , j = 1 , , d i,j=1,\dots,d
  4. d s 2 = - 2 d x + d x - + δ i j d x i d x j ds^{2}=-2dx^{+}dx^{-}+\delta_{ij}dx^{i}dx^{j}
  5. i , j = 1 , , d - 1 i,j=1,\dots,d-1
  6. x + = t + x 2 x^{+}=\frac{t+x}{\sqrt{2}}
  7. x - = t - x 2 x^{-}=\frac{t-x}{\sqrt{2}}
  8. x + e + β x + x^{+}\to e^{+\beta}x^{+}
  9. x - e - β x - x^{-}\to e^{-\beta}x^{-}
  10. x i x i x^{i}\to x^{i}
  11. x x_{\perp}
  12. x + x + x^{+}\to x^{+}
  13. x - x - + δ i j α i x j + α 2 2 x + x^{-}\to x^{-}+\delta_{ij}\alpha^{i}x^{j}+\frac{\alpha^{2}}{2}x^{+}
  14. x i x i + α i x + x^{i}\to x^{i}+\alpha^{i}x^{+}
  15. x + x + + δ i j α i x j + α 2 2 x - x^{+}\to x^{+}+\delta_{ij}\alpha^{i}x^{j}+\frac{\alpha^{2}}{2}x^{-}
  16. x - x - x^{-}\to x^{-}
  17. x i x i + α i x - x^{i}\to x^{i}+\alpha^{i}x^{-}

Light_cone_gauge.html

  1. A + = 0 A^{+}=0
  2. A + ( x 0 , x 1 , x 2 , x 3 ) = A 0 ( x 0 , x 1 , x 2 , x 3 ) + A 3 ( x 0 , x 1 , x 2 , x 3 ) A^{+}(x^{0},x^{1},x^{2},x^{3})=A^{0}(x^{0},x^{1},x^{2},x^{3})+A^{3}(x^{0},x^{1% },x^{2},x^{3})
  3. X + ( σ , τ ) = p + τ X^{+}(\sigma,\tau)=p^{+}\tau
  4. p + p^{+}
  5. τ \tau

Light_dark_matter.html

  1. 2 \approx 2
  2. m 2 / M 4 \approx m^{2}/M^{4}
  3. 2 \sim 2

Light_front_quantization.html

  1. x + c t + z x^{+}\equiv ct+z
  2. x - c t - z x^{-}\equiv ct-z
  3. t t
  4. z z
  5. c c
  6. x x
  7. y y
  8. x = ( x , y ) \vec{x}_{\perp}=(x,y)
  9. t t
  10. z z
  11. z z
  12. x - x^{-}
  13. x + x^{+}
  14. t t
  15. H ψ = E ψ H\psi=E\psi
  16. H H
  17. - 2 2 m 2 -\frac{\hbar^{2}}{2m}\nabla^{2}
  18. V ( r ) V(\vec{r})
  19. ψ \psi
  20. r \vec{r}
  21. E E
  22. p ¯ i = ( p i + , p i ) \underline{p}_{i}=(p_{i}^{+},\vec{p}_{\perp i})
  23. i i
  24. p i + p i 2 + m i 2 + p i z p_{i}^{+}\equiv\sqrt{p_{i}^{2}+m_{i}^{2}}+p_{iz}
  25. p i = ( p i x , p i y ) \vec{p}_{\perp i}=(p_{ix},p_{iy})
  26. m i m_{i}
  27. p i - p i 2 + m i 2 - p i z p_{i}^{-}\equiv\sqrt{p_{i}^{2}+m_{i}^{2}}-p_{iz}
  28. m i 2 = p i + p i - - p i 2 m_{i}^{2}=p_{i}^{+}p_{i}^{-}-\vec{p}_{\perp i}^{2}
  29. H H
  30. 𝒫 - \mathcal{P}^{-}
  31. P ¯ ( P + , P ) \underline{P}\equiv(P^{+},\vec{P}_{\perp})
  32. P - P^{-}
  33. ( M 2 + P 2 ) / P + (M^{2}+P_{\perp}^{2})/P^{+}
  34. M M
  35. 𝒫 - ψ = M 2 + P 2 P + ψ \mathcal{P}^{-}\psi=\frac{M^{2}+P_{\perp}^{2}}{P^{+}}\psi
  36. t t
  37. x b j x_{bj}
  38. x x
  39. x x
  40. log Q 2 \log Q^{2}
  41. x i = k i + / P + x_{i}={k^{+}_{i}/P^{+}}
  42. x x
  43. J z J^{z}
  44. x x
  45. T T
  46. ϕ 4 \phi^{4}
  47. [ K j , P k ] = i δ j k H [K^{j},P^{k}]=i\delta^{jk}H
  48. P k P^{k}
  49. K j K^{j}
  50. H = H 0 + V H=H_{0}+V
  51. x + = x 0 + x n ^ = 0 , x^{+}=x^{0}+\vec{x}\cdot\hat{n}=0,
  52. x 0 = c t x^{0}=ct
  53. n ^ = z ^ \hat{n}=\hat{z}
  54. x - = x 0 - x n ^ and x := x - n ^ ( n ^ x ) . x^{-}=x^{0}-\vec{x}\cdot\hat{n}\quad\mbox{and}~{}\quad\vec{x}_{\perp}:=\vec{x}% -\hat{n}(\hat{n}\cdot\vec{x}).
  55. x x
  56. y y
  57. x y = 1 2 ( x - y + + x + y - ) - x y . x\cdot y=\frac{1}{2}(x^{-}y^{+}+x^{+}y^{-})-\vec{x}_{\perp}\cdot\vec{y}_{\perp}.
  58. P - := H - P n ^ P^{-}:=H-\vec{P}\cdot\hat{n}
  59. J := J - n ^ ( n ^ J ) \vec{J}_{\perp}:=\vec{J}-\hat{n}(\hat{n}\cdot\vec{J})
  60. P - P^{-}
  61. P + := H + P n ^ P^{+}:=H+\vec{P}\cdot\hat{n}
  62. P := P - n ^ ( n ^ P ) \vec{P}_{\perp}:=\vec{P}-\hat{n}(\hat{n}\cdot\vec{P})
  63. J 3 := n ^ J J_{3}:=\hat{n}\cdot\vec{J}
  64. n ^ \hat{n}
  65. K 3 := n ^ K K_{3}:=\hat{n}\cdot\vec{K}
  66. E 1 E_{1}
  67. E 2 E_{2}
  68. ( p + , p ) ( p + , p ) = ( v + p + , p + v x + ) , (p^{+},\vec{p}_{\perp})\to(p^{+\prime},\vec{p}_{\perp}^{\prime})=(v^{+}p^{+},% \vec{p}_{\perp}+\vec{v}_{\perp}x^{+}),
  69. P + P^{+}
  70. S S
  71. j j = 1 2 ϵ j k l Λ - 1 ( p ) k Λ - 1 μ ( p ) l J μ ν ν , j^{j}=\frac{1}{2}\sum\epsilon_{jkl}\Lambda^{-1}(p)^{k}{}_{\mu}\Lambda^{-1}(p)^% {l}{}_{\nu}J^{\mu\nu},
  72. Λ - 1 ( p ) μ ν \Lambda^{-1}(p)^{\mu}{}_{\nu}
  73. p μ p^{\mu}
  74. ( m , 0 ) (m,\vec{0})
  75. j \vec{j}
  76. S U ( 2 ) SU(2)
  77. Λ - 1 ( P ) μ ν \Lambda^{-1}(P)^{\mu}{}_{\nu}
  78. Λ - 1 ( P ) k μ \Lambda^{-1}(P)^{k}{}_{\mu}
  79. n ^ \hat{n}
  80. P + P^{+}
  81. P + P^{+}
  82. P i + P_{i}^{+}
  83. P + = i P i + ) P^{+}=\sum_{i}P_{i}^{+})
  84. P + P^{+}
  85. P i + P_{i}^{+}
  86. P + P^{+}
  87. P + = 0 P^{+}=0
  88. ( - P 3 H ) (-P^{3}\to H)
  89. P + P^{+}
  90. S S
  91. ( P + = 0 ) (P^{+}=0)
  92. S S
  93. x + x^{+}
  94. x + 0 x^{+}\not=0
  95. x + x^{+}
  96. P - P^{-}
  97. U ( R ) U(R)
  98. U 0 ( R ) U ( R ) U_{0}(R)U^{\dagger}(R)
  99. S S
  100. n ^ = R n ^ \hat{n}^{\prime}=R\hat{n}
  101. S S
  102. U ( R ) U(R)
  103. U ( R ) = Ω ± ( n ^ ) Ω ± ( R n ^ ) U 0 ( R ) . U(R)=\Omega_{\pm}(\hat{n})\Omega^{\dagger}_{\pm}(R\hat{n})U_{0}(R).
  104. S S
  105. P - P^{-}
  106. S S
  107. S U ( 2 ) SU(2)
  108. n ^ \hat{n}
  109. J = 1 P + [ 1 2 ( P + - P - ) ( n ^ × E ) - ( n ^ × P ) ( K n ^ ) + P ( n ^ j ) + M j ] , \vec{J}_{\perp}=\frac{1}{P^{+}}[\frac{1}{2}(P^{+}-P^{-})(\hat{n}\times\vec{E}_% {\perp})-(\hat{n}\times\vec{P}_{\perp})(\vec{K}\cdot\hat{n})+\vec{P}_{\perp}(% \hat{n}\cdot\vec{j})+M\vec{j}_{\perp}],
  110. P - P^{-}
  111. M M
  112. j \vec{j}_{\perp}
  113. U 0 ( Λ , a ) U_{0}(\Lambda,a)
  114. U ( Λ , a ) U(\Lambda,a)
  115. U ( g ) = U 0 ( g ) U(g)=U_{0}(g)
  116. g g
  117. P - = ( P 2 + M 2 ) / P + P^{-}=(\vec{P}_{\perp}^{2}+M^{2})/P^{+}
  118. P - P^{-}
  119. n ^ {\hat{n}}
  120. x b j = Q 2 2 M ν x_{bj}=\frac{Q^{2}}{2M\nu}
  121. p X \ell p\to\ell^{\prime}X
  122. Q 2 = - q 2 Q^{2}=-q^{2}
  123. ν = E - E \nu=E_{\ell}-E_{\ell^{\prime}}
  124. P P\to\infty
  125. z ^ \hat{z}
  126. x b j x_{bj}
  127. x = k z P z x=\frac{k^{z}}{P^{z}}
  128. t t
  129. τ = t + z / c \tau=t+z/c
  130. 1 / [ E i n i t i a l - E i n t e r m e d i a t e + i ϵ ] {1/[E_{initial}-E_{intermediate}+i\epsilon]}
  131. E i n t e r m e d i a t e = j E j = j m 2 + k j 2 E_{intermediate}=\sum_{j}E_{j}=\sum_{j}\sqrt{m^{2}+{\vec{k}}^{2}_{j}}
  132. P P
  133. z ^ \hat{z}
  134. P P
  135. 2 P / [ 2 - j [ k 2 + m 2 x i ] j + i ϵ ] 2P/[\mathcal{M}^{2}-\sum_{j}\big[{k^{2}_{\perp}+\frac{m^{2}}{x_{i}}}\big]_{j}+% i\epsilon]
  136. 2 \mathcal{M}^{2}
  137. 1 P \frac{1}{P}
  138. t t
  139. e + e - γ e^{+}e^{-}\gamma
  140. 1 / P 2 1/P^{2}
  141. P P\to\infty
  142. + +
  143. + +
  144. P P\to\infty
  145. x + = c t + z x^{+}=ct+z
  146. x - = c t - z x^{-}=ct-z
  147. x , y , z x,y,z
  148. z z
  149. x x
  150. y y
  151. z z
  152. x , y x,y
  153. x , y , z x,y,z
  154. ω x = ω 0 c t - ω x = ω 0 t - ω x x - ω y y - ω z z = 0 \omega\cdot x=\omega_{0}ct-\vec{\omega}\cdot\vec{x}=\omega_{0}t-\omega_{x}x-% \omega_{y}y-\omega_{z}z=0
  155. c t + z = 0 ct+z=0
  156. x = ( c t , x ) x=(ct,\vec{x})
  157. ω = ( ω 0 , ω ) \omega=(\omega_{0},\vec{\omega})
  158. ω 2 = ω 0 2 - ω 2 = 0 \omega^{2}=\omega_{0}^{2}-\vec{\omega}^{2}=0
  159. ω = ( 1 / c , 0 , 0 , - 1 / c ) \omega=(1/c,0,0,-1/c)
  160. ω \omega
  161. σ = ω x \sigma=\omega\cdot x
  162. x + x^{+}
  163. t t
  164. k i k_{i}
  165. k i 2 = m i 2 k_{i}^{2}=m_{i}^{2}
  166. x - x^{-}
  167. ω \omega
  168. ω \omega
  169. ω = ( 1 / c , 0 , 0 , - 1 / c ) \omega=(1/c,0,0,-1/c)
  170. k i , x i \vec{k}_{\perp i},x_{i}
  171. s , t s,t
  172. ω \omega
  173. ω \omega
  174. ω \omega
  175. k \vec{k}
  176. l l
  177. Y l m ( k ^ ) Y_{lm}(\hat{\vec{k}})
  178. ψ l m ( k ) = f ( k ) Y l m ( k ^ ) \psi_{lm}(\vec{k})=f(k)Y_{lm}(\hat{k})
  179. k ^ = k / k \hat{k}=\vec{k}/k
  180. f ( k ) f(k)
  181. k = | k | k=|\vec{k}|
  182. J = - i [ k × k ] \vec{J}=-i[\vec{k}\times\partial\vec{k}]
  183. ψ l m ( k , n ^ ) = f 1 ( k , k n ^ ) Y l m ( k ^ ) + f 2 ( k , k n ^ ) Y l m ( n ^ ) , \psi_{lm}(\vec{k},\hat{n})=f_{1}(k,\vec{k}\cdot\hat{n})Y_{lm}(\hat{k})+f_{2}(k% ,\vec{k}\cdot\hat{n})Y_{lm}(\hat{n}),
  184. n ^ = ω / | ω | \hat{n}=\vec{\omega}/|\vec{\omega}|
  185. f 1 , 2 ( k , k n ^ ) f_{1,2}(k,\vec{k}\cdot\hat{n})
  186. k k
  187. k n ^ \vec{k}\cdot\hat{n}
  188. k k
  189. k n ^ \vec{k}\cdot\hat{n}
  190. k \vec{k}
  191. n ^ \hat{n}
  192. k k
  193. ω \omega
  194. Y l m ( n ^ ) \propto Y_{lm}(\hat{n})
  195. J = - i [ k × k ] - i [ n ^ × n ^ ] \vec{J}=-i[\vec{k}\times\partial\vec{k}]-i[\hat{n}\times\partial\hat{n}]
  196. J = - i [ k × k ] - i [ n ^ × n ^ ] + s 1 + s 2 . \vec{J}=-i[\vec{k}\times\partial\vec{k}]-i[\hat{n}\times\partial\hat{n}]+\vec{% s}_{1}+\vec{s}_{2}.
  197. f 1 , 2 f_{1,2}
  198. k n ^ \vec{k}\cdot\hat{n}
  199. n ^ \hat{n}
  200. k \vec{k}
  201. k k
  202. A = ( n ^ J ) 2 A=(\hat{n}\cdot\vec{J})^{2}
  203. J 2 , J z \vec{J}^{2},J_{z}
  204. a a
  205. A A
  206. ψ = ψ l m a ( k , n ^ ) \psi=\psi_{lma}(\vec{k},\hat{n})
  207. l l
  208. l + 1 l+1
  209. ψ l m a ( k , n ^ ) \psi_{lma}(\vec{k},\hat{n})
  210. a a
  211. - i [ n ^ × n ^ ] -i[\hat{n}\times\partial\hat{n}]
  212. S S
  213. D D
  214. ϕ 1 + 1 4 \phi^{4}_{1+1}
  215. P + P^{+}
  216. z z
  217. x x
  218. y y
  219. z z
  220. 5 {}_{5}
  221. ζ \zeta
  222. Q 2 Q^{2}
  223. ϕ 4 \phi^{4}
  224. ϕ 4 \phi^{4}
  225. ϕ 4 \phi^{4}

Like_terms.html

  1. a x + b x ax+bx\,\!
  2. x x\,\!
  3. ( a + b ) x (a+b)x\,\!
  4. a + b a+b\,\!
  5. a a\,\!
  6. b b\,\!
  7. a = 5 a=5\,\!
  8. b = 3 b=3\,\!
  9. 5 x + 3 x 5x+3x\,\!
  10. ( 5 + 3 ) x (5+3)x\,\!
  11. 8 x 8x\,\!
  12. 5 x + 3 x = 8 x 5x+3x=8x\,\!
  13. 3 ( 4 x 2 y - 6 y ) + 7 x 2 y - 3 y 2 + 2 ( 8 y - 4 y 2 - 4 x 2 y ) 3(4x^{2}y-6y)+7x^{2}y-3y^{2}+2(8y-4y^{2}-4x^{2}y)\,\!
  14. 12 x 2 y - 18 y + 7 x 2 y - 3 y 2 + 16 y - 8 y 2 - 8 x 2 y 12x^{2}y-18y+7x^{2}y-3y^{2}+16y-8y^{2}-8x^{2}y\,\!
  15. x 2 y , x^{2}y,\,\!
  16. y 2 , y^{2},\,\!
  17. y . y.\,\!
  18. 11 x 2 y - 2 y - 11 y 2 11x^{2}y-2y-11y^{2}\,\!

Limaçon_trisectrix.html

  1. r = a sin 3 2 θ sin 1 2 θ = a ( 3 cos 2 1 2 θ - sin 2 1 2 θ ) = a ( 1 + 2 cos θ ) r=a\frac{\sin\tfrac{3}{2}\theta}{\sin\tfrac{1}{2}\theta}=a(3\cos^{2}\tfrac{1}{% 2}\theta-\sin^{2}\tfrac{1}{2}\theta)=a(1+2\cos\theta)
  2. r = 2 a cos θ 3 r=2a\cos{\theta\over 3}
  3. r = a sin θ 3 r=a\sin{\theta\over 3}

Limiting_density_of_discrete_points.html

  1. H ( X ) = - p ( x ) log p ( x ) d x . H(X)=-\int p(x)\log p(x)\,dx.
  2. n n
  3. { x i } \{x_{i}\}
  4. n n\to\infty
  5. m ( x ) m(x)
  6. lim n 1 n ( number of points in a < x < b ) = a b m ( x ) d x \lim_{n\to\infty}\frac{1}{n}\,(\mbox{number of points in }~{}a<x<b)=\int_{a}^{% b}m(x)\,dx
  7. H ( X ) = - p ( x ) log p ( x ) m ( x ) d x . H(X)=-\int p(x)\log\frac{p(x)}{m(x)}\,dx.
  8. m ( x ) m(x)
  9. m ( x ) m(x)
  10. m ( x ) m(x)
  11. p ( x ) p(x)

Lindeberg's_condition.html

  1. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  2. X k : Ω , k X_{k}:\Omega\to\mathbb{R},\,\,k\in\mathbb{N}
  3. 𝔼 [ X k ] = μ k \mathbb{E}\,[X_{k}]=\mu_{k}
  4. Var [ X k ] = σ k 2 \mathrm{Var}\,[X_{k}]=\sigma_{k}^{2}
  5. s n 2 := k = 1 n σ k 2 . s_{n}^{2}:=\sum_{k=1}^{n}\sigma_{k}^{2}.
  6. X k X_{k}
  7. lim n 1 s n 2 k = 1 n 𝔼 [ ( X k - μ k ) 2 𝟏 { | X k - μ k | > ε s n } ] = 0 \lim_{n\to\infty}\frac{1}{s_{n}^{2}}\sum_{k=1}^{n}\mathbb{E}\big[(X_{k}-\mu_{k% })^{2}\cdot\mathbf{1}_{\{|X_{k}-\mu_{k}|>\varepsilon s_{n}\}}\big]=0
  8. ε > 0 \varepsilon>0
  9. Z n := k = 1 n ( X k - μ k ) s n Z_{n}:=\frac{\sum_{k=1}^{n}\left(X_{k}-\mu_{k}\right)}{s_{n}}
  10. n . n\to\infty.
  11. max k = 1 , , n σ k 2 s n 2 0 , as n , \max_{k=1,\ldots,n}\frac{\sigma_{k}^{2}}{s_{n}^{2}}\to 0,\quad\,\text{ as }n% \to\infty,
  12. max k = 1 , , n σ k 2 s n 2 0 \max_{k=1,\ldots,n}\frac{\sigma^{2}_{k}}{s_{n}^{2}}\to 0
  13. n n\to\infty
  14. X k X_{k}
  15. 1 k n 1\leq k\leq n
  16. s n 2 s_{n}^{2}
  17. n n

Lindemann_index.html

  1. q i = 1 N - 1 j i r i j 2 - r i j 2 r i j q_{i}=\frac{1}{N-1}\sum_{j\neq i}\frac{\sqrt{\langle r_{ij}^{2}\rangle-\langle r% _{ij}\rangle^{2}}}{\langle r_{ij}\rangle}

Line_coordinates.html

  1. l 1 x + m 1 y + 1 = 0 l_{1}x+m_{1}y+1=0\,
  2. l 2 x + m 2 y + 1 = 0. l_{2}x+m_{2}y+1=0.\,
  3. x = m 1 - m 2 l 1 m 2 - l 2 m 1 , y = - l 1 - l 2 l 1 m 2 - l 2 m 1 . x=\frac{m_{1}-m_{2}}{l_{1}m_{2}-l_{2}m_{1}},\,y=-\frac{l_{1}-l_{2}}{l_{1}m_{2}% -l_{2}m_{1}}.
  4. | l 1 m 1 1 l 2 m 2 1 l 3 m 3 1 | = 0. \begin{vmatrix}l_{1}&m_{1}&1\\ l_{2}&m_{2}&1\\ l_{3}&m_{3}&1\end{vmatrix}=0.
  5. ( m 1 n 2 - m 2 n 1 , l 2 n 1 - l 1 n 2 , l 1 m 2 - l 2 m 1 ) . (m_{1}n_{2}-m_{2}n_{1},\,l_{2}n_{1}-l_{1}n_{2},\,l_{1}m_{2}-l_{2}m_{1}).\,
  6. | l 1 m 1 n 1 l 2 m 2 n 2 l 3 m 3 n 3 | = 0. \begin{vmatrix}l_{1}&m_{1}&n_{1}\\ l_{2}&m_{2}&n_{2}\\ l_{3}&m_{3}&n_{3}\end{vmatrix}=0.
  7. ( y 1 z 2 - y 2 z 1 , x 2 z 1 - x 1 z 2 , x 1 y 2 - x 2 y 1 ) . (y_{1}z_{2}-y_{2}z_{1},\,x_{2}z_{1}-x_{1}z_{2},\,x_{1}y_{2}-x_{2}y_{1}).\,
  8. x 1 y 2 - x 2 y 1 , x 1 z 2 - x 2 z 1 , y 1 z 2 - y 2 z 1 x_{1}y_{2}-x_{2}y_{1},\,x_{1}z_{2}-x_{2}z_{1},\,y_{1}z_{2}-y_{2}z_{1}
  9. x 1 y 2 - x 2 y 1 , x 1 z 2 - x 1 z 2 , y 1 z 2 - y 2 z 1 , x 1 w 2 - x 2 w 1 , y 1 w 2 - y 2 w 1 , z 1 w 2 - z 2 w 1 . x_{1}y_{2}-x_{2}y_{1},\,x_{1}z_{2}-x_{1}z_{2},\,y_{1}z_{2}-y_{2}z_{1},\,x_{1}w% _{2}-x_{2}w_{1},\,y_{1}w_{2}-y_{2}w_{1},\,z_{1}w_{2}-z_{2}w_{1}.
  10. z = ( tan θ 2 ) ( 1 + s ϵ ) z=(\tan\frac{\theta}{2})(1+s\epsilon)
  11. z = ( tan θ 2 ) ( cosh s + j sinh s ) z=(\tan\frac{\theta}{2})(\cosh s+j\sinh s)
  12. z = ( tanh d 2 ) ( sinh s + j cosh s ) z=(\tanh\frac{d}{2})(\sinh s+j\cosh s)

Line_integral.html

  1. W = 𝐅 · 𝐬 W=\mathbf{F}·\mathbf{s}
  2. C f d s = a b f ( 𝐫 ( t ) ) | 𝐫 ( t ) | d t . \int\limits_{C}f\,ds=\int_{a}^{b}f\left(\mathbf{r}(t)\right)\,\,|\mathbf{r}^{% \prime}(t)|\,dt.
  3. a < b a<b
  4. I I
  5. I = lim Δ s i 0 i = 1 n f ( 𝐫 ( t i ) ) Δ s i . I=\lim_{\Delta s_{i}\rightarrow 0}\sum_{i=1}^{n}f(\mathbf{r}(t_{i}))\Delta s_{% i}.
  6. Δ s i = | 𝐫 ( t i + Δ t ) - 𝐫 ( t i ) | | 𝐫 ( t i ) | Δ t . \Delta s_{i}=|\mathbf{r}(t_{i}+\Delta t)-\mathbf{r}(t_{i})|\approx|\mathbf{r}^% {\prime}(t_{i})|\Delta t.
  7. I = lim Δ t 0 i = 1 n f ( 𝐫 ( t i ) ) | 𝐫 ( t i ) | Δ t I=\lim_{\Delta t\rightarrow 0}\sum_{i=1}^{n}f(\mathbf{r}(t_{i}))|\mathbf{r}^{% \prime}(t_{i})|\Delta t
  8. I = a b f ( 𝐫 ( t ) ) | 𝐫 ( t ) | d t . I=\int_{a}^{b}f(\mathbf{r}(t))|\mathbf{r}^{\prime}(t)|\,dt.
  9. C 𝐅 ( 𝐫 ) d 𝐫 = a b 𝐅 ( 𝐫 ( t ) ) 𝐫 ( t ) d t . \int\limits_{C}\mathbf{F}(\mathbf{r})\cdot\,d\mathbf{r}=\int_{a}^{b}\mathbf{F}% (\mathbf{r}(t))\cdot\mathbf{r}^{\prime}(t)\,dt.
  10. I = lim Δ t 0 i = 1 n 𝐅 ( 𝐫 ( t i ) ) Δ 𝐫 i I=\lim_{\Delta t\rightarrow 0}\sum_{i=1}^{n}\mathbf{F}(\mathbf{r}(t_{i}))\cdot% \Delta\mathbf{r}_{i}
  11. Δ 𝐫 i = 𝐫 ( t i + Δ t ) - 𝐫 ( t i ) 𝐫 ( t i ) Δ t \Delta\mathbf{r}_{i}=\mathbf{r}(t_{i}+\Delta t)-\mathbf{r}(t_{i})\approx% \mathbf{r}^{\prime}(t_{i})\Delta t
  12. I = lim Δ t 0 i = 1 n 𝐅 ( 𝐫 ( t i ) ) 𝐫 ( t i ) Δ t I=\lim_{\Delta t\rightarrow 0}\sum_{i=1}^{n}\mathbf{F}(\mathbf{r}(t_{i}))\cdot% \mathbf{r}^{\prime}(t_{i})\Delta t
  13. G = 𝐅 , \nabla G=\mathbf{F},
  14. d G ( 𝐫 ( t ) ) d t = G ( 𝐫 ( t ) ) 𝐫 ( t ) = 𝐅 ( 𝐫 ( t ) ) 𝐫 ( t ) \frac{dG(\mathbf{r}(t))}{dt}=\nabla G(\mathbf{r}(t))\cdot\mathbf{r}^{\prime}(t% )=\mathbf{F}(\mathbf{r}(t))\cdot\mathbf{r}^{\prime}(t)
  15. C 𝐅 ( 𝐫 ) d 𝐫 = a b 𝐅 ( 𝐫 ( t ) ) 𝐫 ( t ) d t = a b d G ( 𝐫 ( t ) ) d t d t = G ( 𝐫 ( b ) ) - G ( 𝐫 ( a ) ) . \int_{C}\mathbf{F}(\mathbf{r})\cdot\,d\mathbf{r}=\int_{a}^{b}\mathbf{F}(% \mathbf{r}(t))\cdot\mathbf{r}^{\prime}(t)\,dt=\int_{a}^{b}\frac{dG(\mathbf{r}(% t))}{dt}\,dt=G(\mathbf{r}(b))-G(\mathbf{r}(a)).
  16. F ( x , y ) = ( P ( x , y ) , Q ( x , y ) ) F(x,y)=(P(x,y),Q(x,y))
  17. C 𝐅 ( 𝐫 ) d 𝐫 t = a b - Q ( x , y ) d x + P ( x , y ) d y = a b ( P ( 𝐫 ( t ) ) , Q ( 𝐫 ( t ) ) ) ( r 2 ( t ) , - r 1 ( t ) ) d t . \int\limits_{C}\mathbf{F}(\mathbf{r})\cdot\,d\mathbf{r}^{t}=\int_{a}^{b}-Q(x,y% )dx+P(x,y)dy=\int_{a}^{b}(P(\mathbf{r}(t)),Q(\mathbf{r}(t)))\cdot(r^{\prime}_{% 2}(t),-r^{\prime}_{1}(t))dt.
  18. r ( t ) = ( r 1 ( t ) , r 2 ( t ) ) r(t)=(r_{1}(t),r_{2}(t))
  19. L U L\subset U
  20. γ : [ a , b ] L \gamma:[a,b]\to L
  21. γ ( t ) = x ( t ) + i y ( t ) . \gamma(t)=x(t)+iy(t).
  22. L f ( z ) d z \int_{L}f(z)\,dz
  23. k = 1 n f ( γ ( t k ) ) [ γ ( t k ) - γ ( t k - 1 ) ] = k = 1 n f ( γ k ) Δ γ k . \sum_{k=1}^{n}f(\gamma(t_{k}))[\gamma(t_{k})-\gamma(t_{k-1})]=\sum_{k=1}^{n}f(% \gamma_{k})\Delta\gamma_{k}.
  24. γ \gamma
  25. L f ( z ) d z = a b f ( γ ( t ) ) γ ( t ) d t . \int_{L}f(z)\,dz=\int_{a}^{b}f(\gamma(t))\,\gamma\,^{\prime}(t)\,dt.
  26. L L
  27. L f ( z ) d z \oint_{L}f(z)\,dz
  28. L L
  29. d z ¯ \overline{dz}
  30. L f d z ¯ := L f ¯ d z ¯ = a b f ( γ ( t ) ) γ ( t ) ¯ d t . \int_{L}f\overline{dz}:=\overline{\int_{L}\overline{f}dz}=\int_{a}^{b}f(\gamma% (t))\,\overline{\gamma^{\prime}(t)}\,dt.
  31. L f ( z ) d z = 0 2 π 1 e i t i e i t d t = i 0 2 π e - i t e i t d t = i 0 2 π d t = i ( 2 π - 0 ) = 2 π i . \begin{aligned}\displaystyle\oint_{L}f(z)\,dz&\displaystyle=\int_{0}^{2\pi}{1% \over e^{it}}ie^{it}\,dt=i\int_{0}^{2\pi}e^{-it}e^{it}\,dt\\ &\displaystyle=i\int_{0}^{2\pi}\,dt=i(2\pi-0)=2\pi i.\end{aligned}
  32. 𝐫 ( t ) = ( x ( t ) , y ( t ) ) \mathbf{r}(t)=(x(t),y(t))
  33. f ( z ) = u ( z ) + i v ( z ) f(z)=u(z)+iv(z)
  34. L f ( z ) ¯ d z = L f ¯ d x + i L f ¯ d y = L ( u , v ) d 𝐫 + i L ( - v , u ) d 𝐫 , \int_{L}\overline{f(z)}\,dz=\int_{L}\bar{f}\,dx+i\int_{L}\bar{f}\,dy=\int_{L}(% u,v)\cdot d\mathbf{r}+i\int_{L}(-v,u)\cdot d\mathbf{r},
  35. γ \gamma
  36. 𝐫 \mathbf{r}
  37. L L
  38. 1 2 i L z ¯ d z \frac{1}{2i}\int_{L}\overline{z}\;dz

Line_plot_survey.html

  1. n = A × 10 B × D n=\frac{A\times 10}{B\times D}

Linear-nonlinear-Poisson_cascade_model.html

  1. 𝐱 \mathbf{x}
  2. 𝐤 \mathbf{k}
  3. 𝐱 \mathbf{x}
  4. f f
  5. P ( spike ) f ( 𝐤 𝐱 ) P(\textrm{spike})\propto f(\mathbf{k}\cdot\mathbf{x})
  6. P ( y spikes ) = ( Δ λ ) y y ! e - Δ λ P(y\textrm{~{}spikes})=\frac{\left(\Delta\lambda\right)^{y}}{y!}e^{-\Delta\lambda}
  7. λ = f ( 𝐤 𝐱 ) \lambda=f(\mathbf{k}\cdot\mathbf{x})
  8. Δ \Delta
  9. 𝐤 𝟏 , 𝐤 𝟐 , , 𝐤 𝐧 \mathbf{k_{1}},\mathbf{k_{2}},\ldots,\mathbf{k_{n}}
  10. P ( spike ) f ( 𝐤 𝟏 𝐱 , 𝐤 𝟐 𝐱 , , 𝐤 𝐧 𝐱 ) P(\textrm{spike})\propto f(\mathbf{k_{1}}\!\cdot\!\mathbf{x},\;\mathbf{k_{2}}% \!\cdot\!\mathbf{x},\;\ldots,\;\mathbf{k_{n}}\!\cdot\!\mathbf{x})
  11. P ( spike ) f ( K 𝐱 ) , P(\textrm{spike})\propto f(K\mathbf{x}),
  12. K K
  13. 𝐤 𝐢 \mathbf{k_{i}}
  14. { k i } \{{k_{i}}\}
  15. f f
  16. f f
  17. f f

Linear_flow_on_the_torus.html

  1. 𝕋 n = S 1 × S 1 × × S 1 n \mathbb{T}^{n}=\underbrace{S^{1}\times S^{1}\times\cdots\times S^{1}}_{n}
  2. d θ 1 d t = ω 1 , d θ 2 d t = ω 2 , , d θ n d t = ω n . \frac{d\theta_{1}}{dt}=\omega_{1},\quad\frac{d\theta_{2}}{dt}=\omega_{2},\quad% \cdots,\quad\frac{d\theta_{n}}{dt}=\omega_{n}.
  3. Φ ω t ( θ 1 , θ 2 , , θ n ) = ( θ 1 + ω 1 t , θ 2 + ω 2 t , , θ n + ω n t ) mod 2 π . \Phi_{\omega}^{t}(\theta_{1},\theta_{2},\dots,\theta_{n})=(\theta_{1}+\omega_{% 1}t,\theta_{2}+\omega_{2}t,\dots,\theta_{n}+\omega_{n}t)\mod 2\pi.

Linear_inequality.html

  1. a x + b y < c and a x + b y c ax+by<c\,\text{ and }ax+by\geq c
  2. f ( x ¯ ) < b f(\bar{x})<b
  3. f ( x ¯ ) b f(\bar{x})\leq b
  4. x ¯ = ( x 1 , x 2 , , x n ) \bar{x}=(x_{1},x_{2},\ldots,x_{n})
  5. a 1 x 1 + a 2 x 2 + + a n x n < b a_{1}x_{1}+a_{2}x_{2}+\cdots+a_{n}x_{n}<b
  6. a 1 x 1 + a 2 x 2 + + a n x n b a_{1}x_{1}+a_{2}x_{2}+\cdots+a_{n}x_{n}\leq b
  7. x 1 , x 2 , , x n x_{1},x_{2},...,x_{n}
  8. a 1 , a 2 , , a n a_{1},a_{2},...,a_{n}
  9. g ( x ) < 0 g(x)<0\,
  10. g ( x ) 0 g(x)\leq 0
  11. a 0 + a 1 x 1 + a 2 x 2 + + a n x n < 0 a_{0}+a_{1}x_{1}+a_{2}x_{2}+\cdots+a_{n}x_{n}<0
  12. a 0 + a 1 x 1 + a 2 x 2 + + a n x n 0 a_{0}+a_{1}x_{1}+a_{2}x_{2}+\cdots+a_{n}x_{n}\leq 0
  13. a 11 x 1 \displaystyle a_{11}x_{1}
  14. x 1 , x 2 , , x n x_{1},\ x_{2},...,x_{n}
  15. a 11 , a 12 , , a m n a_{11},\ a_{12},...,\ a_{mn}
  16. b 1 , b 2 , , b m b_{1},\ b_{2},...,b_{m}
  17. A x b Ax\leq b

Linear_motion.html

  1. x x
  2. t t
  3. x 1 \,x_{1}
  4. x 2 \,x_{2}
  5. Δ x = x 2 - x 1 \Delta x=x_{2}-x_{1}
  6. θ \theta
  7. m s - 1 ms^{-1}
  8. Δ x \Delta x
  9. Δ t \Delta t
  10. 𝐯 𝐚𝐯 = Δ x Δ t = x 2 - x 1 t 2 - t 1 \mathbf{v_{av}}=\frac{\Delta x}{\Delta t}=\frac{x_{2}-x_{1}}{t_{2}-t_{1}}
  11. t 1 t_{1}
  12. x 1 x_{1}
  13. t 2 t_{2}
  14. x 2 x_{2}
  15. 𝐯 = lim Δ t 0 Δ x Δ t \mathbf{v}=\lim_{\Delta t\to 0}{\Delta x\over\Delta t}
  16. = d x d t =\frac{dx}{dt}
  17. v v
  18. v = | 𝐯 | = | d x d t | v=\left|\mathbf{v}\right|=\left|{\frac{dx}{dt}}\right|
  19. m s - 2 ms^{-2}
  20. 𝐚 𝐚𝐯 \mathbf{a_{av}}
  21. Δ 𝐯 = 𝐯 𝟐 - 𝐯 𝟏 \Delta\mathbf{v}=\mathbf{v_{2}}-\mathbf{v_{1}}
  22. Δ t \Delta t
  23. 𝐚 𝐚𝐯 = Δ 𝐯 Δ t = 𝐯 𝟐 - 𝐯 𝟏 t 2 - t 1 \mathbf{a_{av}}=\frac{\Delta\mathbf{v}}{\Delta t}=\frac{\mathbf{v_{2}}-\mathbf% {v_{1}}}{t_{2}-t_{1}}
  24. Δ 𝐯 \Delta\mathbf{v}
  25. Δ t \Delta t
  26. Δ t \Delta t
  27. 𝐚 = lim Δ t 0 Δ 𝐯 Δ t \mathbf{a}=\lim_{\Delta t\to 0}{\Delta\mathbf{v}\over\Delta t}
  28. = d 𝐯 d t = d 2 x d t 2 =\frac{d\mathbf{v}}{dt}=\frac{d^{2}x}{dt^{2}}
  29. m s - 3 ms^{-3}
  30. m s - 4 ms^{-4}
  31. 𝐯 = 𝐮 + 𝐚𝐭 \mathbf{v}=\mathbf{u}+\mathbf{a}\mathbf{t}\;\!
  32. 𝐬 = 𝐮𝐭 + 1 2 𝐚𝐭 2 \mathbf{s}=\mathbf{u}\mathbf{t}+\begin{matrix}\frac{1}{2}\end{matrix}\mathbf{a% }\mathbf{t}^{2}
  33. 𝐯 2 = 𝐮 2 + 2 𝐚𝐬 {\mathbf{v}}^{2}={\mathbf{u}}^{2}+2{\mathbf{a}}\mathbf{s}
  34. 𝐬 = 1 2 ( 𝐯 + 𝐮 ) 𝐭 \mathbf{s}=\tfrac{1}{2}\left(\mathbf{v}+\mathbf{u}\right)\mathbf{t}
  35. 𝐮 \mathbf{u}
  36. 𝐯 \mathbf{v}
  37. 𝐚 \mathbf{a}
  38. 𝐬 \mathbf{s}
  39. 𝐭 \mathbf{t}
  40. 𝐬 \mathbf{s}
  41. 𝐫 \mathbf{r}
  42. 𝐚 𝐭 \mathbf{a}_{\mathbf{t}}
  43. 𝐚 𝐜 = v 2 / r = ω 2 r \mathbf{a}_{\mathbf{c}}=v^{2}/r=\omega^{2}r
  44. 𝐅 \mathbf{F}_{\perp}
  45. 𝐣 = 1 𝐭𝐨 N \mathbf{j}\ =1\ \mathbf{to}\ N
  46. 𝐱 \mathbf{x}
  47. θ \theta
  48. θ = 𝐬 / 𝐫 \theta=\mathbf{s}/\mathbf{r}
  49. 𝐯 \mathbf{v}
  50. ω \omega
  51. ω = 𝐯 / 𝐫 \omega=\mathbf{v}/\mathbf{r}
  52. 𝐚 \mathbf{a}
  53. α \alpha
  54. α = 𝐚 𝐭 / 𝐫 \alpha=\mathbf{a_{\mathbf{t}}}/\mathbf{r}
  55. 𝐦 \mathbf{m}
  56. 𝐈 \mathbf{I}
  57. 𝐈 = 𝐦 𝐣 𝐫 𝐣 2 \mathbf{I}=\sum\mathbf{m_{j}}\mathbf{r_{j}}^{2}
  58. 𝐅 = 𝐦𝐚 \mathbf{F}=\mathbf{m}\mathbf{a}
  59. τ = 𝐈 α \tau=\mathbf{I}\alpha
  60. τ = 𝐫 𝐣 𝐅 𝐣 \tau=\sum\mathbf{r_{j}}\mathbf{F}_{\perp}\mathbf{{}_{j}}
  61. 𝐩 = 𝐦𝐯 \mathbf{p}=\mathbf{m}\mathbf{v}
  62. 𝐋 = 𝐈 ω \mathbf{L}=\mathbf{I}\omega
  63. 𝐋 = 𝐫 𝐣 𝐩 𝐣 \mathbf{L}=\sum\mathbf{r_{j}}\mathbf{p}\mathbf{{}_{j}}
  64. 1 2 𝐦𝐯 2 \frac{1}{2}\mathbf{m}\mathbf{v}^{2}
  65. 1 2 𝐈 ω 2 \frac{1}{2}\mathbf{I}\omega^{2}
  66. 1 2 𝐦 𝐣 𝐯 2 = 1 2 𝐦 𝐣 𝐫 𝐣 2 ω 2 \frac{1}{2}\sum\mathbf{m_{j}}\mathbf{v}^{2}=\frac{1}{2}\sum\mathbf{m_{j}}% \mathbf{r_{j}}^{2}\omega^{2}

Linear_production_game.html

  1. a k j a^{j}_{k}
  2. c \vec{c}
  3. b i = ( b 1 i , , b m i ) \vec{b^{i}}=(b^{i}_{1},...,b^{i}_{m})
  4. P ( S ) P(S)
  5. v ( S ) = max x 0 ( c 1 x 1 + + c n x n ) v(S)=\max_{x\geq 0}(c_{1}x_{1}+...+c_{n}x_{n})
  6. s . t . a j 1 x 1 + a j 2 x 2 + + a j n x n i S b j i j = 1 , 2 , , m s.t.\quad a^{1}_{j}x_{1}+a^{2}_{j}x_{2}+...+a^{n}_{j}x_{n}\leq\sum_{i\in S}b^{% i}_{j}\quad\forall j=1,2,...,m
  7. P ( N ) P(N)
  8. α \alpha
  9. P ( N ) P(N)
  10. x i = k = 1 m α k b k i x^{i}=\sum_{k=1}^{m}\alpha_{k}b^{i}_{k}
  11. x \vec{x}
  12. x \vec{x}
  13. α j \alpha_{j}

Link_concordance.html

  1. L 0 S n L_{0}\subset S^{n}
  2. L 1 S n L_{1}\subset S^{n}
  3. f : L 0 × [ 0 , 1 ] S n × [ 0 , 1 ] f:L_{0}\times[0,1]\to S^{n}\times[0,1]
  4. f ( L 0 × { 0 } ) = L 0 × { 0 } f(L_{0}\times\{0\})=L_{0}\times\{0\}
  5. f ( L 0 × { 1 } ) = L 1 × { 1 } f(L_{0}\times\{1\})=L_{1}\times\{1\}
  6. M 0 , M 1 N M_{0},M_{1}\subset N
  7. N × [ 0 , 1 ] , N\times[0,1],
  8. W N × [ 0 , 1 ] W\subset N\times[0,1]
  9. M 0 × { 0 } M_{0}\times\{0\}
  10. M 1 × { 1 } . M_{1}\times\{1\}.

Lin–Tsien_equation.html

  1. 2 u t x + u x u x x - u y y = 0. 2u_{tx}+u_{x}u_{xx}-u_{yy}=0.\,

Lions–Lax–Milgram_theorem.html

  1. inf v V = 1 sup h H 1 | B ( h , v ) | c ; \inf_{\|v\|_{V}=1}\sup_{\|h\|_{H}\leq 1}|B(h,v)|\geq c;
  2. B ( h , v ) = f , v for all v V . B(h,v)=\langle f,v\rangle\mbox{ for all }~{}v\in V.
  3. v H c v V ; \|v\|_{H}\leq c\|v\|_{V};
  4. B ( v , v ) α v V 2 . B(v,v)\geq\alpha\|v\|_{V}^{2}.
  5. t u ( t , x ) = Δ u ( t , x ) , \partial_{t}u(t,x)=\Delta u(t,x),
  6. [ 0 , T ) × Ω [ 0 , + ) × 𝐑 n . [0,T)\times\Omega\subseteq[0,+\infty)\times\mathbf{R}^{n}.

Liouville's_formula.html

  1. n n
  2. y = A ( x ) y y^{\prime}=A(x)y\,
  3. I I
  4. A ( x ) A(x)
  5. x I x∈I
  6. n n
  7. Φ Φ
  8. I I
  9. Φ ( x ) Φ(x)
  10. n n
  11. Φ ( x ) = A ( x ) Φ ( x ) , x I . \Phi^{\prime}(x)=A(x)\Phi(x),\qquad x\in I.
  12. tr A ( ξ ) = i = 1 n a i , i ( ξ ) , ξ I , \mathrm{tr}\,A(\xi)=\sum_{i=1}^{n}a_{i,i}(\xi),\qquad\xi\in I,
  13. A A
  14. Φ Φ
  15. det Φ ( x ) = det Φ ( x 0 ) exp ( x 0 x tr A ( ξ ) d ξ ) \det\Phi(x)=\det\Phi(x_{0})\,\exp\biggl(\int_{x_{0}}^{x}\mathrm{tr}\,A(\xi)\,% \textrm{d}\xi\biggr)
  16. x x
  17. I I
  18. y = ( 1 - 1 / x 1 + x - 1 ) = A ( x ) y y^{\prime}=\underbrace{\begin{pmatrix}1&-1/x\\ 1+x&-1\end{pmatrix}}_{=\,A(x)}y
  19. I = I=
  20. ( 0 , ) (0, ∞)
  21. y ( x ) = ( 1 x ) , x I , y(x)=\begin{pmatrix}1\\ x\end{pmatrix},\qquad x\in I,
  22. y ( x ) = ( y 1 ( x ) y 2 ( x ) ) y(x)=\begin{pmatrix}y_{1}(x)\\ y_{2}(x)\end{pmatrix}
  23. Φ ( x ) = ( y 1 ( x ) 1 y 2 ( x ) x ) , x I , \Phi(x)=\begin{pmatrix}y_{1}(x)&1\\ y_{2}(x)&x\end{pmatrix},\qquad x\in I,
  24. A ( x ) A(x)
  25. x I x∈I
  26. x x
  27. y y
  28. y 1 ( x ) = y 1 ( x ) - y 2 ( x ) x = x y 1 ( x ) - y 2 ( x ) x = c 1 x , x I . y^{\prime}_{1}(x)=y_{1}(x)-\frac{y_{2}(x)}{x}=\frac{x\,y_{1}(x)-y_{2}(x)}{x}=% \frac{c_{1}}{x},\qquad x\in I.
  29. y 1 ( x ) = c 1 ln x + c 2 , x I , y_{1}(x)=c_{1}\ln x+c_{2},\qquad x\in I,
  30. y 2 ( x ) = x y 1 ( x ) - c 1 = c 1 x ln x + c 2 x - c 1 , x I , y_{2}(x)=x\,y_{1}(x)-c_{1}=\,c_{1}x\ln x+c_{2}x-c_{1},\qquad x\in I,
  31. y y
  32. Φ ( x ) = ( ln x 1 x ln x - 1 x ) , x I , \Phi(x)=\begin{pmatrix}\ln x&1\\ x\ln x-1&x\end{pmatrix},\qquad x\in I,
  33. x x
  34. ( det Φ ) = i = 1 n det ( Φ 1 , 1 Φ 1 , 2 Φ 1 , n Φ i , 1 Φ i , 2 Φ i , n Φ n , 1 Φ n , 2 Φ n , n ) . (\det\Phi)^{\prime}=\sum_{i=1}^{n}\det\begin{pmatrix}\Phi_{1,1}&\Phi_{1,2}&% \cdots&\Phi_{1,n}\\ \vdots&\vdots&&\vdots\\ \Phi^{\prime}_{i,1}&\Phi^{\prime}_{i,2}&\cdots&\Phi^{\prime}_{i,n}\\ \vdots&\vdots&&\vdots\\ \Phi_{n,1}&\Phi_{n,2}&\cdots&\Phi_{n,n}\end{pmatrix}.
  35. Φ Φ
  36. Φ = A Φ Φ=AΦ
  37. Φ Φ
  38. Φ i , k = j = 1 n a i , j Φ j , k , i , k { 1 , , n } , \Phi^{\prime}_{i,k}=\sum_{j=1}^{n}a_{i,j}\Phi_{j,k}\,,\qquad i,k\in\{1,\ldots,% n\},
  39. ( Φ i , 1 , , Φ i , n ) = j = 1 n a i , j ( Φ j , 1 , , Φ j , n ) , i { 1 , , n } . (\Phi^{\prime}_{i,1},\dots,\Phi^{\prime}_{i,n})=\sum_{j=1}^{n}a_{i,j}(\Phi_{j,% 1},\ldots,\Phi_{j,n}),\qquad i\in\{1,\ldots,n\}.
  40. i i
  41. j = 1 j i n a i , j ( Φ j , 1 , , Φ j , n ) , \sum_{\scriptstyle j=1\atop\scriptstyle j\not=i}^{n}a_{i,j}(\Phi_{j,1},\ldots,% \Phi_{j,n}),
  42. det ( Φ 1 , 1 Φ 1 , 2 Φ 1 , n Φ i , 1 Φ i , 2 Φ i , n Φ n , 1 Φ n , 2 Φ n , n ) = det ( Φ 1 , 1 Φ 1 , 2 Φ 1 , n a i , i Φ i , 1 a i , i Φ i , 2 a i , i Φ i , n Φ n , 1 Φ n , 2 Φ n , n ) = a i , i det Φ \det\begin{pmatrix}\Phi_{1,1}&\Phi_{1,2}&\cdots&\Phi_{1,n}\\ \vdots&\vdots&&\vdots\\ \Phi^{\prime}_{i,1}&\Phi^{\prime}_{i,2}&\cdots&\Phi^{\prime}_{i,n}\\ \vdots&\vdots&&\vdots\\ \Phi_{n,1}&\Phi_{n,2}&\cdots&\Phi_{n,n}\end{pmatrix}=\det\begin{pmatrix}\Phi_{% 1,1}&\Phi_{1,2}&\cdots&\Phi_{1,n}\\ \vdots&\vdots&&\vdots\\ a_{i,i}\Phi_{i,1}&a_{i,i}\Phi_{i,2}&\cdots&a_{i,i}\Phi_{i,n}\\ \vdots&\vdots&&\vdots\\ \Phi_{n,1}&\Phi_{n,2}&\cdots&\Phi_{n,n}\end{pmatrix}=a_{i,i}\det\Phi
  43. ( det Φ ) = i = 1 n a i , i det Φ = tr A det Φ (\det\Phi)^{\prime}=\sum_{i=1}^{n}a_{i,i}\det\Phi=\mathrm{tr}\,A\,\det\Phi
  44. A A
  45. I I
  46. I I
  47. g ( x ) := det Φ ( x ) exp ( - x 0 x tr A ( ξ ) d ξ ) , x I , g(x):=\det\Phi(x)\exp\left(-\int_{x_{0}}^{x}\mathrm{tr}\,A(\xi)\,\textrm{d}\xi% \right),\qquad x\in I,
  48. g ( x ) = ( ( det Φ ( x ) ) - det Φ ( x ) tr A ( x ) ) exp ( - x 0 x tr A ( ξ ) d ξ ) = 0 , x I , g^{\prime}(x)=\bigl((\det\Phi(x))^{\prime}-\det\Phi(x)\,\mathrm{tr}\,A(x)\bigr% )\exp\biggl(-\int_{x_{0}}^{x}\mathrm{tr}\,A(\xi)\,\textrm{d}\xi\biggr)=0,% \qquad x\in I,
  49. g g
  50. I I
  51. g ( x < s u b > 0 ) = d e t Φ ( x 0 ) g(x<sub>0)=detΦ(x_{0})

Liouville_surface.html

  1. z = f ( x , y ) z=f(x,y)
  2. d s 2 = ( f 1 ( x ) + f 2 ( y ) ) ( d x 2 + d y 2 ) . ds^{2}=(f_{1}(x)+f_{2}(y))(dx^{2}+dy^{2}).\,

Liquid.html

  1. p = ρ g z p=\rho gz\,
  2. ρ \rho\,
  3. g g\,
  4. c = K / ρ c=\sqrt{K/\rho}
  5. c = K / ρ c=\sqrt{K/\rho}
  6. K 0 K_{0}
  7. K K_{\infty}
  8. G 0 = 0 G_{0}=0
  9. G 0 G_{0}
  10. G G_{\infty}

Liquid_junction_potential.html

  1. E nt = R T F ln a 2 a 1 E_{\mathrm{nt}}=\frac{RT}{F}\ln\frac{a_{2}}{a_{1}}
  2. E wt = t M R T F ln a 2 a 1 E_{\mathrm{wt}}=t_{M}\frac{RT}{F}\ln\frac{a_{2}}{a_{1}}

Liquid_water_content.html

  1. n = N / V n=N/V
  2. L W C = ( m w * n ) / N LWC=(m_{w}*n)/N
  3. L W C = m w / V c LWC=m_{w}/V_{c}
  4. m w = - m a c p Δ T a L c ( T ) m_{w}=\frac{-m_{a}\cdot c_{p}\cdot\Delta T_{a}}{L_{c}(T)}
  5. Δ T a \Delta T_{a}

List_of_Banach_spaces.html

  1. 1 q + 1 p = 1 , \frac{1}{q}+\frac{1}{p}=1,
  2. q = p p - 1 . q=\frac{p}{p-1}.
  3. x 2 = ( i = 1 n | x i | 2 ) 1 / 2 \|x\|_{2}=\left(\sum_{i=1}^{n}|x_{i}|^{2}\right)^{1/2}
  4. x p = ( i = 1 n | x i | p ) 1 / p \|x\|_{p}=\left(\sum_{i=1}^{n}|x_{i}|^{p}\right)^{1/p}
  5. x = max 1 i n | x i | \|x\|_{\infty}=\max_{1\leq i\leq n}|x_{i}|
  6. x p = ( i = 1 | x i | p ) 1 / p \|x\|_{p}=\left(\sum_{i=1}^{\infty}|x_{i}|^{p}\right)^{1/p}
  7. x 1 = i = 1 | x i | \|x\|_{1}=\sum_{i=1}^{\infty}|x_{i}|
  8. x = sup i | x i | \|x\|_{\infty}=\sup_{i}|x_{i}|
  9. x = sup i | x i | \|x\|_{\infty}=\sup_{i}|x_{i}|
  10. x = sup i | x i | \|x\|_{\infty}=\sup_{i}|x_{i}|
  11. x b v = | x 1 | + i = 1 | x i + 1 - x i | \|x\|_{bv}=|x_{1}|+\sum_{i=1}^{\infty}|x_{i+1}-x_{i}|
  12. x b v 0 = i = 1 | x i + 1 - x i | \|x\|_{bv_{0}}=\sum_{i=1}^{\infty}|x_{i+1}-x_{i}|
  13. x b s = sup n | i = 1 n x i | \|x\|_{bs}=\sup_{n}\left|\sum_{i=1}^{n}x_{i}\right|
  14. x b s = sup n | i = 1 n x i | \|x\|_{bs}=\sup_{n}\left|\sum_{i=1}^{n}x_{i}\right|
  15. f B = sup x X | f ( x ) | \|f\|_{B}=\sup_{x\in X}|f(x)|
  16. f B = sup x X | f ( x ) | \|f\|_{B}=\sup_{x\in X}\left|f(x)\right|
  17. μ b a = sup A Σ | μ | ( A ) \|\mu\|_{ba}=\sup_{A\in\Sigma}|\mu|(A)
  18. μ b a = sup A Σ | μ | ( A ) \|\mu\|_{ba}=\sup_{A\in\Sigma}|\mu|(A)
  19. μ b a = sup A Σ | μ | ( A ) \|\mu\|_{ba}=\sup_{A\in\Sigma}|\mu|(A)
  20. f p = { | f | p d μ } 1 / p \|f\|_{p}=\left\{\int|f|^{p}\,d\mu\right\}^{1/p}
  21. f 1 = | f | d μ \|f\|_{1}=\int|f|\,d\mu
  22. N μ N_{\mu}^{\perp}
  23. f inf { C 0 : | f ( x ) | C for almost every x } . \|f\|_{\infty}\equiv\inf\{C\geq 0:|f(x)|\leq C\mbox{ for almost every }~{}x\}.
  24. N μ = { σ b a ( Σ ) : λ μ } N_{\mu}^{\perp}=\{\sigma\in ba(\Sigma):\lambda\ll\mu\}
  25. f B V = lim x a + f ( x ) + V f ( I ) \|f\|_{BV}=\lim_{x\to a^{+}}f(x)+V_{f}(I)
  26. f B V = V f ( I ) \|f\|_{BV}=V_{f}(I)
  27. lim x a + f ( x ) = 0 \lim_{x\to a^{+}}f(x)=0
  28. f B V = lim x a + f ( x ) + V f ( I ) \|f\|_{BV}=\lim_{x\to a^{+}}f(x)+V_{f}(I)
  29. f = i = 0 n sup x [ a , b ] | f ( i ) ( x ) | . \|f\|=\sum_{i=0}^{n}\sup_{x\in[a,b]}|f^{(i)}(x)|.
  30. X X X X\oplus X\oplus X
  31. X X X\oplus X

List_of_Chicago_Bears_head_coaches.html

  1. W i n s + 1 2 T i e s G a m e s \frac{Wins+\frac{1}{2}Ties}{Games}

List_of_Chinese_discoveries.html

  1. 177147 176776 \tfrac{177147}{176776}
  2. 355 113 \tfrac{355}{113}
  3. 736 232 \tfrac{736}{232}
  4. 355 113 \tfrac{355}{113}
  5. 52163 16604 \tfrac{52163}{16604}
  6. K 𝔭 K_{\mathfrak{p}}
  7. 𝔭 \mathfrak{p}
  8. a - ( a - 1 + ( b - 1 - a ) - 1 ) - 1 = a b a a-(a^{-1}+(b^{-1}-a)^{-1})^{-1}=aba
  9. a b 0 , 1 ab\neq 0,1
  10. b b
  11. - b - 1 -b^{-1}
  12. ( a + a b - 1 a ) - 1 + ( a + b ) - 1 = a - 1 . (a+ab^{-1}a)^{-1}+(a+b)^{-1}=a^{-1}.
  13. ( x + m + 1 ) i = 0 m ( - 1 ) i ( x + y + i m - i ) ( y + 2 i i ) - i = 0 m ( x + i m - i ) ( - 4 ) i = ( x - m ) ( x m ) . (x+m+1)\sum_{i=0}^{m}(-1)^{i}{\displaystyle\left({{x+y+i}\atop{m-i}}\right)}{% \displaystyle\left({{y+2i}\atop{i}}\right)}-\sum_{i=0}^{m}{\displaystyle\left(% {{x+i}\atop{m-i}}\right)}(-4)^{i}=(x-m){\displaystyle\left({{x}\atop{m}}\right% )}.

List_of_convolutions_of_probability_distributions.html

  1. i = 1 n X i Y \sum_{i=1}^{n}X_{i}\sim Y
  2. X 1 , X 2 , , X n X_{1},X_{2},\dots,X_{n}\,
  3. X i X_{i}
  4. Y Y
  5. i = 1 n Bernoulli ( p ) Binomial ( n , p ) 0 < p < 1 n = 1 , 2 , \sum_{i=1}^{n}\mathrm{Bernoulli}(p)\sim\mathrm{Binomial}(n,p)\qquad 0<p<1\quad n% =1,2,\dots\,\!
  6. i = 1 n Binomial ( n i , p ) Binomial ( i = 1 n n i , p ) 0 < p < 1 n i = 1 , 2 , \sum_{i=1}^{n}\mathrm{Binomial}(n_{i},p)\sim\mathrm{Binomial}\left(\sum_{i=1}^% {n}n_{i},p\right)\qquad 0<p<1\quad n_{i}=1,2,\dots\,\!
  7. i = 1 n NegativeBinomial ( n i , p ) NegativeBinomial ( i = 1 n n i , p ) 0 < p < 1 n i = 1 , 2 , \sum_{i=1}^{n}\mathrm{NegativeBinomial}(n_{i},p)\sim\mathrm{NegativeBinomial}% \left(\sum_{i=1}^{n}n_{i},p\right)\qquad 0<p<1\quad n_{i}=1,2,\dots\,\!
  8. i = 1 n Geometric ( p ) NegativeBinomial ( n , p ) 0 < p < 1 n = 1 , 2 , \sum_{i=1}^{n}\mathrm{Geometric}(p)\sim\mathrm{NegativeBinomial}(n,p)\qquad 0<% p<1\quad n=1,2,\dots\,\!
  9. i = 1 n Poisson ( λ i ) Poisson ( i = 1 n λ i ) λ i > 0 \sum_{i=1}^{n}\mathrm{Poisson}(\lambda_{i})\sim\mathrm{Poisson}\left(\sum_{i=1% }^{n}\lambda_{i}\right)\qquad\lambda_{i}>0\,\!
  10. i = 1 n Normal ( μ i , σ i 2 ) Normal ( i = 1 n μ i , i = 1 n σ i 2 ) - < μ i < σ i 2 > 0 \sum_{i=1}^{n}\mathrm{Normal}(\mu_{i},\sigma_{i}^{2})\sim\mathrm{Normal}\left(% \sum_{i=1}^{n}\mu_{i},\sum_{i=1}^{n}\sigma_{i}^{2}\right)\qquad-\infty<\mu_{i}% <\infty\quad\sigma_{i}^{2}>0
  11. i = 1 n Cauchy ( a i , γ i ) Cauchy ( i = 1 n a i , i = 1 n γ i ) - < a i < γ i > 0 \sum_{i=1}^{n}\mathrm{Cauchy}(a_{i},\gamma_{i})\sim\mathrm{Cauchy}\left(\sum_{% i=1}^{n}a_{i},\sum_{i=1}^{n}\gamma_{i}\right)\qquad-\infty<a_{i}<\infty\quad% \gamma_{i}>0
  12. i = 1 n Gamma ( α i , β ) Gamma ( i = 1 n α i , β ) α i > 0 β > 0 \sum_{i=1}^{n}\mathrm{Gamma}(\alpha_{i},\beta)\sim\mathrm{Gamma}\left(\sum_{i=% 1}^{n}\alpha_{i},\beta\right)\qquad\alpha_{i}>0\quad\beta>0
  13. i = 1 n Exponential ( θ ) Gamma ( n , θ ) θ > 0 n = 1 , 2 , \sum_{i=1}^{n}\mathrm{Exponential}(\theta)\sim\mathrm{Gamma}(n,\theta)\qquad% \theta>0\quad n=1,2,\dots
  14. i = 1 n χ 2 ( r i ) χ 2 ( i = 1 n r i ) r i = 1 , 2 , \sum_{i=1}^{n}\chi^{2}(r_{i})\sim\chi^{2}\left(\sum_{i=1}^{n}r_{i}\right)% \qquad r_{i}=1,2,\dots
  15. i = 1 r N 2 ( 0 , 1 ) χ r 2 r = 1 , 2 , \sum_{i=1}^{r}N^{2}(0,1)\sim\chi^{2}_{r}\qquad r=1,2,\dots
  16. i = 1 n ( X i - X ¯ ) 2 σ 2 χ n - 1 2 , \sum_{i=1}^{n}(X_{i}-\bar{X})^{2}\sim\sigma^{2}\chi^{2}_{n-1},\quad
  17. X 1 , , X n X_{1},\dots,X_{n}
  18. N ( μ , σ 2 ) N(\mu,\sigma^{2})
  19. X ¯ = 1 n i = 1 n X i . \bar{X}=\frac{1}{n}\sum_{i=1}^{n}X_{i}.\,\!

List_of_countries_by_Sen_social_welfare_function.html

  1. W Gini = Income ¯ ( 1 - G ) W_{\mathrm{Gini}}=\overline{\,\text{Income}}\cdot\left(1-G\right)

List_of_mathematical_series.html

  1. 0 0 0^{0}
  2. B n ( x ) B_{n}(x)
  3. B n B_{n}
  4. B 1 = - 1 2 . B_{1}=-\frac{1}{2}.
  5. E n E_{n}
  6. ζ ( s ) \zeta(s)
  7. Γ ( z ) \Gamma(z)
  8. ψ n ( z ) \psi_{n}(z)
  9. Li s ( z ) \operatorname{Li}_{s}(z)
  10. k = 0 m k n - 1 = B n ( m + 1 ) - B n n \sum_{k=0}^{m}k^{n-1}=\frac{B_{n}(m+1)-B_{n}}{n}\,\!
  11. k = 1 m k = m ( m + 1 ) 2 \sum_{k=1}^{m}k=\frac{m(m+1)}{2}\,\!
  12. k = 1 m k 2 = m ( m + 1 ) ( 2 m + 1 ) 6 = m 3 3 + m 2 2 + m 6 \sum_{k=1}^{m}k^{2}=\frac{m(m+1)(2m+1)}{6}=\frac{m^{3}}{3}+\frac{m^{2}}{2}+% \frac{m}{6}\,\!
  13. k = 1 m k 3 = [ m ( m + 1 ) 2 ] 2 = m 4 4 + m 3 2 + m 2 4 \sum_{k=1}^{m}k^{3}=\left[\frac{m(m+1)}{2}\right]^{2}=\frac{m^{4}}{4}+\frac{m^% {3}}{2}+\frac{m^{2}}{4}\,\!
  14. ζ ( 2 n ) = k = 1 1 k 2 n = ( - 1 ) n + 1 B 2 n ( 2 π ) 2 n 2 ( 2 n ) ! \zeta(2n)=\sum^{\infty}_{k=1}\frac{1}{k^{2n}}=(-1)^{n+1}\frac{B_{2n}(2\pi)^{2n% }}{2(2n)!}
  15. ζ ( 2 ) = k = 1 1 k 2 = π 2 6 \zeta(2)=\sum^{\infty}_{k=1}\frac{1}{k^{2}}=\frac{\pi^{2}}{6}\,\!
  16. ζ ( 4 ) = k = 1 1 k 4 = π 4 90 \zeta(4)=\sum^{\infty}_{k=1}\frac{1}{k^{4}}=\frac{\pi^{4}}{90}\,\!
  17. ζ ( 6 ) = k = 1 1 k 6 = π 6 945 \zeta(6)=\sum^{\infty}_{k=1}\frac{1}{k^{6}}=\frac{\pi^{6}}{945}\,\!
  18. k = 0 n z k = 1 - z n + 1 1 - z \sum_{k=0}^{n}z^{k}=\frac{1-z^{n+1}}{1-z}\,\!
  19. k = 1 n k z k = z 1 - ( n + 1 ) z n + n z n + 1 ( 1 - z ) 2 \sum_{k=1}^{n}kz^{k}=z\frac{1-(n+1)z^{n}+nz^{n+1}}{(1-z)^{2}}\,\!
  20. k = 1 n k 2 z k = z 1 + z - ( n + 1 ) 2 z n + ( 2 n 2 + 2 n - 1 ) z n + 1 - n 2 z n + 2 ( 1 - z ) 3 \sum_{k=1}^{n}k^{2}z^{k}=z\frac{1+z-(n+1)^{2}z^{n}+(2n^{2}+2n-1)z^{n+1}-n^{2}z% ^{n+2}}{(1-z)^{3}}\,\!
  21. k = 1 n k m z k = ( z d d z ) m z - z n + 1 1 - z \sum_{k=1}^{n}k^{m}z^{k}=\left(z\frac{d}{dz}\right)^{m}\frac{z-z^{n+1}}{1-z}
  22. | z | < 1 |z|<1
  23. Li n ( z ) = k = 1 z k k n \operatorname{Li}_{n}(z)=\sum_{k=1}^{\infty}\frac{z^{k}}{k^{n}}\,\!
  24. d d z Li n ( z ) = Li n - 1 ( z ) z \frac{d}{dz}\operatorname{Li}_{n}(z)=\frac{\operatorname{Li}_{n-1}(z)}{z}\,\!
  25. Li 1 ( z ) = k = 1 z k k = - ln ( 1 - z ) \operatorname{Li}_{1}(z)=\sum_{k=1}^{\infty}\frac{z^{k}}{k}=-\ln(1-z)\!
  26. Li 0 ( z ) = k = 1 z k = z 1 - z \operatorname{Li}_{0}(z)=\sum_{k=1}^{\infty}z^{k}=\frac{z}{1-z}\!
  27. Li - 1 ( z ) = k = 1 k z k = z ( 1 - z ) 2 \operatorname{Li}_{-1}(z)=\sum_{k=1}^{\infty}kz^{k}=\frac{z}{(1-z)^{2}}\,\!
  28. Li - 2 ( z ) = k = 1 k 2 z k = z ( 1 + z ) ( 1 - z ) 3 \operatorname{Li}_{-2}(z)=\sum_{k=1}^{\infty}k^{2}z^{k}=\frac{z(1+z)}{(1-z)^{3% }}\,\!
  29. Li - 3 ( z ) = k = 1 k 3 z k = z ( 1 + 4 z + z 2 ) ( 1 - z ) 4 \operatorname{Li}_{-3}(z)=\sum_{k=1}^{\infty}k^{3}z^{k}=\frac{z(1+4z+z^{2})}{(% 1-z)^{4}}\,\!
  30. Li - 4 ( z ) = k = 1 k 4 z k = z ( 1 + z ) ( 1 + 10 z + z 2 ) ( 1 - z ) 5 \operatorname{Li}_{-4}(z)=\sum_{k=1}^{\infty}k^{4}z^{k}=\frac{z(1+z)(1+10z+z^{% 2})}{(1-z)^{5}}\,\!
  31. k = 0 z k k ! = e z \sum_{k=0}^{\infty}\frac{z^{k}}{k!}=e^{z}\,\!
  32. k = 0 k z k k ! = z e z \sum_{k=0}^{\infty}k\frac{z^{k}}{k!}=ze^{z}\,\!
  33. k = 0 k 2 z k k ! = ( z + z 2 ) e z \sum_{k=0}^{\infty}k^{2}\frac{z^{k}}{k!}=(z+z^{2})e^{z}\,\!
  34. k = 0 k 3 z k k ! = ( z + 3 z 2 + z 3 ) e z \sum_{k=0}^{\infty}k^{3}\frac{z^{k}}{k!}=(z+3z^{2}+z^{3})e^{z}\,\!
  35. k = 0 k 4 z k k ! = ( z + 7 z 2 + 6 z 3 + z 4 ) e z \sum_{k=0}^{\infty}k^{4}\frac{z^{k}}{k!}=(z+7z^{2}+6z^{3}+z^{4})e^{z}\,\!
  36. k = 0 k n z k k ! = z d d z k = 0 k n - 1 z k k ! = e z T n ( z ) \sum_{k=0}^{\infty}k^{n}\frac{z^{k}}{k!}=z\frac{d}{dz}\sum_{k=0}^{\infty}k^{n-% 1}\frac{z^{k}}{k!}\,\!=e^{z}T_{n}(z)
  37. T n ( z ) T_{n}(z)
  38. k = 0 ( - 1 ) k z 2 k + 1 ( 2 k + 1 ) ! = sin z \sum_{k=0}^{\infty}\frac{(-1)^{k}z^{2k+1}}{(2k+1)!}=\sin z\,\!
  39. k = 0 z 2 k + 1 ( 2 k + 1 ) ! = sinh z \sum_{k=0}^{\infty}\frac{z^{2k+1}}{(2k+1)!}=\sinh z\,\!
  40. k = 0 ( - 1 ) k z 2 k ( 2 k ) ! = cos z \sum_{k=0}^{\infty}\frac{(-1)^{k}z^{2k}}{(2k)!}=\cos z\,\!
  41. k = 0 z 2 k ( 2 k ) ! = cosh z \sum_{k=0}^{\infty}\frac{z^{2k}}{(2k)!}=\cosh z\,\!
  42. k = 1 ( - 1 ) k - 1 ( 2 2 k - 1 ) 2 2 k B 2 k z 2 k - 1 ( 2 k ) ! = tan z , | z | < π 2 \sum_{k=1}^{\infty}\frac{(-1)^{k-1}(2^{2k}-1)2^{2k}B_{2k}z^{2k-1}}{(2k)!}=\tan z% ,|z|<\frac{\pi}{2}\,\!
  43. k = 1 ( 2 2 k - 1 ) 2 2 k B 2 k z 2 k - 1 ( 2 k ) ! = tanh z , | z | < π 2 \sum_{k=1}^{\infty}\frac{(2^{2k}-1)2^{2k}B_{2k}z^{2k-1}}{(2k)!}=\tanh z,|z|<% \frac{\pi}{2}\,\!
  44. k = 0 ( - 1 ) k 2 2 k B 2 k z 2 k - 1 ( 2 k ) ! = cot z , | z | < π \sum_{k=0}^{\infty}\frac{(-1)^{k}2^{2k}B_{2k}z^{2k-1}}{(2k)!}=\cot z,|z|<\pi\,\!
  45. k = 0 2 2 k B 2 k z 2 k - 1 ( 2 k ) ! = coth z , | z | < π \sum_{k=0}^{\infty}\frac{2^{2k}B_{2k}z^{2k-1}}{(2k)!}=\coth z,|z|<\pi\,\!
  46. k = 0 ( - 1 ) k - 1 ( 2 2 k - 2 ) B 2 k z 2 k - 1 ( 2 k ) ! = csc z , | z | < π \sum_{k=0}^{\infty}\frac{(-1)^{k-1}(2^{2k}-2)B_{2k}z^{2k-1}}{(2k)!}=\csc z,|z|% <\pi\,\!
  47. k = 0 - ( 2 2 k - 2 ) B 2 k z 2 k - 1 ( 2 k ) ! = csch z , | z | < π \sum_{k=0}^{\infty}\frac{-(2^{2k}-2)B_{2k}z^{2k-1}}{(2k)!}=\operatorname{csch}% z,|z|<\pi\,\!
  48. k = 0 ( - 1 ) k E 2 k z 2 k ( 2 k ) ! = sec z , | z | < π 2 \sum_{k=0}^{\infty}\frac{(-1)^{k}E_{2k}z^{2k}}{(2k)!}=\sec z,|z|<\frac{\pi}{2}\,\!
  49. k = 0 E 2 k z 2 k ( 2 k ) ! = sech z , | z | < π 2 \sum_{k=0}^{\infty}\frac{E_{2k}z^{2k}}{(2k)!}=\operatorname{sech}z,|z|<\frac{% \pi}{2}\,\!
  50. k = 0 ( 2 k ) ! z 2 k + 1 2 2 k ( k ! ) 2 ( 2 k + 1 ) = arcsin z , | z | 1 \sum_{k=0}^{\infty}\frac{(2k)!z^{2k+1}}{2^{2k}(k!)^{2}(2k+1)}=\arcsin z,|z|% \leq 1\,\!
  51. k = 0 ( - 1 ) k ( 2 k ) ! z 2 k + 1 2 2 k ( k ! ) 2 ( 2 k + 1 ) = arcsinh z , | z | 1 \sum_{k=0}^{\infty}\frac{(-1)^{k}(2k)!z^{2k+1}}{2^{2k}(k!)^{2}(2k+1)}=% \operatorname{arcsinh}{z},|z|\leq 1\,\!
  52. k = 0 ( - 1 ) k z 2 k + 1 2 k + 1 = arctan z , | z | < 1 \sum_{k=0}^{\infty}\frac{(-1)^{k}z^{2k+1}}{2k+1}=\arctan z,|z|<1\,\!
  53. k = 0 z 2 k + 1 2 k + 1 = arctanh z , | z | < 1 \sum_{k=0}^{\infty}\frac{z^{2k+1}}{2k+1}=\operatorname{arctanh}z,|z|<1\,\!
  54. ln 2 + k = 1 ( - 1 ) k - 1 ( 2 k ) ! z 2 k 2 2 k + 1 k ( k ! ) 2 = ln ( 1 + 1 + z 2 ) , | z | 1 \ln 2+\sum_{k=1}^{\infty}\frac{(-1)^{k-1}(2k)!z^{2k}}{2^{2k+1}k(k!)^{2}}=\ln% \left(1+\sqrt{1+z^{2}}\right),|z|\leq 1\,\!
  55. k = 0 ( 4 k ) ! 2 4 k 2 ( 2 k ) ! ( 2 k + 1 ) ! z k = 1 - 1 - z z , | z | < 1 \sum^{\infty}_{k=0}\frac{(4k)!}{2^{4k}\sqrt{2}(2k)!(2k+1)!}z^{k}=\sqrt{\frac{1% -\sqrt{1-z}}{z}},|z|<1
  56. k = 0 2 2 k ( k ! ) 2 ( k + 1 ) ( 2 k + 1 ) ! z 2 k + 2 = ( arcsin z ) 2 , | z | 1 \sum^{\infty}_{k=0}\frac{2^{2k}(k!)^{2}}{(k+1)(2k+1)!}z^{2k+2}=\left(\arcsin{z% }\right)^{2},|z|\leq 1
  57. n = 0 k = 0 n - 1 ( 4 k 2 + α 2 ) ( 2 n ) ! z 2 n + n = 0 α k = 0 n - 1 [ ( 2 k + 1 ) 2 + α 2 ] ( 2 n + 1 ) ! z 2 n + 1 = e α arcsin z , | z | 1 \sum^{\infty}_{n=0}\frac{\prod_{k=0}^{n-1}(4k^{2}+\alpha^{2})}{(2n)!}z^{2n}+% \sum^{\infty}_{n=0}\frac{\alpha\prod_{k=0}^{n-1}[(2k+1)^{2}+\alpha^{2}]}{(2n+1% )!}z^{2n+1}=e^{\alpha\arcsin{z}},|z|\leq 1
  58. ( 1 + z ) α = k = 0 ( α k ) z k , | z | < 1 (1+z)^{\alpha}=\sum_{k=0}^{\infty}{\alpha\choose k}z^{k},|z|<1
  59. k = 0 ( α + k - 1 k ) z k = 1 ( 1 - z ) α , | z | < 1 \sum_{k=0}^{\infty}{{\alpha+k-1}\choose k}z^{k}=\frac{1}{(1-z)^{\alpha}},|z|<1
  60. k = 0 1 k + 1 ( 2 k k ) z k = 1 - 1 - 4 z 2 z , | z | 1 4 \sum_{k=0}^{\infty}\frac{1}{k+1}{2k\choose k}z^{k}=\frac{1-\sqrt{1-4z}}{2z},|z% |\leq\frac{1}{4}
  61. k = 0 ( 2 k k ) z k = 1 1 - 4 z , | z | < 1 4 \sum_{k=0}^{\infty}{2k\choose k}z^{k}=\frac{1}{\sqrt{1-4z}},|z|<\frac{1}{4}
  62. k = 0 ( 2 k + α k ) z k = 1 1 - 4 z ( 1 - 1 - 4 z 2 z ) α , | z | < 1 4 \sum_{k=0}^{\infty}{2k+\alpha\choose k}z^{k}=\frac{1}{\sqrt{1-4z}}\left(\frac{% 1-\sqrt{1-4z}}{2z}\right)^{\alpha},|z|<\frac{1}{4}
  63. k = 1 H k z k = - ln ( 1 - z ) 1 - z , | z | < 1 \sum_{k=1}^{\infty}H_{k}z^{k}=\frac{-\ln(1-z)}{1-z},|z|<1
  64. k = 1 H k k + 1 z k + 1 = 1 2 [ ln ( 1 - z ) ] 2 , | z | < 1 \sum_{k=1}^{\infty}\frac{H_{k}}{k+1}z^{k+1}=\frac{1}{2}\left[\ln(1-z)\right]^{% 2},\qquad|z|<1
  65. k = 1 ( - 1 ) k - 1 H 2 k 2 k + 1 z 2 k + 1 = 1 2 arctan z log ( 1 + z 2 ) , | z | < 1 \sum_{k=1}^{\infty}\frac{(-1)^{k-1}H_{2k}}{2k+1}z^{2k+1}=\frac{1}{2}\arctan{z}% \log{(1+z^{2})},\qquad|z|<1
  66. n = 0 k = 0 2 n ( - 1 ) k 2 k + 1 z 4 n + 2 4 n + 2 = 1 4 arctan z log 1 + z 1 - z , | z | < 1 \sum_{n=0}^{\infty}\sum_{k=0}^{2n}\frac{(-1)^{k}}{2k+1}\frac{z^{4n+2}}{4n+2}=% \frac{1}{4}\arctan{z}\log{\frac{1+z}{1-z}},\qquad|z|<1
  67. k = 0 n ( n k ) = 2 n \sum_{k=0}^{n}{n\choose k}=2^{n}
  68. k = 0 n ( - 1 ) k ( n k ) = 0 \sum_{k=0}^{n}(-1)^{k}{n\choose k}=0
  69. k = 0 n ( k m ) = ( n + 1 m + 1 ) \sum_{k=0}^{n}{k\choose m}={n+1\choose m+1}
  70. k = 0 n ( m + k - 1 k ) = ( n + m n ) \sum_{k=0}^{n}{m+k-1\choose k}={n+m\choose n}
  71. k = 0 n ( α k ) ( β n - k ) = ( α + β n ) \sum_{k=0}^{n}{\alpha\choose k}{\beta\choose n-k}={\alpha+\beta\choose n}
  72. k = 1 sin ( k θ ) k = π - θ 2 , 0 < θ < 2 π \sum_{k=1}^{\infty}\frac{\sin(k\theta)}{k}=\frac{\pi-\theta}{2},0<\theta<2\pi\,\!
  73. k = 1 cos ( k θ ) k = - 1 2 ln ( 2 - 2 cos θ ) , θ \sum_{k=1}^{\infty}\frac{\cos(k\theta)}{k}=-\frac{1}{2}\ln(2-2\cos\theta),% \theta\in\mathbb{R}\,\!
  74. k = 0 sin [ ( 2 k + 1 ) θ ] 2 k + 1 = π 4 , 0 < θ < π \sum_{k=0}^{\infty}\frac{\sin[(2k+1)\theta]}{2k+1}=\frac{\pi}{4},0<\theta<\pi\,\!
  75. B n ( x ) = - n ! 2 n - 1 π n k = 1 1 k n cos ( 2 π k x - π n 2 ) , 0 < x < 1 B_{n}(x)=-\frac{n!}{2^{n-1}\pi^{n}}\sum_{k=1}^{\infty}\frac{1}{k^{n}}\cos\left% (2\pi kx-\frac{\pi n}{2}\right),0<x<1\,\!
  76. k = 0 n sin ( θ + k α ) = sin ( n + 1 ) α 2 sin ( θ + n α 2 ) sin α 2 \sum_{k=0}^{n}\sin(\theta+k\alpha)=\frac{\sin\frac{(n+1)\alpha}{2}\sin(\theta+% \frac{n\alpha}{2})}{\sin\frac{\alpha}{2}}\,\!
  77. k = 1 n - 1 sin π k n = cot π 2 n \sum_{k=1}^{n-1}\sin\frac{\pi k}{n}=\cot\frac{\pi}{2n}\,\!
  78. k = 1 n - 1 sin 2 π k n = 0 \sum_{k=1}^{n-1}\sin\frac{2\pi k}{n}=0\,\!
  79. k = 0 n - 1 csc 2 ( θ + π k n ) = n 2 csc 2 ( n θ ) \sum_{k=0}^{n-1}\csc^{2}\left(\theta+\frac{\pi k}{n}\right)=n^{2}\csc^{2}(n% \theta)\,\!
  80. k = 1 n - 1 csc 2 π k n = n 2 - 1 3 \sum_{k=1}^{n-1}\csc^{2}\frac{\pi k}{n}=\frac{n^{2}-1}{3}\,\!
  81. k = 1 n - 1 csc 4 π k n = n 4 + 10 n 2 - 11 45 \sum_{k=1}^{n-1}\csc^{4}\frac{\pi k}{n}=\frac{n^{4}+10n^{2}-11}{45}\,\!
  82. m = b + 1 b m 2 - b 2 = 1 2 H 2 b \sum_{m=b+1}^{\infty}\frac{b}{m^{2}-b^{2}}=\frac{1}{2}H_{2b}
  83. m = 1 y m 2 + y 2 = - 1 2 y + π 2 coth ( π y ) \sum^{\infty}_{m=1}\frac{y}{m^{2}+y^{2}}=-\frac{1}{2y}+\frac{\pi}{2}\coth(\pi y)
  84. n n

List_of_New_York_Jets_head_coaches.html

  1. W i n s + 1 2 T i e s G a m e s \frac{Wins+\frac{1}{2}Ties}{Games}

List_of_nonlinear_partial_differential_equations.html

  1. d L d t = L A - A L \frac{dL}{dt}=LA-AL
  2. u t + u x + u u x - u x x t = 0 \displaystyle u_{t}+u_{x}+uu_{x}-u_{xxt}=0
  3. u t + H u x x + u u x = 0 \displaystyle u_{t}+Hu_{xx}+uu_{x}=0
  4. u t = 𝐛 𝐯 x \displaystyle u_{t}=\mathbf{b}\cdot\mathbf{v}_{x}
  5. 𝐯 x t = u x x 𝐛 + 𝐚 × 𝐯 x - 2 𝐯 × ( 𝐯 × 𝐛 ) \displaystyle\mathbf{v}_{xt}=u_{xx}\mathbf{b}+\mathbf{a}\times\mathbf{v}_{x}-2% \mathbf{v}\times(\mathbf{v}\times\mathbf{b})
  6. ( 1 - u t 2 ) u x x + 2 u x u t u x t - ( 1 + u x 2 ) u t t = 0 \displaystyle(1-u_{t}^{2})u_{xx}+2u_{x}u_{t}u_{xt}-(1+u_{x}^{2})u_{tt}=0
  7. u t t - u x x - u x x x x - 3 ( u 2 ) x x = 0 \displaystyle u_{tt}-u_{xx}-u_{xxxx}-3(u^{2})_{xx}=0
  8. u t t - u x x - 2 * α * ( u * u x ) x - β * u x x t t = 0 = 0 \displaystyle u_{tt}-u_{xx}-2*\alpha*(u*u_{x})_{x}-\beta*u_{xxtt}=0=0
  9. u t = ( u 4 ) x x + ( u 3 ) x \displaystyle u_{t}=(u^{4})_{xx}+(u^{3})_{x}
  10. u t + u u x = ν u x x \displaystyle u_{t}+uu_{x}=\nu u_{xx}
  11. c t = D 2 ( c 3 - c - γ 2 c ) \displaystyle\frac{\partial c}{\partial t}=D\nabla^{2}\left(c^{3}-c-\gamma% \nabla^{2}c\right)
  12. g i j t = ( Δ R ) g i j \frac{\partial g_{ij}}{\partial t}=(\Delta R)g_{ij}
  13. u t + 2 κ u x - u x x t + 3 u u x = 2 u x u x x + u u x x x u_{t}+2\kappa u_{x}-u_{xxt}+3uu_{x}=2u_{x}u_{xx}+uu_{xxx}\,
  14. u t + u x = v 2 - u 2 = v x - v t \displaystyle u_{t}+u_{x}=v^{2}-u^{2}=v_{x}-v_{t}
  15. ρ ( 𝐯 t + 𝐯 𝐯 ) = σ + 𝐟 \displaystyle\rho\left(\frac{\partial\mathbf{v}}{\partial t}+\mathbf{v}\cdot% \nabla\mathbf{v}\right)=\nabla\cdot\sigma+\mathbf{f}
  16. x D u + f ( D u ) = u x\cdot Du+f(Du)=u
  17. det ( i j ¯ ϕ ) = \displaystyle\det(\partial_{i\bar{j}}\phi)=
  18. i u t + c 0 u x x + u y y = c 1 | u | 2 u + c 2 u ϕ x \displaystyle iu_{t}+c_{0}u_{xx}+u_{yy}=c_{1}|u|^{2}u+c_{2}u\phi_{x}
  19. ϕ x x + c 3 ϕ y y = ( | u | 2 ) x \displaystyle\phi_{xx}+c_{3}\phi_{yy}=(|u|^{2})_{x}
  20. u t - u x x t + 4 u u x = 3 u x u x x + u u x x x \displaystyle u_{t}-u_{xxt}+4uu_{x}=3u_{x}u_{xx}+uu_{xxx}
  21. u t = ( u 2 - u x + 2 w ) x \displaystyle u_{t}=(u^{2}-u_{x}+2w)_{x}
  22. w t = ( 2 u w + w x ) x w_{t}=(2uw+w_{x})_{x}
  23. u t = 3 w w x \displaystyle u_{t}=3ww_{x}
  24. w t = 2 w x x x + 2 u w x + u x w \displaystyle w_{t}=2w_{xxx}+2uw_{x}+u_{x}w
  25. u t = u 3 u x x x . \displaystyle u_{t}=u^{3}u_{xxx}.\,
  26. i u t + u x x + 2 | u | x 2 u + | u | 4 u = 0 iu_{t}+u_{xx}+2|u|^{2}_{x}u+|u|^{4}u=0
  27. | u ( x ) | = F ( x ) , x Ω \displaystyle|\nabla u(x)|=F(x),\ x\in\Omega
  28. R a b - 1 2 R g a b = κ T a b \displaystyle R_{ab}-{\textstyle 1\over 2}R\,g_{ab}=\kappa T_{ab}
  29. ( u ) ( u r r + u r / r + u z z ) = ( u r ) 2 + ( u z ) 2 \displaystyle\Re(u)(u_{rr}+u_{r}/r+u_{zz})=(u_{r})^{2}+(u_{z})^{2}
  30. ρ t + ( ρ 𝐮 ) = 0 ρ 𝐮 t + ( 𝐮 ( ρ 𝐮 ) ) + p = 0 E t + ( 𝐮 ( E + p ) ) = 0 , \begin{aligned}&\displaystyle\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho% \mathbf{u})=0\\ &\displaystyle\frac{\partial\rho\mathbf{u}}{\partial t}+\nabla\cdot(\mathbf{u}% \otimes(\rho\mathbf{u}))+\nabla p=0\\ &\displaystyle\frac{\partial E}{\partial t}+\nabla\cdot(\mathbf{u}(E+p))=0,% \end{aligned}
  31. u t = u ( 1 - u ) + 2 u x 2 . \displaystyle\frac{\partial u}{\partial t}=u(1-u)+\frac{\partial^{2}u}{% \partial x^{2}}.\,
  32. u t = u x x + u ( u - a ) ( 1 - u ) + w \displaystyle u_{t}=u_{xx}+u(u-a)(1-u)+w
  33. w t = ϵ u \displaystyle w_{t}=\epsilon u
  34. u t = 6 ( u + ε 2 u 2 ) u x + u x x x \displaystyle u_{t}=6\left(u+\varepsilon^{2}u^{2}\right)u_{x}+u_{xxx}
  35. α ψ + β | ψ | 2 ψ + 1 2 m ( - i - 2 e 𝐀 ) 2 ψ = 0 \displaystyle\alpha\psi+\beta|\psi|^{2}\psi+\tfrac{1}{2m}\left(-i\hbar\nabla-2% e\mathbf{A}\right)^{2}\psi=0
  36. 1 + n 1+n
  37. i t ψ = ( - 1 2 Δ 2 + V ( x ) + g | ψ | 2 ) ψ \displaystyle i\partial_{t}\psi=\left(-\tfrac{1}{2}\Delta^{2}+V(x)+g|\psi|^{2}% \right)\psi
  38. i t u + Δ u = ( ± | x | - n * | u | 2 ) u \displaystyle i\partial_{t}u+\Delta u=\left(\pm|x|^{-n}*|u|^{2}\right)u
  39. 0 = t ( 2 ϕ - ϕ ) - [ ( ϕ × 𝐳 ^ ) ] [ 2 ϕ - ln ( n 0 ω c i ) ] \displaystyle 0=\frac{\partial}{\partial t}\left(\nabla^{2}\phi-\phi\right)-% \left[\left(\nabla\phi\times\hat{\mathbf{z}}\right)\cdot\nabla\right]\left[% \nabla^{2}\phi-\ln\left(\frac{n_{0}}{\omega_{ci}}\right)\right]
  40. 𝐒 t = 𝐒 𝐒 x x . \displaystyle\mathbf{S}_{t}=\mathbf{S}\wedge\mathbf{S}_{xx}.
  41. u t = 1 2 u x x x + 3 u u x - 6 w w x , w t + w x x x + 3 u w x = 0 \displaystyle u_{t}=\tfrac{1}{2}u_{xxx}+3uu_{x}-6ww_{x},\quad w_{t}+w_{xxx}+3% uw_{x}=0
  42. ( u t + u u x ) x = 1 2 u x 2 \displaystyle\left(u_{t}+uu_{x}\right)_{x}=\tfrac{1}{2}u_{x}^{2}
  43. 𝐒 t = 𝐒 ( 2 𝐒 x 2 + 2 𝐒 y 2 ) + u x 𝐒 y + u y 𝐒 x \displaystyle\frac{\partial\mathbf{S}}{\partial t}=\mathbf{S}\wedge\left(\frac% {\partial^{2}\mathbf{S}}{\partial x^{2}}+\frac{\partial^{2}\mathbf{S}}{% \partial y^{2}}\right)+\frac{\partial u}{\partial x}\frac{\partial\mathbf{S}}{% \partial y}+\frac{\partial u}{\partial y}\frac{\partial\mathbf{S}}{\partial x}
  44. 2 u x 2 - α 2 2 u y 2 = - 2 α 2 𝐒 ( 𝐒 x 𝐒 y ) \displaystyle\frac{\partial^{2}u}{\partial x^{2}}-\alpha^{2}\frac{\partial^{2}% u}{\partial y^{2}}=-2\alpha^{2}\mathbf{S}\cdot\left(\frac{\partial\mathbf{S}}{% \partial x}\wedge\frac{\partial\mathbf{S}}{\partial y}\right)
  45. x ( t u + u x u + ϵ 2 x x x u ) + λ y y u = 0 \displaystyle\partial_{x}\left(\partial_{t}u+u\partial_{x}u+\epsilon^{2}% \partial_{xxx}u\right)+\lambda\partial_{yy}u=0
  46. 4 u = E ( w x y 2 - w x x w y y ) , 4 w = a + b ( u y y w x x + u x x w y y - 2 u x y w x y ) \displaystyle\nabla^{4}u=E\left(w_{xy}^{2}-w_{xx}w_{yy}\right),\quad\nabla^{4}% w=a+b\left(u_{yy}w_{xx}+u_{xx}w_{yy}-2u_{xy}w_{xy}\right)
  47. f x = 2 f g c ( x - t ) = g t \displaystyle f_{x}=2fgc(x-t)=g_{t}
  48. u t = u x x x x x + 10 u x x x u + 25 u x x u x + 20 u 2 u x \displaystyle u_{t}=u_{xxxxx}+10u_{xxx}u+25u_{xx}u_{x}+20u^{2}u_{x}
  49. 2 s = ( | 𝐚 | 2 + 1 ) s , 2 𝐚 = ( 𝐚 ) + s 2 𝐚 \displaystyle\nabla^{2}s=\left(|\mathbf{a}|^{2}+1\right)s,\quad\nabla^{2}% \mathbf{a}=\nabla(\nabla\cdot\mathbf{a})+s^{2}\mathbf{a}
  50. 2 u + λ u p = 0 \nabla^{2}u+\lambda u^{p}=0
  51. u x t - ( u u x ) x = u y y \displaystyle u_{xt}-(uu_{x})_{x}=u_{yy}
  52. t u + x 3 u + 6 u x u = 0 \displaystyle\partial_{t}u+\partial^{3}_{x}u+6u\partial_{x}u=0
  53. u t = 6 u u x - u x x x + 3 w w x x , w t = 3 u x w + 6 u w x - 4 w x x x \displaystyle u_{t}=6uu_{x}-u_{xxx}+3ww_{xx},\quad w_{t}=3u_{x}w+6uw_{x}-4w_{xxx}
  54. 1 + n 1+n
  55. u t + 4 u + 2 u + 1 2 | u | 2 = 0 \displaystyle u_{t}+\nabla^{4}u+\nabla^{2}u+\tfrac{1}{2}|\nabla u|^{2}=0
  56. 𝐒 t = 𝐒 i 2 𝐒 x i 2 + 𝐒 J 𝐒 \displaystyle\frac{\partial\mathbf{S}}{\partial t}=\mathbf{S}\wedge\sum_{i}% \frac{\partial^{2}\mathbf{S}}{\partial x_{i}^{2}}+\mathbf{S}\wedge J\mathbf{S}
  57. 2 u t x + u x u x x - u y y \displaystyle 2u_{tx}+u_{x}u_{xx}-u_{yy}
  58. 2 u + e λ u = 0 \displaystyle\nabla^{2}u+e^{\lambda u}=0
  59. div ( D u / 1 + | D u | 2 ) = 0 \displaystyle\operatorname{div}(Du/\sqrt{1+|Du|^{2}})=0
  60. det ( i j ϕ ) = \displaystyle\det(\partial_{ij}\phi)=
  61. ρ ( v i t + v j v i x j ) = - p x i + x j [ μ ( v i x j + v j x i ) + λ v k x k ] + f i \displaystyle\rho\left(\frac{\partial v_{i}}{\partial t}+v_{j}\frac{\partial v% _{i}}{\partial x_{j}}\right)=-\frac{\partial p}{\partial x_{i}}+\frac{\partial% }{\partial x_{j}}\left[\mu\left(\frac{\partial v_{i}}{\partial x_{j}}+\frac{% \partial v_{j}}{\partial x_{i}}\right)+\lambda\frac{\partial v_{k}}{\partial x% _{k}}\right]+f_{i}
  62. ρ t + ( ρ v i ) x i = 0 \frac{\partial\rho}{\partial t}+\frac{\partial\left(\rho\,v_{i}\right)}{% \partial x_{i}}=0
  63. v i x i = 0 \frac{\partial v_{i}}{\partial x_{i}}=0
  64. i t ψ = - 1 2 x 2 ψ + κ | ψ | 2 ψ \displaystyle i\partial_{t}\psi=-{1\over 2}\partial^{2}_{x}\psi+\kappa|\psi|^{% 2}\psi
  65. i t ψ = - 1 2 x 2 ψ + x ( i κ | ψ | 2 ψ ) \displaystyle i\partial_{t}\psi=-{1\over 2}\partial^{2}_{x}\psi+\partial_{x}(i% \kappa|\psi|^{2}\psi)
  66. 2 ω + f 2 σ 2 ω p 2 \displaystyle\nabla^{2}\omega+\frac{f^{2}}{\sigma}\frac{\partial^{2}\omega}{% \partial p^{2}}
  67. = f σ p 𝐕 g p ( ζ g + f ) + R σ p p 2 ( 𝐕 g p T ) \displaystyle=\frac{f}{\sigma}\frac{\partial}{\partial p}\mathbf{V}_{g}\cdot% \nabla_{p}(\zeta_{g}+f)+\frac{R}{\sigma p}\nabla^{2}_{p}(\mathbf{V}_{g}\cdot% \nabla_{p}T)
  68. ( 1 + u y 2 ) u x x - 2 u x u y u x y + ( 1 + u x 2 ) u y y = 0 \displaystyle(1+u_{y}^{2})u_{xx}-2u_{x}u_{y}u_{xy}+(1+u_{x}^{2})u_{yy}=0
  69. u x x - u y y ± sin u cos u + cos u sin 3 u ( v x 2 - v y 2 ) = 0 \displaystyle u_{xx}-u_{yy}\pm\sin u\cos u+\frac{\cos u}{\sin^{3}u}(v_{x}^{2}-% v_{y}^{2})=0
  70. ( v x cot 2 u ) x = ( v y cot 2 u ) y \displaystyle(v_{x}\cot^{2}u)_{x}=(v_{y}\cot^{2}u)_{y}
  71. u t = Δ ( u γ ) \displaystyle u_{t}=\Delta(u^{\gamma})
  72. u t + u u x + v u y = U t + U U x + μ ρ u y y \displaystyle u_{t}+uu_{x}+vu_{y}=U_{t}+UU_{x}+\frac{\mu}{\rho}u_{yy}
  73. u x + v y = 0 \displaystyle u_{x}+v_{y}=0
  74. u t t - u x x = ϵ ( u t - u t 3 ) \displaystyle u_{tt}-u_{xx}=\epsilon(u_{t}-u_{t}^{3})
  75. t g i j = - 2 R i j \displaystyle\partial_{t}g_{ij}=-2R_{ij}
  76. θ t = z [ K ( ψ ) ( ψ z + 1 ) ] \displaystyle\frac{\partial\theta}{\partial t}=\frac{\partial}{\partial z}% \left[K(\psi)\left(\frac{\partial\psi}{\partial z}+1\right)\right]
  77. u t + a x ( u n ) + 3 x 3 ( u n ) = 0 \frac{\partial u}{\partial t}+a\frac{\partial}{\partial x}\left(u^{n}\right)+% \frac{\partial^{3}}{\partial x^{3}}\left(u^{n}\right)=0
  78. u t + 45 u 2 u x + 15 u x u x x + 15 u u x x x + u x x x x x = 0 \displaystyle u_{t}+45u^{2}u_{x}+15u_{x}u_{xx}+15uu_{xxx}+u_{xxxxx}=0
  79. A i t j = [ A i , A j ] t i - t j , i j A i t i = - j = 1 j i n [ A i , A j ] t i - t j , 1 i , j n \displaystyle\begin{aligned}\displaystyle{\partial A_{i}\over\partial t_{j}}&% \displaystyle={\left[A_{i},\ A_{j}\right]\over t_{i}-t_{j}},\quad i\neq j\\ \displaystyle{\partial A_{i}\over\partial t_{i}}&\displaystyle=-\sum_{j=1\atop j% \neq i}^{n}{\left[A_{i},\ A_{j}\right]\over t_{i}-t_{j}},\quad 1\leq i,j\leq n% \end{aligned}
  80. D A ϕ = 0 , F A + = σ ( ϕ ) \displaystyle D^{A}\phi=0,\qquad F^{+}_{A}=\sigma(\phi)
  81. η t + ( η u ) x + ( η v ) y = 0 ( η u ) t + x ( η u 2 + 1 2 g η 2 ) + ( η u v ) y = 0 ( η v ) t + ( η u v ) x + y ( η v 2 + 1 2 g η 2 ) = 0 \displaystyle\begin{aligned}\displaystyle\frac{\partial\eta}{\partial t}+\frac% {\partial(\eta u)}{\partial x}+\frac{\partial(\eta v)}{\partial y}=0\\ \displaystyle\frac{\partial(\eta u)}{\partial t}+\frac{\partial}{\partial x}% \left(\eta u^{2}+\frac{1}{2}g\eta^{2}\right)+\frac{\partial(\eta uv)}{\partial y% }=0\\ \displaystyle\frac{\partial(\eta v)}{\partial t}+\frac{\partial(\eta uv)}{% \partial x}+\frac{\partial}{\partial y}\left(\eta v^{2}+\frac{1}{2}g\eta^{2}% \right)=0\end{aligned}
  82. ϕ t t - ϕ x x + sin ϕ = 0 \displaystyle\,\phi_{tt}-\phi_{xx}+\sin\phi=0
  83. u x t = sinh u \displaystyle u_{xt}=\sinh u
  84. 2 u + sinh u = 0 \displaystyle\nabla^{2}u+\sinh u=0
  85. u t = r u - ( 1 + 2 ) 2 u + N ( u ) \displaystyle\frac{\partial u}{\partial t}=ru-(1+\nabla^{2})^{2}u+N(u)
  86. u x y + α u x + β u y + γ u x u y = 0 \displaystyle u_{xy}+\alpha u_{x}+\beta u_{y}+\gamma u_{x}u_{y}=0
  87. i u x + v + u | v | 2 = 0 \displaystyle iu_{x}+v+u|v|^{2}=0
  88. i v t + u + v | u | 2 = 0 \displaystyle iv_{t}+u+v|u|^{2}=0
  89. 2 log u n = u n + 1 - 2 u n + u n - 1 \displaystyle\nabla^{2}\log u_{n}=u_{n+1}-2u_{n}+u_{n-1}
  90. ( t + z 3 + z ¯ 3 ) v + z ( u v ) + z ¯ ( u w ) = 0 \displaystyle(\partial_{t}+\partial_{z}^{3}+\partial_{\bar{z}}^{3})v+\partial_% {z}(uv)+\partial_{\bar{z}}(uw)=0
  91. z ¯ u = 3 z v \displaystyle\partial_{\bar{z}}u=3\partial_{z}v
  92. z w = 3 z ¯ v \displaystyle\partial_{z}w=3\partial_{\bar{z}}v
  93. i u t + ( ( 1 + | u | 2 ) - 1 / 2 u ) x x = 0 \displaystyle iu_{t}+((1+|u|^{2})^{-1/2}u)_{xx}=0
  94. σ , τ = 1 n ( 3 F t α t β t σ η σ τ 3 F t μ t ν t τ ) \displaystyle\sum_{\sigma,\tau=1}^{n}\left({\partial^{3}F\over\partial t^{% \alpha}t^{\beta}t^{\sigma}}\eta^{\sigma\tau}{\partial^{3}F\over\partial t^{\mu% }t^{\nu}t^{\tau}}\right)
  95. = σ , τ = 1 n ( 3 F t α t ν t σ η σ τ 3 F t μ t β t τ ) \displaystyle=\sum_{\sigma,\tau=1}^{n}\left({\partial^{3}F\over\partial t^{% \alpha}t^{\nu}t^{\sigma}}\eta^{\sigma\tau}{\partial^{3}F\over\partial t^{\mu}t% ^{\beta}t^{\tau}}\right)
  96. S k ( γ ) = - k 8 π S 2 d 2 x 𝒦 ( γ - 1 μ γ , γ - 1 μ γ ) + 2 π k S W Z ( γ ) S_{k}(\gamma)=-\,\frac{k}{8\pi}\int_{S^{2}}d^{2}x\,\mathcal{K}(\gamma^{-1}% \partial^{\mu}\gamma\,,\,\gamma^{-1}\partial_{\mu}\gamma)+2\pi k\,S^{\mathrm{W% }Z}(\gamma)
  97. S W Z ( γ ) = - 1 48 π 2 B 3 d 3 y ϵ i j k 𝒦 ( γ - 1 γ y i , [ γ - 1 γ y j , γ - 1 γ y k ] ) S^{\mathrm{W}Z}(\gamma)=-\,\frac{1}{48\pi^{2}}\int_{B^{3}}d^{3}y\,\epsilon^{% ijk}\mathcal{K}\left(\gamma^{-1}\,\frac{\partial\gamma}{\partial y^{i}}\,,\,% \left[\gamma^{-1}\,\frac{\partial\gamma}{\partial y^{j}}\,,\,\gamma^{-1}\,% \frac{\partial\gamma}{\partial y^{k}}\right]\right)
  98. η t + α η η x + - + K ( x - ξ ) η ξ ( ξ , t ) d ξ = 0 \displaystyle\eta_{t}+\alpha\eta\eta_{x}+\int_{-\infty}^{+\infty}K(x-\xi)\,% \eta_{\xi}(\xi,t)\,\,\text{d}\xi=0
  99. Δ ϕ + h ( x ) ϕ = λ f ( x ) ϕ ( n + 2 ) / ( n - 2 ) \displaystyle\Delta\phi+h(x)\phi=\lambda f(x)\phi^{(n+2)/(n-2)}
  100. D μ F μ ν = 0 , F μ ν = A μ , ν - A ν , μ + [ A μ , A ν ] \displaystyle D_{\mu}F^{\mu\nu}=0,\quad F_{\mu\nu}=A_{\mu,\nu}-A_{\nu,\mu}+[A_% {\mu},\,A_{\nu}]
  101. F α β = ± ϵ α β μ ν F μ ν , F μ ν = A μ , ν - A ν , μ + [ A μ , A ν ] F_{\alpha\beta}=\pm\epsilon_{\alpha\beta\mu\nu}F^{\mu\nu},\quad F_{\mu\nu}=A_{% \mu,\nu}-A_{\nu,\mu}+[A_{\mu},\,A_{\nu}]
  102. i t u + Δ u = - A u \displaystyle i\partial_{t}u+\Delta u=-Au
  103. A = m 2 A + | u | 2 \displaystyle\Box A=m^{2}A+|u|^{2}
  104. i t u + Δ u = u n \displaystyle i\partial_{t}u+\Delta u=un
  105. n = - Δ ( | u | 2 ) \displaystyle\Box n=-\Delta(|u|^{2})
  106. i u t + L 1 u = ϕ u \displaystyle iu_{t}+L_{1}u=\phi u
  107. L 2 ϕ = L 3 ( | u | 2 ) \displaystyle L_{2}\phi=L_{3}(|u|^{2})
  108. ( u x t / u ) t t - ( u x t / u ) x x + 2 ( u 2 ) x t = 0 \displaystyle(u_{xt}/u)_{tt}-(u_{xt}/u)_{xx}+2(u^{2})_{xt}=0
  109. ϕ t t - ϕ x x - ϕ + ϕ 3 = 0 \displaystyle\phi_{tt}-\phi_{xx}-\phi+\phi^{3}=0
  110. 𝐯 x t + ( 𝐯 x 𝐯 t ) 𝐯 = 0 \displaystyle{\mathbf{v}}_{xt}+({\mathbf{v}}_{x}{\mathbf{v}}_{t}){\mathbf{v}}=0

List_of_Oklahoma_City_Thunder_seasons.html

  1. Games behind = ( TeamA s wins - TeamB s wins ) + ( TeamB s losses - TeamA s losses ) 2 \mathrm{Games}\ \mathrm{behind}=\frac{(\mathrm{TeamA^{\prime}s}\ \mathrm{wins}% -\mathrm{TeamB^{\prime}s}\ \mathrm{wins})+(\mathrm{TeamB^{\prime}s}\ \mathrm{% losses}-\mathrm{TeamA^{\prime}s}\ \mathrm{losses})}{\mathrm{2}}

List_of_probabilistic_proofs_of_non-probabilistic_theorems.html

  1. p > 2 p>2
  2. n = 1 1 n 2 = π 2 6 , \qquad\sum_{n=1}^{\infty}\frac{1}{n^{2}}=\frac{\pi^{2}}{6},

List_of_San_Francisco_49ers_head_coaches.html

  1. W i n s + 1 2 T i e s G a m e s \frac{Wins+\frac{1}{2}Ties}{Games}

List_of_Saxon_locomotives_and_railcars.html

  1. 𝔙 \mathfrak{V}
  2. \mathfrak{H}

List_of_types_of_functions.html

  1. \mapsto
  2. λ \lambda
  3. f : A B f:A\rightarrow B

Littlewood_polynomial.html

  1. p ( x ) = i = 0 n a i x i p(x)=\sum_{i=0}^{n}a_{i}x^{i}\,
  2. a i = ± 1 a_{i}=\pm 1
  3. c 1 n + 1 | p n ( z ) | c 2 n + 1 . c_{1}\sqrt{n+1}\leq|p_{n}(z)|\leq c_{2}\sqrt{n+1}.\,
  4. z z
  5. c 2 = 2 c_{2}=\sqrt{2}

Littlewood–Richardson_rule.html

  1. λ , μ , ν \lambda,\mu,\nu
  2. λ \lambda
  3. μ \mu
  4. ν \nu
  5. c λ , μ ν c_{\lambda,\mu}^{\nu}
  6. s λ s μ = ν c λ , μ ν s ν . s_{\lambda}s_{\mu}=\sum_{\nu}c_{\lambda,\mu}^{\nu}s_{\nu}.
  7. c λ , μ ν c_{\lambda,\mu}^{\nu}
  8. ν / λ \nu/\lambda
  9. μ \mu
  10. i i
  11. i + 1 i+1
  12. λ = ( 2 , 1 ) \lambda=(2,1)
  13. μ = ( 3 , 2 , 1 ) \mu=(3,2,1)
  14. ν = ( 4 , 3 , 2 ) \nu=(4,3,2)
  15. c λ , μ ν = 2 c_{\lambda,\mu}^{\nu}=2
  16. ν / λ \nu/\lambda
  17. μ \mu
  18. i > 1 i>1
  19. ( i - 1 ) [ j ] (i-1)[j]
  20. ( i - 1 ) [ j ] (i-1)[j]
  21. λ , μ , ν \lambda,\mu,\nu
  22. ν / λ \nu/\lambda
  23. μ \mu
  24. | λ | + | μ | = | ν | |\lambda|+|\mu|=|\nu|
  25. c λ , μ ν c_{\lambda,\mu}^{\nu}
  26. i [ j + 1 ] i[j+1]
  27. i - 1 [ j ] i-1[j]
  28. i [ j ] i[j]
  29. c λ , μ ν c_{\lambda,\mu}^{\nu}
  30. λ μ ν {}^{ν}_{λμ}
  31. s λ s μ = c λ μ ν s ν s_{\lambda}s_{\mu}=\sum c_{\lambda\mu}^{\nu}s_{\nu}
  32. λ μ ν {}^{ν}_{λμ}
  33. s ν / λ = μ c λ μ ν s μ . s_{\nu/\lambda}=\sum_{\mu}c_{\lambda\mu}^{\nu}s_{\mu}.
  34. λ μ ν {}^{ν}_{λμ}
  35. σ λ σ μ = c λ μ ν σ ν \sigma_{\lambda}\sigma_{\mu}=\sum c_{\lambda\mu}^{\nu}\sigma_{\nu}
  36. λ μ ν {}^{ν}_{λμ}
  37. λ μ ν {}^{ν}_{λμ}
  38. E λ E μ = ν ( E ν ) c λ μ ν . E^{\lambda}\otimes E^{\mu}=\bigoplus_{\nu}(E^{\nu})^{\oplus c_{\lambda\mu}^{% \nu}}.
  39. λ μ ν {}^{ν}_{λμ}
  40. λ μ ν {}^{ν}_{λμ}
  41. λ μ ν {}^{ν}_{λμ}
  42. s λ s μ / ν = λ + ω ( T j ) P s λ + ω ( T ) s_{\lambda}s_{\mu/\nu}=\sum_{\lambda+\omega(T_{\geq j})\in P}s_{\lambda+\omega% (T)}
  43. S μ S n = λ S λ \displaystyle S_{\mu}S_{n}=\sum_{\lambda}S_{\lambda}
  44. \geq
  45. ( a p ) (a^{p})
  46. s ( a p ) s ( b q ) s_{(a^{p})}s_{(b^{q})}
  47. λ \lambda
  48. p + q \leq p+q
  49. λ q + 1 = λ q + 2 = = λ p = a , \lambda_{q+1}=\lambda_{q+2}=\cdots=\lambda_{p}=a,
  50. λ q max ( a , b ) \lambda_{q}\geq\mathrm{max}(a,b)
  51. λ i + λ p + q - i + 1 = a + b , i = 1 , , q . \lambda_{i}+\lambda_{p+q-i+1}=a+b,\quad{i=1,\dots,q}.
  52. S λ S μ = c λ μ ν S ν . S_{\lambda}S_{\mu}=\sum c_{\lambda\mu}^{\nu}S_{\nu}.
  53. λ μ ν {}^{ν}_{λμ}

Lituus_(mathematics).html

  1. r 2 θ = k r^{2}\theta=k\,
  2. r r
  3. x x
  4. ( θ , r ) = ( 1 2 , 2 k ) (\theta,r)=(\tfrac{1}{2},\sqrt{2k})
  5. ( 1 2 , - 2 k ) (\tfrac{1}{2},-\sqrt{2k})

Liu_Hui's_π_algorithm.html

  1. π \pi
  2. 92 / 29 92/29
  3. π 10 3.162 \pi\approx\sqrt{10}\approx 3.162
  4. π 142 / 45 3.156 π≈142/45≈3.156
  5. π \pi
  6. π \pi
  7. π 3.1416 π≈3.1416
  8. π \pi
  9. π \pi
  10. π \pi
  11. π \pi
  12. π 3.1416 π≈3.1416
  13. π \pi
  14. A A
  15. r r
  16. C C
  17. r r
  18. r r
  19. π \pi
  20. lim N area of N -gon = area of circle . \lim_{N\to\infty}\,\text{area of }N\,\text{-gon}=\,\text{area of circle}.\,
  21. d d
  22. d d
  23. A B C D ABCD
  24. N N→∞
  25. d 0 d→0
  26. A B C D 0 ABCD→0
  27. N N→∞
  28. N N
  29. L L
  30. R R
  31. R L RL
  32. N N
  33. N N
  34. π \pi
  35. π \pi
  36. π \pi
  37. π \pi
  38. N N
  39. N N
  40. N N
  41. A N A_{N}
  42. N N
  43. A 2 N A_{2N}
  44. N N
  45. D 2 N = A 2 N - A N . D_{2N}=A_{2N}-A_{N}.
  46. D 2 N D_{2N}
  47. A 2 N + D 2 N . A_{2N}+D_{2N}.
  48. A 2 N < C < A 2 N + D 2 N . A_{2N}<C<A_{2N}+D_{2N}.
  49. π \pi
  50. A 2 N < π < A 2 N + D 2 N . A_{2N}<\pi<A_{2N}+D_{2N}.
  51. M M
  52. A B AB
  53. r r
  54. A B AB
  55. O P C OPC
  56. A C AC
  57. m m
  58. A O P AOP
  59. A P C APC
  60. G 2 = r 2 - ( M 2 ) 2 {}G^{2}=r^{2}-\left(\tfrac{M}{2}\right)^{2}
  61. G = r 2 - M 2 4 {}G=\sqrt{r^{2}-\tfrac{M^{2}}{4}}
  62. j = r - G = r - r 2 - M 2 4 {}j=r-G=r-\sqrt{r^{2}-\tfrac{M^{2}}{4}}
  63. m 2 = ( M 2 ) 2 + j 2 {}m^{2}=\left(\tfrac{M}{2}\right)^{2}+j^{2}
  64. m = m 2 . {}m=\sqrt{m^{2}}.
  65. r r
  66. A 96 = 313 584 625 {}A_{96}=313{584\over 625}
  67. A 192 = 314 64 625 {}A_{192}=314{64\over 625}
  68. D 192 = 314 64 625 - 313 584 625 = 105 625 {}D_{192}=314\frac{64}{625}-313\frac{584}{625}=\frac{105}{625}
  69. π \pi
  70. A 2 N < C < A 2 N + D 2 N . A_{2N}<C<A_{2N}+D_{2N}.
  71. r r
  72. C C
  73. 100 × π 100\times\pi
  74. 314 64 625 < 100 × π < 314 64 625 + 105 625 {}314\frac{64}{625}<100\times\pi<314\frac{64}{625}+\frac{105}{625}
  75. 314 64 625 < 100 × π < 314 169 625 {}314\frac{64}{625}<100\times\pi<314\frac{169}{625}
  76. 3.141024 < π < 3.142704. {}3.141024<\pi<3.142704.
  77. π \pi
  78. π \pi
  79. 157 50 \tfrac{157}{50}
  80. π \pi
  81. 2 - m 2 = 2 + ( 2 - M 2 ) , 2-m^{2}=\sqrt{2+(2-M^{2})}\,,
  82. m m
  83. M M
  84. D D
  85. N N
  86. N N
  87. N N
  88. D N = A N - A N / 2 D_{N}=A_{N}-A_{N/2}\,
  89. D 96 1 4 D 48 D_{96}\approx\tfrac{1}{4}D_{48}
  90. D 192 1 4 D 96 D_{192}\approx\tfrac{1}{4}D_{96}
  91. D 384 \displaystyle D_{384}
  92. π = A 192 + D 384 + D 768 + D 1536 + D 3072 + A 192 + F D 192 . {}\pi=A_{192}+D_{384}+D_{768}+D_{1536}+D_{3072}+\cdots\approx A_{192}+F\cdot D% _{192}.\,
  93. F = 1 4 + ( 1 4 ) 2 + ( 1 4 ) 3 + ( 1 4 ) 4 + = 1 3 . F=\tfrac{1}{4}+\left(\tfrac{1}{4}\right)^{2}+\left(\tfrac{1}{4}\right)^{3}+% \left(\tfrac{1}{4}\right)^{4}+\cdots=\tfrac{1}{3}.
  94. D 192 D_{192}
  95. = π A 192 + ( 1 3 ) D 192 3927 1250 3.1416. {}=\pi\approx A_{192}+\left(\tfrac{1}{3}\right)D_{192}\approx{3927\over 1250}% \approx 3.1416.\,
  96. A A
  97. π \pi
  98. π \pi
  99. π = 3927 1250 . \pi={3927\over 1250}.
  100. π \pi
  101. N N
  102. A 2 N = m N × r A_{2N}=m_{N}\times r
  103. A 24576 = 3.14159261864 < π A_{24576}=3.14159261864<\pi
  104. π \pi
  105. A 24576 < π < A 24576 + D 24576 A_{24576}<\pi<A_{24576}+D_{24576}
  106. D 24576 = A 24576 - A 12288 = 0.0000001021 D_{24576}=A_{24576}-A_{12288}=0.0000001021
  107. A 24576 = 3.14159261864 < π < 3.14159261864 + 0.0000001021 A_{24576}=3.14159261864<\pi<3.14159261864+0.0000001021
  108. 3.14159261864 < π < 3.141592706934 3.14159261864<\pi<3.141592706934
  109. 3.1415926 < π < 3.1415927 3.1415926<\pi<3.1415927
  110. π \pi
  111. π 355 113 \pi\approx{355\over 113}
  112. π \pi
  113. π \pi
  114. 22 7 {22\over 7}
  115. π = 10 ) \pi=\sqrt{10})
  116. π \pi
  117. π 355 113 . \pi\approx{355\over 113}.
  118. π \pi
  119. N N
  120. π < 22 / 7 = 3.142857 \pi<22/7=3.142857
  121. 223 / 71 = 3.140845 223/71=3.140845
  122. π \pi
  123. A 192 = 3.1410319509 A_{192}=3.1410319509
  124. D 192 = 0.0016817478 D_{192}=0.0016817478
  125. π A 192 + 1 3 D 192 3.1410319509 + 0.0016817478 / 3 \pi\approx A_{192}+\frac{1}{3}D_{192}\approxeq 3.1410319509+0.0016817478/3
  126. π 3.1410319509 + 0.0005605826 \pi\approx 3.1410319509+0.0005605826
  127. π 3.1415925335. \pi\approx 3.1415925335.

Load_factor_(electrical).html

  1. f L o a d = Average load Maximum load in given time period f_{Load}=\frac{\,\text{Average load}}{\,\text{Maximum load in given time % period}}
  2. f D e m a n d = Maximum load in given time period Maximum possible load f_{Demand}=\frac{\,\text{Maximum load in given time period}}{\,\text{Maximum % possible load}}

Loanable_funds.html

  1. P S + Δ B = P I , PS+\Delta B=PI,
  2. P , S , I P,S,I
  3. Δ B \Delta B

Local_parameter.html

  1. \mathbb{C}
  2. \mathbb{C}
  3. f f
  4. P C P\in C
  5. f m P 𝒪 C , P f\in m_{P}\subset\mathcal{O}_{C,P}
  6. 𝒪 C , P \mathcal{O}_{C,P}
  7. ord P ( f ) = max { d = 0 , 1 , 2 , : f m P d } ; \operatorname{ord}_{P}(f)=\max\{d=0,1,2,\ldots:f\in m^{d}_{P}\};
  8. 𝒪 C , P \mathcal{O}_{C,P}
  9. f K ( C ) f\in K(C)
  10. m P d m_{P}^{d}
  11. 𝒪 C , P \mathcal{O}_{C,P}
  12. m P m_{P}
  13. P C P\in C
  14. ord P ( f / g ) = ord P ( f ) - ord P ( g ) \operatorname{ord}_{P}(f/g)=\operatorname{ord}_{P}(f)-\operatorname{ord}_{P}(g)
  15. ord P \operatorname{ord}_{P}
  16. 𝒪 C , P \mathcal{O}_{C,P}
  17. m P m_{P}
  18. t K ( C ) t\in K(C)
  19. ord P ( t ) = 1 \operatorname{ord}_{P}(t)=1

Locally_finite_poset.html

  1. ( f * g ) ( x , y ) := x z y f ( x , z ) g ( z , y ) . (f*g)(x,y):=\sum_{x\leq z\leq y}f(x,z)g(z,y).

Location_estimation_in_sensor_networks.html

  1. θ \theta
  2. N N
  3. x n = θ + w n x_{n}=\theta+w_{n}
  4. w n w_{n}
  5. n n
  6. x n x_{n}
  7. m n ( x n ) m_{n}(x_{n})
  8. θ ^ = f ( m 1 ( x 1 ) , , m N ( x N ) ) \hat{\theta}=f(m_{1}(x_{1}),\cdot,m_{N}(x_{N}))
  9. m n , 1 n N m_{n},\,1\leq n\leq N
  10. f ( m 1 ( x 1 ) , , m N ( x N ) ) f(m_{1}(x_{1}),\cdot,m_{N}(x_{N}))
  11. 𝔼 θ - θ ^ 2 \mathbb{E}\|\theta-\hat{\theta}\|^{2}
  12. x n x_{n}
  13. m n ( x n ) = x n m_{n}(x_{n})=x_{n}
  14. θ ^ = 1 N n = 1 N x n \hat{\theta}=\frac{1}{N}\sum_{n=1}^{N}x_{n}
  15. 𝔼 θ - θ ^ 2 = var ( θ ^ ) = σ 2 N \mathbb{E}\|\theta-\hat{\theta}\|^{2}=\,\text{var}(\hat{\theta})=\frac{\sigma^% {2}}{N}
  16. w n 𝒩 ( 0 , σ 2 ) w_{n}\sim\mathcal{N}(0,\sigma^{2})
  17. m n ( x n ) m_{n}(x_{n})
  18. w n 𝒩 ( 0 , σ 2 ) w_{n}\sim\mathcal{N}(0,\sigma^{2})
  19. m n ( x n ) = I ( x n - τ ) = { 1 x n > τ 0 x n τ m_{n}(x_{n})=I(x_{n}-\tau)=\begin{cases}1&x_{n}>\tau\\ 0&x_{n}\leq\tau\end{cases}
  20. θ ^ = τ - F - 1 ( 1 N n = 1 N m n ( x n ) ) , F ( x ) = 1 2 π σ x e - w 2 / 2 σ 2 d w \hat{\theta}=\tau-F^{-1}\left(\frac{1}{N}\sum\limits_{n=1}^{N}m_{n}(x_{n})% \right),\quad F(x)=\frac{1}{\sqrt{2\pi}\sigma}\int\limits_{x}^{\infty}e^{-w^{2% }/2\sigma^{2}}\,dw
  21. τ \tau
  22. θ \theta
  23. m n ( x n ) m_{n}(x_{n})
  24. ( q = F ( τ - θ ) ) (q=F(\tau-\theta))
  25. q ^ \hat{q}
  26. q q
  27. θ \theta
  28. τ = θ \tau=\theta
  29. π σ 2 4 \frac{\pi\sigma^{2}}{4}
  30. π / 2 \pi/2
  31. τ \tau
  32. θ \theta
  33. | τ - θ | σ |\tau-\theta|\sim\sigma
  34. τ \tau
  35. θ \theta
  36. θ \theta
  37. w n w_{n}
  38. [ - U , U ] [-U,U]
  39. σ 2 N \frac{\sigma^{2}}{N}
  40. U U
  41. τ \tau
  42. σ \sigma
  43. τ 1 , τ 2 \tau_{1},\tau_{2}
  44. N / 2 N/2
  45. m A ( x ) = I ( x - τ 1 ) m_{A}(x)=I(x-\tau_{1})
  46. N / 2 N/2
  47. m B ( x ) = I ( x - τ 2 ) m_{B}(x)=I(x-\tau_{2})
  48. q ^ 1 = 2 N n = 1 N / 2 m A ( x n ) , q ^ 2 = 2 N n = 1 + N / 2 N m B ( x n ) \hat{q}_{1}=\frac{2}{N}\sum\limits_{n=1}^{N/2}m_{A}(x_{n}),\quad\hat{q}_{2}=% \frac{2}{N}\sum\limits_{n=1+N/2}^{N}m_{B}(x_{n})
  49. θ ^ = F - 1 ( q ^ 2 ) τ 1 - F - 1 ( q ^ 1 ) τ 2 F - 1 ( q ^ 2 ) - F - 1 ( q ^ 1 ) , F ( x ) = 1 2 π x e - v 2 / 2 d w \hat{\theta}=\frac{F^{-1}(\hat{q}_{2})\tau_{1}-F^{-1}(\hat{q}_{1})\tau_{2}}{F^% {-1}(\hat{q}_{2})-F^{-1}(\hat{q}_{1})},\quad F(x)=\frac{1}{\sqrt{2\pi}}\int% \limits_{x}^{\infty}e^{-v^{2}/2}dw
  50. τ 1 , τ 2 \tau_{1},\tau_{2}
  51. x n = θ + w n , n = 1 , , N x_{n}=\theta+w_{n},\quad n=1,\dots,N
  52. θ [ - U , U ] \theta\in[-U,U]
  53. w n 𝒫 , that is : w n is bounded to [ - U , U ] , 𝔼 ( w n ) = 0 w_{n}\in\mathcal{P},\,\text{ that is }:w_{n}\,\text{ is bounded to }[-U,U],% \mathbb{E}(w_{n})=0
  54. m n ( x n ) = { 1 x S n 0 x S n m_{n}(x_{n})=\begin{cases}1&x\in S_{n}\\ 0&x\notin S_{n}\end{cases}
  55. S n S_{n}
  56. [ - 2 U , 2 U ] [-2U,2U]
  57. θ ^ = n = 1 N α n m n ( x n ) \hat{\theta}=\sum\limits_{n=1}^{N}\alpha_{n}m_{n}(x_{n})
  58. S n S_{n}
  59. α n \alpha_{n}
  60. N / 2 N/2
  61. θ \theta
  62. [ 0 , 2 U ] [0,2U]
  63. N / 4 N/4
  64. [ - U , 0 ] [ U , 2 U ] [-U,0]\cup[U,2U]
  65. α n \alpha_{n}
  66. δ \delta
  67. | 𝔼 ( θ - θ ^ ) | < δ |\mathbb{E}(\theta-\hat{\theta})|<\delta
  68. θ [ - U , U ] \theta\in[-U,U]
  69. w n 𝒫 w_{n}\in\mathcal{P}
  70. N log 8 U δ N\geq\lceil\log\frac{8U}{\delta}\rceil
  71. δ \delta
  72. N log 2 U δ N\geq\lceil\log\frac{2U}{\delta}\rceil
  73. 𝔼 θ - θ ^ ϵ 2 \mathbb{E}\|\theta-\hat{\theta}\|\leq\epsilon^{2}
  74. ϵ \epsilon
  75. m n ( ) m_{n}(\cdot)

Lochs'_theorem.html

  1. lim n m n = 6 ln 2 ln 10 π 2 0.97027014 \lim_{n\rightarrow\infty}\frac{m}{n}=\frac{6\ln 2\ln 10}{\pi^{2}}\approx 0.970% 27014
  2. ln 10 \ln 10
  3. ln 11 \ln 11
  4. π 2 6 ln 2 ln 10 1.03064083 \frac{\pi^{2}}{6\ln 2\ln 10}\approx 1.03064083

Log-space_transducer.html

  1. M M
  2. O ( log n ) O(\log n)
  3. M M
  4. f : Σ Σ f\colon\Sigma^{\ast}\rightarrow\Sigma^{\ast}
  5. Σ \Sigma
  6. M M
  7. w w
  8. f ( w ) f(w)
  9. A Σ A\subseteq\Sigma^{\ast}
  10. B Σ B\subseteq\Sigma^{\ast}
  11. f f
  12. A A
  13. B B
  14. w A f ( w ) B w\in A\iff f(w)\in B

LogitBoost.html

  1. f = t α t h t f=\sum_{t}\alpha_{t}h_{t}
  2. i log ( 1 + e - y i f ( x i ) ) \sum_{i}\log\left(1+e^{-y_{i}f(x_{i})}\right)

LogP_machine.html

  1. L L
  2. o o
  3. g g
  4. P P
  5. L L
  6. o o
  7. g g
  8. P P
  9. L L
  10. o o
  11. g g

Lommel_polynomial.html

  1. J m + ν ( z ) = J ν ( z ) R m , ν ( z ) - J ν - 1 ( z ) R m - 1 , ν + 1 ( z ) \displaystyle J_{m+\nu}(z)=J_{\nu}(z)R_{m,\nu}(z)-J_{\nu-1}(z)R_{m-1,\nu+1}(z)
  2. R m , ν = n = 0 [ m / 2 ] ( - 1 ) n ( m - n ) ! Γ ( ν + m - n ) n ! ( m - 2 n ) ! Γ ( ν + n ) ( z / 2 ) 2 n - m . R_{m,\nu}=\sum_{n=0}^{[m/2]}\frac{(-1)^{n}(m-n)!\Gamma(\nu+m-n)}{n!(m-2n)!% \Gamma(\nu+n)}(z/2)^{2n-m}.

Lonely_runner_conjecture.html

  1. t d D || t d || 1 k , \exists t\in\mathbb{R}\quad\forall d\in D\quad||td||\geq\frac{1}{k},

Longest_path_problem.html

  1. n / exp ( Ω ( log n ) ) n/\exp(\Omega(\sqrt{\log n}))
  2. ϵ > 0 \epsilon>0
  3. 2 ( log n ) 1 - ϵ 2^{(\log n)^{1-\epsilon}}
  4. ϵ > 0 \epsilon>0
  5. n 1 - ϵ n^{1-\epsilon}
  6. n / log 2 + ϵ n n/\log^{2+\epsilon}n
  7. O ( n / log n ) O(n/\log n)
  8. d d
  9. d d
  10. O ( d ! 2 d n ) O(d!2^{d}n)
  11. n n
  12. d d
  13. O ( ! 2 n ) O(\ell!2^{\ell}n)
  14. \ell
  15. W [ 1 ] W[1]

Looman–Menchoff_theorem.html

  1. f / x \partial f/\partial x
  2. f / y \partial f/\partial y
  3. f z ¯ = 1 2 ( f x + i f y ) = 0. \frac{\partial f}{\partial\bar{z}}=\frac{1}{2}\left(\frac{\partial f}{\partial x% }+i\frac{\partial f}{\partial y}\right)=0.

Loop_variant.html

  1. { I C } S { I } { I } 𝐰𝐡𝐢𝐥𝐞 C 𝐝𝐨 S { I ¬ C } , \frac{\{I\land C\}\;S\;\{I\}}{\{I\}\;\mathbf{while}\;C\;\mathbf{do}\;S\;\{I% \land\lnot C\}},
  2. < is well-founded , [ I C V = z ] S [ I V < z ] [ I ] 𝐰𝐡𝐢𝐥𝐞 C 𝐝𝐨 S [ I ¬ C ] , \frac{<\textrm{\ is\ well-founded},\;[I\land C\land V=z]\;S\;[I\land V<z]}{[I]% \;\mathbf{while}\;C\;\mathbf{do}\;S\;[I\land\lnot C]},
  3. 𝐰𝐡𝐢𝐥𝐞 C 𝐝𝐨 S \mathbf{while}\;C\;\mathbf{do}\;S
  4. [ I C ] S [ I ] . [I\land C]\;S\;[I].
  5. Σ \Sigma
  6. σ \sigma^{\prime}
  7. σ \sigma
  8. σ , \sigma,
  9. σ \sigma^{\prime}
  10. σ . \sigma.
  11. σ σ , \sigma^{\prime}\neq\sigma,
  12. σ \sigma^{\prime}
  13. σ \sigma
  14. σ = σ , \sigma^{\prime}=\sigma,
  15. σ 0 , σ 1 , , σ n \sigma_{0},\sigma_{1},\,\dots\,,\sigma_{n}
  16. σ 0 = σ , \sigma_{0}=\sigma,
  17. σ n = σ \sigma_{n}=\sigma^{\prime}
  18. σ i + 1 \sigma_{i+1}
  19. σ i \sigma_{i}
  20. 0 i < n . 0\leq i<n.
  21. σ \sigma
  22. σ \sigma^{\prime}
  23. σ \sigma^{\prime}
  24. σ \sigma
  25. σ \sigma
  26. σ , \sigma^{\prime},
  27. Σ , \Sigma,
  28. V : Σ ω 1 . V:\Sigma\rightarrow\omega_{1}.

Low-power_electronics.html

  1. E stored = 1 2 C U 2 E_{\mathrm{stored}}={1\over 2}CU^{2}

Luke's_variational_principle.html

  1. \mathcal{L}
  2. = - t 0 t 1 { V ( t ) ρ [ Φ t + 1 2 | s y m b o l Φ | 2 + 1 2 ( Φ z ) 2 + g z ] d x d y d z } d t . \mathcal{L}=-\int_{t_{0}}^{t_{1}}\left\{\iiint_{V(t)}\rho\left[\frac{\partial% \Phi}{\partial t}+\frac{1}{2}\left|symbol{\nabla}\Phi\right|^{2}+\frac{1}{2}% \left(\frac{\partial\Phi}{\partial z}\right)^{2}+g\,z\right]\;\,\text{d}x\;\,% \text{d}y\;\,\text{d}z\;\right\}\;\,\text{d}t.
  3. = - t 0 t 1 { - h ( s y m b o l x ) η ( s y m b o l x , t ) ρ [ Φ t + 1 2 | s y m b o l Φ | 2 + 1 2 ( Φ z ) 2 ] d z + 1 2 ρ g η 2 } d s y m b o l x d t . \mathcal{L}=-\,\int_{t_{0}}^{t_{1}}\iint\left\{\int_{-h(symbol{x})}^{\eta(% symbol{x},t)}\rho\,\left[\frac{\partial\Phi}{\partial t}+\,\frac{1}{2}\left|% symbol{\nabla}\Phi\right|^{2}+\,\frac{1}{2}\left(\frac{\partial\Phi}{\partial z% }\right)^{2}\right]\;\,\text{d}z\;+\,\frac{1}{2}\,\rho\,g\,\eta^{2}\right\}\;% \,\text{d}symbol{x}\;\,\text{d}t.
  4. δ = 0 \delta\mathcal{L}=0
  5. δ Φ \displaystyle\delta_{\Phi}\mathcal{L}
  6. δ Φ = - ρ t 0 t 1 { t - h ( s y m b o l x ) η ( s y m b o l x , t ) δ Φ d z + s y m b o l - h ( s y m b o l x ) η ( s y m b o l x , t ) δ Φ s y m b o l Φ d z } d s y m b o l x d t + ρ t 0 t 1 { - h ( s y m b o l x ) η ( s y m b o l x , t ) δ Φ ( s y m b o l s y m b o l Φ + 2 Φ z 2 ) d z } d s y m b o l x d t + ρ t 0 t 1 [ ( η t + s y m b o l Φ s y m b o l η - Φ z ) δ Φ ] z = η ( s y m b o l x , t ) d s y m b o l x d t - ρ t 0 t 1 [ ( s y m b o l Φ s y m b o l h + Φ z ) δ Φ ] z = - h ( s y m b o l x ) d s y m b o l x d t = 0. \begin{aligned}\displaystyle\delta_{\Phi}\mathcal{L}\,=&\displaystyle-\,\rho\,% \int_{t_{0}}^{t_{1}}\iint\left\{\frac{\partial}{\partial t}\int_{-h(symbol{x})% }^{\eta(symbol{x},t)}\delta\Phi\;\,\text{d}z\;+\,symbol{\nabla}\cdot\int_{-h(% symbol{x})}^{\eta(symbol{x},t)}\delta\Phi\,symbol{\nabla}\Phi\;\,\text{d}z\,% \right\}\;\,\text{d}symbol{x}\;\,\text{d}t\\ &\displaystyle+\,\rho\,\int_{t_{0}}^{t_{1}}\iint\left\{\int_{-h(symbol{x})}^{% \eta(symbol{x},t)}\delta\Phi\;\left(symbol{\nabla}\cdot symbol{\nabla}\Phi\,+% \,\frac{\partial^{2}\Phi}{\partial z^{2}}\right)\;\,\text{d}z\,\right\}\;\,% \text{d}symbol{x}\;\,\text{d}t\\ &\displaystyle+\,\rho\,\int_{t_{0}}^{t_{1}}\iint\left[\left(\frac{\partial\eta% }{\partial t}\,+\,symbol{\nabla}\Phi\cdot symbol{\nabla}\eta\,-\,\frac{% \partial\Phi}{\partial z}\right)\,\delta\Phi\right]_{z=\eta(symbol{x},t)}\;\,% \text{d}symbol{x}\;\,\text{d}t\\ &\displaystyle-\,\rho\,\int_{t_{0}}^{t_{1}}\iint\left[\left(symbol{\nabla}\Phi% \cdot symbol{\nabla}h\,+\,\frac{\partial\Phi}{\partial z}\right)\,\delta\Phi% \right]_{z=-h(symbol{x})}\;\,\text{d}symbol{x}\;\,\text{d}t\\ \displaystyle=&\displaystyle 0.\end{aligned}
  7. Δ Φ = 0 for - h ( s y m b o l x ) < z < η ( s y m b o l x , t ) , \Delta\Phi\,=\,0\qquad\,\text{ for }-h(symbol{x})\,<\,z\,<\,\eta(symbol{x},t),
  8. η t + s y m b o l Φ s y m b o l η - Φ z = 0. at z = η ( s y m b o l x , t ) . \frac{\partial\eta}{\partial t}\,+\,symbol{\nabla}\Phi\cdot symbol{\nabla}\eta% \,-\,\frac{\partial\Phi}{\partial z}\,=\,0.\qquad\,\text{ at }z\,=\,\eta(% symbol{x},t).
  9. s y m b o l Φ s y m b o l h + Φ z = 0 at z = - h ( s y m b o l x ) . symbol{\nabla}\Phi\cdot symbol{\nabla}h\,+\,\frac{\partial\Phi}{\partial z}\,=% \,0\qquad\,\text{ at }z\,=\,-h(symbol{x}).
  10. δ η = ( Φ , η + δ η ) - ( Φ , η ) = - t 0 t 1 [ ρ δ η ( Φ t + 1 2 | s y m b o l Φ | 2 + 1 2 ( Φ z ) 2 + g η ) ] z = η ( s y m b o l x , t ) d s y m b o l x d t = 0. \delta_{\eta}\mathcal{L}\,=\,\mathcal{L}(\Phi,\eta+\delta\eta)\,-\,\mathcal{L}% (\Phi,\eta)=\,-\,\int_{t_{0}}^{t_{1}}\iint\left[\rho\,\delta\eta\,\left(\frac{% \partial\Phi}{\partial t}+\,\frac{1}{2}\,\left|symbol{\nabla}\Phi\right|^{2}\,% +\,\frac{1}{2}\,\left(\frac{\partial\Phi}{\partial z}\right)^{2}+\,g\,\eta% \right)\,\right]_{z=\eta(symbol{x},t)}\;\,\text{d}symbol{x}\;\,\text{d}t\,=\,0.
  11. Φ t + 1 2 | s y m b o l Φ | 2 + 1 2 ( Φ z ) 2 + g η = 0 at z = η ( s y m b o l x , t ) . \frac{\partial\Phi}{\partial t}+\,\frac{1}{2}\,\left|symbol{\nabla}\Phi\right|% ^{2}\,+\,\frac{1}{2}\,\left(\frac{\partial\Phi}{\partial z}\right)^{2}+\,g\,% \eta\,=\,0\qquad\,\text{ at }z\,=\,\eta(symbol{x},t).
  12. ρ η t = + δ δ φ , ρ φ t = - δ δ η , \begin{aligned}\displaystyle\rho\,\frac{\partial\eta}{\partial t}&% \displaystyle=\,+\,\frac{\delta\mathcal{H}}{\delta\varphi},\\ \displaystyle\rho\,\frac{\partial\varphi}{\partial t}&\displaystyle=\,-\,\frac% {\delta\mathcal{H}}{\delta\eta},\end{aligned}
  13. ( φ , η ) \mathcal{H}(\varphi,\eta)
  14. = { - h ( s y m b o l x ) η ( s y m b o l x , t ) 1 2 ρ [ | s y m b o l Φ | 2 + ( Φ z ) 2 ] d z + 1 2 ρ g η 2 } d s y m b o l x . \mathcal{H}\,=\,\iint\left\{\int_{-h(symbol{x})}^{\eta(symbol{x},t)}\frac{1}{2% }\,\rho\,\left[\left|symbol{\nabla}\Phi\right|^{2}\,+\,\left(\frac{\partial% \Phi}{\partial z}\right)^{2}\right]\,\,\text{d}z\,+\,\frac{1}{2}\,\rho\,g\,% \eta^{2}\right\}\;\,\text{d}symbol{x}.
  15. δ / δ Φ = 0. \delta\mathcal{H}/\delta\Phi\,=\,0.
  16. H = t 0 t 1 { φ ( s y m b o l x , t ) η ( s y m b o l x , t ) t - H ( φ , η ; s y m b o l x , t ) } d s y m b o l x d t , \mathcal{L}_{H}=\int_{t_{0}}^{t_{1}}\iint\left\{\varphi(symbol{x},t)\,\frac{% \partial\eta(symbol{x},t)}{\partial t}\,-\,H(\varphi,\eta;symbol{x},t)\right\}% \;\,\text{d}symbol{x}\;\,\text{d}t,
  17. φ ( s y m b o l x , t ) = Φ ( s y m b o l x , η ( s y m b o l x , t ) , t ) \varphi(symbol{x},t)=\Phi(symbol{x},\eta(symbol{x},t),t)
  18. H ( φ , η ; s y m b o l x , t ) H(\varphi,\eta;symbol{x},t)
  19. ( φ , η ) = H ( φ , η ; s y m b o l x , t ) d s y m b o l x . \mathcal{H}(\varphi,\eta)\,=\,\iint H(\varphi,\eta;symbol{x},t)\;\,\text{d}% symbol{x}.
  20. H = 1 2 ρ 1 + | s y m b o l η | 2 φ ( D ( η ) φ ) + 1 2 ρ g η 2 , H\,=\,\frac{1}{2}\,\rho\,\sqrt{1\,+\,\left|symbol{\nabla}\eta\right|^{2}}\;\;% \varphi\,\bigl(D(\eta)\;\varphi\bigr)\,+\,\frac{1}{2}\,\rho\,g\,\eta^{2},
  21. H = 1 2 ρ φ [ w ( 1 + | s y m b o l η | 2 ) - s y m b o l η s y m b o l φ ] + 1 2 ρ g η 2 , H\,=\,\frac{1}{2}\,\rho\,\varphi\,\Bigl[w\,\left(1\,+\,\left|symbol{\nabla}% \eta\right|^{2}\right)-\,symbol{\nabla}\eta\cdot symbol{\nabla}\,\varphi\Bigr]% \,+\,\frac{1}{2}\,\rho\,g\,\eta^{2},
  22. w = W ( η ) φ , w\,=\,W(\eta)\,\varphi,
  23. s y m b o l φ = s y m b o l Φ ( s y m b o l x , η ( s y m b o l x , t ) , t ) = [ s y m b o l Φ + Φ z s y m b o l η ] z = η ( s y m b o l x , t ) = [ s y m b o l Φ ] z = η ( s y m b o l x , t ) + w s y m b o l η . symbol{\nabla}\varphi\,=\,symbol{\nabla}\Phi\bigl(symbol{x},\eta(symbol{x},t),% t\bigr)\,=\,\left[symbol{\nabla}\Phi\,+\,\frac{\partial\Phi}{\partial z}\,% symbol{\nabla}\eta\right]_{z=\eta(symbol{x},t)}\,=\,\Bigl[symbol{\nabla}\Phi% \Bigr]_{z=\eta(symbol{x},t)}\,+\,w\,symbol{\nabla}\eta.
  24. H \mathcal{L}_{H}
  25. φ ( s y m b o l x , t ) \varphi(symbol{x},t)
  26. η ( s y m b o l x , t ) \eta(symbol{x},t)
  27. ρ η t \displaystyle\rho\,\frac{\partial\eta}{\partial t}

Lumpability.html

  1. T = { t 1 , t 2 , } \scriptstyle{T=\{t_{1},t_{2},\ldots\}}
  2. { X i } \{X_{i}\}
  3. m t j q ( n , m ) = m t j q ( n , m ) , \sum_{m\in t_{j}}q(n,m)=\sum_{m\in t_{j}}q(n^{\prime},m),
  4. m t j p ( n , m ) = m t j p ( n , m ) , \sum_{m\in t_{j}}p(n,m)=\sum_{m\in t_{j}}p(n^{\prime},m),
  5. P = ( 1 2 3 8 1 16 1 16 7 16 7 16 0 1 8 1 16 0 1 2 7 16 0 1 16 3 8 9 16 ) P=\begin{pmatrix}\frac{1}{2}&\frac{3}{8}&\frac{1}{16}&\frac{1}{16}\\ \frac{7}{16}&\frac{7}{16}&0&\frac{1}{8}\\ \frac{1}{16}&0&\frac{1}{2}&\frac{7}{16}\\ 0&\frac{1}{16}&\frac{3}{8}&\frac{9}{16}\end{pmatrix}
  6. P t = ( 7 8 1 8 1 16 15 16 ) P_{t}=\begin{pmatrix}\frac{7}{8}&\frac{1}{8}\\ \frac{1}{16}&\frac{15}{16}\end{pmatrix}

Lüroth's_theorem.html

  1. K K
  2. M M
  3. K K
  4. K ( X ) K(X)
  5. f ( X ) K ( X ) f(X)\in K(X)
  6. M = K ( f ( X ) ) M=K(f(X))
  7. K K
  8. K ( X ) K(X)

Lyapunov_redesign.html

  1. V V
  2. x ˙ = f ( t , x ) + G ( t , x ) [ u + δ ( t , x , u ) ] \dot{x}=f(t,x)+G(t,x)[u+\delta(t,x,u)]
  3. x R n x\in R^{n}
  4. u R p u\in R^{p}
  5. f f
  6. G G
  7. δ \delta
  8. ( t , x , u ) [ 0 , inf ) × D × R p (t,x,u)\in[0,\inf)\times D\times R^{p}
  9. D R n D\subset R^{n}
  10. x ˙ = f ( t , x ) + G ( t , x ) u \dot{x}=f(t,x)+G(t,x)u
  11. u = ϕ ( t , x ) + v u=\phi(t,x)+v
  12. v v

M-derived_filter.html

  1. k 2 = Z Y k^{2}=\frac{Z}{Y}
  2. 1 - m 2 m Z . \frac{1-m^{2}}{m}Z.
  3. 1 - m 2 m Y . \frac{1-m^{2}}{m}Y.
  4. ω c = 1 L C . \omega_{c}=\frac{1}{\sqrt{LC}}.
  5. ω = ω c 1 - m 2 . \omega_{\infty}=\frac{\omega_{c}}{\sqrt{1-m^{2}}}.
  6. ω \omega_{\infty}
  7. ω c \omega_{c}\,\!
  8. ω \omega_{\infty}
  9. ω c \omega_{c}\,\!
  10. Z i T = 1 - ω 2 Z_{iT}=\sqrt{1-\omega^{2}}
  11. Z i Π m = 1 - ( ω / ω ) 2 1 - ω 2 . Z_{i\Pi m}=\frac{1-\left(\omega/\omega_{\infty}\right)^{2}}{\sqrt{1-\omega^{2}% }}.
  12. Z i Π = 1 1 - ω 2 Z_{i\Pi}=\frac{1}{\sqrt{1-\omega^{2}}}
  13. Z i T m = 1 - ω 2 1 - ( ω / ω ) 2 Z_{iTm}=\frac{\sqrt{1-\omega^{2}}}{1-\left(\omega/\omega_{\infty}\right)^{2}}
  14. γ = sinh - 1 m Z k 2 + ( 1 - m 2 ) Z 2 \gamma=\sinh^{-1}\frac{mZ}{\sqrt{k^{2}+(1-m^{2})Z^{2}}}
  15. γ n = n γ \gamma_{n}=n\gamma\,\!
  16. 0 < ω < ω c 0<\omega<\omega_{c}\,\!
  17. γ = α + i β = 0 + i 1 2 cos - 1 ( 1 - 2 m 2 ( ω c ω ) 2 - ( ω c ω ) 2 ) \gamma=\alpha+i\beta=0+i\frac{1}{2}\cos^{-1}\left(1-\frac{2m^{2}}{\left(\frac{% \omega_{c}}{\omega}\right)^{2}-\left(\frac{\omega_{c}}{\omega_{\infty}}\right)% ^{2}}\right)
  18. ω c < ω < ω \omega_{c}<\omega<\omega_{\infty}
  19. γ = α + i β = 1 2 cosh - 1 ( 2 m 2 ( ω c ω ) 2 - ( ω c ω ) 2 - 1 ) + i π 2 \gamma=\alpha+i\beta=\frac{1}{2}\cosh^{-1}\left(\frac{2m^{2}}{\left(\frac{% \omega_{c}}{\omega}\right)^{2}-\left(\frac{\omega_{c}}{\omega_{\infty}}\right)% ^{2}}-1\right)+i\frac{\pi}{2}
  20. ω < ω < \omega_{\infty}<\omega<\infty
  21. γ = α + i β = 1 2 cosh - 1 ( 1 - 2 m 2 ( ω c ω ) 2 - ( ω c ω ) 2 ) + i 0 \gamma=\alpha+i\beta=\frac{1}{2}\cosh^{-1}\left(1-\frac{2m^{2}}{\left(\frac{% \omega_{c}}{\omega}\right)^{2}-\left(\frac{\omega_{c}}{\omega_{\infty}}\right)% ^{2}}\right)+i0

Maass_wave_form.html

  1. γ = ( a b c d ) Γ \gamma=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in\Gamma
  2. τ \tau\in\mathbb{H}
  3. f ( a τ + b c τ + d ) = ( c τ + d ) k f ( τ ) f\left(\frac{a\tau+b}{c\tau+d}\right)=(c\tau+d)^{k}f(\tau)
  4. Δ k f = s f \Delta_{k}f=sf
  5. Δ k \Delta_{k}
  6. Δ k = - y 2 ( 2 x 2 + 2 y 2 ) i k y ( x + i y ) \Delta_{k}=-y^{2}\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}% {\partial y^{2}}\right)iky\left(\frac{\partial}{\partial x}+i\frac{\partial}{% \partial y}\right)

Machinability.html

  1. V T n = C VT^{n}=C

Maffei_1.html

  1. ± \pm
  2. ± \pm

Magnetic-coded_lock.html

  1. N = C 4 m N=C\cdot 4^{m}
  2. N N
  3. C C
  4. m m

Magnetic_diffusivity.html

  1. η = 1 μ 0 σ 0 \eta=\frac{1}{\mu_{0}\sigma_{0}}
  2. μ 0 \mu_{0}
  3. σ 0 \sigma_{0}
  4. σ 0 = n e e 2 m e ν c \sigma_{0}=\frac{n_{e}e^{2}}{m_{e}\nu_{c}}
  5. n e n_{e}
  6. e e
  7. m e m_{e}
  8. ν c \nu_{c}

Magnetic_lens.html

  1. H Z H_{Z}
  2. H R H_{R}
  3. H Z H_{Z}
  4. H R H_{R}

Mahler's_inequality.html

  1. k = 1 n ( x k + y k ) 1 / n k = 1 n x k 1 / n + k = 1 n y k 1 / n \prod_{k=1}^{n}(x_{k}+y_{k})^{1/n}\geq\prod_{k=1}^{n}x_{k}^{1/n}+\prod_{k=1}^{% n}y_{k}^{1/n}
  2. k = 1 n ( x k x k + y k ) 1 / n 1 n k = 1 n x k x k + y k , \prod_{k=1}^{n}\left({x_{k}\over x_{k}+y_{k}}\right)^{1/n}\leq{1\over n}\sum_{% k=1}^{n}{x_{k}\over x_{k}+y_{k}},
  3. k = 1 n ( y k x k + y k ) 1 / n 1 n k = 1 n y k x k + y k . \prod_{k=1}^{n}\left({y_{k}\over x_{k}+y_{k}}\right)^{1/n}\leq{1\over n}\sum_{% k=1}^{n}{y_{k}\over x_{k}+y_{k}}.
  4. k = 1 n ( x k x k + y k ) 1 / n + k = 1 n ( y k x k + y k ) 1 / n 1 n n = 1. \prod_{k=1}^{n}\left({x_{k}\over x_{k}+y_{k}}\right)^{1/n}+\prod_{k=1}^{n}% \left({y_{k}\over x_{k}+y_{k}}\right)^{1/n}\leq{1\over n}n=1.

Maintenance_dose.html

  1. MD = C p C L F \mbox{MD}~{}=\frac{C_{p}CL}{F}

MANCOVA.html

  1. Σ m o d e l \Sigma_{model}
  2. Σ r e s - 1 \Sigma_{res}^{-1}
  3. A = Σ m o d e l × Σ r e s - 1 A=\Sigma_{model}\times\Sigma_{res}^{-1}
  4. Σ m o d e l = Σ r e s i d u a l \Sigma_{model}=\Sigma_{residual}
  5. A I A\sim I
  6. λ p \lambda_{p}
  7. A A
  8. Λ W i l k s = 1... p ( 1 / ( 1 + λ p ) ) = det ( I + A ) - 1 = det ( Σ r e s ) / det ( Σ r e s + Σ m o d e l ) \Lambda_{Wilks}=\prod_{1...p}(1/(1+\lambda_{p}))=\det(I+A)^{-1}=\det(\Sigma_{% res})/\det(\Sigma_{res}+\Sigma_{model})
  9. Λ P i l l a i = 1... p ( 1 / ( 1 + λ p ) ) = tr ( ( I + A ) - 1 ) \Lambda_{Pillai}=\sum_{1...p}(1/(1+\lambda_{p}))=\mathrm{tr}((I+A)^{-1})
  10. Λ L H = 1... p ( λ p ) = tr ( A ) \Lambda_{LH}=\sum_{1...p}(\lambda_{p})=\mathrm{tr}(A)
  11. Λ R o y = m a x p ( λ p ) = A \Lambda_{Roy}=max_{p}(\lambda_{p})=\|A\|_{\infty}

Mangler_Transformation.html

  1. U U
  2. x , y , u , v x,y,u,v
  3. U ¯ \bar{U}
  4. x ¯ , y ¯ , u ¯ , v ¯ \bar{x},\bar{y},\bar{u},\bar{v}
  5. x ¯ = 1 L 2 0 x r 2 ( x ) d x , y ¯ = r ( x ) L y , u ¯ = u , v ¯ = L r ( v + r r y u ) , U ¯ = U , \begin{array}[]{l}\displaystyle\bar{x}=\frac{1}{L^{2}}\int\limits_{0}^{x}r^{2}% (x)dx,\quad\bar{y}=\frac{r(x)}{L}y,\\ \displaystyle\bar{u}=u,\quad\bar{v}=\frac{L}{r}\left(v+\frac{r^{\prime}}{r}yu% \right),\\ \displaystyle\bar{U}=U,\end{array}
  6. L L
  7. r ( x ) r(x)

Manipulation_of_atoms_by_optical_field.html

  1. 2 k 2\hbar k
  2. k k
  3. v = 2 k / m v=2\hbar k/m

Mapping_torus.html

  1. M f = ( I × X ) ( 1 , x ) ( 0 , f ( x ) ) M_{f}=\frac{(I\times X)}{(1,x)\sim(0,f(x))}

Margin_classifier.html

  1. ( x , y ) (x,y)
  2. x X x\in X
  3. y Y = { - 1 , + 1 } y\in Y=\{-1,+1\}
  4. h j C h_{j}\in C
  5. j j
  6. C C
  7. α j R \alpha_{j}\in R
  8. t t
  9. x x
  10. y j t α j h j ( x ) | α j | \frac{y\sum_{j}^{t}\alpha_{j}h_{j}(x)}{\sum|\alpha_{j}|}
  11. S S
  12. m m
  13. D D
  14. d d
  15. m d 1 m\geq d\geq 1
  16. 1 - δ 1-\delta
  17. P D ( y j t α j h j ( x ) | α j | 0 ) P S ( y j t α j h j ( x ) | α j | θ ) + O ( 1 m d log 2 ( m / d ) / θ 2 + log ( 1 / δ ) ) P_{D}\left(\frac{y\sum_{j}^{t}\alpha_{j}h_{j}(x)}{\sum|\alpha_{j}|}\leq 0% \right)\leq P_{S}\left(\frac{y\sum_{j}^{t}\alpha_{j}h_{j}(x)}{\sum|\alpha_{j}|% }\leq\theta\right)+O\left(\frac{1}{\sqrt{m}}\sqrt{d\log^{2}(m/d)/\theta^{2}+% \log(1/\delta)}\right)
  18. θ > 0 \theta>0

Marine_snow.html

  1. d C 1 d t = g C 1 - 20.8 α r 3 G C 1 2 | \frac{\mathrm{d}C_{1}}{\mathrm{d}t}=gC_{1}-20.8\alpha r^{3}GC_{1}^{2}|

Markov_information_source.html

  1. f : S Γ f:S\to\Gamma
  2. f ( s k ) f(s_{k})
  3. s k s_{k}

Markov_kernel.html

  1. ( X , 𝒜 ) (X,\mathcal{A})
  2. ( Y , ) (Y,\mathcal{B})
  3. ( X , 𝒜 ) (X,\mathcal{A})
  4. ( Y , ) (Y,\mathcal{B})
  5. κ : X × [ 0 , 1 ] \kappa\colon X\times\mathcal{B}\to[0,1]
  6. x κ ( x , B ) x\mapsto\kappa(x,B)
  7. 𝒜 \mathcal{A}
  8. B B\in\mathcal{B}
  9. B κ ( x , B ) B\mapsto\kappa(x,B)
  10. ( Y , ) (Y,\mathcal{B})
  11. x X x\in X
  12. x X x\in X
  13. κ ( x , . ) \kappa(x,.)
  14. ( Y , ) (Y,\mathcal{B})
  15. B B\in\mathcal{B}
  16. x κ ( x , B ) x\mapsto\kappa(x,B)
  17. σ \sigma
  18. 𝒜 \mathcal{A}
  19. X = Y = \Z X=Y=\Z
  20. 𝒜 = = 𝒫 ( \Z ) \mathcal{A}=\mathcal{B}=\mathcal{P}(\Z)
  21. κ \kappa
  22. κ ( x , B ) = 1 2 𝟏 x - 1 ( B ) + 1 2 𝟏 x + 1 ( B ) , x \Z , B 𝒫 ( \Z ) \kappa(x,B)=\frac{1}{2}\mathbf{1}_{x-1}(B)+\frac{1}{2}\mathbf{1}_{x+1}(B),% \quad\forall x\in\Z,\quad\forall B\in\mathcal{P}(\Z)
  23. \Z \Z
  24. 𝟏 \mathbf{1}
  25. S = Y = 𝒩 S=Y=\mathcal{N}
  26. 𝒜 = = 𝒫 ( 𝒩 ) \mathcal{A}=\mathcal{B}=\mathcal{P}(\mathcal{N})
  27. κ ( x , B ) = { 𝟏 0 ( B ) x = 0 , P [ ξ 1 + + ξ x B ] else, \kappa(x,B)=\begin{cases}\mathbf{1}_{0}(B)&\quad x=0,\\ P[\xi_{1}+\dots+\xi_{x}\in B]&\quad\,\text{else,}\\ \end{cases}
  28. ξ i \xi_{i}
  29. X = Y X=Y
  30. 𝒜 = = 𝒫 ( X ) = 𝒫 ( Y ) \mathcal{A}=\mathcal{B}=\mathcal{P}(X)=\mathcal{P}(Y)
  31. | X | = | Y | = n |X|=|Y|=n
  32. ( K i j ) 1 i , j n (K_{ij})_{1\leq i,j\leq n}
  33. Σ j Y K i j = 1 \Sigma_{j\in Y}K_{ij}=1
  34. i X i\in X
  35. κ ( i , B ) = Σ j B K i j , i X , B \kappa(i,B)=\Sigma_{j\in B}K_{ij},\quad\forall i\in X,\quad\forall B\in% \mathcal{B}
  36. ( X , 𝒜 , P ) (X,\mathcal{A},P)
  37. κ \kappa
  38. ( X , 𝒜 ) (X,\mathcal{A})
  39. ( Y , ) (Y,\mathcal{B})
  40. Q Q
  41. ( X × Y , 𝒜 ) (X\times Y,\mathcal{A}\otimes\mathcal{B})
  42. Q ( A × B ) = A κ ( x , B ) d P ( x ) , A 𝒜 , B Q(A\times B)=\int_{A}\kappa(x,B)dP(x),\quad\forall A\in\mathcal{A},\quad% \forall B\in\mathcal{B}
  43. ( S , Y ) (S,Y)
  44. X X
  45. ( S , Y ) (S,Y)
  46. ( Ω , , P ) (\Omega,\mathcal{F},P)
  47. 𝒢 \mathcal{G}\subseteq\mathcal{F}
  48. σ \sigma
  49. κ \kappa
  50. ( Ω , 𝒢 ) (\Omega,\mathcal{G})
  51. ( S , Y ) (S,Y)
  52. κ ( . , B ) \kappa(.,B)
  53. E [ 𝟏 { X B } | 𝒢 ] E[\mathbf{1}_{\{X\in B\}}|\mathcal{G}]
  54. B Y B\in Y
  55. P [ X B | 𝒢 ] = E [ 𝟏 { X B } | 𝒢 ] = κ ( ω , B ) , P - a . s . B 𝒢 P[X\in B|\mathcal{G}]=E[\mathbf{1}_{\{X\in B\}}|\mathcal{G}]=\kappa(\omega,B),% \quad P-a.s.\forall B\in\mathcal{G}
  56. X X
  57. 𝒢 \mathcal{G}

Markov_spectrum.html

  1. ξ \xi
  2. | ξ - m n | < 1 5 n 2 , \left|\xi-\frac{m}{n}\right|<\frac{1}{\sqrt{5}\,n^{2}},
  3. ξ \xi
  4. | ξ - m n | < c n 2 \left|\xi-\frac{m}{n}\right|<\frac{c}{n^{2}}
  5. lim inf n n 2 | ξ - m n | , \liminf_{n\to\infty}n^{2}\left|\xi-\frac{m}{n}\right|,
  6. ξ \xi
  7. F = 2 221 564 096 + 283 748 462 491 993 569 = 4.5278295661 . F=\frac{2\,221\,564\,096+283\,748\sqrt{462}}{491\,993\,569}=4.5278295661\dots.

Martin's_maximum.html

  1. 1 \aleph_{1}
  2. 1 \aleph_{1}
  3. 1 \aleph_{1}
  4. 2 \aleph_{2}
  5. 2 \aleph_{2}

Mass_fraction_(chemistry).html

  1. w i w_{i}
  2. m i m_{i}
  3. m t o t m_{tot}
  4. w i = m i m t o t w_{i}=\frac{m_{i}}{m_{tot}}
  5. i = 1 N m i = m t o t ; i = 1 N w i = 1 \sum_{i=1}^{N}m_{i}=m_{tot};\sum_{i=1}^{N}w_{i}=1
  6. ρ i \rho_{i}
  7. ρ \rho
  8. w i = ρ i ρ w_{i}=\frac{\rho_{i}}{\rho}
  9. w i = ρ i ρ = c i M i ρ w_{i}=\frac{\rho_{i}}{\rho}=\frac{c_{i}M_{i}}{\rho}
  10. x i x_{i}
  11. x i = w i M M i x_{i}=w_{i}\cdot\frac{M}{M_{i}}
  12. M i M_{i}
  13. i i
  14. M M
  15. x i = w i M i i w i M i x_{i}=\frac{\frac{w_{i}}{M_{i}}}{\sum_{i}\frac{w_{i}}{M_{i}}}

Mass_spectral_interpretation.html

  1. M + e - M + + 2 e - M+e^{-}\to M^{+\bullet}+2e^{-}
  2. M + M^{+\bullet}
  3. R i n g s + π B o n d s = u = C - H 2 - X 2 + N 2 + 1 Rings+\pi Bonds=u=C-\frac{H}{2}-\frac{X}{2}+\frac{N}{2}+1\,
  4. M 13 = n + r 13 \frac{M}{13}=n+\frac{r}{13}\,
  5. C n H n + r C_{n}H_{n+r}\,
  6. u = ( n - r + 2 ) 2 u=\frac{\left(n-r+2\right)}{2}\,

Massera's_lemma.html

  1. V ( ζ ) = 0 G ( | φ ( t , ζ ) | ) d t V(\zeta)=\int_{0}^{\infty}G(|\varphi(t,\zeta)|)dt
  2. ζ is φ ( t , ζ ) \zeta\,\text{ is }\varphi(t,\zeta)
  3. g : [ 0 , ) R g:[0,\infty)\rightarrow R
  4. g ( t ) 0 g(t)\rightarrow 0
  5. t t\rightarrow\infty
  6. h : [ 0 , ) R h:[0,\infty)\rightarrow R
  7. G : [ 0 , ) [ 0 , ) G:[0,\infty)\rightarrow[0,\infty)
  8. G G
  9. G G^{\prime}
  10. 0 G ( u ( t ) ) d t k 1 ; 0 G ( u ( t ) ) h ( t ) d t k 2 . \int_{0}^{\infty}G(u(t))\,dt\leq k_{1};\quad\int_{0}^{\infty}G^{\prime}(u(t))h% (t)\,dt\leq k_{2}.
  11. g : [ 0 , ) R g:[0,\infty)\rightarrow R
  12. g ( t ) 0 g(t)\rightarrow 0
  13. t t\rightarrow\infty
  14. h : [ 0 , ) R h:[0,\infty)\rightarrow R
  15. G : [ 0 , ) [ 0 , ) G:[0,\infty)\rightarrow[0,\infty)
  16. G G
  17. G G^{\prime}
  18. [ 0 , ) [0,\infty)
  19. l l
  20. u : l [ 0 , ) u:\mathbb{R}^{l}\rightarrow[0,\infty)
  21. 0 u ( t 1 , , t l ) g ( t 1 + + t l ) 0\leq u(t_{1},\ldots,t_{l})\leq g(t_{1}+\cdots+t_{l})
  22. t i 0 t_{i}\geq 0
  23. i = 1 , , l i=1,\ldots,l
  24. 0 0 G ( u ( s 1 , , s l ) ) d s 1 d s l < k 1 \int_{0}^{\infty}\cdots\int_{0}^{\infty}G(u(s_{1},\ldots,s_{l}))ds_{1}\ldots ds% _{l}<k_{1}
  25. 0 0 G ( u ( s 1 , , s l ) ) × h ( s 1 + + s l ) d s 1 d s l < k 2 \int_{0}^{\infty}\cdots\int_{0}^{\infty}G^{\prime}(u(s_{1},\ldots,s_{l}))% \times h(s_{1}+\cdots+s_{l})ds_{1}\ldots ds_{l}<k_{2}

Matching_distance.html

  1. 1 = r + a + b \ell_{1}=r+a+b
  2. 2 = r + a \ell_{2}=r^{\prime}+a^{\prime}
  3. d match ( 1 , 2 ) = max { δ ( r , r ) , δ ( b , a ) , δ ( a , Δ ) } = 4 d\text{match}(\ell_{1},\ell_{2})=\max\{\delta(r,r^{\prime}),\delta(b,a^{\prime% }),\delta(a,\Delta)\}=4
  4. 1 \ell_{1}
  5. 2 \ell_{2}
  6. C 1 C_{1}
  7. C 2 C_{2}
  8. 1 \ell_{1}
  9. 2 \ell_{2}
  10. { ( x , y ) \R 2 : x = y } \{(x,y)\in\R^{2}:x=y\}
  11. 1 \ell_{1}
  12. 2 \ell_{2}
  13. d match ( 1 , 2 ) = min σ max p C 1 δ ( p , σ ( p ) ) d\text{match}(\ell_{1},\ell_{2})=\min_{\sigma}\max_{p\in C_{1}}\delta(p,\sigma% (p))
  14. σ \sigma
  15. C 1 C_{1}
  16. C 2 C_{2}
  17. δ ( ( x , y ) , ( x , y ) ) = min { max { | x - x | , | y - y | } , max { y - x 2 , y - x 2 } } . \delta\left((x,y),(x^{\prime},y^{\prime})\right)=\min\left\{\max\{|x-x^{\prime% }|,|y-y^{\prime}|\},\max\left\{\frac{y-x}{2},\frac{y^{\prime}-x^{\prime}}{2}% \right\}\right\}.
  18. d match d\text{match}
  19. L L_{\infty}
  20. Δ \Delta
  21. δ \delta

Matching_theory_(economics).html

  1. m t = M ( u t , v t ) = μ u t a v t b m_{t}\;=\;M(u_{t},v_{t})\;=\;\mu u_{t}^{a}v_{t}^{b}
  2. μ \,\mu\,
  3. a \,a\,
  4. b \,b\,
  5. u t \,u_{t}\,
  6. t \,t\,
  7. v t \,v_{t}\,
  8. m t \,m_{t}\,
  9. a + b 1 a+b\approx 1
  10. δ \,\delta\,
  11. t \,t\,
  12. n t = L t - u t \,n_{t}=L_{t}-u_{t}\,
  13. L t \,L_{t}\,
  14. t \,t\,
  15. n t + 1 = μ u t a v t b + ( 1 - δ ) n t n_{t+1}\;=\mu u_{t}^{a}v_{t}^{b}+(1-\delta)n_{t}
  16. δ \,\delta\,

Material_failure_theory.html

  1. σ 1 \sigma_{1}
  2. σ 3 \sigma_{3}
  3. σ t \sigma_{t}
  4. σ c \sigma_{c}
  5. σ c < σ 3 < σ 1 < σ t \sigma_{c}<\sigma_{3}<\sigma_{1}<\sigma_{t}\,
  6. ε c < ε 3 < ε 1 < ε t \varepsilon_{c}<\varepsilon_{3}<\varepsilon_{1}<\varepsilon_{t}\,
  7. σ \sigma
  8. σ = 2 E γ π a \sigma=\sqrt{\cfrac{2E\gamma}{\pi a}}
  9. E E
  10. γ \gamma
  11. a a
  12. 2 a 2a
  13. σ π a \sigma\sqrt{\pi a}
  14. K Ic = Y σ c π a K_{\rm Ic}=Y\sigma_{c}\sqrt{\pi a}
  15. σ c \sigma_{c}
  16. Y Y
  17. K Ic K_{\rm Ic}
  18. K IIc K_{\rm IIc}
  19. K IIIc K_{\rm IIIc}
  20. G I := P 2 t d u d a G_{I}:=\cfrac{P}{2t}~{}\cfrac{du}{da}
  21. P P
  22. t t
  23. u u
  24. a a
  25. 2 a 2a
  26. G Ic G_{\rm Ic}
  27. G Ic = 1 E K Ic 2 G_{\rm Ic}=\cfrac{1}{E}~{}K_{\rm Ic}^{2}
  28. E E

Mathematical_beauty.html

  1. e i π + 1 = 0 . \displaystyle e^{i\pi}+1=0\,.

Mathematical_descriptions_of_opacity.html

  1. 𝐄 ( z , t ) = Re [ 𝐄 0 e i ( k z - ω t ) ] , \mathbf{E}(z,t)=\operatorname{Re}\!\left[\mathbf{E}_{0}e^{i(kz-\omega t)}% \right]\!,
  2. λ = 2 π k . \lambda=\frac{2\pi}{k}.
  3. λ 0 = 2 π c ω , \lambda_{0}=\frac{2\pi\mathrm{c}}{\omega},
  4. n = λ 0 λ = c k ω . n=\frac{\lambda_{0}}{\lambda}=\frac{\mathrm{c}k}{\omega}.
  5. I ( z ) | 𝐄 0 e i ( k z - ω t ) | 2 = | 𝐄 0 | 2 . I(z)\propto\left|\mathbf{E}_{0}e^{i(kz-\omega t)}\right|^{2}=|\mathbf{E}_{0}|^% {2}.
  6. I ( z ) = I 0 | 𝐄 0 | 2 . I(z)=I_{0}\propto|\mathbf{E}_{0}|^{2}.
  7. Re [ 𝐄 0 e i ( k z - ω t ) ] = Re [ 𝐄 0 * e - i ( k z - ω t ) ] , \operatorname{Re}\!\left[\mathbf{E}_{0}e^{i(kz-\omega t)}\right]=\operatorname% {Re}\!\left[\mathbf{E}_{0}^{*}e^{-i(kz-\omega t)}\right]\!,
  8. 𝐄 ( z , t ) = e - α z / 2 Re [ 𝐄 0 e i ( k z - ω t ) ] , \mathbf{E}(z,t)=e^{-\alpha z/2}\operatorname{Re}\!\left[\mathbf{E}_{0}e^{i(kz-% \omega t)}\right]\!,
  9. I ( z ) | e - α z / 2 𝐄 0 e i ( k z - ω t ) | 2 = | 𝐄 0 | 2 e - α z , I(z)\propto\left|e^{-\alpha z/2}\mathbf{E}_{0}e^{i(kz-\omega t)}\right|^{2}=|% \mathbf{E}_{0}|^{2}e^{-\alpha z},
  10. I ( z ) = I 0 e - α z . I(z)=I_{0}e^{-\alpha z}.
  11. 𝐄 ( z , t ) = e - z / ( 2 δ pen ) Re [ 𝐄 0 e i ( k z - ω t ) ] , \mathbf{E}(z,t)=e^{-z/(2\delta_{\mathrm{pen}})}\operatorname{Re}\!\left[% \mathbf{E}_{0}e^{i(kz-\omega t)}\right]\!,
  12. I ( z ) = I 0 e - z / δ pen , I(z)=I_{0}e^{-z/\delta_{\mathrm{pen}}},
  13. 𝐄 ( z , t ) = e - z / δ skin Re [ 𝐄 0 e i ( k z - ω t ) ] , \mathbf{E}(z,t)=e^{-z/\delta_{\mathrm{skin}}}\operatorname{Re}\!\left[\mathbf{% E}_{0}e^{i(kz-\omega t)}\right]\!,
  14. I ( z ) = I 0 e - 2 z / δ skin , I(z)=I_{0}e^{-2z/\delta_{\mathrm{skin}}},
  15. {}\approx{}
  16. α = 1 / δ pen = 2 / δ skin . \alpha=1/\delta_{\mathrm{pen}}=2/\delta_{\mathrm{skin}}.
  17. 𝐄 ( z , t ) = Re [ 𝐄 0 e i ( k ¯ z - ω t ) ] , \mathbf{E}(z,t)=\operatorname{Re}\!\left[\mathbf{E}_{0}e^{i(\underline{k}z-% \omega t)}\right]\!,
  18. I ( z ) | 𝐄 0 e i ( k ¯ z - ω t ) | 2 = | 𝐄 0 | 2 e - 2 Im ( k ¯ ) z , I(z)\propto\left|\mathbf{E}_{0}e^{i(\underline{k}z-\omega t)}\right|^{2}=|% \mathbf{E}_{0}|^{2}e^{-2\operatorname{Im}(\underline{k})z},
  19. I ( z ) = I 0 e - 2 Im ( k ¯ ) z . I(z)=I_{0}e^{-2\operatorname{Im}(\underline{k})z}.
  20. Re ( k ¯ ) = k , \operatorname{Re}(\underline{k})=k,
  21. Im ( k ¯ ) = α / 2. \operatorname{Im}(\underline{k})=\alpha/2.
  22. Re ( k ¯ ) = k , \operatorname{Re}(\underline{k})=k,
  23. Im ( k ¯ ) = - α / 2. \operatorname{Im}(\underline{k})=-\alpha/2.
  24. 𝐄 ( z , t ) = Re [ 𝐄 0 e - γ z + i ω t ] , \mathbf{E}(z,t)=\operatorname{Re}\!\left[\mathbf{E}_{0}e^{-\gamma z+i\omega t}% \right]\!,
  25. I ( z ) | 𝐄 0 e - γ z + i ω t | 2 = | 𝐄 0 | 2 e - 2 Re ( γ ) z , I(z)\propto\left|\mathbf{E}_{0}e^{-\gamma z+i\omega t}\right|^{2}=|\mathbf{E}_% {0}|^{2}e^{-2\operatorname{Re}(\gamma)z},
  26. I ( z ) = I 0 e - 2 Re ( γ ) z . I(z)=I_{0}e^{-2\operatorname{Re}(\gamma)z}.
  27. γ = i k ¯ * , \gamma=i\underline{k}^{*},
  28. Re ( γ ) = Im ( k ¯ ) = α / 2. \operatorname{Re}(\gamma)=\operatorname{Im}(\underline{k})=\alpha/2.
  29. Im ( γ ) = Re ( k ¯ ) = k . \operatorname{Im}(\gamma)=\operatorname{Re}(\underline{k})=k.
  30. k ¯ \underline{k}
  31. n = c v = c k ω , n=\frac{\mathrm{c}}{v}=\frac{\mathrm{c}k}{\omega},
  32. n ¯ = c k ¯ ω . \underline{n}=\frac{\mathrm{c}\underline{k}}{\omega}.
  33. 𝐄 ( z , t ) = Re [ 𝐄 0 e i ω ( n ¯ z / c - t ) ] . \mathbf{E}(z,t)=\operatorname{Re}\!\left[\mathbf{E}_{0}e^{i\omega(\underline{n% }z/\mathrm{c}-t)}\right]\!.
  34. I ( z ) | 𝐄 0 e i ω ( n ¯ z / c - t ) | 2 = | 𝐄 0 | 2 e - 2 ω Im ( n ¯ ) z / c , I(z)\propto\left|\mathbf{E}_{0}e^{i\omega(\underline{n}z/\mathrm{c}-t)}\right|% ^{2}=|\mathbf{E}_{0}|^{2}e^{-2\omega\operatorname{Im}(\underline{n})z/\mathrm{% c}},
  35. I ( z ) = I 0 e - 2 ω Im ( n ¯ ) z / c . I(z)=I_{0}e^{-2\omega\operatorname{Im}(\underline{n})z/\mathrm{c}}.
  36. Re ( n ¯ ) = c k ω . \operatorname{Re}(\underline{n})=\frac{\mathrm{c}k}{\omega}.
  37. Im ( n ¯ ) = c α 2 ω = λ 0 α 4 π . \operatorname{Im}(\underline{n})=\frac{\mathrm{c}\alpha}{2\omega}=\frac{% \lambda_{0}\alpha}{4\pi}.
  38. n ¯ \underline{n}
  39. n = c μ ε (SI) , n = μ ε (cgs) , n=\mathrm{c}\sqrt{\mu\varepsilon}\quad\,\text{(SI)},\qquad n=\sqrt{\mu% \varepsilon}\quad\,\text{(cgs)},
  40. n ¯ = c μ ε ¯ (SI) , n ¯ = μ ε ¯ (cgs) , \underline{n}=\mathrm{c}\sqrt{\mu\underline{\varepsilon}}\quad\,\text{(SI)},% \qquad\underline{n}=\sqrt{\mu\underline{\varepsilon}}\quad\,\text{(cgs)},
  41. Re ( ε ¯ ) = c 2 ε 0 ω 2 μ / μ 0 ( k 2 - α 2 4 ) (SI) , Re ( ε ¯ ) = c 2 ω 2 μ ( k 2 - α 2 4 ) (cgs) , \operatorname{Re}(\underline{\varepsilon})=\frac{\mathrm{c}^{2}\varepsilon_{0}% }{\omega^{2}\mu/\mu_{0}}\!\left(k^{2}-\frac{\alpha^{2}}{4}\right)\quad\,\text{% (SI)},\qquad\operatorname{Re}(\underline{\varepsilon})=\frac{\mathrm{c}^{2}}{% \omega^{2}\mu}\!\left(k^{2}-\frac{\alpha^{2}}{4}\right)\quad\,\text{(cgs)},
  42. Im ( ε ¯ ) = c 2 ε 0 ω 2 μ / μ 0 k α (SI) , Im ( ε ¯ ) = c 2 ω 2 μ k α (cgs) . \operatorname{Im}(\underline{\varepsilon})=\frac{\mathrm{c}^{2}\varepsilon_{0}% }{\omega^{2}\mu/\mu_{0}}k\alpha\quad\,\text{(SI)},\qquad\operatorname{Im}(% \underline{\varepsilon})=\frac{\mathrm{c}^{2}}{\omega^{2}\mu}k\alpha\quad\,% \text{(cgs)}.
  43. × 𝐇 = 𝐉 + d 𝐃 d t (SI) , × 𝐇 = 4 π c 𝐉 + 1 c d 𝐃 d t (cgs) , \nabla\times\mathbf{H}=\mathbf{J}+\frac{\mathrm{d}\mathbf{D}}{\mathrm{d}t}% \quad\,\text{(SI)},\qquad\nabla\times\mathbf{H}=\frac{4\pi}{\mathrm{c}}\mathbf% {J}+\frac{1}{\mathrm{c}}\frac{\mathrm{d}\mathbf{D}}{\mathrm{d}t}\quad\,\text{(% cgs)},
  44. × 𝐇 = σ 𝐄 + ε d 𝐄 d t (SI) , × 𝐇 = 4 π σ c 𝐄 + ε c d 𝐄 d t (cgs) , \nabla\times\mathbf{H}=\sigma\mathbf{E}+\varepsilon\frac{\mathrm{d}\mathbf{E}}% {\mathrm{d}t}\quad\,\text{(SI)},\qquad\nabla\times\mathbf{H}=\frac{4\pi\sigma}% {\mathrm{c}}\mathbf{E}+\frac{\varepsilon}{\mathrm{c}}\frac{\mathrm{d}\mathbf{E% }}{\mathrm{d}t}\quad\,\text{(cgs)},
  45. 𝐇 = Re [ 𝐇 0 e - i ω t ] , \mathbf{H}=\operatorname{Re}\!\left[\mathbf{H}_{0}e^{-i\omega t}\right]\!,
  46. 𝐄 = Re [ 𝐄 0 e - i ω t ] , \mathbf{E}=\operatorname{Re}\!\left[\mathbf{E}_{0}e^{-i\omega t}\right]\!,
  47. × 𝐇 0 = - i ω 𝐄 0 ( ε + i σ ω ) (SI) , × 𝐇 0 = - i ω c 𝐄 0 ( ε + i 4 π σ ω ) (cgs) . \nabla\times\mathbf{H}_{0}=-i\omega\mathbf{E}_{0}\!\left(\varepsilon+i\frac{% \sigma}{\omega}\right)\quad\,\text{(SI)},\qquad\nabla\times\mathbf{H}_{0}=% \frac{-i\omega}{\mathrm{c}}\mathbf{E}_{0}\!\left(\varepsilon+i\frac{4\pi\sigma% }{\omega}\right)\quad\,\text{(cgs)}.
  48. ε ¯ = ε + i σ ω (SI) , ε ¯ = ε + i 4 π σ ω (cgs) . \underline{\varepsilon}=\varepsilon+i\frac{\sigma}{\omega}\quad\,\text{(SI)},% \qquad\underline{\varepsilon}=\varepsilon+i\frac{4\pi\sigma}{\omega}\quad\,% \text{(cgs)}.
  49. σ = k α ω μ (SI) , σ = k α c 2 4 π ω μ (cgs) . \sigma=\frac{k\alpha}{\omega\mu}\quad\,\text{(SI)},\qquad\sigma=\frac{k\alpha% \mathrm{c}^{2}}{4\pi\omega\mu}\quad\,\text{(cgs)}.

Mathematical_diagram.html

  1. λ \lambda
  2. λ \lambda
  3. λ \lambda

Mathematical_economics.html

  1. f f
  2. x x
  3. g g
  4. x x
  5. h h
  6. x x
  7. f f
  8. g g
  9. j j
  10. m m
  11. m m
  12. h h
  13. j j
  14. l l
  15. l l

Mathematical_principles_of_reinforcement.html

  1. b ( t ) = b 1 × e - t τ b(t)=b1\times e^{\frac{-t}{\tau}}
  2. τ \tau
  3. R = A × ( e - t C - e - t I ) R=A\times(e-\frac{t}{C}-e-\frac{t}{I})
  4. b b
  5. b b
  6. δ \delta
  7. τ \tau
  8. b = 1 δ + τ b=\frac{1}{\delta+\tau}
  9. b = 1 τ b=\frac{1}{\tau}
  10. 1 / τ 1/\tau
  11. 1 τ = a r \frac{1}{\tau}=ar
  12. τ = 1 / a r \tau=1/ar
  13. b = 1 ( δ + 1 a r ) b=\frac{1}{(\delta+\frac{1}{ar})}
  14. b = r δ r + 1 / a b=\frac{r}{\delta r+1/a}
  15. A = a r A=ar
  16. b b
  17. b = c . r δ r + 1 / a b=\frac{c.r}{\delta r+1/a}
  18. c c
  19. r = b n r=\frac{b}{n}
  20. β \beta
  21. 1 - β 1-\beta
  22. β ( 1 - β ) \beta(1-\beta)
  23. β ( 1 - β ) 2 \beta(1-\beta)^{2}
  24. n n
  25. β ( 1 - β ) n - 1 \beta(1-\beta)^{n-1}
  26. F R n = 1 - ( 1 - β ) n FR_{n}=1-(1-\beta)^{n}
  27. c F R n = 1 - e - λ n c_{FR_{n}}=1-e^{-\lambda n}
  28. b = c . δ - n / δ a b=\frac{c.}{\delta-n/{\delta a}}
  29. n 0 n0
  30. c F R n = 1 - ( 1 - β ) n + n 0 = 1 - ϵ ( 1 - β ) n c_{FR_{n}}=1-(1-\beta)n+n0=1-\epsilon(1-\beta)n
  31. ϵ \epsilon
  32. ϵ = ( 1 - β ) n 0 \epsilon=(1-\beta)n0
  33. ϵ = 1 \epsilon=1
  34. β \beta
  35. 1 - e - λ δ 1-e^{-\lambda\delta}
  36. 1 - ϵ ( 1 - β ) δ n = 1 - ϵ e - λ δ n 1-\epsilon(1-\beta)\delta n=1-\epsilon e^{-\lambda\delta n}
  37. n n
  38. 1 / n 1/n
  39. β \beta
  40. 1 - p 1-p
  41. β ( 1 - β ) \beta(1-\beta)
  42. C ( n ) = j = 1 β ( 1 - β ) j - 1 ( 1 - p ) j - 1 C(n)=\sum_{j=1}^{\infty}\beta(1-\beta)^{j-1}(1-p)^{j-1}

Matrix_(mathematics).html

  1. [ 1 9 - 13 20 5 - 6 ] . \begin{bmatrix}1&9&-13\\ 20&5&-6\end{bmatrix}.
  2. 𝐀 = [ - 1.3 0.6 20.4 5.5 9.7 - 6.2 ] . \mathbf{A}=\begin{bmatrix}-1.3&0.6\\ 20.4&5.5\\ 9.7&-6.2\end{bmatrix}.
  3. [ 3 7 2 ] \begin{bmatrix}3&7&2\end{bmatrix}
  4. [ 4 1 8 ] \begin{bmatrix}4\\ 1\\ 8\end{bmatrix}
  5. [ 9 13 5 1 11 7 2 6 3 ] \begin{bmatrix}9&13&5\\ 1&11&7\\ 2&6&3\end{bmatrix}
  6. 𝐀 = [ a 11 a 12 a 1 n a 21 a 22 a 2 n a m 1 a m 2 a m n ] = ( a 11 a 12 a 1 n a 21 a 22 a 2 n a m 1 a m 2 a m n ) m × n . \mathbf{A}=\begin{bmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{m1}&a_{m2}&\cdots&a_{mn}\end{bmatrix}=\left(\begin{array}[]{rrrr}a_{11}&a_{% 12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{m1}&a_{m2}&\cdots&a_{mn}\end{array}\right)\in\mathbb{R}^{m\times n}.
  7. A ¯ ¯ \underline{\underline{A}}
  8. 𝐀 = [ 4 - 7 \color r e d 5 0 - 2 0 11 8 19 1 - 3 12 ] \mathbf{A}=\begin{bmatrix}4&-7&\color{red}{5}&0\\ -2&0&11&8\\ 19&1&-3&12\end{bmatrix}
  9. 𝐀 = [ 0 - 1 - 2 - 3 1 0 - 1 - 2 2 1 0 - 1 ] \mathbf{A}=\begin{bmatrix}0&-1&-2&-3\\ 1&0&-1&-2\\ 2&1&0&-1\end{bmatrix}
  10. [ 1 3 1 1 0 0 ] + [ 0 0 5 7 5 0 ] = [ 1 + 0 3 + 0 1 + 5 1 + 7 0 + 5 0 + 0 ] = [ 1 3 6 8 5 0 ] \begin{bmatrix}1&3&1\\ 1&0&0\end{bmatrix}+\begin{bmatrix}0&0&5\\ 7&5&0\end{bmatrix}=\begin{bmatrix}1+0&3+0&1+5\\ 1+7&0+5&0+0\end{bmatrix}=\begin{bmatrix}1&3&6\\ 8&5&0\end{bmatrix}
  11. 2 [ 1 8 - 3 4 - 2 5 ] = [ 2 1 2 8 2 - 3 2 4 2 - 2 2 5 ] = [ 2 16 - 6 8 - 4 10 ] 2\cdot\begin{bmatrix}1&8&-3\\ 4&-2&5\end{bmatrix}=\begin{bmatrix}2\cdot 1&2\cdot 8&2\cdot-3\\ 2\cdot 4&2\cdot-2&2\cdot 5\end{bmatrix}=\begin{bmatrix}2&16&-6\\ 8&-4&10\end{bmatrix}
  12. [ 1 2 3 0 - 6 7 ] T = [ 1 0 2 - 6 3 7 ] \begin{bmatrix}1&2&3\\ 0&-6&7\end{bmatrix}^{\mathrm{T}}=\begin{bmatrix}1&0\\ 2&-6\\ 3&7\end{bmatrix}
  13. [ 𝐀𝐁 ] i , j = A i , 1 B 1 , j + A i , 2 B 2 , j + + A i , n B n , j = r = 1 n A i , r B r , j [\mathbf{AB}]_{i,j}=A_{i,1}B_{1,j}+A_{i,2}B_{2,j}+\cdots+A_{i,n}B_{n,j}=\sum_{% r=1}^{n}A_{i,r}B_{r,j}
  14. ( 2 × 1000 ) + ( 3 × 100 ) + ( 4 × 10 ) = 2340 : (2×1000)+(3×100)+(4×10)=2340:
  15. [ 2 ¯ 3 ¯ 4 ¯ 1 0 0 ] [ 0 1000 ¯ 1 100 ¯ 0 10 ¯ ] = [ 3 2340 ¯ 0 1000 ] . \begin{aligned}\displaystyle\begin{bmatrix}\underline{2}&\underline{3}&% \underline{4}\\ 1&0&0\\ \end{bmatrix}\begin{bmatrix}0&\underline{1000}\\ 1&\underline{100}\\ 0&\underline{10}\\ \end{bmatrix}&\displaystyle=\begin{bmatrix}3&\underline{2340}\\ 0&1000\\ \end{bmatrix}.\end{aligned}
  16. [ 1 2 3 4 ] [ 0 1 0 0 ] = [ 0 1 0 3 ] , \begin{bmatrix}1&2\\ 3&4\\ \end{bmatrix}\begin{bmatrix}0&1\\ 0&0\\ \end{bmatrix}=\begin{bmatrix}0&1\\ 0&3\\ \end{bmatrix},
  17. [ 0 1 0 0 ] [ 1 2 3 4 ] = [ 3 4 0 0 ] . \begin{bmatrix}0&1\\ 0&0\\ \end{bmatrix}\begin{bmatrix}1&2\\ 3&4\\ \end{bmatrix}=\begin{bmatrix}3&4\\ 0&0\\ \end{bmatrix}.
  18. 𝐀 = [ 1 \color r e d 2 3 4 5 \color r e d 6 7 8 \color r e d 9 \color r e d 10 \color r e d 11 \color r e d 12 ] [ 1 3 4 5 7 8 ] . \mathbf{A}=\begin{bmatrix}1&\color{red}{2}&3&4\\ 5&\color{red}{6}&7&8\\ \color{red}{9}&\color{red}{10}&\color{red}{11}&\color{red}{12}\end{bmatrix}% \rightarrow\begin{bmatrix}1&3&4\\ 5&7&8\end{bmatrix}.
  19. 𝐀 = [ a c b d ] \mathbf{A}=\begin{bmatrix}a&c\\ b&d\end{bmatrix}\,
  20. [ 0 0 ] , [ 1 0 ] , [ 1 1 ] \begin{bmatrix}0\\ 0\end{bmatrix},\begin{bmatrix}1\\ 0\end{bmatrix},\begin{bmatrix}1\\ 1\end{bmatrix}
  21. [ 0 1 ] \begin{bmatrix}0\\ 1\end{bmatrix}
  22. [ 1 1.25 0 1 ] \begin{bmatrix}1&1.25\\ 0&1\end{bmatrix}
  23. [ - 1 0 0 1 ] \begin{bmatrix}-1&0\\ 0&1\end{bmatrix}
  24. [ 3 / 2 0 0 2 / 3 ] \begin{bmatrix}3/2&0\\ 0&2/3\end{bmatrix}
  25. [ 3 / 2 0 0 3 / 2 ] \begin{bmatrix}3/2&0\\ 0&3/2\end{bmatrix}
  26. [ cos ( π / 6 R ) - sin ( π / 6 R ) sin ( π / 6 R ) cos ( π / 6 R ) ] \begin{bmatrix}\cos(\pi/6^{R})&-\sin(\pi/6^{R})\\ \sin(\pi/6^{R})&\cos(\pi/6^{R})\end{bmatrix}
  27. [ a 11 0 0 0 a 22 0 0 0 a 33 ] \begin{bmatrix}a_{11}&0&0\\ 0&a_{22}&0\\ 0&0&a_{33}\\ \end{bmatrix}
  28. [ a 11 0 0 a 21 a 22 0 a 31 a 32 a 33 ] \begin{bmatrix}a_{11}&0&0\\ a_{21}&a_{22}&0\\ a_{31}&a_{32}&a_{33}\\ \end{bmatrix}
  29. [ a 11 a 12 a 13 0 a 22 a 23 0 0 a 33 ] \begin{bmatrix}a_{11}&a_{12}&a_{13}\\ 0&a_{22}&a_{23}\\ 0&0&a_{33}\\ \end{bmatrix}
  30. I 1 = [ 1 ] , I 2 = [ 1 0 0 1 ] , , I n = [ 1 0 0 0 1 0 0 0 1 ] I_{1}=\begin{bmatrix}1\end{bmatrix},\ I_{2}=\begin{bmatrix}1&0\\ 0&1\end{bmatrix},\ \cdots,\ I_{n}=\begin{bmatrix}1&0&\cdots&0\\ 0&1&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&1\end{bmatrix}
  31. [ 1 / 4 0 0 1 ] \begin{bmatrix}1/4&0\\ 0&1\\ \end{bmatrix}
  32. [ 1 / 4 0 0 - 1 / 4 ] \begin{bmatrix}1/4&0\\ 0&-1/4\end{bmatrix}
  33. A T = A - 1 , A^{\mathrm{T}}=A^{-1},\,
  34. A T A = A A T = I , A^{\mathrm{T}}A=AA^{\mathrm{T}}=I,\,
  35. tr ( 𝖠𝖡 ) = i = 1 m j = 1 n A i j B j i = tr ( 𝖡𝖠 ) . \scriptstyle\operatorname{tr}(\mathsf{AB})=\sum_{i=1}^{m}\sum_{j=1}^{n}A_{ij}B% _{ji}=\operatorname{tr}(\mathsf{BA}).
  36. det [ a b c d ] = a d - b c . \det\begin{bmatrix}a&b\\ c&d\end{bmatrix}=ad-bc.
  37. det ( 𝖠 - λ 𝖨 ) = 0. \det(\mathsf{A}-\lambda\mathsf{I})=0.
  38. f ( 𝐯 j ) = i = 1 m a i , j 𝐰 i for j = 1 , , n . f(\mathbf{v}_{j})=\sum_{i=1}^{m}a_{i,j}\mathbf{w}_{i}\qquad\mbox{for }~{}j=1,% \ldots,n.
  39. k k
  40. M = i I R M=\bigoplus_{i\in I}R
  41. 𝔽 𝕄 I ( R ) \mathbb{CFM}_{I}(R)
  42. I × I I\times I
  43. 𝔽 𝕄 I ( R ) \mathbb{RFM}_{I}(R)
  44. a + i b [ a - b b a ] , a+ib\leftrightarrow\begin{bmatrix}a&-b\\ b&a\end{bmatrix},
  45. H ( f ) = [ 2 f x i x j ] . H(f)=\left[\frac{\partial^{2}f}{\partial x_{i}\,\partial x_{j}}\right].
  46. f / x i \partial f/\partial x_{i}
  47. J ( f ) = [ f i x j ] 1 i m , 1 j n . J(f)=\left[\frac{\partial f_{i}}{\partial x_{j}}\right]_{1\leq i\leq m,1\leq j% \leq n}.
  48. a 1 a 2 a n i < j ( a j - a i ) a_{1}a_{2}\cdots a_{n}\prod_{i<j}(a_{j}-a_{i})\;

Matrix_differential_equation.html

  1. 𝐱 ( t ) = 𝐀𝐱 ( t ) \mathbf{x^{\prime}}(t)=\mathbf{Ax}(t)
  2. t t
  3. 𝐱 ( t ) = c 1 e λ 1 t 𝐮 1 + c 2 e λ 2 t 𝐮 2 + + c n e λ n t 𝐮 n \mathbf{x}(t)=c_{1}e^{\lambda_{1}t}\mathbf{u}_{1}+c_{2}e^{\lambda_{2}t}\mathbf% {u}_{2}+\cdots+c_{n}e^{\lambda_{n}t}\mathbf{u}_{n}
  4. 𝐱 ( t ) = 𝐀𝐱 ( t ) + 𝐛 \mathbf{x}^{\prime}(t)=\mathbf{Ax}(t)+\mathbf{b}
  5. 𝐱 ( t ) = 𝟎 \mathbf{x}^{\prime}(t)=\mathbf{0}
  6. 𝐱 * = - 𝐀 - 1 𝐛 \mathbf{x}^{*}=-\mathbf{A}^{-1}\mathbf{b}
  7. 𝐱 ( t ) = 𝐀 [ 𝐱 ( t ) - 𝐱 * ] \mathbf{x}^{\prime}(t)=\mathbf{A}[\mathbf{x}(t)-\mathbf{x}^{*}]
  8. 𝐱 h + 𝐱 * \mathbf{x}_{h}+\mathbf{x}^{*}
  9. 𝐱 h \mathbf{x}_{h}
  10. 𝐱 ( t ) = 𝐀 [ 𝐱 ( t ) - 𝐱 * ] \mathbf{x}^{\prime}(t)=\mathbf{A}[\mathbf{x}(t)-\mathbf{x}^{*}]
  11. 𝐱 ( t ) = 𝐱 * + e 𝐀 t [ 𝐱 ( 0 ) - 𝐱 * ] \mathbf{x}(t)=\mathbf{x}^{*}+e^{\mathbf{A}t}[\mathbf{x}(0)-\mathbf{x}^{*}]
  12. 𝐀 \mathbf{A}
  13. λ 1 , λ 2 , , λ n \lambda_{1},\lambda_{2},\dots,\lambda_{n}
  14. e 𝐀 t = j = 0 n - 1 r j + 1 ( t ) 𝐏 j e^{\mathbf{A}t}=\sum_{j=0}^{n-1}r_{j+1}{\left(t\right)}\mathbf{P}_{j}
  15. 𝐏 0 = 𝐈 \mathbf{P}_{0}=\mathbf{I}
  16. 𝐏 j = k = 1 j ( 𝐀 - λ k 𝐈 ) = 𝐏 j - 1 ( 𝐀 - λ j 𝐈 ) , j = 1 , 2 , , n - 1 \mathbf{P}_{j}=\prod_{k=1}^{j}\left(\mathbf{A}-\lambda_{k}\mathbf{I}\right)=% \mathbf{P}_{j-1}\left(\mathbf{A}-\lambda_{j}\mathbf{I}\right),j=1,2,\dots,n-1
  17. r ˙ 1 = λ 1 r 1 \dot{r}_{1}=\lambda_{1}r_{1}
  18. r 1 ( 0 ) = 1 r_{1}{\left(0\right)}=1
  19. r ˙ j = λ j r j + r j - 1 , j = 2 , 3 , , n \dot{r}_{j}=\lambda_{j}r_{j}+r_{j-1},j=2,3,\dots,n
  20. r j ( 0 ) = 0 , j = 2 , 3 , , n r_{j}{\left(0\right)}=0,j=2,3,\dots,n
  21. r i ( t ) r_{i}{\left(t\right)}
  22. 𝐀 \mathbf{A}
  23. d x d t = a 1 x + b 1 y , d y d t = a 2 x + b 2 y \frac{dx}{dt}=a_{1}x+b_{1}y,\quad\frac{dy}{dt}=a_{2}x+b_{2}y
  24. a 1 , a 2 , b 1 a_{1},a_{2},b_{1}\,\!
  25. b 2 b_{2}\,\!
  26. x x\,\!
  27. y y\,\!
  28. d x d t = 3 x - 4 y , d y d t = 4 x - 7 y . \frac{dx}{dt}=3x-4y,\quad\frac{dy}{dt}=4x-7y.
  29. x ( 0 ) = y ( 0 ) = 1. x(0)=y(0)=1.\,\!
  30. ( x y ) = ( 3 - 4 4 - 7 ) ( x y ) . \begin{pmatrix}x^{\prime}\\ y^{\prime}\end{pmatrix}=\begin{pmatrix}3&-4\\ 4&-7\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}.
  31. I n I_{n}\,\!
  32. det ( [ 3 - 4 4 - 7 ] - λ [ 1 0 0 1 ] ) \det\left(\begin{bmatrix}3&-4\\ 4&-7\end{bmatrix}-\lambda\begin{bmatrix}1&0\\ 0&1\end{bmatrix}\right)
  33. det [ 3 - λ - 4 4 - 7 - λ ] \det\begin{bmatrix}3-\lambda&-4\\ 4&-7-\lambda\end{bmatrix}
  34. det [ 3 - λ - 4 4 - 7 - λ ] = 0 \det\begin{bmatrix}3-\lambda&-4\\ 4&-7-\lambda\end{bmatrix}=0
  35. - 21 - 3 λ + 7 λ + λ 2 + 16 = 0 -21-3\lambda+7\lambda+\lambda^{2}+16=0\,\!
  36. λ 2 + 4 λ - 5 = 0 \lambda^{2}+4\lambda-5=0\,\!
  37. λ 1 \lambda_{1}\,\!
  38. λ 2 \lambda_{2}\,\!
  39. λ 2 + 5 λ - λ - 5 = 0 \lambda^{2}+5\lambda-\lambda-5=0\,\!
  40. λ ( λ + 5 ) - 1 ( λ + 5 ) = 0 \lambda(\lambda+5)-1(\lambda+5)=0\,\!
  41. ( λ - 1 ) ( λ + 5 ) = 0 (\lambda-1)(\lambda+5)=0\,\!
  42. λ = 1 , - 5 \lambda=1,-5\,\!
  43. λ 1 = 1 \lambda_{1}=1\,\!
  44. λ 2 = - 5 \lambda_{2}=-5\,\!
  45. λ 1 = 1 \lambda_{1}=1\,\!
  46. ( 3 - 4 4 - 7 ) ( α β ) = 1 ( α β ) . \begin{pmatrix}3&-4\\ 4&-7\end{pmatrix}\begin{pmatrix}\alpha\\ \beta\end{pmatrix}=1\begin{pmatrix}\alpha\\ \beta\end{pmatrix}.
  47. 3 α - 4 β = α 3\alpha-4\beta=\alpha
  48. α = 2 β . \alpha=2\beta.
  49. α = 2 β \alpha=2\beta\,\!
  50. α \alpha\,\!
  51. β \beta\,\!
  52. α = 2 β \alpha=2\beta\,\!
  53. α = 2 \alpha=2\,\!
  54. β = 1 \beta=1\,\!
  55. 𝐯 ^ 1 = ( 2 1 ) . \mathbf{\hat{v}}_{1}=\begin{pmatrix}2\\ 1\end{pmatrix}.
  56. λ = - 5 , \,\!\,\lambda=-5\,,
  57. 𝐯 ^ 2 = ( 1 2 ) . \mathbf{\hat{v}}_{2}=\begin{pmatrix}1\\ 2\end{pmatrix}.
  58. ( x y ) = A e λ 1 t 𝐯 ^ 1 + B e λ 2 t 𝐯 ^ 2 . \begin{pmatrix}x\\ y\end{pmatrix}=Ae^{\lambda_{1}t}\mathbf{\hat{v}}_{1}+Be^{\lambda_{2}t}\mathbf{% \hat{v}}_{2}.
  59. ( x y ) = A e t ( 2 1 ) + B e - 5 t ( 1 2 ) . \begin{pmatrix}x\\ y\end{pmatrix}=Ae^{t}\begin{pmatrix}2\\ 1\end{pmatrix}+Be^{-5t}\begin{pmatrix}1\\ 2\end{pmatrix}.
  60. ( 2 1 1 2 ) ( A e t B e - 5 t ) = ( x y ) . \begin{pmatrix}2&1\\ 1&2\end{pmatrix}\begin{pmatrix}Ae^{t}\\ Be^{-5t}\end{pmatrix}=\begin{pmatrix}x\\ y\end{pmatrix}.
  61. x x\,\!
  62. y y\,\!
  63. x = 2 A e t + B e - 5 t x=2Ae^{t}+Be^{-5t}\,\!
  64. y = A e t + 2 B e - 5 t . y=Ae^{t}+2Be^{-5t}.\,\!
  65. x ( 0 ) = y ( 0 ) = 1 x(0)=y(0)=1\,\!
  66. x ( 0 ) = y ( 0 ) = 1 x(0)=y(0)=1\,\!
  67. t = 0 t=0\,\!
  68. 1 = 2 A + B 1=2A+B\,\!
  69. 1 = A + 2 B . 1=A+2B.\,\!
  70. x = 2 3 e t + 1 3 e - 5 t x=\frac{2}{3}e^{t}+\frac{1}{3}e^{-5t}\,\!
  71. y = 1 3 e t + 2 3 e - 5 t . y=\frac{1}{3}e^{t}+\frac{2}{3}e^{-5t}.\,\!

Mattis–Bardeen_theory.html

  1. α s = | M s | 2 N s ( E ) N s ( E + ω ) × [ f ( E ) - f ( E + ω ) ] d E \alpha_{s}=\int{\left|M_{s}\right|^{2}N_{s}(E)N_{s}(E+\hbar\omega)\times[f(E)-% f(E+\hbar\omega)]{\rm{}}}dE
  2. N s N_{s}
  3. M s M_{s}
  4. H 1 H_{1}
  5. H 1 = k σ , k σ B k σ , k σ c k σ * c k σ H_{1}=\sum\limits_{k\sigma,k^{\prime}\sigma^{\prime}}{B_{k^{\prime}\sigma^{% \prime},k\sigma}c_{k^{\prime}\sigma^{\prime}}^{*}}c_{k^{\prime}\sigma^{\prime}}
  6. M s M_{s}
  7. M M
  8. F ( Δ , E , E ) = 1 2 ( 1 ± < m t p l > Δ 2 E E ) F(\Delta,E,E^{\prime})=\frac{1}{2}(1\pm\frac{<}{m}tpl>{{\Delta^{2}}}{{EE^{% \prime}}})
  9. α s = | M | 2 F ( Δ , E , E + ω ) N s ( E ) N s ( E + ω ) × [ f ( E ) - f ( E + ω ) ] d E \alpha_{s}=\int{\left|M\right|^{2}F(\Delta,E,E+\hbar\omega)N_{s}(E)N_{s}(E+% \hbar\omega)\times[f(E)-f(E+\hbar\omega)]{\rm{}}}dE
  10. σ 1 \sigma_{1}
  11. σ 1 E 2 \sigma_{1}E^{2}
  12. < m t p l > α s α n = σ 1 s σ n \frac{<}{m}tpl>{{\alpha_{s}}}{{\alpha_{n}}}=\frac{{\sigma_{1s}}}{{\sigma_{n}}}
  13. < m t p l > α s α n = 2 ω Δ | E(E + ω ) + Δ 2 | [ f ( E ) - f ( E + ω ) ] ( E 2 - Δ 2 ) 1 / 2 [ ( E + ω ) 2 - Δ 2 ] 1 / 2 d E + 1 ω Δ - ω - Δ | E(E + ω ) + Δ 2 | [ 1 - 2 f ( E + ω ) ] ( E 2 - Δ 2 ) 1 / 2 [ ( E + ω ) 2 - Δ 2 ] 1 / 2 d E \frac{<}{m}tpl>{{\alpha_{s}}}{{\alpha_{n}}}=\frac{2}{{\hbar\omega}}\int_{% \Delta}^{\infty}{\frac{{\left|{{\,\text{E(E + }}\hbar\omega{\,\text{)}}+\Delta% ^{2}}\right|[f(E)-f(E+\hbar\omega)]}}{{(E^{2}-\Delta^{2})^{1/2}[(E+\hbar\omega% )^{2}-\Delta^{2}]^{1/2}}}dE}{\,\text{ + }}\frac{1}{{\hbar\omega}}\int_{\Delta-% \hbar\omega}^{-\Delta}{\frac{{\left|{{\,\text{E(E + }}\hbar\omega{\,\text{)}}+% \Delta^{2}}\right|[1-2f(E+\hbar\omega)]}}{{(E^{2}-\Delta^{2})^{1/2}[(E+\hbar% \omega)^{2}-\Delta^{2}]^{1/2}}}dE}

Maximal_function.html

  1. ( M f ) ( x ) = sup x B 1 | B | B | f | (Mf)(x)=\sup_{x\in B}\frac{1}{|B|}\int_{B}|f|
  2. | { x : ( M f ) ( x ) > α } | c α 𝐑 n | f | . |\{x\ :\ (Mf)(x)>\alpha\}|\leq\frac{c}{\alpha}\int_{\mathbf{R}^{n}}|f|.
  3. M f L p A f L p , \|Mf\|_{L^{p}}\leq A\|f\|_{L^{p}},
  4. 𝐑 + n + 1 := { ( x , t ) : x 𝐑 n , t > 0 } \mathbf{R}^{n+1}_{+}:=\left\{(x,t)\ :\ x\in\mathbf{R}^{n},t>0\right\}
  5. F * ( x ) = sup | x - y | < t | F ( y , t ) | . F^{*}(x)=\sup_{|x-y|<t}|F(y,t)|.
  6. { ( y , t ) : | x - y | < t } \{(y,t)\ :\ |x-y|<t\}
  7. 𝐑 + n + 1 \mathbf{R}^{n+1}_{+}
  8. 𝐑 n Φ = 1 \int_{\mathbf{R}^{n}}\Phi=1
  9. Φ t ( x ) = t - n Φ ( x t ) \Phi_{t}(x)=t^{-n}\Phi(\tfrac{x}{t})
  10. F ( x , t ) = f Φ t ( x ) := 𝐑 n f ( x - y ) Φ t ( y ) d y . F(x,t)=f\ast\Phi_{t}(x):=\int_{\mathbf{R}^{n}}f(x-y)\Phi_{t}(y)\,dy.
  11. sup t > 0 | f Φ t ( x ) | ( M f ) ( x ) 𝐑 n Φ \sup_{t>0}|f\ast\Phi_{t}(x)|\leq(Mf)(x)\int_{\mathbf{R}^{n}}\Phi
  12. f Φ t ( x ) f\ast\Phi_{t}(x)
  13. f f^{\sharp}
  14. f ( x ) = sup x B 1 | B | B | f ( y ) - f B | d y f^{\sharp}(x)=\sup_{x\in B}\frac{1}{|B|}\int_{B}|f(y)-f_{B}|\,dy
  15. T ( f ) L 2 C f L 2 , \|T(f)\|_{L^{2}}\leq C\|f\|_{L^{2}},
  16. g ( x ) T ( f ) ( x ) d x = g ( x ) K ( x - y ) f ( y ) d y d x . \int g(x)T(f)(x)\,dx=\iint g(x)K(x-y)f(y)\,dy\,dx.
  17. | K ( x - y ) - K ( x ) | C | y | γ | x | n + γ , |K(x-y)-K(x)|\leq C\frac{|y|^{\gamma}}{|x|^{n+\gamma}},
  18. | x | 2 | y | |x|\geq 2|y|
  19. ( T ( f ) ) ( x ) C ( M ( | f | r ) ) 1 r ( x ) (T(f))^{\sharp}(x)\leq C(M(|f|^{r}))^{\frac{1}{r}}(x)
  20. ( X , , m ) (X,\mathcal{B},m)
  21. f * ( x ) := sup n 1 1 n i n - 1 | f ( T i ( x ) ) | . f^{*}(x):=\sup_{n\geq 1}\frac{1}{n}\sum_{i}^{n-1}|f(T^{i}(x))|.
  22. m ( { x X : f * ( x ) > α } ) f 1 α , m\left(\{x\in X\ :\ f^{*}(x)>\alpha\}\right)\leq\frac{\|f\|_{1}}{\alpha},
  23. { f n } \{f_{n}\}
  24. f * ( x ) = sup n | f n ( x ) | f^{*}(x)=\sup_{n}|f_{n}(x)|
  25. f ( x ) = lim n f n ( x ) f(x)=\lim_{n\rightarrow\infty}f_{n}(x)
  26. L p , 1 < p L^{p},1<p\leq\infty
  27. L 1 L^{1}
  28. f f
  29. f * f^{*}

Maximising_measure.html

  1. β ( f ) := sup { X f d ν | ν Inv ( T ) } . \beta(f):=\sup\left.\left\{\int_{X}f\,\mathrm{d}\nu\right|\nu\in\mathrm{Inv}(T% )\right\}.
  2. X f d μ = β ( f ) . \int_{X}f\,\mathrm{d}\mu=\beta(f).

Maximum-minimums_identity.html

  1. max { x 1 , x 2 , , x n } = i = 1 n x i - i < j min { x i , x j } + i < j < k min { x i , x j , x k } - + ( - 1 ) n + 1 min { x 1 , x 2 , , x n } , \begin{aligned}\displaystyle\max\{x_{1},x_{2},\ldots,x_{n}\}&\displaystyle=% \sum_{i=1}^{n}x_{i}-\sum_{i<j}\min\{x_{i},x_{j}\}+\sum_{i<j<k}\min\{x_{i},x_{j% },x_{k}\}-\cdots\\ &\displaystyle\qquad\cdots+\left(-1\right)^{n+1}\min\{x_{1},x_{2},\ldots,x_{n}% \},\end{aligned}
  2. min { x 1 , x 2 , , x n } = i = 1 n x i - i < j max { x i , x j } + i < j < k max { x i , x j , x k } - + ( - 1 ) n + 1 max { x 1 , x 2 , , x n } . \begin{aligned}\displaystyle\min\{x_{1},x_{2},\ldots,x_{n}\}&\displaystyle=% \sum_{i=1}^{n}x_{i}-\sum_{i<j}\max\{x_{i},x_{j}\}+\sum_{i<j<k}\max\{x_{i},x_{j% },x_{k}\}-\cdots\\ &\displaystyle\qquad\cdots+\left(-1\right)^{n+1}\max\{x_{1},x_{2},\ldots,x_{n}% \}.\end{aligned}

Maximum_bubble_pressure_method.html

  1. σ = Δ P max × R cap 2 \sigma=\frac{\Delta P_{\rm{max}}\times R_{\rm{cap}}}{2}

Maximum_cut.html

  1. G = ( V , E ) G=(V,E)
  2. | E | |E|
  3. | E | / 2 |E|/2
  4. α 0.878 \alpha\approx 0.878
  5. α = 2 π min 0 θ π θ 1 - cos θ \alpha=\tfrac{2}{\pi}\min_{0\leq\theta\leq\pi}\tfrac{\theta}{1-\cos\theta}
  6. 16 17 0.941 \tfrac{16}{17}\approx 0.941
  7. G = ( V , E ) G=(V,E)
  8. H = ( V , X ) H=(V,X)
  9. G = ( V , E ) G=(V,E)
  10. H = ( V , X ) H=(V,X)

Maximum_likelihood_sequence_estimation.html

  1. L ( x ) = p ( r x ) , L(x)=p(r\mid x),
  2. P ( x ) = p ( x r ) , P(x)=p(x\mid r),
  3. P ( x ) = p ( x r ) = p ( r x ) p ( x ) p ( r ) . P(x)=p(x\mid r)=\frac{p(r\mid x)p(x)}{p(r)}.

Maximum_spacing_estimation.html

  1. D i ( θ ) = F ( x ( i ) ; θ ) - F ( x ( i - 1 ) ; θ ) , i = 1 , , n + 1. D_{i}(\theta)=F(x_{(i)};\,\theta)-F(x_{(i-1)};\,\theta),\quad i=1,\ldots,n+1.
  2. θ ^ = arg max θ Θ S n ( θ ) , where S n ( θ ) = ln D 1 D 2 D n + 1 n + 1 = 1 n + 1 i = 1 n + 1 ln D i ( θ ) . \hat{\theta}=\underset{\theta\in\Theta}{\operatorname{arg\,max}}\;S_{n}(\theta% ),\quad\,\text{where }\ S_{n}(\theta)=\ln\!\!\sqrt[n+1]{D_{1}D_{2}\cdots D_{n+% 1}}=\frac{1}{n+1}\sum_{i=1}^{n+1}\ln{D_{i}}(\theta).
  3. n / + 1 {n}/{+1}
  4. μ = 0.6 λ MSE = ln 0.6 - 2 0.255 , \mu=0.6\quad\Rightarrow\quad\lambda_{\,\text{MSE}}=\frac{\ln 0.6}{-2}\approx 0% .255,
  5. 1 / λ {1}/{λ}
  6. D 1 = x ( 1 ) - a b - a , D i = x ( i ) - x ( i - 1 ) b - a for i = 2 , , n , D n + 1 = b - x ( n ) b - a D_{1}=\frac{x_{(1)}-a}{b-a},\ \ D_{i}=\frac{x_{(i)}-x_{(i-1)}}{b-a}\ \,\text{% for }i=2,\ldots,n,\ \ D_{n+1}=\frac{b-x_{(n)}}{b-a}\
  7. S n ( a , b ) = 1 n + 1 ln ( x ( 1 ) - a ) + 1 n + 1 ln ( b - x ( n ) ) - ln ( b - a ) + i = 2 n ln ( x ( i ) - x ( i - 1 ) ) S_{n}(a,b)=\tfrac{1}{n+1}\ln(x_{(1)}-a)+\tfrac{1}{n+1}\ln(b-x_{(n)})-\ln(b-a)+% \sum_{i=2}^{n}\ln(x_{(i)}-x_{(i-1)})
  8. a ^ = x ( 1 ) \scriptstyle\hat{a}=x_{(1)}
  9. b ^ = x ( n ) \scriptstyle\hat{b}=x_{(n)}
  10. X i + k = X i + k - 1 = = X i , X_{i+k}=X_{i+k-1}=\cdots=X_{i},\,
  11. D i + k ( θ ) = D i + k - 1 ( θ ) = = D i + 1 ( θ ) = 0. D_{i+k}(\theta)=D_{i+k-1}(\theta)=\cdots=D_{i+1}(\theta)=0.\,
  12. f i ( θ ) f_{i}(\theta)
  13. D i ( θ ) D_{i}(\theta)
  14. lim x i x i - 1 x i - 1 x i f ( t ; θ ) d t x i - x i - 1 = f ( x i - 1 , θ ) = f ( x i , θ ) , \lim_{x_{i}\to x_{i-1}}\frac{\int_{x_{i-1}}^{x_{i}}f(t;\theta)\,dt}{x_{i}-x_{i% -1}}=f(x_{i-1},\theta)=f(x_{i},\theta),
  15. x i = x i - 1 x_{i}=x_{i-1}
  16. x ± δ x\pm\delta
  17. y L = F ( x - δ , θ ^ ) y_{L}=F(x-\delta,\hat{\theta})
  18. y U = F ( x + δ , θ ^ ) y_{U}=F(x+\delta,\hat{\theta})
  19. D j = y U - y L r - 1 ( j = i + 1 , , i + r - 1 ) . D_{j}=\frac{y_{U}-y_{L}}{r-1}\quad(j=i+1,\ldots,i+r-1).
  20. S n ( θ ) = M n ( θ ) = - j = 1 n + 1 ln D j ( θ ) , S_{n}(\theta)=M_{n}(\theta)=-\sum_{j=1}^{n+1}\ln{D_{j}(\theta)},
  21. θ 0 \theta^{0}
  22. M n ( θ ) \scriptstyle M_{n}(\theta)
  23. μ M \displaystyle\mu_{M}
  24. A A
  25. A = C 1 + C 2 χ n 2 A=C_{1}+C_{2}\chi^{2}_{n}\,
  26. C 1 \displaystyle C_{1}
  27. χ n 2 \chi^{2}_{n}
  28. n n
  29. H 0 H_{0}
  30. n n
  31. F ( x , θ ) F(x,\theta)
  32. T ( θ ) = M ( θ ) - C 1 C 2 T(\theta)=\frac{M(\theta)-C_{1}}{C_{2}}
  33. H 0 H_{0}
  34. α \alpha
  35. θ ^ \hat{\theta}
  36. S n ( θ ^ ) = M n ( θ ^ ) S_{n}(\hat{\theta})=M_{n}(\hat{\theta})
  37. T ( θ ^ ) = M ( θ ^ ) + k 2 - C 1 C 2 , T(\hat{\theta})=\frac{M(\hat{\theta})+\frac{k}{2}-C_{1}}{C_{2}},
  38. k k
  39. F ( X j + m ) - F ( X j ) F(X_{j+m})-F(X_{j})
  40. k ( k > 1 ) \mathbb{R}^{k}(k>1)
  41. D j D_{j}
  42. D j D_{j}
  43. M ( θ ) M(\theta)
  44. D j D_{j}
  45. D j D_{j}
  46. σ ~ \textstyle\tilde{\sigma}
  47. M ( θ ) = - j = 1 n + 1 log D i ( θ ) \scriptstyle M(\theta)=-\sum_{j=1}^{n+1}\log{D_{i}(\theta)}
  48. D i ( θ ) \scriptstyle D_{i}(\theta)
  49. M n = - j = 0 n ln ( ( n + 1 ) ( X n , i + 1 - X n , i ) ) \scriptstyle M_{n}=-\sum_{j=0}^{n}\ln{((n+1)(X_{n,i+1}-X_{n,i}))}
  50. ( n + 1 ) (n+1)