wpmath0000015_4

Interval_propagation.html

  1. x 1 + x 2 = x 3 , x_{1}+x_{2}=x_{3},
  2. [ x 3 ] := [ x 3 ] ( [ x 1 ] + [ x 2 ] ) [x_{3}]:=[x_{3}]\cap([x_{1}]+[x_{2}])
  3. [ x 1 ] := [ x 1 ] ( [ x 3 ] - [ x 2 ] ) [x_{1}]:=[x_{1}]\cap([x_{3}]-[x_{2}])
  4. [ x 2 ] := [ x 2 ] ( [ x 3 ] - [ x 1 ] ) [x_{2}]:=[x_{2}]\cap([x_{3}]-[x_{1}])
  5. x 1 [ - , 5 ] , x_{1}\in[-\infty,5],
  6. x 2 [ - , 4 ] , x_{2}\in[-\infty,4],
  7. x 3 [ 6 , ] x_{3}\in[6,\infty]
  8. x 3 = x 1 + x 2 x 3 [ 6 , ] ( [ - , 5 ] + [ - , 4 ] ) = [ 6 , ] [ - , 9 ] = [ 6 , 9 ] . x_{3}=x_{1}+x_{2}\Rightarrow x_{3}\in[6,\infty]\cap([-\infty,5]+[-\infty,4])=[% 6,\infty]\cap[-\infty,9]=[6,9].
  9. x 1 = x 3 - x 2 x 1 [ - , 5 ] ( [ 6 , ] - [ - , 4 ] ) = [ - , 5 ] [ 2 , ] = [ 2 , 5 ] . x_{1}=x_{3}-x_{2}\Rightarrow x_{1}\in[-\infty,5]\cap([6,\infty]-[-\infty,4])=[% -\infty,5]\cap[2,\infty]=[2,5].
  10. x 2 = x 3 - x 1 x 2 [ - , 4 ] ( [ 6 , ] - [ - , 5 ] ) = [ - , 4 ] [ 1 , ] = [ 1 , 4 ] . x_{2}=x_{3}-x_{1}\Rightarrow x_{2}\in[-\infty,4]\cap([6,\infty]-[-\infty,5])=[% -\infty,4]\cap[1,\infty]=[1,4].
  11. x 2 = sin ( x 1 ) , x_{2}=\sin(x_{1}),
  12. x + sin ( x y ) 0 , x+\sin(xy)\leq 0,
  13. a = x y a=xy
  14. b = sin ( a ) b=\sin(a)
  15. c = x + b . c=x+b.
  16. a [ - , ] , a\in[-\infty,\infty],
  17. b [ - 1 , 1 ] , b\in[-1,1],
  18. c [ - , 0 ] . c\in[-\infty,0].
  19. E [ 23 V , 26 V ] E\in[23V,26V]
  20. I [ 4 A , 8 A ] I\in[4A,8A]
  21. U 1 [ 10 V , 11 V ] U_{1}\in[10V,11V]
  22. U 2 [ 14 V , 17 V ] U_{2}\in[14V,17V]
  23. P [ 124 W , 130 W ] P\in[124W,130W]
  24. R 1 [ 0 Ω , [ R_{1}\in[0\Omega,\infty[
  25. R 2 [ 0 Ω , [ . R_{2}\in[0\Omega,\infty[.
  26. P = E I P=EI
  27. U 1 = R 1 I U_{1}=R_{1}I
  28. U 2 = R 2 I U_{2}=R_{2}I
  29. E = U 1 + U 2 . E=U_{1}+U_{2}.
  30. E [ 24 V , 26 V ] E\in[24V,26V]
  31. I [ 4.769 A , 5.417 A ] I\in[4.769A,5.417A]
  32. U 1 [ 10 V , 11 V ] U_{1}\in[10V,11V]
  33. U 2 [ 14 V , 16 V ] U_{2}\in[14V,16V]
  34. P [ 124 W , 130 W ] P\in[124W,130W]
  35. R 1 [ 1.846 Ω , 2.307 Ω ] R_{1}\in[1.846\Omega,2.307\Omega]
  36. R 2 [ 2.584 Ω , 3.355 Ω ] . R_{2}\in[2.584\Omega,3.355\Omega].

Intraocular_lens_power_calculation.html

  1. P = A - 0.9 K - 2.5 L P=A-0.9K-2.5L
  2. P P
  3. A A
  4. K K
  5. L L

Invariant_convex_cone.html

  1. J = ( 0 I - I 0 ) . \displaystyle{J=\begin{pmatrix}0&I\\ -I&0\end{pmatrix}.}
  2. { f , g } = i f x i g y i - f y i g x i . \displaystyle{\{f,g\}=\sum_{i}{\partial f\over\partial x_{i}}{\partial g\over% \partial y_{i}}-{\partial f\over\partial y_{i}}{\partial g\over\partial x_{i}}.}
  3. Q ( x , y ) = a i ( x i 2 + y i 2 ) , \displaystyle{Q(x,y)=\sum a_{i}(x_{i}^{2}+y_{i}^{2}),}
  4. h = ( a b c d ) h=\begin{pmatrix}a&b\\ c&d\end{pmatrix}
  5. h ( z ) = ( a z + b ) ( c z + d ) - 1 . \displaystyle{h(z)=(az+b)(cz+d)^{-1}.}
  6. H = G exp ( C ) = exp ( C ) G . \displaystyle{H=G\cdot\exp(C)=\exp(C)\cdot G.}
  7. H ¯ = G exp ( C ¯ ) = exp ( C ¯ ) G . \displaystyle{\overline{H}=G\cdot\exp(\overline{C})=\exp(\overline{C})\cdot G.}
  8. exp C ¯ \exp\overline{C}
  9. exp C ¯ \exp\overline{C}
  10. G exp C ¯ G\exp\overline{C}
  11. H ¯ \overline{H}
  12. H ¯ \overline{H}
  13. H ¯ \overline{H}
  14. H ¯ \overline{H}
  15. H ¯ \overline{H}
  16. B ( ( z , w ) , ( z , w ) ) = z i w i - w i z i . \displaystyle{B((z,w),(z^{\prime},w^{\prime}))=\sum z_{i}w_{i}^{\prime}-w_{i}z% _{i}^{\prime}.}
  17. ( Z I ) \begin{pmatrix}Z\\ I\end{pmatrix}
  18. g = ( A B C D ) g=\begin{pmatrix}A&B\\ C&D\end{pmatrix}

Inverse_distribution.html

  1. G ( y ) = Pr ( Y y ) = Pr ( X 1 y ) = 1 - Pr ( X < 1 y ) = 1 - F ( 1 y ) . G(y)=\Pr(Y\leq y)=\Pr\left(X\geq\frac{1}{y}\right)=1-\Pr\left(X<\frac{1}{y}% \right)=1-F\left(\frac{1}{y}\right).
  2. g ( y ) = 1 y 2 f ( 1 y ) . g(y)=\frac{1}{y^{2}}f\left(\frac{1}{y}\right).
  3. f ( x ) x - 1 for 0 < a < x < b , f(x)\propto x^{-1}\quad\,\text{ for }0<a<x<b,
  4. \propto\!\,
  5. g ( y ) y - 1 for 0 b - 1 < y < a - 1 , g(y)\propto y^{-1}\quad\,\text{ for }0\leq b^{-1}<y<a^{-1},
  6. g ( y ) = y - 2 1 b - a , g(y)=y^{-2}\frac{1}{b-a},
  7. G ( y ) = b - y - 1 b - a . G(y)=\frac{b-y^{-1}}{b-a}.
  8. f ( x ) = 1 k π Γ ( k + 1 2 ) Γ ( k 2 ) 1 ( 1 + x 2 k ) 1 + k 2 . f(x)=\frac{1}{\sqrt{k\pi}}\frac{\Gamma\left(\frac{k+1}{2}\right)}{\Gamma\left(% \frac{k}{2}\right)}\frac{1}{\left(1+\frac{x^{2}}{k}\right)^{\frac{1+k}{2}}}.
  9. g ( y ) = 1 k π Γ ( k + 1 2 ) Γ ( k 2 ) 1 y 2 ( 1 + 1 y 2 k ) 1 + k 2 . g(y)=\frac{1}{\sqrt{k\pi}}\frac{\Gamma\left(\frac{k+1}{2}\right)}{\Gamma\left(% \frac{k}{2}\right)}\frac{1}{y^{2}\left(1+\frac{1}{y^{2}k}\right)^{\frac{1+k}{2% }}}.

Inverse_matrix_gamma_distribution.html

  1. Γ p \Gamma_{p}
  2. β = 2 , α = n 2 \beta=2,\alpha=\frac{n}{2}

Iribarren_number.html

  1. ξ = tan α H / L 0 , \xi=\frac{\tan\alpha}{\sqrt{H/L_{0}}},
  2. L 0 = g 2 π T 2 , L_{0}=\frac{g}{2\pi}\,T^{2},
  3. ξ 0 = tan α H 0 / L 0 \xi_{0}=\frac{\tan\alpha}{\sqrt{H_{0}/L_{0}}}
  4. ξ b = tan α H b / L 0 , \xi_{b}=\frac{\tan\alpha}{\sqrt{H_{b}/L_{0}}},

Iron:rusticyanin_reductase.html

  1. \rightleftharpoons

Irrationality_sequence.html

  1. n = 1 1 a n x n \sum_{n=1}^{\infty}\frac{1}{a_{n}x_{n}}
  2. 2 2 n 2^{2^{n}}
  3. x n = 1 x_{n}=1
  4. 1 2 + 1 3 + 1 7 + 1 43 + = 1 , \frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{43}+\cdots=1,
  5. n ! n!
  6. x n = n + 2 x_{n}=n+2
  7. n = 0 1 ( n + 2 ) n ! = 1 2 + 1 3 + 1 8 + 1 30 + 1 144 + = 1. \sum_{n=0}^{\infty}\frac{1}{(n+2)n!}=\frac{1}{2}+\frac{1}{3}+\frac{1}{8}+\frac% {1}{30}+\frac{1}{144}+\cdots=1.
  8. lim sup n log log a n n > log 2 \limsup_{n}\frac{\log\log a_{n}}{n}>\log 2
  9. lim n a n 1 / n = . \lim_{n\to\infty}a_{n}^{1/n}=\infty.
  10. lim n a n 1 / 2 n < . \lim_{n\to\infty}a_{n}^{1/2^{n}}<\infty.
  11. n = 1 1 a n x n \sum_{n=1}^{\infty}\frac{1}{a_{n}x_{n}}

Irrigation_game.html

  1. G = ( V ; A ) G=(V;A)
  2. V V
  3. A A
  4. i j i\leq j
  5. j j
  6. i i
  7. e A , e\in A,
  8. e = i j ¯ e=\overline{ij}
  9. e e
  10. i ; j V i;j\in V
  11. i j i\leq j
  12. c : A + c:A\rightarrow\mathbb{R}_{+}
  13. c c
  14. ( G ; c (G;c
  15. e A , e\in A,
  16. e = i j ¯ , e=\overline{ij},
  17. c e c_{e}
  18. j j
  19. i i
  20. # V 3 , # N 2 \#V\geq 3,\#N\geq 2
  21. ( G ; c ) (G;c)
  22. N N
  23. S N S\subseteq N
  24. S S
  25. S S
  26. ( G ; c ) (G;c)
  27. N = V { r o o t } N=V\setminus\{root\}
  28. S S
  29. v ( G ; c ) ( S ) = e A s c ( e ) . v_{(G;c)}(S)=\sum\limits_{e\in A_{s}}c(e).
  30. G = ( V , A ) G=(V,A)
  31. V = { r o o t , 1 , 2 , 3 } , A = { r o o t 1 ¯ , r o o t 2 ¯ , 23 ¯ } V=\{root,1,2,3\},A=\{\overline{root1},\overline{root2},\overline{23}\}
  32. c c
  33. c ( r o o t 1 ¯ ) = 11 , c ( r o o t 2 ¯ ) = 5 c(\overline{root1})=11,\ \ c(\overline{root2})=5
  34. c ( 23 ¯ ) = 7 c(\overline{23})=7
  35. v ( G ; c ) ( ) = 0 v_{(G;c)}(\emptyset)=0
  36. v ( G ; c ) ( { 1 } ) = 11 v_{(G;c)}(\{1\})=11
  37. v ( G ; c ) ( { 2 } ) = 5 v_{(G;c)}(\{2\})=5
  38. v ( G ; c ) ( { 3 } ) = 7 v_{(G;c)}(\{3\})=7
  39. v ( G ; c ) ( { 1 , 2 } ) = 16 v_{(G;c)}(\{1,2\})=16
  40. v ( G ; c ) ( { 1 , 3 } ) = 23 v_{(G;c)}(\{1,3\})=23
  41. v ( G ; c ) ( { 2 , 3 } ) = 12 v_{(G;c)}(\{2,3\})=12
  42. v ( G ; c ) ( { 1 , 2 , 3 } ) = 23. v_{(G;c)}(\{1,2,3\})=23.

IS::MP_model.html

  1. i ¯ \overline{i}

Isoeugenol_synthase.html

  1. \rightleftharpoons

Isoflavonoid_synthase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Isolated_horizon.html

  1. ( Δ , [ ] ) (\Delta\,,[\ell])
  2. [ ] [\ell]
  3. Δ \Delta
  4. [ ] [\ell]
  5. Δ \Delta
  6. S 2 × S^{2}\times\mathbb{R}
  7. l l
  8. Δ \Delta
  9. θ ( l ) := h ^ a b ^ a l b \displaystyle\theta_{(l)}:=\hat{h}^{ab}\hat{\nabla}_{a}l_{b}
  10. Δ \Delta
  11. T a b T_{ab}
  12. Δ \Delta
  13. V a := - T b a l b V^{a}:=-T^{a}_{b}l^{b}
  14. V a V a 0 V^{a}V_{a}\leq 0
  15. l a l^{a}
  16. [ , 𝒟 a ] = 0 [\mathcal{L}_{\ell},\mathcal{D}_{a}]=0
  17. 𝒟 a \mathcal{D}_{a}
  18. = ^ \hat{=}
  19. h ^ a b \hat{h}^{ab}
  20. ^ \hat{\nabla}
  21. κ = ^ 0 \kappa\,\hat{=}\,0
  22. Im ( ρ ) = ^ 0 \,\text{Im}(\rho)\,\hat{=}\,0
  23. Re ( ρ ) = ^ 0 \,\text{Re}(\rho)\,\hat{=}\,0
  24. σ = ^ 0 \sigma\,\hat{=}\,0
  25. Φ 00 = ^ 0 , Φ 10 = Φ 01 ¯ = ^ 0 \Phi_{00}\,\hat{=}\,0\,,\quad\Phi_{10}=\overline{\Phi_{01}}\,\hat{=}\,0
  26. Ψ 0 = ^ 0 , Ψ 1 = ^ 0 \Psi_{0}\,\hat{=}\,0\,,\quad\Psi_{1}\,\hat{=}\,0
  27. ϕ 0 = ^ 0 , Φ 02 = Φ 20 ¯ = 2 ϕ 0 ϕ 2 ¯ = ^ 0 . \phi_{0}\,\hat{=}\,0\,,\quad\Phi_{02}=\overline{\Phi_{20}}=\,2\,\phi_{0}\,% \overline{\phi_{2}}\,\hat{=}\,0\,.
  28. π = ^ α + β ¯ , ε = ^ ε ¯ , μ ¯ = ^ μ . \pi\,\hat{=}\,\alpha+\bar{\beta}\,,\quad\varepsilon\,\hat{=}\,\bar{\varepsilon% }\,,\quad\bar{\mu}\,\hat{=}\,\mu\,.
  29. l a a = v + U r + X 3 y + X 4 z , l^{a}\partial_{a}=\partial_{v}+U\partial_{r}+X^{3}\partial_{y}+X^{4}\partial_{% z}\,,
  30. n a a = - r , n^{a}\partial_{a}=-\partial_{r}\,,
  31. m a a = Ω r + ξ 3 y + ξ 4 z , m^{a}\partial_{a}=\Omega\partial_{r}+\xi^{3}\partial_{y}+\xi^{4}\partial_{z}\,,
  32. m ¯ a a = Ω ¯ r + ξ ¯ 3 y + ξ ¯ 4 z . \bar{m}^{a}\partial_{a}=\bar{\Omega}\partial_{r}+\bar{\xi}^{3}\partial_{y}+% \bar{\xi}^{4}\partial_{z}\,.
  33. { y , z } \{y,z\}
  34. ν = τ = γ = 0 , μ = μ ¯ , π = α + β ¯ . \nu=\tau=\gamma=0\,,\quad\mu=\bar{\mu}\,,\quad\pi=\alpha+\bar{\beta}\,.

Isoleucine_N-monooxygenase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons
  4. \rightleftharpoons

Isopentenyl_phosphate_kinase.html

  1. \rightleftharpoons

Isotuberculosinol_synthase.html

  1. \rightleftharpoons

Italo_Jose_Dejter.html

  1. 𝒞 {\mathcal{C}}
  2. 𝒞 {\mathcal{C}}
  3. 𝒞 {\mathcal{C}}
  4. C 4 \vec{C}_{4}
  5. ± \pm
  6. C \vec{C}

Iterated_limit.html

  1. lim y q ( lim x p f ( x , y ) ) . \lim_{y\to q}\big(\lim_{x\to p}f(x,y)\big).\,
  2. lim ( x , y ) ( p , q ) f ( x , y ) , \lim_{(x,y)\to(p,q)}f(x,y),\,
  3. lim ( x , y ) ( p , q ) f ( x , y ) = lim x p lim y q f ( x , y ) = lim y q lim x p f ( x , y ) . \lim_{(x,y)\to(p,q)}f(x,y)=\lim_{x\to p}\lim_{y\to q}f(x,y)=\lim_{y\to q}\lim_% {x\to p}f(x,y).
  4. f ( x , y ) = x 2 x 2 + y 2 f(x,y)=\frac{x^{2}}{x^{2}+y^{2}}
  5. f ( x , y ) = x y x 2 + y 2 , f(x,y)=\frac{xy}{x^{2}+y^{2}},
  6. lim y 0 ( lim x 0 x 2 x 2 + y 2 ) = lim y 0 0 = 0 , \lim_{y\to 0}\left(\lim_{x\to 0}\frac{x^{2}}{x^{2}+y^{2}}\right)=\lim_{y\to 0}% 0=0,
  7. lim x 0 ( lim y 0 x 2 x 2 + y 2 ) = lim x 0 1 = 1. \lim_{x\to 0}\left(\lim_{y\to 0}\frac{x^{2}}{x^{2}+y^{2}}\right)=\lim_{x\to 0}% 1=1.
  8. lim x 0 ( lim y 0 x y x 2 + y 2 ) = lim x 0 0 = 0 \lim_{x\to 0}\left(\lim_{y\to 0}\frac{xy}{x^{2}+y^{2}}\right)=\lim_{x\to 0}0=0
  9. lim y 0 ( lim x 0 x y x 2 + y 2 ) = lim y 0 0 = 0 , \lim_{y\to 0}\left(\lim_{x\to 0}\frac{xy}{x^{2}+y^{2}}\right)=\lim_{y\to 0}0=0,
  10. lim ( ( x , y ) ( 0 , 0 ) : y = x ) x y x 2 + y 2 = lim x 0 x 2 x 2 + x 2 = 1 2 . \lim_{\Big((x,y)\to(0,0)\,:\,y=x\Big)}\frac{xy}{x^{2}+y^{2}}=\lim_{x\to 0}% \frac{x^{2}}{x^{2}+x^{2}}=\frac{1}{2}.
  11. lim ( x , y ) ( 0 , 0 ) x y x 2 + y 2 \lim_{(x,y)\to(0,0)}\frac{xy}{x^{2}+y^{2}}

J-structure.html

  1. j ( e + x ) + j ( e + j ( x ) ) = e j(e+x)+j(e+j(x))=e
  2. P ( x ) = - ( d j ) x - 1 . P(x)=-(dj)_{x}^{-1}.
  3. x * = Q ( x , e ) e - x x^{*}=Q(x,e)e-x
  4. j ( x ) = Q ( x ) - 1 x * . j(x)=Q(x)^{-1}x^{*}.
  5. x ( x 2 y ) = x 2 ( x y ) . x(x^{2}y)=x^{2}(xy).
  6. x - 1 ( x y ) = x ( x - 1 y ) . x^{-1}(xy)=x(x^{-1}y).
  7. ϕ a ( t , u ) = Q ( t a + u ( e - a ) ) . \phi_{a}(t,u)=Q(ta+u(e-a)).
  8. V a = { x V : ϕ a ( t , u ) x = t 2 x } V_{a}=\left\{{x\in V:\phi_{a}(t,u)x=t^{2}x}\right\}
  9. V a = { x V : ϕ a ( t , u ) x = t u x } V^{\prime}_{a}=\left\{{x\in V:\phi_{a}(t,u)x=tux}\right\}
  10. V e - a = { x V : ϕ a ( t , u ) x = u 2 x } V_{e-a}=\left\{{x\in V:\phi_{a}(t,u)x=u^{2}x}\right\}

Jacobson's_conjecture.html

  1. n J n = { 0 } . \bigcap_{n\in\mathbb{N}}J^{n}=\{0\}.

Jacobsthal_sum.html

  1. ϕ n ( a ) = m mod p ( m ( m n + a ) p ) \phi_{n}(a)=\sum_{m\bmod p}{\left({{m(m^{n}+a)}\atop{p}}\right)}

Jacques_Laskar.html

  1. 1 / 5 Myr - 1 1/5\;\,\text{Myr}^{-1}

Jagiellonian_Compromise.html

  1. j j
  2. N j N_{j}
  3. N j \sqrt{N_{j}}
  4. M M
  5. q q
  6. q = 1 2 ( 1 + i = 1 M N i i = 1 M N i ) q=\frac{1}{2}\cdot\left(1+\frac{\sqrt{\sum_{i=1}^{M}N_{i}}}{\sum_{i=1}^{M}% \sqrt{N_{i}}}\right)
  7. M M
  8. q * q_{*}
  9. M M
  10. q * 1 / 2 + 1 / π M q_{*}\approx 1/2+1/\sqrt{\pi M}

Jaynes–Cummings–Hubbard_model.html

  1. κ \kappa
  2. = 1 \hbar=1
  3. H = n = 1 N ω c a n a n + n = 1 N ω a σ n + σ n - + κ n = 1 N ( a n + 1 a n + a n a n + 1 ) + η n = 1 N ( a n σ n + + a n σ n - ) H=\sum_{n=1}^{N}\omega_{c}a_{n}^{\dagger}a_{n}+\sum_{n=1}^{N}\omega_{a}\sigma_% {n}^{+}\sigma_{n}^{-}+\kappa\sum_{n=1}^{N}\left(a_{n+1}^{\dagger}a_{n}+a_{n}^{% \dagger}a_{n+1}\right)+\eta\sum_{n=1}^{N}\left(a_{n}\sigma_{n}^{+}+a_{n}^{% \dagger}\sigma_{n}^{-}\right)
  4. σ n ± \sigma_{n}^{\pm}
  5. κ \kappa
  6. η \eta
  7. ω c \omega_{c}
  8. ω a \omega_{a}
  9. N ^ c n = 1 N a n a n \hat{N}_{c}\equiv\sum_{n=1}^{N}a_{n}^{\dagger}a_{n}
  10. N ^ a n = 1 N σ n + σ n - \hat{N}_{a}\equiv\sum_{n=1}^{N}\sigma_{n}^{+}\sigma_{n}^{-}
  11. [ H , N ^ c + N ^ a ] = 0 [H,\hat{N}_{c}+\hat{N}_{a}]=0

Jim_Stasheff.html

  1. A A_{\infty}

Johnson_SU_distribution.html

  1. Φ ( γ + δ sinh - 1 ( x - ξ λ ) ) \Phi(\gamma+\delta\sinh^{-1}(\frac{x-\xi}{\lambda}))
  2. ξ - λ exp δ - 2 2 sinh ( γ δ ) \xi-\lambda\exp{\frac{\delta^{-2}}{2}}\sinh(\frac{\gamma}{\delta})
  3. λ 2 2 ( exp ( δ - 2 ) - 1 ) ( exp ( δ - 2 ) cosh ( 2 γ δ ) + 1 ) \frac{\lambda^{2}}{2}(\exp{(\delta^{-2})}-1)(\exp{(\delta^{-2})}\cosh{(\frac{2% \gamma}{\delta})}+1)
  4. S U S_{U}
  5. z = γ + δ sinh - 1 ( x - ξ λ ) z=\gamma+\delta\sinh^{-1}(\frac{x-\xi}{\lambda})
  6. z 𝒩 ( 0 , 1 ) z\sim\mathcal{N}(0,1)
  7. S U S_{U}
  8. x = λ sinh ( Φ - 1 ( U ) - γ δ ) + ξ x=\lambda\sinh\left(\frac{\Phi^{-1}(U)-\gamma}{\delta}\right)+\xi

Jonckheere's_trend_test.html

  1. H 0 : θ 1 = θ 2 = = θ k H_{0}:\theta_{1}=\theta_{2}=\cdots=\theta_{k}
  2. H A : θ 1 H_{A}:\theta_{1}
  3. θ 2 \theta_{2}
  4. \cdots
  5. θ k \theta_{k}
  6. VAR ( S ) = 2 ( n 3 - t i 3 ) + 3 ( n 2 - t i 2 ) 18 \operatorname{VAR}(S)=\frac{2(n^{3}-\sum t^{3}_{i})+3(n^{2}-\sum t^{2}_{i})}{18}
  7. z = S c VAR ( S ) z=\frac{S_{c}}{\sqrt{\operatorname{VAR}(S)}}
  8. VAR ( S ) = 2 ( n 3 - t i 3 - u i 3 ) + 3 ( n 2 - t i 2 - u i 2 ) + 5 n 18 + ( t i 3 - 3 t i 2 + 2 n ) ( u i 3 - 3 u i 2 + 2 n ) 9 n ( n - 1 ) ( n - 2 ) + ( t i 2 - n ) ( u i 2 - n ) 2 n ( n - 1 ) \begin{aligned}\displaystyle\operatorname{VAR}(S)=&\displaystyle\frac{2\left(n% ^{3}-\sum t^{3}_{i}-\sum u^{3}_{i}\right)+3\left(n^{2}-\sum t^{2}_{i}-\sum u^{% 2}_{i}\right)+5n}{18}\\ &\displaystyle{}+\frac{\left(\sum t^{3}_{i}-3\sum t^{2}_{i}+2n\right)\left(% \sum u^{3}_{i}-3\sum u^{2}_{i}+2n\right)}{9n(n-1)(n-2)}\\ &\displaystyle{}+\frac{\left(\sum t^{2}_{i}-n\right)\left(\sum u^{2}_{i}-n% \right)}{2n(n-1)}\end{aligned}
  9. z = S VAR ( S ) z=\frac{S}{\sqrt{\operatorname{VAR}(S)}}
  10. As n = 12 , n 2 = 144 and n 3 = 1728. Also \,\text{As }n=12\,\text{, }n^{2}=144\,\text{ and }n^{3}=1728.\,\text{ Also}
  11. t i 2 = 16 t^{2}_{i}=16
  12. t i 3 = 24 t^{3}_{i}=24
  13. u i 2 = 48 u^{2}_{i}=48
  14. u i 3 = 192 u^{3}_{i}=192
  15. VAR ( S ) = 2 ( 1728 - 24 - 192 ) + 3 ( 144 - 16 - 48 ) + 60 18 + ( 24 - 48 + 24 ) ( 192 - 144 + 24 ) 9 × 12 × 11 × 10 + ( 16 - 12 ) ( 48 - 12 ) 2 × 12 × 11 = 185.212 \begin{aligned}\displaystyle\operatorname{VAR}(S)=&\displaystyle\frac{2(1728-2% 4-192)+3(144-16-48)+60}{18}\\ &\displaystyle+\frac{(24-48+24)(192-144+24)}{9\times 12\times 11\times 10}\\ &\displaystyle+\frac{(16-12)(48-12)}{2\times 12\times 11}\\ &\displaystyle=185.212\end{aligned}
  16. z = S VAR ( S ) = 40 185.212 = 2.939 z=\frac{S}{\sqrt{\operatorname{VAR}(S)}}=\frac{40}{\sqrt{185.212}}=2.939

Jordan_operator_algebra.html

  1. a 2 = a 2 , a 2 a 2 + b 2 . \displaystyle{\|a^{2}\|=\|a\|^{2},\,\,\,\|a^{2}\|\leq\|a^{2}+b^{2}\|.}

Joy's_Law_(computing).html

  1. S = 2 Y - 1984 , S=2^{Y-1984},
  2. S S
  3. Y Y

Jun_Maekawa.html

  1. 2 \scriptstyle\sqrt{2}

K-graph_C*-algebra.html

  1. Λ \Lambda
  2. r r
  3. s s
  4. d : Λ k d:\Lambda\to\mathbb{N}^{k}
  5. d ( λ ) = m + n d(\lambda)=m+n
  6. μ , ν Λ \mu,\nu\in\Lambda
  7. d ( μ ) = m , d ( ν ) = n d(\mu)=m,d(\nu)=n
  8. λ = μ ν \lambda=\mu\nu
  9. E E
  10. \mathbb{N}
  11. T k T^{k}
  12. f 1 , , f k {f_{1},...,f_{k}}
  13. d : T k k d:T^{k}\to\mathbb{N}^{k}
  14. d ( f 1 n 1 f k n k ) = ( n 1 , , n k ) d(f_{1}^{n_{1}}...f_{k}^{n_{k}})=(n_{1},\ldots,n_{k})
  15. Ω k = { ( m , n ) : m , n k , m n } \Omega_{k}=\{(m,n):m,n\in\mathbb{Z}^{k},m\leq n\}
  16. Ω k \Omega_{k}
  17. r ( m , n ) = ( m , m ) r(m,n)=(m,m)
  18. s ( m , n ) = ( n , n ) s(m,n)=(n,n)
  19. ( m , n ) ( n , p ) = ( m , p ) (m,n)(n,p)=(m,p)
  20. d ( m , n ) = n - m d(m,n)=n-m
  21. n k n\in\mathbb{N}^{k}
  22. Λ n = d - 1 ( n ) \Lambda^{n}=d^{-1}(n)
  23. Λ 0 = Obj ( Λ ) \Lambda^{0}=\operatorname{Obj}(\Lambda)
  24. v , w Λ 0 v,w\in\Lambda^{0}
  25. X Λ X\subseteq\Lambda
  26. v X = { λ X : r ( λ ) = v } vX=\{\lambda\in X:r(\lambda)=v\}
  27. X w = { λ X : s ( λ ) = w } Xw=\{\lambda\in X:s(\lambda)=w\}
  28. v X w = v X X w vXw=vX\cap Xw
  29. 0 < # v Λ n < 0<\#v\Lambda^{n}<\infty
  30. v Λ 0 v\in\Lambda^{0}
  31. n k n\in\mathbb{N}^{k}
  32. Λ \Lambda
  33. E = ( E 0 , E 1 , r , s , c ) E=(E^{0},E^{1},r,s,c)
  34. E 0 = Λ 0 E^{0}=\Lambda^{0}
  35. E 1 = i = 1 k Λ e i E^{1}=\cup_{i=1}^{k}\Lambda^{e_{i}}
  36. r , s r,s
  37. Λ \Lambda
  38. c : E 1 { 1 , , k } c:E^{1}\to\{1,\ldots,k\}
  39. c ( e ) = i c(e)=i
  40. e Λ e i e\in\Lambda^{e_{i}}
  41. e 1 , , e n e_{1},\ldots,e_{n}
  42. k \mathbb{N}^{k}
  43. Λ \Lambda
  44. e i + e j e_{i}+e_{j}
  45. i j i\neq j
  46. E E
  47. Λ \Lambda
  48. Λ \Lambda
  49. { s λ : λ Λ } \{s_{\lambda}:\lambda\in\Lambda\}
  50. s λ s μ = s λ μ s_{\lambda}s_{\mu}=s_{\lambda\mu}
  51. λ , μ , λ μ Λ \lambda,\mu,\lambda\mu\in\Lambda
  52. { s v : v Λ 0 } \{s_{v}:v\in\Lambda^{0}\}
  53. d ( μ ) = d ( ν ) d(\mu)=d(\nu)
  54. s μ * s ν = δ μ , ν s s ( μ ) s_{\mu}^{*}s_{\nu}=\delta_{\mu,\nu}s_{s(\mu)}
  55. s v = λ v Λ n s λ s λ * s_{v}=\sum_{\lambda\in v\Lambda^{n}}s_{\lambda}s_{\lambda}^{*}
  56. n k n\in\mathbb{N}^{k}
  57. v Λ 0 v\in\Lambda^{0}
  58. C * ( Λ ) C^{*}(\Lambda)
  59. Λ \Lambda

K-noid.html

  1. f ( z ) = 1 / ( z k - 1 ) 2 , g ( z ) = z k - 1 f(z)=1/(z^{k}-1)^{2},g(z)=z^{k-1}\,\!
  2. X ( z ) = 1 2 { ( - 1 k z ( z k - 1 ) ) [ \displaystyle X(z)=\frac{1}{2}\Re\Bigg\{\Big(\frac{-1}{kz(z^{k}-1)}\Big)\Big[
  3. Y ( z ) = 1 2 { ( i k z ( z k - 1 ) ) [ \displaystyle Y(z)=\frac{1}{2}\Re\Bigg\{\Big(\frac{i}{kz(z^{k}-1)}\Big)\Big[
  4. Z ( z ) = { 1 k - k z k } Z(z)=\Re\left\{\frac{1}{k-kz^{k}}\right\}
  5. F 1 2 ( a , b ; c ; z ) {}_{2}F_{1}(a,b;c;z)

Kalman's_conjecture.html

  1. d x d t = P x + q f ( e ) , e = r * x x R n , \frac{dx}{dt}=Px+qf(e),\quad e=r^{*}x\quad x\in R^{n},
  2. k 1 < f ( e ) < k 2 . k_{1}<f^{\prime}(e)<k_{2}.\,

Kansa_method.html

  1. Ω d \Omega\subseteq^{d}
  2. L u ( X ) = f ( X ) , X Ω , ( 1 ) Lu(X)=f(X),\quad X\in\Omega,\qquad(1)
  3. u ( X ) = g ( X ) , X Ω D , ( 2 ) u(X)=g(X),\quad X\in\partial\Omega_{D},\qquad(2)
  4. u ( X ) n = h ( X ) , X Ω N , ( 3 ) \frac{\partial u(X)}{\partial n}=h(X),\quad X\in\partial\Omega_{N},\qquad(3)
  5. Ω D , Ω N \partial\Omega_{D},\quad\partial\Omega_{N}
  6. Ω D Ω N = Ω \partial\Omega_{D}\cup\partial\Omega_{N}=\partial\Omega
  7. u ( X ) * = i = 1 N α i ϕ ( r i ) , ( 4 ) {{u(X)}^{*}}=\sum\limits_{i=1}^{N}\alpha_{i}\phi\left(r_{i}\right),\qquad(4)
  8. α i {{\alpha}_{i}}
  9. ϕ ( r i ) \phi\left(r_{i}\right)
  10. u ( X ) * = i = 1 N α i ϕ ( r i ) + k = 1 M α k + N γ k ( X ) , ( 5 ) {{u(X)}^{*}}=\sum\limits_{i=1}^{N}\alpha_{i}\phi\left(r_{i}\right)+\sum\limits% _{k=1}^{M}\alpha_{k+N}\gamma_{k}\left(X\right),\qquad(5)
  11. γ k ( X ) {{\gamma}_{k}}(X)
  12. 𝐀 α = b , ( 6 ) \mathbf{A}\alpha=b,\qquad(6)
  13. 𝐀 = ( L ( ϕ ) L ( γ ) ϕ γ ϕ n γ n γ 0 ) , 𝐛 = ( f g h 0 ) , ϕ = ϕ ( x i , x j ) , γ = γ k ( X i ) . ( 7 ) \mathbf{A}=\left(\begin{matrix}L(\phi)&L(\gamma)\\ \phi&\gamma\\ \frac{\partial{\phi}}{\partial n}&\frac{\partial{\gamma}}{\partial n}\\ {\gamma}&0\\ \end{matrix}\right),\quad\mathbf{b}=\left(\begin{matrix}f\\ g\\ h\\ 0\\ \end{matrix}\right),\quad\phi=\phi\left(x_{i},x_{j}\right),\quad\gamma=\gamma_% {k}\left(X_{i}\right).\qquad(7)
  14. α i \alpha_{i}

Kantowski-Sachs_metric.html

  1. × S 2 \mathbb{R}\times S^{2}
  2. d s 2 = - d t 2 + e 2 Λ t d z 2 + 1 Λ ( d θ 2 + sin 2 θ d ϕ 2 ) ds^{2}=-dt^{2}+e^{2\sqrt{\Lambda}t}dz^{2}+\frac{1}{\Lambda}(d\theta^{2}+\sin^{% 2}\theta d\phi^{2})
  3. × S O ( 3 ) \mathbb{R}\times SO(3)

Kellogg's_theorem.html

  1. k 2 k\geq 2
  2. C k C^{k}
  3. C k C^{k}
  4. C k , α C^{k,\alpha}
  5. C k , α C^{k,\alpha}

Kelly's_lemma.html

  1. j i π i q i j \displaystyle\sum_{j\neq i}\pi_{i}q^{\prime}_{ij}
  2. i j π i q i j = i j π j q j i = π j i j q j i = - π j q j j \sum_{i\neq j}\pi_{i}q_{ij}=\sum_{i\neq j}\pi_{j}q^{\prime}_{ji}=\pi_{j}\sum_{% i\neq j}q_{ji}=-\pi_{j}q_{jj}

Kemeny's_constant.html

  1. K = j π j m i j K=\sum_{j}\pi_{j}m_{ij}
  2. K i + w i 1 / w i = j P i j K j + 1 K_{i}+w_{i}\cdot 1/w_{i}=\sum_{j}P_{ij}K_{j}+1
  3. K i = j P i j K j . K_{i}=\sum_{j}P_{ij}K_{j}.

Kermack–McKendrick_theory.html

  1. d S d t = - λ S , \frac{dS}{dt}=-\lambda S,
  2. d i d t + d i d a = δ ( a ) λ S - γ ( a ) i , \frac{di}{dt}+\frac{di}{da}=\delta(a)\lambda S-\gamma(a)i,
  3. d R d t = 0 γ ( a ) i ( a , t ) d a , \frac{dR}{dt}=\int_{0}^{\infty}\gamma(a)i(a,t)\,da,
  4. δ ( a ) \delta(a)
  5. λ = 0 β ( a ) i ( a , t ) d a . \lambda=\int_{0}^{\infty}\beta(a)i(a,t)\,da.
  6. γ ( a ) \gamma(a)
  7. β ( a ) \beta(a)
  8. I ( t ) = 0 i ( a , t ) d a I(t)=\int_{0}^{\infty}i(a,t)\,da
  9. d S d t = b 0 + b S S + b I I + b R R - λ S - m S S , \frac{dS}{dt}=b_{0}+b_{S}S+b_{I}I+b_{R}R-\lambda S-m_{S}S,
  10. d i d t + d i d a = δ ( a ) λ ( S + σ R ) - γ ( a ) i - μ ( a ) i - m i ( a ) i , \frac{di}{dt}+\frac{di}{da}=\delta(a)\lambda(S+\sigma R)-\gamma(a)i-\mu(a)i-m_% {i}(a)i,
  11. I ( t ) = 0 i ( a , t ) d a I(t)=\int_{0}^{\infty}i(a,t)\,da
  12. d R d t = 0 γ ( a ) i ( a , t ) d a - σ λ R - m R R , \frac{dR}{dt}=\int_{0}^{\infty}\gamma(a)i(a,t)\,da-\sigma\lambda R-m_{R}R,
  13. b 0 b_{0}
  14. σ \sigma
  15. λ = 0 β ( a ) i ( a , t ) d a . \lambda=\int_{0}^{\infty}\beta(a)i(a,t)\,da.

Kernel_(image_processing).html

  1. [ 0 0 0 0 1 0 0 0 0 ] \begin{bmatrix}0&0&0\\ 0&1&0\\ 0&0&0\end{bmatrix}
  2. [ 1 0 - 1 0 0 0 - 1 0 1 ] \begin{bmatrix}\ \ 1&0&-1\\ \ \ 0&0&\ \ 0\\ -1&0&\ \ 1\end{bmatrix}
  3. [ 0 1 0 1 - 4 1 0 1 0 ] \begin{bmatrix}0&\ \ 1&0\\ 1&-4&1\\ 0&\ \ 1&0\end{bmatrix}
  4. [ - 1 - 1 - 1 - 1 8 - 1 - 1 - 1 - 1 ] \begin{bmatrix}-1&-1&-1\\ -1&\ \ 8&-1\\ -1&-1&-1\end{bmatrix}
  5. [ 0 - 1 0 - 1 5 - 1 0 - 1 0 ] \begin{bmatrix}\ \ 0&-1&\ \ 0\\ -1&\ \ 5&-1\\ \ \ 0&-1&\ \ 0\end{bmatrix}
  6. 1 9 [ 1 1 1 1 1 1 1 1 1 ] \frac{1}{9}\begin{bmatrix}1&1&1\\ 1&1&1\\ 1&1&1\end{bmatrix}
  7. 1 16 [ 1 2 1 2 4 2 1 2 1 ] \frac{1}{16}\begin{bmatrix}1&2&1\\ 2&4&2\\ 1&2&1\end{bmatrix}
  8. - 1 256 [ 1 4 6 4 1 4 16 24 16 4 6 24 - 476 24 6 4 16 24 16 4 1 4 6 4 1 ] \frac{-1}{256}\begin{bmatrix}1&4&\ \ 6&4&1\\ 4&16&\ \ 24&16&4\\ 6&24&-476&24&6\\ 4&16&\ \ 24&16&4\\ 1&4&\ \ 6&4&1\end{bmatrix}

Kervaire_semi-characteristic.html

  1. i = 0 n dim H 2 i ( M , R ) mod 2 \sum_{i=0}^{n}\dim H^{2i}(M,R)\bmod 2

Ketosteroid_monooxygenase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons
  4. \rightleftharpoons

Khinchin_integral.html

  1. f ( x ) = a x g ( t ) d t f(x)=\int_{a}^{x}g(t)\,dt
  2. k | y k - x k | < δ \sum_{k}\left|y_{k}-x_{k}\right|<\delta
  3. k | f ( y k ) - f ( x k ) | < ε . \sum_{k}|f(y_{k})-f(x_{k})|<\varepsilon.
  4. lim ε 0 μ ( E [ x - ε , x + ε ] ) 2 ε = 1 \lim_{\varepsilon\to 0}\frac{\mu(E\cap[x-\varepsilon,x+\varepsilon])}{2% \varepsilon}=1
  5. g - 1 ( [ y - ε , y + ε ] ) g^{-1}([y-\varepsilon,y+\varepsilon])
  6. f ( x ) - f ( x ) x - x \frac{f(x^{\prime})-f(x)}{x^{\prime}-x}

Kinematics_equations.html

  1. [ Z i ] = [ cos θ i - sin θ i 0 0 sin θ i cos θ i 0 0 0 0 1 d i 0 0 0 1 ] , [Z_{i}]=\begin{bmatrix}\cos\theta_{i}&-\sin\theta_{i}&0&0\\ \sin\theta_{i}&\cos\theta_{i}&0&0\\ 0&0&1&d_{i}\\ 0&0&0&1\end{bmatrix},
  2. [ X i ] = [ 1 0 0 a i , i + 1 0 cos α i , i + 1 - sin α i , i + 1 0 0 sin α i , i + 1 cos α i , i + 1 0 0 0 0 1 ] . [X_{i}]=\begin{bmatrix}1&0&0&a_{i,i+1}\\ 0&\cos\alpha_{i,i+1}&-\sin\alpha_{i,i+1}&0\\ 0&\sin\alpha_{i,i+1}&\cos\alpha_{i,i+1}&0\\ 0&0&0&1\end{bmatrix}.
  3. [ Z 1 ] [ X 1 ] [ Z 2 ] [ X 2 ] [ X n - 1 ] [ Z n ] = [ I ] . [Z_{1}][X_{1}][Z_{2}][X_{2}]\ldots[X_{n-1}][Z_{n}]=[I].\!
  4. [ T ] = [ Z 1 ] [ X 1 ] [ Z 2 ] [ X 2 ] [ X n - 1 ] [ Z n ] , [T]=[Z_{1}][X_{1}][Z_{2}][X_{2}]\ldots[X_{n-1}][Z_{n}],\!
  5. [ T ] = [ Z 1 , j ] [ X 1 , j ] [ Z 2 , j ] [ X 2 , j ] [ X n - 1 , j ] [ Z n , j ] , j = 1 , , m . [T]=[Z_{1,j}][X_{1,j}][Z_{2,j}][X_{2,j}]\ldots[X_{n-1,j}][Z_{n,j}],\quad j=1,% \ldots,m.\!

Kinetic_convex_hull.html

  1. < x <_{x}
  2. < y <_{y}
  3. < s <_{s}
  4. x [ a b ] x[ab]
  5. a b < x a b . n e x t ab<_{x}ab.next
  6. a b . n e x t ab.next
  7. a b ab
  8. a b ab
  9. a b . n e x t ab.next
  10. y l i [ a b ] yli[ab]
  11. a b < y c e ( a b ) ab<_{y}ce(ab)
  12. a b > y c e ( a b ) ab>_{y}ce(ab)
  13. a b ab
  14. b b
  15. c e ( a b ) ce(ab)
  16. c e ( a b ) ce(ab)
  17. a b ab
  18. a b ab
  19. y r i [ a b ] yri[ab]
  20. a b < y c e ( a b ) ab<_{y}ce(ab)
  21. a b > y c e ( a b ) ab>_{y}ce(ab)
  22. a b ab
  23. a a
  24. c e ( a b ) ce(ab)
  25. y t [ a b ] yt[ab]
  26. c e ( a b ) < y a b ce(ab)<_{y}ab
  27. a b ab
  28. a < s c e ( a b ) < s b a<_{s}ce(ab)<_{s}b
  29. c e ( a b ) < y a b ce(ab)<_{y}ab
  30. s r t [ a b ] srt[ab]
  31. a < s c e ( a b ) a<_{s}ce(ab)
  32. a b ab
  33. a < s c e ( a b ) < s b a<_{s}ce(ab)<_{s}b
  34. c e ( a b ) < y a b ce(ab)<_{y}ab
  35. s r t [ a b ] srt[ab]
  36. c e ( a b ) < s b ce(ab)<_{s}b
  37. a b ab
  38. a < s c e ( a b ) < s b a<_{s}ce(ab)<_{s}b
  39. c e ( a b ) < y a b ce(ab)<_{y}ab
  40. s l [ a b ] sl[ab]
  41. b < s c e ( a b ) b<_{s}ce(ab)
  42. a b ab
  43. b < s c e ( a b ) b<_{s}ce(ab)
  44. a b < y c e ( a b ) ab<_{y}ce(ab)
  45. s r [ a b ] sr[ab]
  46. c e ( a b ) < s a ce(ab)<_{s}a
  47. a b ab
  48. c e ( a b ) < s a ce(ab)<_{s}a
  49. a b < y c e ( a b ) ab<_{y}ce(ab)
  50. x [ a b ] x[ab]
  51. y l i [ a b ] yli[ab]
  52. y r i [ a b ] yri[ab]
  53. y t [ a b ] yt[ab]
  54. s r t [ a b ] srt[ab]
  55. s r t [ a b ] srt[ab]
  56. s l [ a b ] sl[ab]
  57. s r [ a b ] sr[ab]
  58. s l sl
  59. s r sr
  60. s l t slt
  61. s r t srt
  62. y t yt
  63. O ( n 2 + ϵ ) O(n^{2+\epsilon})
  64. ϵ > 0 \epsilon>0
  65. Ω ( n 2 ) \Omega(n^{2})

Kinetic_data_structure.html

  1. n n
  2. v v
  3. v = f ( t ) v=f(t)
  4. t t
  5. advance ( t ) \textrm{advance}(t)
  6. t t
  7. change ( v , f ( t ) ) \textrm{change}(v,f(t))
  8. v v
  9. f ( t ) f(t)
  10. t t
  11. t t
  12. t t
  13. t t
  14. t t
  15. t t
  16. n n
  17. O ( n ϵ ) O(n^{\epsilon})
  18. ϵ \epsilon
  19. n n
  20. O ( n polylog n ) O(n\textrm{polylog}n)
  21. O ( n 1 + ϵ ) O(n^{1+\epsilon})
  22. ϵ \epsilon
  23. t = t=\infty
  24. t = t=\infty
  25. f ( t ) = a t + b f(t)=at+b
  26. f ( t ) = i = 0 n a i t i f(t)=\sum_{i=0}^{n}a_{i}t^{i}
  27. n n
  28. O ( 1 ) O(1)

Kinetic_diameter.html

  1. O ( n ) O(n)
  2. O ( n ) O(n)
  3. O ( log 2 n ) O(\log^{2}n)
  4. O ( log 2 n ) O(\log^{2}n)
  5. O ( n 2 + ϵ ) O(n^{2+\epsilon})
  6. ϵ > 0 \epsilon>0
  7. Ω ( n 2 ) \Omega(n^{2})

Kinetic_Euclidean_minimum_spanning_tree.html

  1. O ( n * m ) O(n*m)

Kinetic_heap.html

  1. B B
  2. Y , Z Y,Z
  3. A A
  4. B , C B,C
  5. X X
  6. B B
  7. A A
  8. m a x max
  9. m i n min
  10. m i n - h e a p min-heap
  11. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  12. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  13. O ( n n log n ) O(n\sqrt{n\log n})
  14. Ω ( n log n ) \Omega(n\log n)

Kinetic_minimum_box.html

  1. O ( n 2 + ϵ ) O(n^{2+\epsilon})

Kinetic_minimum_spanning_tree.html

  1. O ( n 2 ) O(n^{2})
  2. O ( p n 1 2 log 3 2 n ) O(pn^{\frac{1}{2}}\log^{\frac{3}{2}}n)
  3. p p
  4. O ( n 19 12 log 3 2 n ) O(n^{\frac{19}{12}}\log^{\frac{3}{2}}n)

Kinetic_priority_queue.html

  1. m a x max
  2. m i n min
  3. m i n - q u e u e min-queue
  4. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  5. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  6. O ( n log 2 n ) O(n\log^{2}n)
  7. O ( n log 2 n ) O(n\log^{2}n)
  8. O ( n log n ) O(n\log n)
  9. O ( m n log 3 2 n ) O(m\sqrt{n}\log^{\frac{3}{2}}n)
  10. O ( m α ( n ) log 2 n ) O(m\alpha(n)\log^{2}n)
  11. O ( m log n log log n ) O(m\log n\log\log n)
  12. δ δ
  13. O ( n 2 log n ) O(n^{2}\log n)
  14. O ( λ δ ( n ) log n ) O(\lambda_{\delta}(n)\log n)
  15. α ( x ) \alpha(x)
  16. δ \delta
  17. δ \delta
  18. λ δ ( n ) \lambda_{\delta}(n)
  19. n n
  20. δ - \delta-
  21. n n
  22. m m

Kinetic_smallest_enclosing_disk.html

  1. O ( log 2 n ) O(\log^{2}n)
  2. Θ ( n ) \Theta(n)
  3. O ( n 3 + ϵ ) O(n^{3+\epsilon})
  4. ϵ > 0 \epsilon>0
  5. Ω ( n 2 ) \Omega(n^{2})
  6. O ( n 1 + ϵ ) O(n^{1+\epsilon})
  7. o ( n 3 + ϵ ) o(n^{3+\epsilon})
  8. O ( 1 / ϵ 5 / 2 ) O(1/\epsilon^{5/2})
  9. O ( ( n / ϵ ) log n ) O((n/\sqrt{\epsilon})\log n)

Kinetic_sorted_list.html

  1. O ( n log m ) O(n\log m)
  2. O ( n 2 m ) O(\frac{n^{2}}{m})
  3. m m
  4. O ( n m ) O(\frac{n}{m})
  5. O ( n 2 m 2 ) O(\frac{n^{2}}{m^{2}})
  6. O ( 1 ) O(1)
  7. O ( n 2 m 2 ) O(\frac{n^{2}}{m^{2}})
  8. O ( n 2 m ) O(\frac{n^{2}}{m})
  9. O ( n log m ) O(n\log m)

Kinetic_tournament.html

  1. O ( lg 2 n ) O(\lg^{2}n)
  2. < v a r > s <var>s

Kinetic_triangulation.html

  1. Ω ( n 2 ) \Omega(n^{2})
  2. Ω ( n 2 ) \Omega(n^{2})
  3. O ( n 3 ) O(n^{3})
  4. O ( 1 ) O(1)
  5. O ( n 2 β s + 2 ( n ) log 2 n ) O(n^{2}\beta_{s+2}(n)\log^{2}n)
  6. s s
  7. β s + 2 ( q ) = λ s + 2 ( q ) q \beta_{s+2}(q)=\frac{\lambda_{s+2}(q)}{q}
  8. λ s + 2 ( q ) \lambda_{s+2}(q)
  9. O ( n 2 2 log n log log n ) O(n^{2}2^{\sqrt{\log n\log\log n}})
  10. O ( lg n ) O(\lg n)

Kinetic_width.html

  1. a b ab
  2. a b ab
  3. O ( n ) O(n)
  4. O ( n ) O(n)
  5. O ( log 2 n ) O(\log^{2}n)
  6. O ( log 2 n ) O(\log^{2}n)
  7. O ( n 2 + ϵ ) O(n^{2+\epsilon})
  8. ϵ > 0 \epsilon>0
  9. Ω ( n 2 ) \Omega(n^{2})

Klein_graphs.html

  1. ( x - 7 ) ( x + 1 ) 7 ( x 2 - 7 ) 8 (x-7)(x+1)^{7}(x^{2}-7)^{8}

Kneser's_theorem_(combinatorics).html

  1. | A + B | | A + H | + | B + H | - | H | | A | + | B | - | H | . \begin{aligned}\displaystyle|A+B|&\displaystyle\geq|A+H|+|B+H|-|H|\\ &\displaystyle\geq|A|+|B|-|H|.\end{aligned}
  2. H = { g G : g + ( A + B ) = ( A + B ) } . H=\{g\in G:g+(A+B)=(A+B)\}.

Kodaira–Spencer_map.html

  1. δ : T 0 S H 1 ( X , T X ) \delta:T_{0}S\to H^{1}(X,TX)
  2. 𝒳 S \mathcal{X}\to S
  3. X = 𝒳 0 X=\mathcal{X}_{0}
  4. δ \delta
  5. T 𝒳 | X T 0 S 𝒪 X T\mathcal{X}|_{X}\to T_{0}S\otimes\mathcal{O}_{X}
  6. T X TX
  7. v v
  8. T 0 S T_{0}S
  9. δ ( v ) \delta(v)
  10. 𝒳 S = Spec ( [ t ] / t 2 ) \mathcal{X}\to S=\operatorname{Spec}(\mathbb{C}[t]/t^{2})
  11. H 1 ( X , T X ) H^{1}(X,TX)

Koecher–Vinberg_theorem.html

  1. C C
  2. a = 0 a=0
  3. a a
  4. - a -a
  5. C ¯ \overline{C}
  6. C C
  7. A A
  8. C * = { a A : b C a , b > 0 } C^{*}=\{a\in A\colon\forall b\in C\langle a,b\rangle>0\}
  9. C = C * C=C^{*}
  10. a , b C a,b\in C
  11. T : A A T\colon A\to A
  12. C C C\to C
  13. T ( a ) = b T(a)=b
  14. A A
  15. A + = { a 2 : a A } A_{+}=\{a^{2}\colon a\in A\}

Kolmogorov_automorphism.html

  1. ( X , , μ ) (X,\mathcal{B},\mu)
  2. T T
  3. T T
  4. 𝒦 \mathcal{K}\subset\mathcal{B}
  5. (1) 𝒦 T 𝒦 \mbox{(1) }~{}\mathcal{K}\subset T\mathcal{K}
  6. (2) n = 0 T n 𝒦 = \mbox{(2) }~{}\bigvee_{n=0}^{\infty}T^{n}\mathcal{K}=\mathcal{B}
  7. (3) n = 0 T - n 𝒦 = { X , } \mbox{(3) }~{}\bigcap_{n=0}^{\infty}T^{-n}\mathcal{K}=\{X,\varnothing\}
  8. \vee
  9. \cap
  10. { X , } \mathcal{B}\neq\{X,\varnothing\}
  11. 𝒦 T 𝒦 . \mathcal{K}\neq T\mathcal{K}.

Koopman–von_Neumann_classical_mechanics.html

  1. i t ρ ( x , p ) = L ^ ρ ( x , p ) i\frac{\partial}{\partial t}\rho(x,p)=\hat{L}\rho(x,p)
  2. L ^ = - i H ( x , p ) p x + i H ( x , p ) x p , \hat{L}=-i\frac{\partial H(x,p)}{\partial p}\frac{\partial}{\partial x}+i\frac% {\partial H(x,p)}{\partial x}\frac{\partial}{\partial p},
  3. H ( x , p ) H(x,p)
  4. i t ψ ( x , p ) = L ^ ψ ( x , p ) , i\frac{\partial}{\partial t}\psi(x,p)=\hat{L}\psi(x,p),
  5. t ψ ( x , p ) = [ - H ( x , p ) p x + H ( x , p ) x p ] ψ ( x , p ) , \frac{\partial}{\partial t}\psi(x,p)=\left[-\frac{\partial H(x,p)}{\partial p}% \frac{\partial}{\partial x}+\frac{\partial H(x,p)}{\partial x}\frac{\partial}{% \partial p}\right]\psi(x,p),
  6. t ψ * ( x , p ) = [ - H ( x , p ) p x + H ( x , p ) x p ] ψ * ( x , p ) . \frac{\partial}{\partial t}\psi^{*}(x,p)=\left[-\frac{\partial H(x,p)}{% \partial p}\frac{\partial}{\partial x}+\frac{\partial H(x,p)}{\partial x}\frac% {\partial}{\partial p}\right]\psi^{*}(x,p).
  7. ρ ( x , p ) = ψ * ( x , p ) ψ ( x , p ) \rho(x,p)=\psi^{*}(x,p)\psi(x,p)
  8. t ρ ( x , p ) = [ - H ( x , p ) p x + H ( x , p ) x p ] ρ ( x , p ) \frac{\partial}{\partial t}\rho(x,p)=\left[-\frac{\partial H(x,p)}{\partial p}% \frac{\partial}{\partial x}+\frac{\partial H(x,p)}{\partial x}\frac{\partial}{% \partial p}\right]\rho(x,p)
  9. ψ ( t ) | ψ ( t ) = 1 \langle\psi(t)|\psi(t)\rangle=1
  10. A ^ \hat{A}
  11. t t
  12. A ( t ) = Ψ ( t ) | A ^ | Ψ ( t ) . \langle A(t)\rangle=\langle\Psi(t)|\hat{A}|\Psi(t)\rangle.
  13. A ^ \hat{A}
  14. t t
  15. A A
  16. | A | Ψ ( t ) | 2 \left|\langle A|\Psi(t)\rangle\right|^{2}
  17. A ^ | A = A | A \hat{A}|A\rangle=A|A\rangle
  18. m d d t x = p , d d t p = - U ( x ) , m\frac{d}{dt}\langle x\rangle=\langle p\rangle,\qquad\frac{d}{dt}\langle p% \rangle=\langle-U^{\prime}(x)\rangle,
  19. m d d t Ψ ( t ) | x ^ | Ψ ( t ) = Ψ ( t ) | p ^ | Ψ ( t ) , d d t Ψ ( t ) | p ^ | Ψ ( t ) = Ψ ( t ) | - U ( x ^ ) | Ψ ( t ) . \begin{aligned}\displaystyle m\frac{d}{dt}\langle\Psi(t)|\hat{x}|\Psi(t)% \rangle&\displaystyle=\langle\Psi(t)|\hat{p}|\Psi(t)\rangle,\\ \displaystyle\frac{d}{dt}\langle\Psi(t)|\hat{p}|\Psi(t)\rangle&\displaystyle=% \langle\Psi(t)|-U^{\prime}(\hat{x})|\Psi(t)\rangle.\end{aligned}
  20. d Ψ / d t | x ^ | Ψ + Ψ | x ^ | d Ψ / d t = Ψ | p ^ / m | Ψ , d Ψ / d t | p ^ | Ψ + Ψ | p ^ | d Ψ / d t = Ψ | - U ( x ^ ) | Ψ , \begin{aligned}\displaystyle\langle d\Psi/dt|\hat{x}|\Psi\rangle+\langle\Psi|% \hat{x}|d\Psi/dt\rangle&\displaystyle=\langle\Psi|\hat{p}/m|\Psi\rangle,\\ \displaystyle\langle d\Psi/dt|\hat{p}|\Psi\rangle+\langle\Psi|\hat{p}|d\Psi/dt% \rangle&\displaystyle=\langle\Psi|-U^{\prime}(\hat{x})|\Psi\rangle,\end{aligned}
  21. i | d Ψ ( t ) / d t = L ^ | Ψ ( t ) i|d\Psi(t)/dt\rangle=\hat{L}|\Psi(t)\rangle
  22. i m Ψ ( t ) | [ L ^ , x ^ ] | Ψ ( t ) = Ψ ( t ) | p ^ | Ψ ( t ) , i Ψ ( t ) | [ L ^ , p ^ ] | Ψ ( t ) = - Ψ ( t ) | U ( x ^ ) | Ψ ( t ) . \begin{aligned}\displaystyle im\langle\Psi(t)|[\hat{L},\hat{x}]|\Psi(t)\rangle% &\displaystyle=\langle\Psi(t)|\hat{p}|\Psi(t)\rangle,\\ \displaystyle i\langle\Psi(t)|[\hat{L},\hat{p}]|\Psi(t)\rangle&\displaystyle=-% \langle\Psi(t)|U^{\prime}(\hat{x})|\Psi(t)\rangle.\end{aligned}
  23. L ^ \hat{L}
  24. i m [ L ^ , x ^ ] = p ^ , i [ L ^ , p ^ ] = - U ( x ^ ) . im[\hat{L},\hat{x}]=\hat{p},\qquad i[\hat{L},\hat{p}]=-U^{\prime}(\hat{x}).
  25. [ x ^ , p ^ ] = 0 [\hat{x},\hat{p}]=0
  26. L ^ \hat{L}
  27. L ^ = L ( x ^ , p ^ ) \hat{L}=L(\hat{x},\hat{p})
  28. i m [ L ( x ^ , p ^ ) , x ^ ] = 0 = p ^ im[L(\hat{x},\hat{p}),\hat{x}]=0=\hat{p}
  29. i [ L ( x ^ , p ^ ) , p ^ ] = 0 = - U ( x ^ ) i[L(\hat{x},\hat{p}),\hat{p}]=0=-U^{\prime}(\hat{x})
  30. λ ^ x \hat{\lambda}_{x}
  31. λ ^ p \hat{\lambda}_{p}
  32. [ x ^ , λ ^ x ] = [ p ^ , λ ^ p ] = i , [ x ^ , p ^ ] = [ x ^ , λ ^ p ] = [ p ^ , λ ^ x ] = [ λ ^ x , λ ^ p ] = 0. [\hat{x},\hat{\lambda}_{x}]=[\hat{p},\hat{\lambda}_{p}]=i,\quad[\hat{x},\hat{p% }]=[\hat{x},\hat{\lambda}_{p}]=[\hat{p},\hat{\lambda}_{x}]=[\hat{\lambda}_{x},% \hat{\lambda}_{p}]=0.
  33. L ^ \hat{L}
  34. L ^ = L ( x ^ , λ ^ x , p ^ , λ ^ p ) \hat{L}=L(\hat{x},\hat{\lambda}_{x},\hat{p},\hat{\lambda}_{p})
  35. m L λ x ( x , λ x , p , λ p ) = p , L λ p ( x , λ x , p , λ p ) = - U ( x ) . mL^{\prime}_{\lambda_{x}}(x,\lambda_{x},p,\lambda_{p})=p,\qquad L^{\prime}_{% \lambda_{p}}(x,\lambda_{x},p,\lambda_{p})=-U^{\prime}(x).
  36. | Ψ ( t ) |\Psi(t)\rangle
  37. i d d t | Ψ ( t ) = L ^ | Ψ ( t ) , L ^ = p ^ m λ ^ x - U ( x ^ ) λ ^ p . i\frac{d}{dt}|\Psi(t)\rangle=\hat{L}|\Psi(t)\rangle,\qquad\hat{L}=\frac{\hat{p% }}{m}\hat{\lambda}_{x}-U^{\prime}(\hat{x})\hat{\lambda}_{p}.
  38. x ^ \hat{x}
  39. p ^ \hat{p}
  40. x ^ | x p = x | x p , p ^ | x p = p | x p , A ( x ^ , p ^ ) | x p = A ( x , p ) | x p , \hat{x}|x\,p\rangle=x|x\,p\rangle,\quad\hat{p}|x\,p\rangle=p|x\,p\rangle,\quad A% (\hat{x},\hat{p})|x\,p\rangle=A(x,p)|x\,p\rangle,
  41. 1 = d x d p | x p x p | . 1=\int dxdp\,|x\,p\rangle\langle x\,p|.
  42. x p | λ ^ x | Ψ = - i x x p | Ψ , x p | λ ^ p | Ψ = - i p x p | Ψ . \langle x\,p|\hat{\lambda}_{x}|\Psi\rangle=-i\frac{\partial}{\partial x}% \langle x\,p|\Psi\rangle,\qquad\langle x\,p|\hat{\lambda}_{p}|\Psi\rangle=-i% \frac{\partial}{\partial p}\langle x\,p|\Psi\rangle.
  43. x p | \langle x\,p|
  44. [ t + p m x - U ( x ) p ] x p | Ψ ( t ) = 0. \left[\frac{\partial}{\partial t}+\frac{p}{m}\frac{\partial}{\partial x}-U^{% \prime}(x)\frac{\partial}{\partial p}\right]\langle x\,p|\Psi(t)\rangle=0.
  45. x , p | Ψ ( t ) \langle x,\,p|\Psi(t)\rangle
  46. x x
  47. p p
  48. t t
  49. ρ ( x , p ; t ) = | x , p | Ψ ( t ) | 2 \rho(x,p;t)=\left|\langle x,\,p|\Psi(t)\rangle\right|^{2}
  50. t ρ ( x , p ; t ) = Ψ ( t ) | x , p t x , p | Ψ ( t ) + x , p | Ψ ( t ) ( t x , p | Ψ ( t ) ) * \frac{\partial}{\partial t}\rho(x,p;t)=\langle\Psi(t)|x,\,p\rangle\frac{% \partial}{\partial t}\langle x,\,p|\Psi(t)\rangle+\langle x,\,p|\Psi(t)\rangle% \left(\frac{\partial}{\partial t}\langle x,\,p|\Psi(t)\rangle\right)^{*}
  51. [ t + p m x - U ( x ) p ] ρ ( x , p ; t ) = 0. \left[\frac{\partial}{\partial t}+\frac{p}{m}\frac{\partial}{\partial x}-U^{% \prime}(x)\frac{\partial}{\partial p}\right]\rho(x,p;t)=0.
  52. A = Ψ ( t ) | A ( x ^ , p ^ ) | Ψ ( t ) = d x d p Ψ ( t ) | x p A ( x , p ) x p | Ψ ( t ) = d x d p A ( x , p ) Ψ ( t ) | x p x p | Ψ ( t ) = d x d p A ( x , p ) ρ ( x , p ; t ) . \begin{aligned}\displaystyle\langle A\rangle&\displaystyle=\langle\Psi(t)|A(% \hat{x},\hat{p})|\Psi(t)\rangle=\int dxdp\,\langle\Psi(t)|x\,p\rangle A(x,p)% \langle x\,p|\Psi(t)\rangle\\ &\displaystyle=\int dxdp\,A(x,p)\langle\Psi(t)|x\,p\rangle\langle x\,p|\Psi(t)% \rangle=\int dxdp\,A(x,p)\rho(x,p;t).\end{aligned}
  53. A ( x , p ) A(x,p)
  54. [ x ^ , p ^ ] = 0 [\hat{x},\hat{p}]=0
  55. x ^ \hat{x}
  56. λ ^ p \hat{\lambda}_{p}
  57. x λ p x\lambda_{p}
  58. m d d t x = p , d d t p = - U ( x ) . m\frac{d}{dt}\langle x\rangle=\langle p\rangle,\qquad\frac{d}{dt}\langle p% \rangle=\langle-U^{\prime}(x)\rangle.
  59. m d d t Ψ ( t ) | x ^ | Ψ ( t ) = Ψ ( t ) | p ^ | Ψ ( t ) , d d t Ψ ( t ) | p ^ | Ψ ( t ) = Ψ ( t ) | - U ( x ^ ) | Ψ ( t ) . \begin{aligned}\displaystyle m\frac{d}{dt}\langle\Psi(t)|\hat{x}|\Psi(t)% \rangle&\displaystyle=\langle\Psi(t)|\hat{p}|\Psi(t)\rangle,\\ \displaystyle\frac{d}{dt}\langle\Psi(t)|\hat{p}|\Psi(t)\rangle&\displaystyle=% \langle\Psi(t)|-U^{\prime}(\hat{x})|\Psi(t)\rangle.\end{aligned}
  60. d Ψ / d t | x ^ | Ψ + Ψ | x ^ | d Ψ / d t = Ψ | p ^ / m | Ψ , d Ψ / d t | p ^ | Ψ + Ψ | p ^ | d Ψ / d t = Ψ | - U ( x ^ ) | Ψ , \begin{aligned}\displaystyle\langle d\Psi/dt|\hat{x}|\Psi\rangle+\langle\Psi|% \hat{x}|d\Psi/dt\rangle&\displaystyle=\langle\Psi|\hat{p}/m|\Psi\rangle,\\ \displaystyle\langle d\Psi/dt|\hat{p}|\Psi\rangle+\langle\Psi|\hat{p}|d\Psi/dt% \rangle&\displaystyle=\langle\Psi|-U^{\prime}(\hat{x})|\Psi\rangle,\end{aligned}
  61. i | d Ψ ( t ) / d t = H ^ | Ψ ( t ) , i\hbar|d\Psi(t)/dt\rangle=\hat{H}|\Psi(t)\rangle,
  62. \hbar
  63. H ^ \hat{H}
  64. i m [ H ^ , x ^ ] = p ^ , i [ H ^ , p ^ ] = - U ( x ^ ) . im[\hat{H},\hat{x}]=\hbar\hat{p},\qquad i[\hat{H},\hat{p}]=-\hbar U^{\prime}(% \hat{x}).
  65. [ x ^ , p ^ ] = i [\hat{x},\hat{p}]=i\hbar
  66. H ^ = H ( x ^ , p ^ ) \hat{H}=H(\hat{x},\hat{p})
  67. m H p ( x , p ) = p , H x ( x , p ) = U ( x ) , mH^{\prime}_{p}(x,p)=p,\qquad H^{\prime}_{x}(x,p)=U^{\prime}(x),
  68. H ^ = p ^ 2 2 m + U ( x ^ ) . \hat{H}=\frac{\hat{p}^{2}}{2m}+U(\hat{x}).
  69. [ x ^ , p ^ ] [\hat{x},\hat{p}]

Kostant's_convexity_theorem.html

  1. 𝔨 \mathfrak{k}
  2. 𝔱 \mathfrak{t}
  3. 𝔨 \mathfrak{k}
  4. 𝔱 \mathfrak{t}
  5. 𝔨 \mathfrak{k}
  6. 𝔱 \mathfrak{t}
  7. 𝔤 \mathfrak{g}
  8. 𝔨 \mathfrak{k}
  9. 𝔤 \mathfrak{g}
  10. 𝔭 \mathfrak{p}
  11. 𝔞 \mathfrak{a}
  12. 𝔭 \mathfrak{p}
  13. 𝔞 \mathfrak{a}
  14. 𝔭 \mathfrak{p}
  15. 𝔞 \mathfrak{a}
  16. 𝔞 \mathfrak{a}
  17. 𝔨 \mathfrak{k}
  18. P ( Y n + 1 ) 2 ( m - 1 m ) P ( Y n ) 2 . \displaystyle{\|P^{\perp}(Y_{n+1})\|^{2}\leq\left({m-1\over m}\right)\|P^{% \perp}(Y_{n})\|^{2}.}
  19. P ( Y n + 1 - Y n ) P ( Y n ) . \displaystyle{\|P(Y_{n+1}-Y_{n})\|\leq\|P^{\perp}(Y_{n})\|.}
  20. 𝔱 \mathfrak{t}
  21. 𝔱 \mathfrak{t}
  22. M 𝔱 * \displaystyle{M\rightarrow\mathfrak{t}^{*}}
  23. 𝔨 * \mathfrak{k}^{*}
  24. M 𝔨 * 𝔱 * . \displaystyle{M\rightarrow\mathfrak{k}^{*}\rightarrow\mathfrak{t}^{*}.}
  25. 𝔨 * \mathfrak{k}^{*}
  26. 𝔨 \mathfrak{k}
  27. Ad ( K ) X 𝔱 , \displaystyle{\mathrm{Ad}(K)\cdot X\rightarrow\mathfrak{t},}
  28. 𝔱 \mathfrak{t}
  29. 𝔞 \mathfrak{a}
  30. 𝔞 \mathfrak{a}
  31. Ad ( G ) X 𝔞 . \displaystyle{\mathrm{Ad}(G)\cdot X\rightarrow\mathfrak{a}.}

Koszul_cohomology.html

  1. p + 1 M q - 1 p M q p - 1 M q + 1 \wedge^{p+1}M_{q-1}\rightarrow\wedge^{p}M_{q}\rightarrow\wedge^{p-1}M_{q+1}

König's_theorem_(complex_analysis).html

  1. | x | < R |x|<R
  2. f ( x ) = n = 0 c n x n , c 0 0. f(x)=\sum_{n=0}^{\infty}c_{n}x^{n},\qquad c_{0}\neq 0.
  3. x = r x=r
  4. 0 < σ < 1 0<\sigma<1
  5. | r | < σ R |r|<\sigma R
  6. c n c n + 1 = r + o ( σ n + 1 ) . \frac{c_{n}}{c_{n+1}}=r+o(\sigma^{n+1}).
  7. lim n c n c n + 1 = r . \lim_{n\rightarrow\infty}\frac{c_{n}}{c_{n+1}}=r.
  8. f ( x ) C x - r = - C r 1 1 - x / r = - C r n = 0 [ x r ] n . f(x)\approx\frac{C}{x-r}=-\frac{C}{r}\,\frac{1}{1-x/r}=-\frac{C}{r}\sum_{n=0}^% {\infty}\left[\frac{x}{r}\right]^{n}.
  9. c n c n + 1 r \frac{c_{n}}{c_{n+1}}\approx r

Kronecker's_congruence.html

  1. Φ p ( x , y ) ( x - y p ) ( x p - y ) mod p , \Phi_{p}(x,y)\equiv(x-y^{p})(x^{p}-y)\bmod p,
  2. Φ n ( x , j ) = τ ( x - j ( τ ) ) \Phi_{n}(x,j)=\prod_{\tau}(x-j(\tau))

Krull–Akizuki_theorem.html

  1. B / I B/I
  2. L = K L=K
  3. 𝔭 i \mathfrak{p}_{i}
  4. K i K_{i}
  5. A / 𝔭 i A/{\mathfrak{p}_{i}}
  6. I i I_{i}
  7. B K K i B\to K\to K_{i}
  8. A / 𝔭 i B / I i K i A/{\mathfrak{p}_{i}}\subset B/{I_{i}}\subset K_{i}
  9. B / I i B/{I_{i}}
  10. B = B / I i B=\prod B/{I_{i}}
  11. 0 I B 0\neq I\subset B
  12. I A I\cap A
  13. I n = a n B A + a A I_{n}=a^{n}B\cap A+aA
  14. A / a A A/aA
  15. I n = I l I_{n}=I_{l}
  16. n l n\geq l
  17. a l B a l + 1 B + A . a^{l}B\subset a^{l+1}B+A.
  18. 𝔪 \mathfrak{m}
  19. 𝔪 n + 1 x - 1 A \mathfrak{m}^{n+1}\subset x^{-1}A
  20. a n + 1 x a n + 1 B A I n + 2 a^{n+1}x\in a^{n+1}B\cap A\subset I_{n+2}
  21. a n x a n + 1 B A + A . a^{n}x\in a^{n+1}B\cap A+A.
  22. n l n\geq l
  23. n > l n>l
  24. a n x I n + 1 a^{n}x\in I_{n+1}
  25. n - 1 n-1
  26. n = l n=l
  27. B / a B a l B / a l + 1 B ( a l + 1 B + A ) / a l + 1 B A / a l + 1 B A . B/{aB}\simeq a^{l}B/a^{l+1}B\subset(a^{l+1}B+A)/a^{l+1}B\simeq A/{a^{l+1}B\cap A}.
  28. B / a B B/{aB}
  29. B / a B B/{aB}
  30. \square

Kubo_formula.html

  1. H 0 H_{0}
  2. A ^ \hat{A}
  3. A ^ = 1 Z 0 T r [ ρ 0 ^ A ^ ] = 1 Z 0 n n | A ^ | n e - β E n \langle\hat{A}\rangle={1\over Z_{0}}Tr[\hat{\rho_{0}}\hat{A}]={1\over Z_{0}}% \sum_{n}\langle n|\hat{A}|n\rangle e^{-\beta E_{n}}
  4. ρ 0 ^ = e - β H ^ 0 = n | n n | e - β E n \hat{\rho_{0}}=e^{-\beta\hat{H}_{0}}=\sum_{n}|n\rangle\langle n|e^{-\beta E_{n}}
  5. Z 0 = T r [ ρ ^ 0 ] Z_{0}=Tr[\hat{\rho}_{0}]
  6. t = t 0 t=t_{0}
  7. H ^ ( t ) = H ^ 0 + V ^ ( t ) θ ( t - t 0 ) , \hat{H}(t)=\hat{H}_{0}+\hat{V}(t)\theta(t-t_{0}),
  8. θ ( t ) \theta(t)
  9. V ^ ( t ) \hat{V}(t)
  10. H ^ ( t ) \hat{H}(t)
  11. t - t 0 t-t_{0}
  12. E n ( t ) . E_{n}(t).
  13. ρ ^ ( t ) \hat{\rho}(t)
  14. Z ( t ) = T r [ ρ ^ ( t ) ] , Z(t)=Tr[\hat{\rho}(t)],
  15. A ^ = T r [ ρ ^ ( t ) A ^ ] / T r [ ρ ^ ( t ) ] . \langle\hat{A}\rangle=Tr[\hat{\rho}(t)\,\hat{A}]/Tr[\hat{\rho}(t)].
  16. | n ( t ) |n(t)\rangle
  17. i t | n ( t ) = H ^ ( t ) | n ( t ) , i\partial_{t}|n(t)\rangle=\hat{H}(t)|n(t)\rangle,
  18. V ^ ( t ) \hat{V}(t)
  19. | n ^ ( t ) , |\hat{n}(t)\rangle,
  20. | n ( t ) = e - i H ^ 0 t | n ^ ( t ) = e - i H ^ 0 t U ^ ( t , t 0 ) | n ^ ( t 0 ) , |n(t)\rangle=e^{-i\hat{H}_{0}t}|\hat{n}(t)\rangle=e^{-i\hat{H}_{0}t}\hat{U}(t,% t_{0})|\hat{n}(t_{0})\rangle,
  21. t 0 t_{0}
  22. | n ^ ( t 0 ) = e i H ^ 0 t 0 | n ( t 0 ) |\hat{n}(t_{0})\rangle=e^{i\hat{H}_{0}t_{0}}|n(t_{0})\rangle
  23. V ^ ( t ) \hat{V}(t)
  24. U ^ ( t , t 0 ) = 1 - i t 0 t d t V ^ ( t ) \hat{U}(t,t_{0})=1-i\int_{t_{0}}^{t}dt^{\prime}\hat{V}(t^{\prime})
  25. A ^ ( t ) \hat{A}(t)
  26. A ^ ( t ) = A ^ 0 - i t 0 t d t 1 Z 0 n e - β E n n ( t 0 ) | A ^ ( t ) V ^ ( t ) - V ^ ( t ) A ^ ( t ) | n ( t 0 ) = A ^ 0 - i t 0 t d t [ A ^ ( t ) , V ^ ( t ) ] 0 \begin{array}[]{rcl}\langle\hat{A}(t)\rangle&=&\langle\hat{A}\rangle_{0}-i\int% _{t_{0}}^{t}dt^{\prime}{1\over Z_{0}}\sum_{n}e^{-\beta E_{n}}\langle n(t_{0})|% \hat{A}(t)\hat{V}(t^{\prime})-\hat{V}(t^{\prime})\hat{A}(t)|n(t_{0})\rangle\\ &=&\langle\hat{A}\rangle_{0}-i\int_{t_{0}}^{t}dt^{\prime}\langle[\hat{A}(t),% \hat{V}(t^{\prime})]\rangle_{0}\end{array}
  27. 0 \langle\rangle_{0}
  28. H 0 . H_{0}.
  29. t > t 0 t>t_{0}

Küpfmüller's_uncertainty_principle.html

  1. Δ f Δ t k \Delta f\Delta t\geq k
  2. k k
  3. 1 1
  4. 1 2 \frac{1}{2}
  5. u ( t ) u(t)
  6. u ^ ( f ) \hat{u}(f)
  7. u ¯ ^ ( f ) \underline{\hat{u}}(f)
  8. u ^ ( f ) = u ¯ ^ ( f ) | Δ f \hat{u}(f)={{\underline{\hat{u}}(f)}}{{\Big|}_{\Delta f}}
  9. Δ f \Delta f
  10. g ^ ( f ) = rect ( f Δ f ) = χ [ - Δ f / 2 , Δ f / 2 ] ( f ) := { 1 | f | Δ f / 2 0 else \hat{g}(f)=\operatorname{rect}\left(\frac{f}{\Delta f}\right)=\chi_{[-\Delta f% /2,\Delta f/2]}(f):=\begin{cases}1&|f|\leq\Delta f/2\\ 0&\,\text{else}\end{cases}
  11. g ^ ( f ) u ^ ( f ) = ( g * u ) ( t ) \hat{g}(f)\cdot\hat{u}(f)=(g*u)(t)
  12. g ( t ) = 1 2 π - Δ f 2 Δ f 2 1 e j 2 π f t d f = 1 2 π Δ f si ( 2 π t Δ f 2 ) g(t)=\frac{1}{\sqrt{2\pi}}\int\limits_{-\frac{\Delta f}{2}}^{\frac{\Delta f}{2% }}1\cdot e^{j2\pi ft}df=\frac{1}{\sqrt{2\pi}}\cdot\Delta f\cdot\operatorname{% si}\left(\frac{2\pi t\cdot\Delta f}{2}\right)
  13. g ( t ) g(t)
  14. ± 1 Δ f \pm\frac{1}{\Delta f}
  15. Δ t \Delta t
  16. g ( t ) g(t)
  17. Δ t = 1 Δ f \Delta t=\frac{1}{\Delta f}
  18. Δ t \Delta t
  19. Δ f \Delta f
  20. 2 Δ f 2\cdot\Delta f
  21. k = 1 2 k=\frac{1}{2}
  22. k = 1 k=1

L-2-hydroxycarboxylate_dehydrogenase_(NAD+).html

  1. \rightleftharpoons

L-galactose_1-dehydrogenase.html

  1. \rightleftharpoons

L-glutamyl-(BtrI_acyl-carrier_protein)_decarboxylase.html

  1. \rightleftharpoons

L-olivosyl-oleandolide_3-O-methyltransferase.html

  1. \rightleftharpoons

L-proline_amide_hydrolase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

L-saccharopine_oxidase.html

  1. \rightleftharpoons

L-serine-phosphatidylethanolamine_phosphatidyltransferase.html

  1. \rightleftharpoons

L-sorbose_1-dehydrogenase.html

  1. \rightleftharpoons

L-threonine_kinase.html

  1. \rightleftharpoons

L-tryptophan—pyruvate_aminotransferase.html

  1. \rightleftharpoons

Lacto-N-biosidase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Lactotriaosylceramide_beta-1,4-galactosyltransferase.html

  1. \rightleftharpoons

Ladyzhenskaya's_inequality.html

  1. u L 4 C u L 2 1 / 2 u L 2 1 / 2 , \|u\|_{L^{4}}\leq C\|u\|_{L^{2}}^{1/2}\|\nabla u\|_{L^{2}}^{1/2},
  2. u L 4 C u L 2 1 / 4 u L 2 3 / 4 . \|u\|_{L^{4}}\leq C\|u\|_{L^{2}}^{1/4}\|\nabla u\|_{L^{2}}^{3/4}.
  3. u L p C u L q α u H 0 s 1 - α , \|u\|_{L^{p}}\leq C\|u\|_{L^{q}}^{\alpha}\|u\|_{H_{0}^{s}}^{1-\alpha},
  4. p > q 1 , s > n ( 1 2 - 1 p ) , and 1 p = α q + ( 1 - α ) ( 1 2 - s n ) . p>q\geq 1,s>n(\tfrac{1}{2}-\tfrac{1}{p}),\,\text{ and }\tfrac{1}{p}=\tfrac{% \alpha}{q}+(1-\alpha)(\tfrac{1}{2}-\tfrac{s}{n}).
  5. u L 2 r C r u L r 1 / 2 u L 2 1 / 2 . \|u\|_{L^{2r}}\leq Cr\|u\|_{L^{r}}^{1/2}\|\nabla u\|_{L^{2}}^{1/2}.
  6. u L 4 { C u L 2 , 1 / 2 u L 2 1 / 2 , n = 2 , C u L 2 , 1 / 4 u L 2 3 / 4 , n = 3. \|u\|_{L^{4}}\leq\begin{cases}C\|u\|_{L^{2,\infty}}^{1/2}\|\nabla u\|_{L^{2}}^% {1/2},&n=2,\\ C\|u\|_{L^{2,\infty}}^{1/4}\|\nabla u\|_{L^{2}}^{3/4},&n=3.\end{cases}

Laguerre_plane.html

  1. y = a x 2 + b x + c y=ax^{2}+bx+c
  2. y = a x 2 + b x + c y=ax^{2}+bx+c
  3. ( , a ) (\infty,a)
  4. 𝒫 := \R 2 ( { } × \R ) , \R , \mathcal{P}:=\R^{2}\cup(\{\infty\}\times\R),\ \infty\notin\R,
  5. 𝒵 := { { ( x , y ) \R 2 y = a x 2 + b x + c } { ( , a ) } a , b , c \R } \mathcal{Z}:=\{\{(x,y)\in\R^{2}\mid y=ax^{2}+bx+c\}\cup\{(\infty,a)\}\mid a,b,% c\in\R\}
  6. ( 𝒫 , 𝒵 , ) (\mathcal{P},\mathcal{Z},\in)
  7. \R 2 \R^{2}
  8. \R \R
  9. y = a x 2 + b x + c y=ax^{2}+bx+c
  10. ( , a ) (\infty,a)
  11. y = a x 2 + b x + c y=ax^{2}+bx+c
  12. A , B A,B
  13. A B A\parallel B
  14. A = B A=B
  15. A A
  16. B B
  17. ( a 1 , a 2 ) , ( b 1 , b 2 ) (a_{1},a_{2}),(b_{1},b_{2})
  18. a 1 = b 1 a_{1}=b_{1}
  19. \parallel
  20. ( 𝒫 , 𝒵 , ) (\mathcal{P},\mathcal{Z},\in)
  21. A , B , C A,B,C
  22. z z
  23. A , B , C A,B,C
  24. P P
  25. z z
  26. P z P^{\prime}\in z
  27. P P P\parallel P^{\prime}
  28. z z
  29. P z P\in z
  30. Q z Q\notin z
  31. P P
  32. z z^{\prime}
  33. P , Q P,Q
  34. z z = { P } z\cap z^{\prime}=\{P\}
  35. z z
  36. z z^{\prime}
  37. P P
  38. ( 𝒫 , 𝒵 , ) (\mathcal{P},\mathcal{Z},\in)
  39. \R 3 \R^{3}
  40. Φ \Phi
  41. ( 0 , 1 , 0 ) (0,1,0)
  42. u 2 + v 2 - v = 0 u^{2}+v^{2}-v=0
  43. ( 0 , 1 2 , . . ) (0,\tfrac{1}{2},..)
  44. r = 1 2 : r=\tfrac{1}{2}\ :
  45. Φ : ( x , z ) ( x 1 + x 2 , x 2 1 + x 2 , z 1 + x 2 ) = ( u , v , w ) . \Phi:\ (x,z)\rightarrow(\frac{x}{1+x^{2}},\frac{x^{2}}{1+x^{2}},\frac{z}{1+x^{% 2}})=(u,v,w)\ .
  46. ( 0 , 1 , a ) (0,1,a)
  47. Φ \Phi
  48. z = a x 2 + b x + c z=ax^{2}+bx+c
  49. w - a = b u + ( a - c ) ( v - 1 ) w-a=bu+(a-c)(v-1)
  50. ( 0 , 1 , a ) (0,1,a)
  51. z = a x 2 + a z=ax^{2}+a
  52. ( 0 , 1 , 0 ) (0,1,0)
  53. a 0 a\neq 0
  54. ( 0 , 1 , 0 ) (0,1,0)
  55. := ( 𝒫 , 𝒵 , ) \mathcal{L}:=(\mathcal{P},\mathcal{Z},\in)
  56. 𝒫 \mathcal{P}
  57. 𝒵 \mathcal{Z}
  58. A , B A,B
  59. A B A\parallel B
  60. A = B A=B
  61. A A
  62. B B
  63. \mathcal{L}
  64. A , B , C A,B,C
  65. z z
  66. A , B , C A,B,C
  67. P P
  68. z z
  69. P z P^{\prime}\in z
  70. P P P\parallel P^{\prime}
  71. z z
  72. P z P\in z
  73. Q z Q\notin z
  74. P P
  75. z z^{\prime}
  76. P , Q P,Q
  77. z z = { P } z\cap z^{\prime}=\{P\}
  78. z z
  79. z z^{\prime}
  80. P P
  81. A , B , C , D A,B,C,D
  82. z z
  83. A , B , C , D z A,B,C,D\in z
  84. \parallel
  85. \parallel
  86. P 𝒫 P\in\mathcal{P}
  87. P ¯ := { Q 𝒫 | P Q } \overline{P}:=\{Q\in\mathcal{P}\ |\ P\parallel Q\}
  88. P ¯ \overline{P}
  89. := ( 𝒫 , 𝒵 , ) \mathcal{L}:=(\mathcal{P},\mathcal{Z},\in)
  90. 𝒜 P := ( 𝒫 { P ¯ } , { z { P ¯ } | P z 𝒵 } { Q ¯ | Q 𝒫 { P ¯ } , ) \mathcal{A}_{P}:=(\mathcal{P}\setminus\{\overline{P}\},\{z\setminus\{\overline% {P}\}\ |\ P\in z\in\mathcal{Z}\}\cup\{\overline{Q}\ |\ Q\in\mathcal{P}% \setminus\{\overline{P}\},\in)
  91. 𝒜 \mathcal{A}_{\infty}
  92. \R 2 \R^{2}
  93. \parallel
  94. 𝒫 \mathcal{P}
  95. P P
  96. 𝒜 P \mathcal{A}_{P}
  97. 𝒫 := { A 1 , A 2 , B 1 , B 2 , C 1 , C 2 } , \mathcal{P}:=\{A_{1},A_{2},B_{1},B_{2},C_{1},C_{2}\}\ ,
  98. 𝒵 := { { A i , B j , C k } | i , j , k = 1 , 2 } , \mathcal{Z}:=\{\{A_{i},B_{j},C_{k}\}\ |\ i,j,k=1,2\}\ ,
  99. A 1 A 2 , B 1 B 2 , C 1 C 2 . A_{1}\parallel A_{2},\ B_{1}\parallel B_{2},\ C_{1}\parallel C_{2}\ .
  100. | 𝒫 | = 6 |\mathcal{P}|=6
  101. | 𝒵 | = 8 . |\mathcal{Z}|=8\ .
  102. | 𝒫 | < |\mathcal{P}|<\infty
  103. z 1 , z 2 z_{1},z_{2}
  104. P ¯ \overline{P}
  105. := ( 𝒫 , 𝒵 , ) \mathcal{L}:=(\mathcal{P},\mathcal{Z},\in)
  106. | z 1 | = | z 2 | = | P ¯ | + 1 |z_{1}|=|z_{2}|=|\overline{P}|+1
  107. := ( 𝒫 , 𝒵 , ) \mathcal{L}:=(\mathcal{P},\mathcal{Z},\in)
  108. z 𝒵 z\in\mathcal{Z}
  109. n := | z | - 1 n:=|z|-1
  110. \mathcal{L}
  111. := ( 𝒫 , 𝒵 , ) \mathcal{L}:=(\mathcal{P},\mathcal{Z},\in)
  112. n n
  113. 𝒜 P \mathcal{A}_{P}
  114. n , n\quad,
  115. | 𝒫 | = n 2 + n , |\mathcal{P}|=n^{2}+n,
  116. | 𝒵 | = n 3 . |\mathcal{Z}|=n^{3}.
  117. \R \R
  118. K K
  119. K K
  120. 𝒫 := K 2 \mathcal{P}:=K^{2}\cup
  121. ( { } × K ) , K (\{\infty\}\times K),\ \infty\notin K
  122. 𝒵 := { { ( x , y ) K 2 | y = a x 2 + b x + c } { ( , a ) } | a , b , c K } \mathcal{Z}:=\{\{(x,y)\in K^{2}\ |\ y=ax^{2}+bx+c\}\cup\{(\infty,a)\}\ |\ a,b,% c\in K\}
  123. ( K ) := ( 𝒫 , 𝒵 , ) \mathcal{L}(K):=(\mathcal{P},\mathcal{Z},\in)
  124. ( a 1 , a 2 ) ( b 1 , b 2 ) (a_{1},a_{2})\parallel(b_{1},b_{2})
  125. a 1 = b 1 a_{1}=b_{1}
  126. ( K ) \mathcal{L}(K)
  127. P 1 , , P 8 P_{1},\ldots,P_{8}
  128. ( K ) \mathcal{L}(K)
  129. ( K ) \mathcal{L}(K)
  130. ( K ) \mathcal{L}(K)
  131. K = G F ( 2 ) K=GF(2)
  132. { 0 , 1 } \{0,1\}
  133. ( K ) \mathcal{L}(K)
  134. K K

Laguerre–Pólya_class.html

  1. E ( z ) E(z)
  2. | E ( x + i y ) | = | E ( x - i y ) | |E(x+iy)|=|E(x-iy)|
  3. | E ( x + i y ) | |E(x+iy)|
  4. n 1 | z n | 2 \sum_{n}\frac{1}{|z_{n}|^{2}}
  5. z m e a + b z + c z 2 n ( 1 - z / z n ) exp ( z / z n ) z^{m}e^{a+bz+cz^{2}}\prod_{n}\left(1-z/z_{n}\right)\exp(z/z_{n})
  6. sin ( z ) , cos ( z ) , exp ( z ) , exp ( - z ) , and exp ( - z 2 ) . \sin(z),\cos(z),\exp(z),\exp(-z),\,\text{and }\exp(-z^{2}).
  7. sinh ( z ) , cosh ( z ) , and exp ( z 2 ) \sinh(z),\cosh(z),\,\text{and }\exp(z^{2})
  8. exp ( - z 2 ) = lim n ( 1 - z 2 / n ) n . \exp(-z^{2})=\lim_{n\to\infty}(1-z^{2}/n)^{n}.
  9. cos z = lim n ( ( 1 + i z / n ) n + ( 1 - i z / n ) n ) / 2 \cos z=\lim_{n\to\infty}((1+iz/n)^{n}+(1-iz/n)^{n})/2
  10. cos z = lim n m = 1 n ( 1 - z 2 ( ( m - 1 2 ) π ) 2 ) \cos z=\lim_{n\to\infty}\prod_{m=1}^{n}\left(1-\frac{z^{2}}{((m-\frac{1}{2})% \pi)^{2}}\right)
  11. cos z = lim n m = 1 n ( 1 - z ( ( m - 1 2 ) π ) 2 ) \cos\sqrt{z}=\lim_{n\to\infty}\prod_{m=1}^{n}\left(1-\frac{z}{((m-\frac{1}{2})% \pi)^{2}}\right)
  12. 1 / Γ ( z ) = lim n 1 n ! ( 1 - ( ln n ) z / n ) n m = 0 n ( z + m ) . 1/\Gamma(z)=\lim_{n\to\infty}\frac{1}{n!}(1-(\ln n)z/n)^{n}\prod_{m=0}^{n}(z+m).

Lake_discharge_problem.html

  1. y c = 2 3 E l a k e y_{c}={2\over 3}E_{lake}\,\!
  2. y c y_{c}\,
  3. E l a k e , E_{lake},
  4. q = y c 3 g q=\sqrt{y_{c}^{3}g}
  5. Q = q b Q=qb
  6. q q\,
  7. y c y_{c}\,
  8. g g\,
  9. Q Q\,
  10. b b\,
  11. v = 1.49 n R 2 3 S 1 2 v={1.49\over n}R^{2\over 3}S^{1\over 2}
  12. v v\,
  13. R R\,
  14. S S\,
  15. n n\,
  16. F r = v g y 0 Fr={v\over\sqrt{gy_{0}}}
  17. F r Fr\,
  18. v v\,
  19. g g\,
  20. y 0 y_{0}\,
  21. Q = v A Q=vA
  22. Q Q\,
  23. v v\,
  24. A A\,
  25. E = v 2 2 g + y E={v^{2}\over 2g}+y
  26. E E\,
  27. v v\,
  28. g g\,
  29. y y\,

Landauer_formula.html

  1. G ( μ ) = G 0 n T n ( μ ) , G(\mu)=G_{0}\sum_{n}T_{n}(\mu)\ ,
  2. G G
  3. G 0 = e 2 / ( π ) 7.75 × 10 - 5 Ω - 1 G_{0}=e^{2}/(\pi\hbar)\approx 7.75\times 10^{-5}\Omega^{-1}
  4. T n T_{n}
  5. E = μ E=\mu

Landweber_iteration.html

  1. y = A x y=Ax
  2. x = A - 1 y x=A^{-1}y
  3. min x A x - y 2 2 / 2 \min_{x}\|Ax-y\|_{2}^{2}/2
  4. x k + 1 = x k - ω A * ( A x k - y ) . x_{k+1}=x_{k}-\omega A^{*}(Ax_{k}-y).
  5. ω \omega
  6. 0 < ω < 2 / σ 1 2 0<\omega<2/\sigma_{1}^{2}
  7. σ 1 \sigma_{1}
  8. A A
  9. f ( x ) = A x - y 2 2 / 2 f(x)=\|Ax-y\|_{2}^{2}/2
  10. x k + 1 = x k - ω f ( x k ) x_{k+1}=x_{k}-\omega\nabla f(x_{k})
  11. x k + 1 = x k - τ f ( x k ) x_{k+1}=x_{k}-\tau\nabla f(x_{k})
  12. f ( x k ) f(x_{k})
  13. τ \tau
  14. 0 < τ < 2 / ( A 2 ) 0<\tau<2/(\|A\|^{2})
  15. \|\cdot\|
  16. min x C f ( x ) \min_{x\in C}f(x)
  17. x k + 1 = 𝒫 C ( x k - τ f ( x k ) ) x_{k+1}=\mathcal{P}_{C}(x_{k}-\tau\nabla f(x_{k}))
  18. 𝒫 \mathcal{P}
  19. 0 < τ < 2 / ( A 2 ) 0<\tau<2/(\|A\|^{2})

Laplacian_of_the_indicator.html

  1. δ δ
  2. δ δ
  3. δ δ
  4. δ δ
  5. δ δ
  6. x 2 𝟏 x > 0 \partial_{x}^{2}\mathbf{1}_{x>0}
  7. δ δ
  8. δ ( x ) - n x x 𝟏 x D , δ ( x ) x 2 𝟏 x D . \begin{aligned}\displaystyle\delta(x)&\displaystyle\to-n_{x}\cdot\nabla_{x}% \mathbf{1}_{x\in D},\\ \displaystyle\delta^{\prime}(x)&\displaystyle\to\nabla_{x}^{2}\mathbf{1}_{x\in D% }.\end{aligned}
  9. δ δ
  10. δ δ
  11. δ δ
  12. δ δ
  13. a b f ( x ) x d x = lim x b f ( x ) - lim x a f ( x ) , \int_{a}^{b}\frac{\partial f(x)}{\partial x}\,dx=\underset{x\nearrow b}{\lim}f% (x)-\underset{x\searrow a}{\lim}f(x),
  14. δ δ
  15. 𝐑 d x 2 ( 𝟏 x D f ( x ) ) d x = 0. \int_{\mathbf{R}^{d}}\nabla_{x}^{2}\left(\mathbf{1}_{x\in D}\,f(x)\right)\;dx=0.
  16. 𝐑 d x 2 𝟏 x D f ( x ) d x + 𝐑 d 𝟏 x D x 2 f ( x ) d x = - 2 𝐑 d x 𝟏 x D x f ( x ) d x . \int_{\mathbf{R}^{d}}\,\nabla_{x}^{2}\mathbf{1}_{x\in D}\,f(x)\;dx+\int_{% \mathbf{R}^{d}}\mathbf{1}_{x\in D}\,\nabla_{x}^{2}f(x)\;dx=-2\int_{\mathbf{R}^% {d}}\nabla_{x}\mathbf{1}_{x\in D}\cdot\nabla_{x}f(x)\;dx.
  17. D lim α β n β α f ( α ) d β = - 𝐑 d x 𝟏 x D x f ( x ) d x . \oint_{\partial D}\,\underset{\alpha\to\beta}{\lim}n_{\beta}\cdot\nabla_{% \alpha}f(\alpha)\;d\beta=-\displaystyle\int_{\mathbf{R}^{d}}\nabla_{x}\mathbf{% 1}_{x\in D}\cdot\nabla_{x}f(x)\;dx.
  18. D g ( β ) d β = - 𝐑 d x 𝟏 x D n x g ( x ) d x . \oint_{\partial D}\,g(\beta)\;d\beta=-\int_{\mathbf{R}^{d}}\,\nabla_{x}\mathbf% {1}_{x\in D}\,\cdot\,n_{x}\,g(x)\;dx.
  19. 0 I ε ( x ) \displaystyle 0\leq I_{\varepsilon}(x)
  20. - lim ε 0 𝐑 d f ( x ) n x x I ε ( x ) d x \displaystyle-\underset{\varepsilon\searrow 0}{\lim}\int_{\mathbf{R}^{d}}\,f(x% )\,n_{x}\cdot\nabla_{x}I_{\varepsilon}(x)\;dx
  21. δ δ

Lattice_(module).html

  1. N W = K N ; W N = W M . N\mapsto W=K\cdot N;W\mapsto N=W\cap M.\,

LCP_array.html

  1. | |
  2. 𝒪 ( n ) \mathcal{O}(n)
  3. 𝒪 ( n ) \mathcal{O}(n)
  4. 𝒪 ( n ) \mathcal{O}(n)
  5. 𝒪 ( n ) \mathcal{O}(n)
  6. A A
  7. S = s 1 , s 2 , s n $ S=s_{1},s_{2},...s_{n}\$
  8. lcp ( v , w ) \operatorname{lcp}(v,w)
  9. v v
  10. w w
  11. S [ i , j ] S[i,j]
  12. S S
  13. i i
  14. j j
  15. H [ 1 , n ] H[1,n]
  16. n n
  17. H [ 1 ] H[1]
  18. H [ i ] = lcp ( S [ A [ i - 1 ] , n ] , S [ A [ i ] , n ] ) H[i]=\operatorname{lcp}(S[A[i-1],n],S[A[i],n])
  19. 1 < i n 1<i\leq n
  20. H [ i ] H[i]
  21. i i
  22. S = b a n a n a $ S=banana\$
  23. A A
  24. H H
  25. \bot
  26. H [ 4 ] = 3 H[4]=3
  27. a n a ana
  28. A [ 3 ] = S [ 4 , 7 ] = a n a $ A[3]=S[4,7]=ana\$
  29. A [ 4 ] = S [ 2 , 7 ] = a n a n a $ A[4]=S[2,7]=anana\$
  30. H [ 1 ] = H[1]=\bot
  31. O ( n log n ) O(n\log n)
  32. O ( n ) O(n)
  33. O ( n ) O(n)
  34. 13 n 13n
  35. 9 n 9n
  36. 9 n 9n
  37. Φ \Phi
  38. O ( n 2 ) O(n^{2})
  39. P P
  40. m m
  41. S S
  42. n n
  43. O ( m log n ) O(m\log n)
  44. O ( m + log n ) O(m+\log n)
  45. O ( m ) O(m)
  46. O ( n ) O(n)
  47. S S
  48. n n
  49. Θ ( n ) \Theta(n)
  50. A A
  51. v m a x v_{max}
  52. i i
  53. v m a x v_{max}
  54. S [ A [ i ] , A [ i ] + v m a x - 1 ] S[A[i],A[i]+v_{max}-1]
  55. A A
  56. H H
  57. S = s 1 , s 2 , s n $ S=s_{1},s_{2},...s_{n}\$
  58. n + 1 n+1
  59. S T ST
  60. O ( n ) O(n)
  61. S T i ST_{i}
  62. 0 i n 0\leq i\leq n
  63. d ( v ) d(v)
  64. S T i ST_{i}
  65. v v
  66. d ( v ) = H [ i + 1 ] d(v)=H[i+1]
  67. a $ a\$
  68. a n a $ ana\$
  69. a n a n a $ anana\$
  70. b a n a n a $ banana\$
  71. S = b a n a n a $ S=banana\$
  72. n a $ na\$
  73. S T 0 ST_{0}
  74. A [ i + 1 ] A[i+1]
  75. S T i ST_{i}
  76. A [ i ] A[i]
  77. v v
  78. d ( v ) H [ i + 1 ] d(v)\leq H[i+1]
  79. d ( v ) = H [ i + 1 ] d(v)=H[i+1]
  80. v v
  81. A [ i ] A[i]
  82. A [ i + 1 ] A[i+1]
  83. A [ i + 1 ] A[i+1]
  84. x x
  85. v v
  86. ( v , x ) (v,x)
  87. S [ A [ i + 1 ] + H [ i + 1 ] , n ] S[A[i+1]+H[i+1],n]
  88. A [ i + 1 ] A[i+1]
  89. v v
  90. S T i + 1 ST_{i+1}
  91. d ( v ) < H [ i + 1 ] d(v)<H[i+1]
  92. v v
  93. A [ i ] A[i]
  94. A [ i + 1 ] A[i+1]
  95. v v
  96. w w
  97. v v
  98. S T i ST_{i}
  99. ( v , w ) (v,w)
  100. y y
  101. ( v , y ) (v,y)
  102. S [ A [ i ] + d ( v ) , A [ i ] + H [ i + 1 ] - 1 ] S[A[i]+d(v),A[i]+H[i+1]-1]
  103. A [ i ] A[i]
  104. A [ i + 1 ] A[i+1]
  105. y y
  106. A [ i ] A[i]
  107. A [ i + 1 ] A[i+1]
  108. w w
  109. y y
  110. ( y , w ) (y,w)
  111. S [ A [ i ] + H [ i + 1 ] , A [ i ] + d ( w ) - 1 ] S[A[i]+H[i+1],A[i]+d(w)-1]
  112. ( v , w ) (v,w)
  113. ( v , y ) (v,y)
  114. A [ i + 1 ] A[i+1]
  115. x x
  116. y y
  117. ( y , x ) (y,x)
  118. S [ A [ i + 1 ] + H [ i + 1 ] , n ] S[A[i+1]+H[i+1],n]
  119. A [ i + 1 ] A[i+1]
  120. v v
  121. S T i + 1 ST_{i+1}
  122. O ( n ) O(n)
  123. i i
  124. S T i ST_{i}
  125. v v
  126. A [ i + 1 ] A[i+1]
  127. j > i j>i
  128. 2 n 2n
  129. H H
  130. A A
  131. A - 1 A^{-1}
  132. A [ i ] = j A - 1 [ j ] = i A[i]=j\Leftrightarrow A^{-1}[j]=i
  133. S [ j , n ] S[j,n]
  134. j j
  135. S S
  136. A - 1 [ j ] A^{-1}[j]
  137. A A
  138. H H
  139. O ( 1 ) O(1)
  140. S [ i , n ] S[i,n]
  141. S [ j , n ] S[j,n]
  142. i i
  143. A - 1 [ i ] A^{-1}[i]
  144. j j
  145. A - 1 [ j ] A^{-1}[j]
  146. H [ A - 1 [ i ] + 1 , A - 1 [ j ] ] H[A^{-1}[i]+1,A^{-1}[j]]
  147. H H
  148. S S
  149. n n
  150. i , j i,j
  151. S S
  152. A - 1 [ i ] < A - 1 [ j ] A^{-1}[i]<A^{-1}[j]
  153. S [ i , n ] S[i,n]
  154. S [ j , n ] S[j,n]
  155. LCP ( i , j ) = H [ RMQ H ( A - 1 [ i ] + 1 , A - 1 [ j ] ) ] \operatorname{LCP}(i,j)=H[\operatorname{RMQ}_{H}(A^{-1}[i]+1,A^{-1}[j])]

Leading_and_lagging_current.html

  1. 1 / j ω C 1/jωC
  2. j ω L jωL
  3. 2 π f 2πf
  4. A \ang θ . A\ang\theta.
  5. \ang θ \ang\theta
  6. ( cos θ , sin θ ) (\cos\theta,\sin\theta)\,
  7. cos θ + j sin θ = e j θ , \cos\theta+j\sin\theta=e^{j\theta},\,
  8. A \ang θ = A \ang δ - β A\ang\theta=A\ang\delta-\beta
  9. β \beta
  10. δ \delta
  11. A \ang θ = A \ang δ - β A\ang\theta=A\ang\delta-\beta
  12. β \beta
  13. δ \delta

Lean_CFP_driven.html

  1. α \alpha
  2. F F = α U U m 1 - U U m + 1 FF=\alpha\frac{UU_{m}}{1-UU_{m}}+1
  3. F F FF
  4. α \alpha
  5. U U m UU_{m}
  6. F F = C T R P T FF=\frac{CT}{RPT}

Lecithinase_C.html

  1. \rightleftharpoons

Left_and_right_derivative.html

  1. + f ( a ) := lim < m t p l > x a + x I f ( x ) - f ( a ) x - a \partial_{+}f(a):=\lim_{<}mtpl>{{\scriptstyle x\to a+\atop\scriptstyle x\in I}% }\frac{f(x)-f(a)}{x-a}
  2. - f ( a ) := lim < m t p l > x a - x I f ( x ) - f ( a ) x - a . \partial_{-}f(a):=\lim_{<}mtpl>{{\scriptstyle x\to a-\atop\scriptstyle x\in I}% }\frac{f(x)-f(a)}{x-a}.
  3. c = inf { x ( a , b ] | f ( x ) - f ( a ) | > ε ( x - a ) } . c=\inf\{\,x\in(a,b]\mid|f(x)-f(a)|>\varepsilon(x-a)\,\}.
  4. | f ( x ) - f ( a ) | | f ( x ) - f ( c ) | + | f ( c ) - f ( a ) | ε ( x - a ) |f(x)-f(a)|\leq|f(x)-f(c)|+|f(c)-f(a)|\leq\varepsilon(x-a)
  5. f x g = f x g f\stackrel{\leftarrow}{\partial}_{x}g=\frac{\partial f}{\partial x}\cdot g
  6. f x g = f g x . f\stackrel{\rightarrow}{\partial}_{x}g=f\cdot\frac{\partial g}{\partial x}.

Lehmer's_totient_problem.html

  1. O ( X 1 / 2 ( log X ) 3 / 4 ) O\left({X^{1/2}(\log X)^{3/4}}\right)

Leighton_relationship.html

  1. [ O 3 ] = J 1 [ N O 2 ] / k 3 [ N O ] [O_{3}]=J_{1}[NO_{2}]/k_{3}[NO]

Lelong_number.html

  1. lim inf z x ϕ ( z ) log | z - x | . \liminf_{z\rightarrow x}\frac{\phi(z)}{\log|z-x|}.

Let_expression.html

  1. a = 3 a=3
  2. b = 4 b=4
  3. a 2 + b 2 \sqrt{a^{2}+b^{2}}
  4. a 2 + b 2 \sqrt{a^{2}+b^{2}}
  5. a = 3 a=3
  6. b = 4 b=4
  7. ( λ f . z ) ( λ x . y ) (\lambda f.z)\ (\lambda x.y)
  8. f f
  9. f x = y f\ x=y
  10. z z
  11. let f x = y in z \operatorname{let}f\ x=y\operatorname{in}z
  12. ( x E and F ) let x : E in F (\exists xE\and F)\iff\operatorname{let}x:E\operatorname{in}F
  13. x FV ( E ) and x FV ( F ) let x : G in E F = E ( let x : G in F ) x\not\in\operatorname{FV}(E)\and x\in\operatorname{FV}(F)\implies\operatorname% {let}x:G\operatorname{in}E\ F=E\ (\operatorname{let}x:G\operatorname{in}F)
  14. x FV ( E ) and x FV ( F ) let x : G in E F = ( let x : G in E ) F x\in\operatorname{FV}(E)\and x\not\in\operatorname{FV}(F)\implies\operatorname% {let}x:G\operatorname{in}E\ F=(\operatorname{let}x:G\operatorname{in}E)\ F
  15. x FV ( E ) and x FV ( F ) let x : G in E F = E F x\not\in\operatorname{FV}(E)\and x\not\in\operatorname{FV}(F)\implies% \operatorname{let}x:G\operatorname{in}E\ F=E\ F
  16. x FV ( y ) ( let x : x = y in z ) = z [ x := y ] = ( λ x . z ) y x\not\in\operatorname{FV}(y)\implies(\operatorname{let}x:x=y\operatorname{in}z% )=z[x:=y]=(\lambda x.z)\ y
  17. x : x:
  18. x FV ( y ) ( let x = y in z ) = z [ x := y ] = ( λ x . z ) y x\not\in\operatorname{FV}(y)\implies(\operatorname{let}x=y\operatorname{in}z)=% z[x:=y]=(\lambda x.z)\ y
  19. x FV ( L ) x\not\in\operatorname{FV}(L)
  20. L ( let x : x = y in z ) L\ (\operatorname{let}x:x=y\operatorname{in}z)
  21. ( let x : x = y in L z ) \iff(\operatorname{let}x:x=y\operatorname{in}L\ z)
  22. x = y and L z \iff x=y\and L\ z
  23. x = y and ( L z ) [ x := y ] \iff x=y\and(L\ z)[x:=y]
  24. x = y and ( L [ x := y ] z [ x := y ] ) \iff x=y\and(L[x:=y]\ z[x:=y])
  25. x = y and L z [ x := y ] \iff x=y\and L\ z[x:=y]
  26. L z [ x := y ] \implies L\ z[x:=y]
  27. L let x : x = y in z L z [ x := y ] L\operatorname{let}x:x=y\operatorname{in}z\implies L\ z[x:=y]
  28. L X = ( X = K ) L\ X=(X=K)
  29. ( let x : x = y in z ) = K z [ x := y ] = K (\operatorname{let}x:x=y\operatorname{in}z)=K\implies z[x:=y]=K
  30. let x : x = y in z = z [ x := y ] \operatorname{let}x:x=y\operatorname{in}z=z[x:=y]
  31. ( λ x . z ) y = z [ x := y ] (\lambda x.z)\ y=z[x:=y]
  32. x FV ( y ) x\not\in\operatorname{FV}(y)
  33. x FV ( y ) let x : x = y in z = ( λ x . z ) y x\not\in\operatorname{FV}(y)\implies\operatorname{let}x:x=y\operatorname{in}z=% (\lambda x.z)\ y
  34. x FV ( y ) ( let x = y in z ) = z [ x := y ] = ( λ x . z ) y x\not\in\operatorname{FV}(y)\implies(\operatorname{let}x=y\operatorname{in}z)=% z[x:=y]=(\lambda x.z)\ y
  35. E = ¬ E=\neg
  36. x FV ( E ) and x FV ( F ) let x : G in E F = E ( let x : G in F ) x\not\in\operatorname{FV}(E)\and x\in\operatorname{FV}(F)\implies\operatorname% {let}x:G\operatorname{in}E\ F=E\ (\operatorname{let}x:G\operatorname{in}F)
  37. let x : G in ¬ F = ¬ ( let x : G in F ) \operatorname{let}x:G\operatorname{in}\neg F=\neg\ (\operatorname{let}x:G% \operatorname{in}F)
  38. ( x E and F ) let x : E in F (\exists xE\and F)\iff\operatorname{let}x:E\operatorname{in}F
  39. ( x G and ¬ F ) = ¬ ( x G and F ) (\exists xG\and\neg F)=\neg\ (\exists xG\and F)
  40. = ( x ¬ G ¬ F ) =(\exists x\neg G\neg F)
  41. x FV ( E ) and x FV ( F ) ( let x : G in E F E ( let x : G in F ) ) x\not\in\operatorname{FV}(E)\and x\in\operatorname{FV}(F)\implies(% \operatorname{let}x:G\operatorname{in}E\ F\to E\ (\operatorname{let}x:G% \operatorname{in}F))
  42. ( x E and F ) let x : E in F (\exists xE\and F)\iff\operatorname{let}x:E\operatorname{in}F
  43. ( v w x E and F ) let v , , w , x : E in F (\exists v\cdots\exists w\exists xE\and F)\iff\operatorname{let}v,\ldots,w,x:E% \operatorname{in}F
  44. x F V ( E ) ( v w x E and F ) ( v w ( E and x F ) ) x\not\in FV(E)\implies(\exists v\cdots\exists w\exists xE\and F)\iff(\exists v% \cdots\exists w(E\and\exists xF))
  45. x F V ( E ) ( let v , , w , x : E and F in L let v , , w : E in let x : F in L ) x\not\in FV(E)\implies(\operatorname{let}v,\ldots,w,x:E\and F\operatorname{in}% L\equiv\operatorname{let}v,\ldots,w:E\operatorname{in}\operatorname{let}x:F% \operatorname{in}L)
  46. f x = y f = λ x . y f\ x=y\equiv f=\lambda x.y
  47. f F V ( E ) ( let f : f = E in L ( λ f . L ) E ) f\not\in FV(E)\implies(\operatorname{let}f:f=E\operatorname{in}L\equiv(\lambda f% .L)\ E)
  48. x F V ( E ) ( let v , , w , x : E and F in L let v , , w : E in let x : F in L ) x\not\in FV(E)\implies(\operatorname{let}v,\ldots,w,x:E\and F\operatorname{in}% L\equiv\operatorname{let}v,\ldots,w:E\operatorname{in}\operatorname{let}x:F% \operatorname{in}L)
  49. ( let s x = y in z ) ( λ x . z ) y (\operatorname{let}_{s}x=y\operatorname{in}z)\equiv(\lambda x.z)\ y
  50. λ f . let x = f x in x \lambda f.\operatorname{let}x=f\ x\operatorname{in}x
  51. λ f . let x x = f ( x x ) in x x \lambda f.\operatorname{let}x\ x=f\ (x\ x)\operatorname{in}x\ x
  52. f x = y f = λ x . y f\ x=y\equiv f=\lambda x.y
  53. λ f . let x = λ x . f ( x x ) in x x \lambda f.\operatorname{let}x=\lambda x.f\ (x\ x)\operatorname{in}x\ x
  54. n F V ( E ) ( let n = E in L ( λ n . L ) E ) n\not\in FV(E)\to(\operatorname{let}n=E\operatorname{in}L\equiv(\lambda n.L)\ E)
  55. λ f . ( λ x . x x ) ( λ x . f ( x x ) ) \lambda f.(\lambda x.x\ x)\ (\lambda x.f\ (x\ x))
  56. λ f . ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) \lambda f.(\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x))
  57. ( let rec x = y in z ) ( λ x . z ) ( Y ( λ x . y ) ) (\operatorname{let\ rec}x=y\operatorname{in}z)\equiv(\lambda x.z)\ (Y\ (% \lambda x.y))
  58. let x X in x \operatorname{let}x\in X\operatorname{in}x
  59. ( let x X in x ) ( let y Y in y ) (\operatorname{let}x\in X\operatorname{in}x)\ (\operatorname{let}y\in Y% \operatorname{in}y)
  60. = let x X and y Y in x y =\operatorname{let}x\in X\and y\in Y\operatorname{in}x\ y
  61. = let ( x , y ) X × Y in x y =\operatorname{let}(x,y)\in X\times Y\operatorname{in}x\ y
  62. ( let x X in x ) ( let x X in x ) (\operatorname{let}x\in X\operatorname{in}x)\ (\operatorname{let}x\in X% \operatorname{in}x)
  63. = let x X in x x =\operatorname{let}x\in X\operatorname{in}x\ x
  64. \equiv
  65. L [ G := S ] L[G:=S]
  66. de - lambda [ V ] V \operatorname{de-lambda}[V]\equiv V
  67. de - lambda [ M N ] de - lambda [ M ] de - lambda [ N ] \operatorname{de-lambda}[M\ N]\equiv\operatorname{de-lambda}[M]\ \operatorname% {de-lambda}[N]
  68. de - lambda [ F = λ P . E ] de - lambda [ F P = E ] \operatorname{de-lambda}[F=\lambda P.E]\equiv\operatorname{de-lambda}[F\ P=E]
  69. de - lambda [ E = F ] de - lambda [ E ] = de - lambda [ F ] \operatorname{de-lambda}[E=F]\equiv\operatorname{de-lambda}[E]=\operatorname{% de-lambda}[F]
  70. de - lambda [ ( λ F . E ) L ] let - combine [ let F : de - lambda [ F = L ] in E ] \operatorname{de-lambda}[(\lambda F.E)L]\equiv\operatorname{let-combine}[% \operatorname{let}F:\operatorname{de-lambda}[F=L]\operatorname{in}E]
  71. V FV [ λ F . E ] de - lambda [ λ F . E ] let - combine [ let V : de - lambda [ V F = E ] in V ] V\not\in\operatorname{FV}[\lambda F.E]\to\operatorname{de-lambda}[\lambda F.E]% \equiv\operatorname{let-combine}[\operatorname{let}V:\operatorname{de-lambda}[% V\ F=E]\operatorname{in}V]
  72. V W let - combine [ let V : E in let W : F in G ] let V , W : E and F in G V\neq W\to\operatorname{let-combine}[\operatorname{let}V:E\operatorname{in}% \operatorname{let}W:F\operatorname{in}G]\equiv\operatorname{let}V,W:E\and F% \operatorname{in}G
  73. let - combine [ let V : E in F ] let V : E in F \operatorname{let-combine}[\operatorname{let}V:E\operatorname{in}F]\equiv% \operatorname{let}V:E\operatorname{in}F
  74. λ f . ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) \lambda f.(\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x))
  75. let p : p f = let x : x q = f ( q q ) in f ( x x ) in p \operatorname{let}p:p\ f=\operatorname{let}x:x\ q=f\ (q\ q)\operatorname{in}f% \ (x\ x)\operatorname{in}p
  76. de - lambda [ λ f . ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) ] \operatorname{de-lambda}[\lambda f.(\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x))]
  77. V FV [ λ F . E ] de - lambda [ λ F . E ] V\not\in\operatorname{FV}[\lambda F.E]\to\operatorname{de-lambda}[\lambda F.E]
  78. V = p , F = f , E = λ x . f ( x x ) ) ( λ x . f ( x x ) V=p,F=f,E=\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x)
  79. let - combine [ let V : de - lambda [ V F = E ] in V ] \operatorname{let-combine}[\operatorname{let}V:\operatorname{de-lambda}[V\ F=E% ]\operatorname{in}V]
  80. let - combine [ let p : de - lambda [ p f = ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) ] in p ] \operatorname{let-combine}[\operatorname{let}p:\operatorname{de-lambda}[p\ f=(% \lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x))]\operatorname{in}p]
  81. de - lambda [ p f = ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) ] \operatorname{de-lambda}[p\ f=(\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x))]
  82. de - lambda [ E = F ] \operatorname{de-lambda}[E=F]
  83. E = p f , F = ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) E=p\ f,F=(\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x))
  84. de - lambda [ E ] = de - lambda [ F ] \operatorname{de-lambda}[E]=\operatorname{de-lambda}[F]
  85. de - lambda [ p f ] = de - lambda [ ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) ] \operatorname{de-lambda}[p\ f]=\operatorname{de-lambda}[(\lambda x.f\ (x\ x))% \ (\lambda x.f\ (x\ x))]
  86. let - combine [ let p : de - lambda [ p f ] = de - lambda [ ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) ] in p ] \operatorname{let-combine}[\operatorname{let}p:\operatorname{de-lambda}[p\ f]=% \operatorname{de-lambda}[(\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x))]% \operatorname{in}p]
  87. let - combine [ let p : de - lambda [ p f ] = de - lambda [ ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) ] in p ] \operatorname{let-combine}[\operatorname{let}p:\operatorname{de-lambda}[p\ f]=% \operatorname{de-lambda}[(\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x))]% \operatorname{in}p]
  88. de - lambda [ ( λ x . f ( x x ) ) ( λ x . f ( x x ) ) ] \operatorname{de-lambda}[(\lambda x.f\ (x\ x))\ (\lambda x.f\ (x\ x))]
  89. de - lambda [ ( λ F . E ) L ] \operatorname{de-lambda}[(\lambda F.E)L]
  90. F = x , E = f ( x x ) , L = ( λ x . f ( x x ) ) F=x,E=f\ (x\ x),L=(\lambda x.f\ (x\ x))
  91. let - combine [ let F : de - lambda [ F = L ] in E ] \operatorname{let-combine}[\operatorname{let}F:\operatorname{de-lambda}[F=L]% \operatorname{in}E]
  92. let - combine [ let x : de - lambda [ x = λ x . f ( x x ) ] in f ( x x ) ] \operatorname{let-combine}[\operatorname{let}x:\operatorname{de-lambda}[x=% \lambda x.f\ (x\ x)]\operatorname{in}f\ (x\ x)]
  93. let - combine [ let p : de - lambda [ p f ] = let - combine [ let x : de - lambda [ x = λ x . f ( x x ) ] in f ( x x ) ] in p ] \operatorname{let-combine}[\operatorname{let}p:\operatorname{de-lambda}[p\ f]=% \operatorname{let-combine}[\operatorname{let}x:\operatorname{de-lambda}[x=% \lambda x.f\ (x\ x)]\operatorname{in}f\ (x\ x)]\operatorname{in}p]
  94. de - lambda [ x = λ x . f ( x x ) ] \operatorname{de-lambda}[x=\lambda x.f\ (x\ x)]
  95. de - lambda [ F = λ P . E ] \operatorname{de-lambda}[F=\lambda P.E]
  96. F = x , P = x , E = f ( x x ) F=x,P=x,E=f\ (x\ x)
  97. de - lambda [ F P = E ] \operatorname{de-lambda}[F\ P=E]
  98. de - lambda [ x x = f ( x x ) ] \operatorname{de-lambda}[x\ x=f\ (x\ x)]
  99. let - combine [ let p : de - lambda [ p f ] = let - combine [ let x : de - lambda [ x x = f ( x x ) ] in f ( x x ) ] in p ] \operatorname{let-combine}[\operatorname{let}p:\operatorname{de-lambda}[p\ f]=% \operatorname{let-combine}[\operatorname{let}x:\operatorname{de-lambda}[x\ x=f% \ (x\ x)]\operatorname{in}f\ (x\ x)]\operatorname{in}p]
  100. let - combine [ let x : de - lambda [ x x = f ( x x ) ] in f ( x x ) ] \operatorname{let-combine}[\operatorname{let}x:\operatorname{de-lambda}[x\ x=f% \ (x\ x)]\operatorname{in}f\ (x\ x)]
  101. let - combine [ Y ] \operatorname{let-combine}[Y]
  102. Y = let x : de - lambda [ x x = f ( x x ) ] in f ( x x ) Y=\operatorname{let}x:\operatorname{de-lambda}[x\ x=f\ (x\ x)]\operatorname{in% }f\ (x\ x)
  103. Y Y
  104. let x : de - lambda [ x x = f ( x x ) ] in f ( x x ) \operatorname{let}x:\operatorname{de-lambda}[x\ x=f\ (x\ x)]\operatorname{in}f% \ (x\ x)
  105. let - combine [ let p : de - lambda [ p f ] = let x : de - lambda [ x x = f ( x x ) ] in f ( x x ) in p ] \operatorname{let-combine}[\operatorname{let}p:\operatorname{de-lambda}[p\ f]=% \operatorname{let}x:\operatorname{de-lambda}[x\ x=f\ (x\ x)]\operatorname{in}f% \ (x\ x)\operatorname{in}p]
  106. let - combine [ Y ] \operatorname{let-combine}[Y]
  107. Y = let p : de - lambda [ p f = let x : de - lambda [ x x = f ( x x ) ] in f ( x x ) ] in p Y=\operatorname{let}p:\operatorname{de-lambda}[p\ f=\operatorname{let}x:% \operatorname{de-lambda}[x\ x=f\ (x\ x)]\operatorname{in}f\ (x\ x)]% \operatorname{in}p
  108. Y Y
  109. let p : p f = let x : de - lambda [ x x = f ( x x ) ] in f ( x x ) in p \operatorname{let}p:p\ f=\operatorname{let}x:\operatorname{de-lambda}[x\ x=f\ % (x\ x)]\operatorname{in}f\ (x\ x)\operatorname{in}p
  110. let p : de - lambda [ p f ] = let x : de - lambda [ x x = f ( x x ) ] in f ( x x ) in p \operatorname{let}p:\operatorname{de-lambda}[p\ f]=\operatorname{let}x:% \operatorname{de-lambda}[x\ x=f\ (x\ x)]\operatorname{in}f\ (x\ x)% \operatorname{in}p
  111. de - lambda [ x x = f ( x x ) ] \operatorname{de-lambda}[x\ x=f\ (x\ x)]
  112. de - lambda [ E = F ] \operatorname{de-lambda}[E=F]
  113. E = x x , F = f ( x x ) E=x\ x,F=f\ (x\ x)
  114. de - lambda [ E ] = de - lambda [ F ] \operatorname{de-lambda}[E]=\operatorname{de-lambda}[F]
  115. de - lambda [ x x ] = de - lambda [ f ( x x ) ] \operatorname{de-lambda}[x\ x]=\operatorname{de-lambda}[f\ (x\ x)]
  116. let p : de - lambda [ p f ] = let x : de - lambda [ x x ] = de - lambda [ f ( x x ) ] in f ( x x ) in p \operatorname{let}p:\operatorname{de-lambda}[p\ f]=\operatorname{let}x:% \operatorname{de-lambda}[x\ x]=\operatorname{de-lambda}[f\ (x\ x)]% \operatorname{in}f\ (x\ x)\operatorname{in}p
  117. de - lambda [ x x ] , de - lambda [ f ( x x ) ] \operatorname{de-lambda}[x\ x],\operatorname{de-lambda}[f\ (x\ x)]
  118. de - lambda [ p f ] , de - lambda [ M 1 N 1 ] , de - lambda [ M 2 N 2 ] , \operatorname{de-lambda}[p\ f],\operatorname{de-lambda}[M_{1}\ N_{1}],% \operatorname{de-lambda}[M_{2}\ N_{2}],
  119. M 1 = p , N 1 = f , M 2 = x , N 2 = x , M 3 = f , N 3 = x x M_{1}=p,N_{1}=f,M_{2}=x,N_{2}=x,M_{3}=f,N_{3}=x\ x
  120. de - lambda [ M 1 ] de - lambda [ N 1 ] , de - lambda [ M 2 ] de - lambda [ N 2 ] , de - lambda [ M 3 ] de - lambda [ N 3 ] \operatorname{de-lambda}[M_{1}]\ \operatorname{de-lambda}[N_{1}],\operatorname% {de-lambda}[M_{2}]\ \operatorname{de-lambda}[N_{2}],\operatorname{de-lambda}[M% _{3}]\ \operatorname{de-lambda}[N_{3}]
  121. de - lambda [ p ] de - lambda [ f ] , de - lambda [ x ] de - lambda [ x ] , de - lambda [ f ] de - lambda [ x ] de - lambda [ x ] \operatorname{de-lambda}[p]\ \operatorname{de-lambda}[f],\operatorname{de-% lambda}[x]\ \operatorname{de-lambda}[x],\operatorname{de-lambda}[f]\ % \operatorname{de-lambda}[x]\ \operatorname{de-lambda}[x]
  122. let p : de - lambda [ p ] de - lambda [ f ] = let x : de - lambda [ x ] de - lambda [ x ] = de - lambda [ f ] ( de - lambda [ x ] de - lambda [ x ] ) in f ( x x ) ] in p \operatorname{let}p:\operatorname{de-lambda}[p]\ \operatorname{de-lambda}[f]=% \operatorname{let}x:\operatorname{de-lambda}[x]\ \operatorname{de-lambda}[x]=% \operatorname{de-lambda}[f]\ (\operatorname{de-lambda}[x]\ \operatorname{de-% lambda}[x])\operatorname{in}f\ (x\ x)]\operatorname{in}p
  123. de - lambda [ V ] \operatorname{de-lambda}[V]
  124. V V
  125. let p : p f = let x : x x = f ( x x ) in f ( x x ) ] in p \operatorname{let}p:p\ f=\operatorname{let}x:x\ x=f\ (x\ x)\operatorname{in}f% \ (x\ x)]\operatorname{in}p
  126. get - lambda [ F , G V = E ] = get - lambda [ F , G = λ V . E ] \operatorname{get-lambda}[F,G\ V=E]=\operatorname{get-lambda}[F,G=\lambda V.E]
  127. get - lambda [ F , F = E ] = de - let [ E ] \operatorname{get-lambda}[F,F=E]=\operatorname{de-let}[E]
  128. de - let [ λ V . E ] λ V . de - let [ E ] \operatorname{de-let}[\lambda V.E]\equiv\lambda V.\operatorname{de-let}[E]
  129. de - let [ M N ] de - let [ M ] de - let [ N ] \operatorname{de-let}[M\ N]\equiv\operatorname{de-let}[M]\ \operatorname{de-% let}[N]
  130. de - let [ V ] V \operatorname{de-let}[V]\equiv V
  131. V F V [ get - lambda [ V , E ] ] de - let [ let V : E in V ] get - lambda [ V , E ] V\not\in FV[\operatorname{get-lambda}[V,E]]\to\operatorname{de-let}[% \operatorname{let}V:E\operatorname{in}V]\equiv\operatorname{get-lambda}[V,E]
  132. V F V [ get - lambda [ V , E ] ] de - let [ let V : E in L ] ( λ V . de - let [ L ] ) get - lambda [ V , E ] V\not\in FV[\operatorname{get-lambda}[V,E]]\to\operatorname{de-let}[% \operatorname{let}V:E\operatorname{in}L]\equiv(\lambda V.\operatorname{de-let}% [L])\ \operatorname{get-lambda}[V,E]
  133. W FV [ get - lambda [ V , E ] ] de - let [ let V , W : E and F in G ] de - let [ let V : E in let W : F in G ] W\not\in\operatorname{FV}[\operatorname{get-lambda}[V,E]]\to\operatorname{de-% let}[\operatorname{let}V,W:E\and F\operatorname{in}G]\equiv\operatorname{de-% let}[\operatorname{let}V:E\operatorname{in}\operatorname{let}W:F\operatorname{% in}G]
  134. V FV [ get - lambda [ V , E ] ] de - let [ let V : E in L ] de - let [ let V : V V = get - lambda [ V , E ] [ V := V V ] in L [ V := V V ] ] V\in\operatorname{FV}[\operatorname{get-lambda}[V,E]]\to\operatorname{de-let}[% \operatorname{let}V:E\operatorname{in}L]\equiv\operatorname{de-let}[% \operatorname{let}V:V\ V=\operatorname{get-lambda}[V,E][V:=V\ V]\operatorname{% in}L[V:=V\ V]]
  135. W FV [ get - lambda [ V , E ] ] de - let [ let V , W : E and F in L ] de - let [ let V : V W = get - lambda [ V , E ] [ V := V W ] in let W : F [ V := V W ] in L [ V := V W ] ] W\in\operatorname{FV}[\operatorname{get-lambda}[V,E]]\to\operatorname{de-let}[% \operatorname{let}V,W:E\and F\operatorname{in}L]\equiv\operatorname{de-let}[% \operatorname{let}V:V\ W=\operatorname{get-lambda}[V,E][V:=V\ W]\operatorname{% in}\operatorname{let}W:F[V:=V\ W]\operatorname{in}L[V:=V\ W]]
  136. lambda - form [ G V = E ] = lambda - form [ G = λ V . E ] \operatorname{lambda-form}[G\ V=E]=\operatorname{lambda-form}[G=\lambda V.E]
  137. lambda - form [ E and F ] = lambda - form [ E ] and lambda - form [ F ] \operatorname{lambda-form}[E\and F]=\operatorname{lambda-form}[E]\and% \operatorname{lambda-form}[F]
  138. lambda - form [ V = E ] = V = E \operatorname{lambda-form}[V=E]=V=E
  139. X FV [ E ] lift - vars [ X , V = E ] = { V } X\in\operatorname{FV}[E]\to\operatorname{lift-vars}[X,V=E]=\{V\}
  140. X FV [ E ] lift - vars [ X , V = E ] = { } X\not\in\operatorname{FV}[E]\to\operatorname{lift-vars}[X,V=E]=\{\}
  141. lift - vars [ X , E and F ] = lift - vars [ X , E ] lift - vars [ X . F ] \operatorname{lift-vars}[X,E\and F]=\operatorname{lift-vars}[X,E]\cup% \operatorname{lift-vars}[X.F]
  142. sub - vars [ E , { V } S , X ] = sub - vars [ E [ V := V X ] , S , X ] \operatorname{sub-vars}[E,\{V\}\cup S,X]=\operatorname{sub-vars}[E[V:=V\ X],S,X]
  143. sub - vars [ E , { } , X ] = E \operatorname{sub-vars}[E,\{\},X]=E
  144. L = lambda - form [ E ] and S = lift - vars [ X , L ] de - let [ let V W , X : E and F in G ] L=\operatorname{lambda-form}[E]\and S=\operatorname{lift-vars}[X,L]\to% \operatorname{de-let}[\operatorname{let}V\ldots W,X:E\and F\operatorname{in}G]
  145. de - let [ let V W : sub - vars [ L , S , X ] in let sub - vars [ lambda - form [ F ] , S , X ] in sub - vars [ G , S , X ] ] \equiv\operatorname{de-let}[\operatorname{let}V\ldots W:\operatorname{sub-vars% }[L,S,X]\operatorname{in}\operatorname{let}\operatorname{sub-vars}[% \operatorname{lambda-form}[F],S,X]\operatorname{in}\operatorname{sub-vars}[G,S% ,X]]
  146. let p : p f = let x : x q = f ( q q ) in f ( x x ) in p \operatorname{let}p:p\ f=\operatorname{let}x:x\ q=f\ (q\ q)\operatorname{in}f% \ (x\ x)\operatorname{in}p
  147. λ f . ( λ x . f ( x x ) ) ( λ q . f ( q q ) ) \lambda f.(\lambda x.f\ (x\ x))\ (\lambda q.f\ (q\ q))
  148. de - let [ let p : p f = let x : x q = f ( q q ) in f ( x x ) in p ] \operatorname{de-let}[\operatorname{let}p:p\ f=\operatorname{let}x:x\ q=f\ (q% \ q)\operatorname{in}f\ (x\ x)\operatorname{in}p]
  149. de - let [ let V : E in V ] \operatorname{de-let}[\operatorname{let}V:E\operatorname{in}V]
  150. V = p , E = p f = let x : x q = f ( q q ) in f ( x x ) V=p,E=p\ f=\operatorname{let}x:x\ q=f\ (q\ q)\operatorname{in}f\ (x\ x)
  151. get - lambda [ V , E ] \operatorname{get-lambda}[V,E]
  152. get - lambda [ p , p f = let x : x q = f ( q q ) in f ( x x ) ] \operatorname{get-lambda}[p,p\ f=\operatorname{let}x:x\ q=f\ (q\ q)% \operatorname{in}f\ (x\ x)]
  153. get - lambda [ F , G V = E ] \operatorname{get-lambda}[F,G\ V=E]
  154. F = p , G = p , V = f , E = let x : x q = f ( q q ) in f ( x x ) F=p,G=p,V=f,E=\operatorname{let}x:x\ q=f\ (q\ q)\operatorname{in}f\ (x\ x)
  155. get - lambda [ F , G = λ V . E ] \operatorname{get-lambda}[F,G=\lambda V.E]
  156. get - lambda [ p , p = λ f . let x : x q = f ( q q ) in f ( x x ) ] \operatorname{get-lambda}[p,p=\lambda f.\operatorname{let}x:x\ q=f\ (q\ q)% \operatorname{in}f\ (x\ x)]
  157. get - lambda [ F , F = E ] \operatorname{get-lambda}[F,F=E]
  158. F = p , E = λ f . let x : x q = f ( q q ) in f ( x x ) F=p,E=\lambda f.\operatorname{let}x:x\ q=f\ (q\ q)\operatorname{in}f\ (x\ x)
  159. de - let [ E ] \operatorname{de-let}[E]
  160. de - let [ λ f . let x : x q = f ( q q ) in f ( x x ) ] \operatorname{de-let}[\lambda f.\operatorname{let}x:x\ q=f\ (q\ q)% \operatorname{in}f\ (x\ x)]
  161. de - let [ λ V . E ] \operatorname{de-let}[\lambda V.E]
  162. V = f , E = let x : x q = f ( q q ) in f ( x x ) V=f,E=\operatorname{let}x:x\ q=f\ (q\ q)\operatorname{in}f\ (x\ x)
  163. λ V . de - let [ E ] \lambda V.\operatorname{de-let}[E]
  164. λ f . de - let [ let x : x q = f ( q q ) in f ( x x ) ] \lambda f.\operatorname{de-let}[\operatorname{let}x:x\ q=f\ (q\ q)% \operatorname{in}f\ (x\ x)]
  165. de - let [ let x : x q = f ( q q ) in f ( x x ) ] \operatorname{de-let}[\operatorname{let}x:x\ q=f\ (q\ q)\operatorname{in}f\ (x% \ x)]
  166. V F V [ get - lambda [ V , E ] ] de - let [ let V : E in L ] V\not\in FV[\operatorname{get-lambda}[V,E]]\to\operatorname{de-let}[% \operatorname{let}V:E\operatorname{in}L]
  167. V = x , E = x q = f ( q q ) , L = f ( x x ) V=x,E=x\ q=f\ (q\ q),L=f\ (x\ x)
  168. ( λ V . de - let [ L ] ) get - lambda [ V , E ] (\lambda V.\operatorname{de-let}[L])\ \operatorname{get-lambda}[V,E]
  169. ( λ x . de - let [ f ( x x ) ] ) get - lambda [ x , x q = f ( q q ) ] (\lambda x.\operatorname{de-let}[f\ (x\ x)])\ \operatorname{get-lambda}[x,x\ q% =f\ (q\ q)]
  170. de - let [ f ( x x ) ] \operatorname{de-let}[f\ (x\ x)]
  171. de - let [ M N ] \operatorname{de-let}[M\ N]
  172. M = f , N = ( x x ) M=f,N=(x\ x)
  173. de - let [ M ] de - let [ N ] \operatorname{de-let}[M]\ \operatorname{de-let}[N]
  174. de - let [ f ] de - let [ x x ] \operatorname{de-let}[f]\ \operatorname{de-let}[x\ x]
  175. ( λ x . de - let [ f ] de - let [ x x ] ) get - lambda [ x , x q = f ( q q ) ] (\lambda x.\operatorname{de-let}[f]\ \operatorname{de-let}[x\ x])\ % \operatorname{get-lambda}[x,x\ q=f\ (q\ q)]
  176. de - let [ x x ] \operatorname{de-let}[x\ x]
  177. de - let [ M N ] \operatorname{de-let}[M\ N]
  178. M = x , N = x M=x,N=x
  179. de - let [ M ] de - let [ N ] \operatorname{de-let}[M]\ \operatorname{de-let}[N]
  180. de - let [ x ] de - let [ x ] \operatorname{de-let}[x]\ \operatorname{de-let}[x]
  181. ( λ x . de - let [ f ] ( de - let [ x ] de - let [ x ] ) ) get - lambda [ x , x q = f ( q q ) ] (\lambda x.\operatorname{de-let}[f]\ (\operatorname{de-let}[x]\ \operatorname{% de-let}[x]))\ \operatorname{get-lambda}[x,x\ q=f\ (q\ q)]
  182. de - let [ V ] \operatorname{de-let}[V]
  183. V V
  184. ( λ x . f ( x x ) ) get - lambda [ x , x q = f ( q q ) ] (\lambda x.f\ (x\ x))\ \operatorname{get-lambda}[x,x\ q=f\ (q\ q)]
  185. get - lambda [ x , x q = f ( q q ) ] \operatorname{get-lambda}[x,x\ q=f\ (q\ q)]
  186. get - lambda [ F , G V = E ] \operatorname{get-lambda}[F,G\ V=E]
  187. F = x , G = x , V = q , E = f ( q q ) F=x,G=x,V=q,E=f\ (q\ q)
  188. get - lambda [ F , G = λ V . E ] \operatorname{get-lambda}[F,G=\lambda V.E]
  189. get - lambda [ x , x = λ q . f ( q q ) ] \operatorname{get-lambda}[x,x=\lambda q.f\ (q\ q)]
  190. ( λ x . f ( x x ) ) get - lambda [ x , x = λ q . f ( q q ) ] (\lambda x.f\ (x\ x))\ \operatorname{get-lambda}[x,x=\lambda q.f\ (q\ q)]
  191. get - lambda [ x , x = λ q . f ( q q ) ] \operatorname{get-lambda}[x,x=\lambda q.f\ (q\ q)]
  192. get - lambda [ F , F = E ] \operatorname{get-lambda}[F,F=E]
  193. F = x , E = λ q . f ( q q ) F=x,E=\lambda q.f\ (q\ q)
  194. de - let [ E ] \operatorname{de-let}[E]
  195. de - let [ λ q . f ( q q ) ] \operatorname{de-let}[\lambda q.f\ (q\ q)]
  196. ( λ x . f ( x x ) ) de - let [ λ q . f ( q q ) ] (\lambda x.f\ (x\ x))\ \operatorname{de-let}[\lambda q.f\ (q\ q)]
  197. de - let [ λ q . f ( q q ) ] \operatorname{de-let}[\lambda q.f\ (q\ q)]
  198. de - let [ λ V . E ] \operatorname{de-let}[\lambda V.E]
  199. V = q , E = f ( q q ) V=q,E=f\ (q\ q)
  200. λ V . de - let [ E ] \lambda V.\operatorname{de-let}[E]
  201. λ q . de - let [ f ( q q ) ] \lambda q.\operatorname{de-let}[f\ (q\ q)]
  202. ( λ x . f ( x x ) ) ( λ q . de - let [ f ( q q ) ] ) (\lambda x.f\ (x\ x))\ (\lambda q.\operatorname{de-let}[f\ (q\ q)])
  203. de - let [ f ( q q ) ] \operatorname{de-let}[f\ (q\ q)]
  204. de - let [ M 1 N 1 ] \operatorname{de-let}[M_{1}\ N_{1}]
  205. M 1 = f , N 1 = q q M_{1}=f,N_{1}=q\ q
  206. de - let [ M 1 ] de - let [ N 1 ] \operatorname{de-let}[M_{1}]\ \operatorname{de-let}[N_{1}]
  207. de - let [ f ] de - let [ q q ] \operatorname{de-let}[f]\ \operatorname{de-let}[q\ q]
  208. de - let [ M 2 N 2 ] \operatorname{de-let}[M_{2}\ N_{2}]
  209. M 2 = q , N 2 = q M_{2}=q,N_{2}=q
  210. de - let [ q ] de - let [ q ] \operatorname{de-let}[q]\ \operatorname{de-let}[q]
  211. ( λ x . f ( x x ) ) ( λ q . de - let [ f ] ( de - let [ q ] de - let [ q ] ) ) (\lambda x.f\ (x\ x))\ (\lambda q.\operatorname{de-let}[f]\ (\operatorname{de-% let}[q]\ \operatorname{de-let}[q]))
  212. de - let [ V ] \operatorname{de-let}[V]
  213. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  214. V V
  215. ( λ x . f ( x x ) ) ( λ q . f ( q q ) ) (\lambda x.f\ (x\ x))\ (\lambda q.f\ (q\ q))
  216. let p , q : p f x = f ( x x ) and q p f = ( p f ) ( p f ) in q p \operatorname{let}p,q:p\ f\ x=f\ (x\ x)\and q\ p\ f=(p\ f)\ (p\ f)% \operatorname{in}q\ p
  217. ( λ p . ( λ q . q p ) λ p . λ f . ( p f ) ( p f ) ) λ f . λ x . f ( x x ) (\lambda p.(\lambda q.q\ p)\ \lambda p.\lambda f.(p\ f)\ (p\ f))\ \lambda f.% \lambda x.f\ (x\ x)
  218. de - let [ let p , q : p f x = f ( x x ) and q p f = ( p f ) ( p f ) in q p ] \operatorname{de-let}[\operatorname{let}p,q:p\ f\ x=f\ (x\ x)\and q\ p\ f=(p\ % f)\ (p\ f)\operatorname{in}q\ p]
  219. de - let [ let p : p f x = f ( x x ) in let q : q p f = ( p f ) ( p f ) in q p ] \operatorname{de-let}[\operatorname{let}p:p\ f\ x=f\ (x\ x)\operatorname{in}% \operatorname{let}q:q\ p\ f=(p\ f)\ (p\ f)\operatorname{in}q\ p]
  220. ( λ p . de - let [ let q : q p f = ( p f ) ( p f ) in q p ] ) get - lambda [ p , p f x = f ( x x ) ] (\lambda p.\operatorname{de-let}[\operatorname{let}q:q\ p\ f=(p\ f)\ (p\ f)% \operatorname{in}q\ p])\ \operatorname{get-lambda}[p,p\ f\ x=f\ (x\ x)]
  221. ( λ p . de - let [ let q : q p f = ( p f ) ( p f ) in q p ] ) λ f . λ x . f ( x x ) (\lambda p.\operatorname{de-let}[\operatorname{let}q:q\ p\ f=(p\ f)\ (p\ f)% \operatorname{in}q\ p])\ \lambda f.\lambda x.f\ (x\ x)
  222. ( λ p . ( λ q . q p ) get - lambda [ q , q p f = ( p f ) ( p f ) ] ) λ f . λ x . f ( x x ) (\lambda p.(\lambda q.q\ p)\ \operatorname{get-lambda}[q,q\ p\ f=(p\ f)\ (p\ f% )])\ \lambda f.\lambda x.f\ (x\ x)
  223. ( λ p . ( λ q . q p ) λ p . λ f . ( p f ) ( p f ) ) λ f . λ x . f ( x x ) (\lambda p.(\lambda q.q\ p)\ \lambda p.\lambda f.(p\ f)\ (p\ f))\ \lambda f.% \lambda x.f\ (x\ x)
  224. let x : x f = f ( x f ) in x \operatorname{let}x:x\ f=f\ (x\ f)\operatorname{in}x
  225. ( λ x . x x ) ( λ x . λ f . f ( x x f ) ) (\lambda x.x\ x)\ (\lambda x.\lambda f.f\ (x\ x\ f))
  226. let x : x f = f ( x f ) in x \operatorname{let}x:x\ f=f\ (x\ f)\operatorname{in}x
  227. let x : get - lambda [ x , x f = f ( x f ) ] [ x := x x ] in x [ x := x x ] \operatorname{let}x:\operatorname{get-lambda}[x,x\ f=f\ (x\ f)][x:=x\ x]% \operatorname{in}x[x:=x\ x]
  228. let x : get - lambda [ x , x = λ f . f ( x f ) ] [ x := x x ] in x x \operatorname{let}x:\operatorname{get-lambda}[x,x=\lambda f.f\ (x\ f)][x:=x\ x% ]\operatorname{in}x\ x
  229. let x : ( x = λ f . f ( x f ) ) [ x := x x ] in x x \operatorname{let}x:(x=\lambda f.f\ (x\ f))[x:=x\ x]\operatorname{in}x\ x
  230. let x : ( x x = λ f . f ( x x f ) ) in x x \operatorname{let}x:(x\ x=\lambda f.f\ (x\ x\ f))\operatorname{in}x\ x
  231. ( λ x . x x ) get - lambda [ x , x x = λ f . f ( x x f ) ] (\lambda x.x\ x)\ \operatorname{get-lambda}[x,x\ x=\lambda f.f\ (x\ x\ f)]
  232. ( λ x . x x ) get - lambda [ x , x = λ x . λ f . f ( x x f ) ] (\lambda x.x\ x)\ \operatorname{get-lambda}[x,x=\lambda x.\lambda f.f\ (x\ x\ % f)]
  233. ( λ x . x x ) ( λ x . λ f . f ( x x f ) ) (\lambda x.x\ x)\ (\lambda x.\lambda f.f\ (x\ x\ f))

Leukotriene-C4_hydrolase.html

  1. \rightleftharpoons

Level_ancestor_problem.html

  1. l l
  2. = log 2 ( depth ( v ) ) \ell=\lfloor\log_{2}(\operatorname{depth}(v))\rfloor

Lidinoid.html

  1. ( 1 / 2 ) [ sin ( 2 x ) cos ( y ) sin ( z ) + sin ( 2 y ) cos ( z ) sin ( x ) + sin ( 2 z ) cos ( x ) sin ( y ) ] - ( 1 / 2 ) [ cos ( 2 x ) cos ( 2 y ) + cos ( 2 y ) cos ( 2 z ) + cos ( 2 z ) cos ( 2 x ) ] + 0.15 = 0 \begin{aligned}\displaystyle(1/2)[&\displaystyle\sin(2x)\cos(y)\sin(z)\\ \displaystyle+&\displaystyle\sin(2y)\cos(z)\sin(x)\\ \displaystyle+&\displaystyle\sin(2z)\cos(x)\sin(y)]\\ \displaystyle-&\displaystyle(1/2)[\cos(2x)\cos(2y)\\ \displaystyle+&\displaystyle\cos(2y)\cos(2z)\\ \displaystyle+&\displaystyle\cos(2z)\cos(2x)]+0.15=0\end{aligned}

Limonene-1,2-diol_dehydrogenase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons

Limonene_1,2-monooxygenase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Linear_Lie_algebra.html

  1. 𝔤 \mathfrak{g}
  2. 𝔤 𝔩 ( V ) \mathfrak{gl}(V)
  3. 𝔤 \mathfrak{g}
  4. 𝔤 \mathfrak{g}
  5. 𝔤 \mathfrak{g}
  6. 𝔤 𝔩 ( V ) \mathfrak{gl}(V)
  7. 𝔤 \mathfrak{g}

Linear_parameter-varying_control.html

  1. u u
  2. y y
  3. x x
  4. x ˙ = f ( x , u , t ) \dot{x}=f(x,u,t)
  5. x ˙ = f ( x ( t ) , u ( t ) , t ) , x ( t 0 ) \dot{x}=f(x(t),u(t),t),x(t_{0})
  6. x ( t 0 ) = x 0 , u ( t 0 ) = u 0 x(t_{0})=x_{0},u(t_{0})=u_{0}
  7. w ( t ) , w w(t),w
  8. x ( t ) x(t)
  9. x ˙ = f ( x ( t ) , w ( t ) , w ˙ ( t ) , u ( t ) ) \dot{x}=f(x(t),w(t),\dot{w}(t),u(t))
  10. w w
  11. x ˙ = A ( w ( t ) ) x ( t ) + B ( w ( t ) ) u ( t ) \dot{x}=A(w(t))x(t)+B(w(t))u(t)
  12. y = C ( w ( t ) ) x ( t ) + D ( w ( t ) ) u ( t ) y=C(w(t))x(t)+D(w(t))u(t)

Linear_predictor_function.html

  1. f ( i ) f(i)
  2. f ( i ) = β 0 + β 1 x i 1 + + β p x i p , f(i)=\beta_{0}+\beta_{1}x_{i1}+\cdots+\beta_{p}x_{ip},
  3. β 0 , , β p \beta_{0},\ldots,\beta_{p}
  4. f ( i ) = s y m b o l β 𝐱 i f(i)=symbol\beta\cdot\mathbf{x}_{i}
  5. f ( i ) = s y m b o l β T 𝐱 i = 𝐱 i T s y m b o l β f(i)=symbol\beta^{\mathrm{T}}\mathbf{x}_{i}=\mathbf{x}^{\mathrm{T}}_{i}symbol\beta
  6. s y m b o l β symbol\beta
  7. 𝐱 i \mathbf{x}_{i}
  8. s y m b o l β T symbol\beta^{\mathrm{T}}
  9. s y m b o l β symbol\beta
  10. s y m b o l β T 𝐱 i symbol\beta^{\mathrm{T}}\mathbf{x}_{i}
  11. y i = f ( i ) + ε i = s y m b o l β T 𝐱 i + ε i , y_{i}=f(i)+\varepsilon_{i}=symbol\beta^{\mathrm{T}}\mathbf{x}_{i}\ +% \varepsilon_{i},
  12. ε i \varepsilon_{i}
  13. 𝐲 = 𝐗 s y m b o l β + s y m b o l ε , \mathbf{y}=\mathbf{X}symbol\beta+symbol\varepsilon,\,
  14. 𝐲 = ( y 1 y 2 y n ) , 𝐗 = ( 𝐱 1 𝐱 2 𝐱 n ) = ( x 11 x 1 p x 21 x 2 p x n 1 x n p ) , s y m b o l β = ( β 1 β p ) , s y m b o l ε = ( ε 1 ε 2 ε n ) . \mathbf{y}=\begin{pmatrix}y_{1}\\ y_{2}\\ \vdots\\ y_{n}\end{pmatrix},\quad\mathbf{X}=\begin{pmatrix}\mathbf{x}^{\prime}_{1}\\ \mathbf{x}^{\prime}_{2}\\ \vdots\\ \mathbf{x}^{\prime}_{n}\end{pmatrix}=\begin{pmatrix}x_{11}&\cdots&x_{1p}\\ x_{21}&\cdots&x_{2p}\\ \vdots&\ddots&\vdots\\ x_{n1}&\cdots&x_{np}\end{pmatrix},\quad symbol\beta=\begin{pmatrix}\beta_{1}\\ \vdots\\ \beta_{p}\end{pmatrix},\quad symbol\varepsilon=\begin{pmatrix}\varepsilon_{1}% \\ \varepsilon_{2}\\ \vdots\\ \varepsilon_{n}\end{pmatrix}.
  15. ε i \varepsilon_{i}
  16. s y m b o l β ^ symbol{\hat{\beta}}
  17. s y m b o l β ^ = ( X T X ) - 1 X T 𝐲 . symbol{\hat{\beta}}=(X^{\mathrm{T}}X)^{-1}X^{\mathrm{T}}\mathbf{y}.
  18. ( X T X ) - 1 X T (X^{\mathrm{T}}X)^{-1}X^{\mathrm{T}}
  19. y i = β 0 + β 1 x i + β 2 x i 2 + + β p x i p , y_{i}=\beta_{0}+\beta_{1}x_{i}+\beta_{2}x_{i}^{2}+\cdots+\beta_{p}x_{i}^{p},
  20. ( x i 1 = x i , x i 2 = x i 2 , , x i p = x i p ) (x_{i1}=x_{i},x_{i2}=x_{i}^{2},\ldots,x_{ip}=x_{i}^{p})
  21. s y m b o l ϕ ( x ) = ( ϕ 1 ( x ) , ϕ 2 ( x ) , , ϕ p ( x ) ) = ( x , x 2 , , x p ) . symbol\phi(x)=(\phi_{1}(x),\phi_{2}(x),\ldots,\phi_{p}(x))=(x,x^{2},\ldots,x^{% p}).
  22. ϕ ( 𝐱 ; 𝐜 ) = ϕ ( || 𝐱 - 𝐜 || ) = ϕ ( ( x 1 - c 1 ) 2 + + ( x K - c K ) 2 ) \phi(\mathbf{x};\mathbf{c})=\phi(||\mathbf{x}-\mathbf{c}||)=\phi(\sqrt{(x_{1}-% c_{1})^{2}+\ldots+(x_{K}-c_{K})^{2}})
  23. ϕ ( 𝐱 ; 𝐜 ) = e - b || 𝐱 - 𝐜 || 2 \phi(\mathbf{x};\mathbf{c})=e^{-b||\mathbf{x}-\mathbf{c}||^{2}}
  24. a + b 2 x i 1 + c x i 2 a+b^{2}x_{i1}+\sqrt{c}x_{i2}
  25. a , b , c a,b,c
  26. b = b 2 , c = c , b^{\prime}=b^{2},c^{\prime}=\sqrt{c},
  27. a + b x i 1 + c x i 2 , a+b^{\prime}x_{i1}+c^{\prime}x_{i2},

Linear_seismic_inversion.html

  1. t = i = 1 n [ ( x i - x i - 1 ) 2 + ( y i - y i - 1 ) 2 + ( z i - z i - 1 ) 2 ] 1 2 v i t=\sum_{i=1}^{n}\frac{\big[(x_{i}-x_{i-1})^{2}+(y_{i}-y_{i-1})^{2}+(z_{i}-z_{i% -1})^{2}\big]^{\frac{1}{2}}}{v_{i}}\!
  2. [ 1 K ( r ) 2 t 2 - ( 1 ρ ( r ) ) ] U ( r , t ) = s ( r , t ) \left[\frac{1}{K(\vec{r})}\frac{\partial^{2}}{\partial t^{2}}-\nabla\cdot\big(% \frac{1}{\rho(\vec{r}\big)}\nabla)\right]U(\vec{r},t)=s(\vec{r},t)\!
  3. K ( r ) K(\vec{r})\!
  4. ρ ( r ) \rho(\vec{r})\!
  5. s ( r , t ) s(\vec{r},t)\!
  6. U ( r , t ) U(\vec{r},t)\!
  7. s ( t ) = w ( t ) * R ( t ) s(t)=w(t)*R(t)\!
  8. m m\!
  9. F j F_{j}\!
  10. j = 1 , , m j=1,\ldots,m\!
  11. n n\!
  12. p i p_{i}\!
  13. i = 1 , , n i=1,\ldots,n\!
  14. F ( p ) \vec{F}\,(\vec{p})\!
  15. F j ( p i ) F_{j}\,(p_{i})\!
  16. p \vec{p}\!
  17. F ( p ) \vec{F}\,(\vec{p})\!
  18. q i q_{i}\!
  19. F ( q ) \vec{F}\,(\vec{q})\!
  20. F ( p ) \vec{F}\,(\vec{p})\!
  21. q \vec{q}\!
  22. F ( p ) = F ( q ) + ( p - q ) F ( q ) p + ( p - q ) 2 2 F ( q ) p 2 + O ( p - q ) 3 \vec{F}\,(\vec{p})=\vec{F}\,(\vec{q})+(\vec{p}-\vec{q})\frac{\partial\vec{F}\,% (\vec{q})}{\partial\vec{p}}+(\vec{p}-\vec{q})^{2}\frac{\partial^{2}\vec{F}\,(% \vec{q})}{\partial\vec{p}^{2}}+O(\vec{p}-\vec{q})^{3}\!
  23. F ( p ) - F ( q ) = ( p - q ) F ( q ) p \vec{F}\,(\vec{p})-\vec{F}\,(\vec{q})=(\vec{p}-\vec{q})\frac{\partial\vec{F}\,% (\vec{q})}{\partial\vec{p}}\!
  24. F \vec{F}\!
  25. m m\!
  26. p \vec{p}\!
  27. q \vec{q}\!
  28. n n\!
  29. m m\!
  30. n n\!
  31. Δ F = 𝐀 Δ p \Delta\vec{F}=\mathbf{A}\,\Delta\vec{p}\!
  32. Δ F = [ F 1 ( p ) - F 1 ( q ) F m ( p ) - F m ( q ) ] \Delta\vec{F}=\begin{bmatrix}F_{1}(\vec{p})-F_{1}(\vec{q})\\ \vdots\\ F_{m}(\vec{p})-F_{m}(\vec{q})\end{bmatrix}\!
  33. Δ p = p - q = [ p 1 - q 1 p n - q n ] \Delta\vec{p}=\vec{p}-\vec{q}=\begin{bmatrix}p_{1}-q_{1}\\ \vdots\\ p_{n}-q_{n}\end{bmatrix}\!
  34. 𝐀 = [ F 1 ( q ) p 1 F 1 ( q ) p 2 F 1 ( q ) p n F 2 ( q ) p 1 F 2 ( q ) p n - 1 F 2 ( q ) p n F j ( q ) p i F m ( q ) p 1 F m ( q ) p 2 F m ( q ) p n ] \mathbf{A}=\begin{bmatrix}\frac{\partial F_{1}(\vec{q})}{\partial p_{1}}&\frac% {\partial F_{1}(\vec{q})}{\partial p_{2}}&\cdots&\frac{\partial F_{1}(\vec{q})% }{\partial p_{n}}\\ \frac{\partial F_{2}(\vec{q})}{\partial p_{1}}&\cdots&\frac{\partial F_{2}(% \vec{q})}{\partial p_{n-1}}&\frac{\partial F_{2}(\vec{q})}{\partial p_{n}}\\ \vdots&\frac{\partial F_{j}(\vec{q})}{\partial p_{i}}&\vdots&\vdots\\ \frac{\partial F_{m}(\vec{q})}{\partial p_{1}}&\frac{\partial F_{m}(\vec{q})}{% \partial p_{2}}&\cdots&\frac{\partial F_{m}(\vec{q})}{\partial p_{n}}\\ \end{bmatrix}\!
  35. Δ F \Delta\vec{F}\!
  36. m × 1 m\times 1\!
  37. Δ p \Delta\vec{p}\!
  38. n × 1 n\times 1\!
  39. 𝐀 \mathbf{A}\!
  40. m × n m\times n\!
  41. F ( q ) \vec{F}\,(\vec{q})\!
  42. F ( p ) \vec{F}\,(\vec{p})\!
  43. Δ F \Delta\vec{F}\!
  44. Δ p \Delta\vec{p}\!
  45. 𝐀 \mathbf{A}\!
  46. m m\!
  47. n n\!
  48. Δ p \Delta\vec{p}\!
  49. Δ p = 𝐀 - 1 Δ F \Delta\vec{p}=\mathbf{A}^{-1}\,\Delta\vec{F}\!
  50. m > n m>n\!
  51. Δ p \Delta\vec{p}\!
  52. e T e \vec{e}\,^{T}\vec{e}\!
  53. e \vec{e}\!
  54. e \vec{e}\!
  55. e = Δ F - 𝐀 Δ p \vec{e}=\Delta\vec{F}-\mathbf{A}\,\Delta\vec{p}\!
  56. e T e \vec{e}\,^{T}\vec{e}\!
  57. 𝐀 Δ p = Δ F 𝐀 T 𝐀 Δ p = 𝐀 T Δ F \begin{aligned}\displaystyle\mathbf{A}\,\Delta\vec{p}&\displaystyle=\Delta\vec% {F}\\ \displaystyle\mathbf{A}^{T}\mathbf{A}\,\Delta\vec{p}&\displaystyle=\mathbf{A}^% {T}\Delta\vec{F}\end{aligned}\!
  58. Δ p = ( 𝐀 T 𝐀 ) - 1 𝐀 T Δ F \Delta\vec{p}=(\mathbf{A}^{T}\mathbf{A})^{-1}\,\mathbf{A}^{T}\Delta\vec{F}\!
  59. L 1 L_{1}\!
  60. L 2 L_{2}\!
  61. Δ p \Delta\vec{p}\!
  62. j = 0 n | Δ p j | \sum_{j=0}^{n}|\Delta p_{j}|\!
  63. j = 0 n | Δ p j | 2 \sum_{j=0}^{n}|\Delta p_{j}|^{2}\!
  64. Δ F \Delta\vec{F}\!
  65. i = 0 m | Δ F i | \sum_{i=0}^{m}|\Delta F_{i}|\!
  66. i = 0 m | Δ F i | 2 \sum_{i=0}^{m}|\Delta F_{i}|^{2}\!
  67. m = n m=n\!
  68. m > n m>n\!
  69. q \vec{q}\!
  70. F ( q ) \vec{F}(\vec{q})\!
  71. F ( p ) - F ( q ) \vec{F}(\vec{p})-\vec{F}(\vec{q})\!
  72. 𝐀 \mathbf{A}\!
  73. 𝐀 \mathbf{A}\!
  74. Δ p \Delta\vec{p}\,\!
  75. p = q + Δ p \vec{p}=\vec{q}+\Delta\vec{p}\!
  76. L 1 L_{1}\!
  77. L 2 L_{2}\!
  78. p \vec{p}\!
  79. F ( q ) = T = a + b z \vec{F}(\vec{q})=\vec{T}=a+bz\!
  80. q = [ a , b ] \vec{q}=[a,b]\!
  81. q \vec{q}\!
  82. p \vec{p}\!
  83. [ a , b ] [a,b]\!
  84. n n\!
  85. T 1 = a + b z 1 T 2 = a + b z 2 T n - 1 = a + b z n - 1 T n = a + b z n \begin{aligned}\displaystyle T_{1}&\displaystyle=a+bz_{1}\\ \displaystyle T_{2}&\displaystyle=a+bz_{2}\\ \displaystyle\vdots\\ \displaystyle T_{n-1}&\displaystyle=a+bz_{n-1}\\ \displaystyle T_{n}&\displaystyle=a+bz_{n}\\ \end{aligned}\!
  86. F ( q ) = T \vec{F}(\vec{q})=T\!
  87. 𝐀 = [ 1 z 1 1 z 2 1 z n - 1 1 z n ] \mathbf{A}=\begin{bmatrix}1&z_{1}\\ 1&z_{2}\\ \vdots&\vdots\\ 1&z_{n-1}\\ 1&z_{n}\\ \end{bmatrix}\!
  88. n = 2 n=2\!
  89. T 1 = 19 C T_{1}=19^{\circ}C\!
  90. z = 2 m z=2m\!
  91. T 2 = 22 C T_{2}=22^{\circ}C\!
  92. z = 8 m z=8m\!
  93. a = 0.5 a=0.5\!
  94. b = 18 C b=18^{\circ}C\!
  95. a a\!
  96. b b\!

Linearised_polynomial.html

  1. L ( x ) = i = 0 n a i x q i , where each a i is in F q m ( = G F ( q m ) ) for some fixed positive integer m . L(x)=\sum_{i=0}^{n}a_{i}x^{q^{i}},\,\text{ where each }a_{i}\,\text{ is in }F_% {q^{m}}(\text{ = }GF(q^{m}))\,\text{ for some fixed positive integer }m.
  2. F q n F_{q^{n}}
  3. F q s F_{q^{s}}
  4. F q n F_{q^{n}}
  5. L 1 ( x ) L 2 ( x ) = L 1 ( L 2 ( x ) ) L_{1}(x)\otimes L_{2}(x)=L_{1}(L_{2}(x))
  6. l ( x ) = i = 0 n a i x i l(x)=\sum_{i=0}^{n}a_{i}x^{i}
  7. x x q x\mapsto x^{q}
  8. Tr ( x ) = i = 0 n - 1 x q i \operatorname{Tr}(x)=\sum_{i=0}^{n-1}x^{q^{i}}
  9. L ( x ) = L 1 ( x ) L 2 ( x ) . L(x)=L_{1}(x)\otimes L_{2}(x).
  10. L ( x ) = L 1 ( x ) L 2 ( x ) , L(x)=L_{1}(x)\otimes L_{2}(x),
  11. L ( x ) = ( x 4 + x 2 + x ) ( x 2 + x ) ( x 2 + x ) . L(x)=(x^{4}+x^{2}+x)\otimes(x^{2}+x)\otimes(x^{2}+x).
  12. F q n F_{q^{n}}
  13. A ( x ) = L ( x ) - α for α F q n , A(x)=L(x)-\alpha\,\text{ for }\alpha\in F_{q^{n}},
  14. F q n F_{q^{n}}
  15. F q n F_{q^{n}}
  16. F q s F_{q^{s}}
  17. F q n F_{q^{n}}

Linked_field.html

  1. q = - a 1 , - a 2 , a 1 a 2 , b 1 , b 2 , - b 1 b 2 . q=\left\langle{-a_{1},-a_{2},a_{1}a_{2},b_{1},b_{2},-b_{1}b_{2}}\right\rangle\ .

Linoleate_10R-lipoxygenase.html

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Linoleate_8R-lipoxygenase.html

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Linoleate_9S-lipoxygenase.html

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Linolenate_9R-lipoxygenase.html

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Lipid_II:glycine_glycyltransferase.html

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Lipid_IVA_3-deoxy-D-manno-octulosonic_acid_transferase.html

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Lipid_IVA_4-amino-4-deoxy-L-arabinosyltransferase.html

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Lipoate—protein_ligase.html

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Lipoyl_amidotransferase.html

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List_of_active_Egyptian_military_aircraft.html

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List_of_definite_integrals.html

  1. a b f ( x ) d x \int_{a}^{b}f(x)\,dx
  2. a f ( x ) d x = lim b a b f ( x ) d x \int_{a}^{\infty}f(x)\,dx=\lim_{b\to\infty}\int_{a}^{b}f(x)\,dx
  3. 0 d x x 2 + a 2 = π 2 a \int_{0}^{\infty}\frac{dx}{x^{2}+a^{2}}=\frac{\pi}{2a}
  4. 0 x m d x x n + a n = π a m - n + 1 n sin [ ( m + 1 ) ( π / n ) ] , 0 < m + 1 < n \int_{0}^{\infty}\frac{x^{m}dx}{x^{n}+a^{n}}=\frac{\pi a^{m-n+1}}{n\sin[(m+1)(% \pi/n)]},0<m+1<n
  5. 0 x p - 1 d x 1 + x = π sin p π , 0 < p < 1 \int_{0}^{\infty}\frac{x^{p-1}dx}{1+x}=\frac{\pi}{\sin p\pi}\ \ ,0<p<1
  6. 0 x m d x 1 + 2 x cos β + x 2 = π sin ( m π ) sin ( m β ) sin β \int_{0}^{\infty}\frac{x^{m}dx}{1+2x\cos\beta+x^{2}}=\frac{\pi}{\sin(m\pi)}% \frac{\sin(m\beta)}{\sin\beta}
  7. 0 d x a 2 - x 2 = π 2 \int_{0}^{\infty}\frac{dx}{\sqrt{a^{2}-x^{2}}}=\frac{\pi}{2}
  8. 0 a a 2 - x 2 d x = π a 2 4 \int_{0}^{a}\sqrt{a^{2}-x^{2}}\,dx=\frac{\pi a^{2}}{4}
  9. 0 a x m ( a n - x n ) p d x = a m + 1 + n p Γ [ ( m + 1 ) / n ] Γ ( p + 1 ) n Γ [ ( ( m + 1 ) / n ) + p + 1 ] \int_{0}^{a}x^{m}(a^{n}-x^{n})^{p}\,dx=\frac{a^{m+1+np}\Gamma[(m+1)/n]\Gamma(p% +1)}{n\Gamma[((m+1)/n)+p+1]}
  10. 0 x m d x ( x n + a n ) r = ( - 1 ) r - 1 π a m + 1 - n r Γ [ ( m + 1 ) / n ] n sin [ ( m + 1 ) π / n ] ( r - 1 ) ! Γ [ ( m + 1 ) / n - r + 1 ] , n ( r - 2 ) < m + 1 < n r \int_{0}^{\infty}\frac{x^{m}\,dx}{({x^{n}+a^{n})}^{r}}=\frac{(-1)^{r-1}\pi a^{% m+1-nr}\Gamma[(m+1)/n]}{n\sin[(m+1)\pi/n](r-1)!\Gamma[(m+1)/n-r+1]}\ \ ,n(r-2)% <m+1<nr
  11. 0 π sin m x sin n x d x = { 0 if m n π 2 if m = n m , n positive integers \int_{0}^{\pi}\sin mx\sin nx\,dx=\begin{cases}0&\,\text{if }m\neq n\\ \dfrac{\pi}{2}&\,\text{if }m=n\end{cases}\ \ m,n\,\text{ positive integers}
  12. 0 π cos m x cos n x d x = { 0 if m n π 2 if m = n m , n positive integers \int_{0}^{\pi}\cos mx\cos nxdx=\begin{cases}0&\,\text{if }m\neq n\\ \dfrac{\pi}{2}&\,\text{if }m=n\end{cases}\ \ m,n\,\text{ positive integers}
  13. 0 π sin m x cos n x d x = { 0 if m + n even 2 m m 2 - n 2 if m + n odd m , n integers . \int_{0}^{\pi}\sin mx\cos nx\,dx=\begin{cases}0&\,\text{if }m+n\,\text{ even}% \\ \dfrac{2m}{m^{2}-n^{2}}&\,\text{if }m+n\,\text{ odd}\end{cases}\ \ m,n\,\text{% integers}.
  14. 0 π 2 sin 2 x d x = 0 π 2 cos 2 x d x = π 4 \int_{0}^{\frac{\pi}{2}}\sin^{2}x\,dx=\int_{0}^{\frac{\pi}{2}}\cos^{2}x\,dx=% \frac{\pi}{4}
  15. 0 π 2 sin 2 m x d x = 0 π 2 cos 2 m x d x = 1 × 3 × 5 × × ( 2 m - 1 ) 2 × 4 × 6 × × 2 m π 2 m = 1 , 2 , 3 , \int_{0}^{\frac{\pi}{2}}\sin^{2m}x\,dx=\int_{0}^{\frac{\pi}{2}}\cos^{2m}x\,dx=% \frac{1\times 3\times 5\times\cdots\times(2m-1)}{2\times 4\times 6\times\cdots% \times 2m}\frac{\pi}{2}\ \ m=1,2,3,\ldots
  16. 0 π 2 sin 2 m + 1 x d x = 0 π 2 cos 2 m + 1 x d x = 2 × 4 × 6 × × 2 m 1 × 3 × 5 × × ( 2 m + 1 ) m = 1 , 2 , 3 , \int_{0}^{\frac{\pi}{2}}\sin^{2m+1}x\,dx=\int_{0}^{\frac{\pi}{2}}\cos^{2m+1}x% \,dx=\frac{2\times 4\times 6\times\cdots\times 2m}{1\times 3\times 5\times% \cdots\times(2m+1)}\ \ m=1,2,3,\ldots
  17. 0 π 2 sin 2 p - 1 x cos 2 q - 1 x d x = Γ ( p ) Γ ( q ) 2 Γ ( p + q ) = 1 2 B ( p , q ) \int_{0}^{\frac{\pi}{2}}\sin^{2p-1}x\cos^{2q-1}x\,dx=\frac{\Gamma(p)\Gamma(q)}% {2\Gamma(p+q)}=\frac{1}{2}\,\text{B}(p,q)
  18. 0 sin p x x d x = { π 2 if p > 0 0 if p = 0 - π 2 if p < 0 \int_{0}^{\infty}\frac{\sin px}{x}\,dx=\begin{cases}\dfrac{\pi}{2}&\,\text{if % }p>0\\ \\ 0&\,\text{if }p=0\\ \\ -\dfrac{\pi}{2}&\text{if }p<0\end{cases}
  19. 0 sin p x cos q x x d x = { 0 if p > q > 0 π 2 if 0 < p < q π 4 if p = q > 0 \int_{0}^{\infty}\frac{\sin px\cos qx}{x}\ dx=\begin{cases}0&\,\text{ if }p>q>% 0\\ \\ \dfrac{\pi}{2}&\,\text{ if }0<p<q\\ \\ \dfrac{\pi}{4}&\,\text{ if }p=q>0\end{cases}
  20. 0 sin p x sin q x x 2 d x = { π p 2 if 0 < p q π q 2 if 0 < q p \int_{0}^{\infty}\frac{\sin px\sin qx}{x^{2}}\ dx=\begin{cases}\dfrac{\pi p}{2% }&\,\text{ if }0<p\leq q\\ \\ \dfrac{\pi q}{2}&\,\text{ if }0<q\leq p\end{cases}
  21. 0 sin 2 p x x 2 d x = π p 2 \int_{0}^{\infty}\frac{\sin^{2}px}{x^{2}}\ dx=\frac{\pi p}{2}
  22. 0 1 - cos p x x 2 d x = π p 2 \int_{0}^{\infty}\frac{1-\cos px}{x^{2}}\ dx=\frac{\pi p}{2}
  23. 0 cos p x - cos q x x d x = ln q p \int_{0}^{\infty}\frac{\cos px-\cos qx}{x}\ dx=\ln\frac{q}{p}
  24. 0 cos p x - cos q x x 2 d x = π ( q - p ) 2 \int_{0}^{\infty}\frac{\cos px-\cos qx}{x^{2}}\ dx=\frac{\pi(q-p)}{2}
  25. 0 cos m x x 2 + a 2 d x = π 2 a e - m a \int_{0}^{\infty}\frac{\cos mx}{x^{2}+a^{2}}\ dx=\frac{\pi}{2a}e^{-ma}
  26. 0 x sin m x x 2 + a 2 d x = π 2 e - m a \int_{0}^{\infty}\frac{x\sin mx}{x^{2}+a^{2}}\ dx=\frac{\pi}{2}e^{-ma}
  27. 0 sin m x x ( x 2 + a 2 ) d x = π 2 a 2 ( 1 - e - m a ) \int_{0}^{\infty}\frac{\sin mx}{x(x^{2}+a^{2})}\ dx=\frac{\pi}{2a^{2}}(1-e^{-% ma})
  28. 0 2 π d x a + b sin x = 2 π a 2 - b 2 \int_{0}^{2\pi}\frac{dx}{a+b\sin x}=\frac{2\pi}{\sqrt{a^{2}-b^{2}}}
  29. 0 2 π d x a + b cos x = 2 π a 2 - b 2 \int_{0}^{2\pi}\frac{dx}{a+b\cos x}=\frac{2\pi}{\sqrt{a^{2}-b^{2}}}
  30. 0 π 2 d x a + b cos x = cos - 1 ( b / a ) a 2 - b 2 \int_{0}^{\frac{\pi}{2}}\frac{dx}{a+b\cos x}=\frac{\cos^{-1}(b/a)}{\sqrt{a^{2}% -b^{2}}}
  31. 0 2 π d x ( a + b sin x ) 2 = 0 2 π d x ( a + b cos x ) 2 = 2 π a ( a 2 - b 2 ) 3 / 2 \int_{0}^{2\pi}\frac{dx}{(a+b\sin x)^{2}}=\int_{0}^{2\pi}\frac{dx}{(a+b\cos x)% ^{2}}=\frac{2\pi a}{(a^{2}-b^{2})^{3/2}}
  32. 0 2 π d x 1 - 2 a cos x + a 2 = 2 π 1 - a 2 , 0 < a < 1 \int_{0}^{2\pi}\frac{dx}{1-2a\cos x+a^{2}}=\frac{2\pi}{1-a^{2}}\ \ \ ,\ 0<a<1
  33. 0 π x sin x d x 1 - 2 a cos x + a 2 = { π a ln | 1 + a | if | a | < 1 π a ln | 1 + 1 a | if | a | > 1 \int_{0}^{\pi}\frac{x\sin x\ dx}{1-2a\cos x+a^{2}}=\begin{cases}\frac{\pi}{a}% \ln\left|1+a\right|&\,\text{if }|a|<1\\ \frac{\pi}{a}\ln\left|1+\frac{1}{a}\right|&\,\text{if }|a|>1\end{cases}
  34. 0 π cos m x d x 1 - 2 a cos x + a 2 = π a m 1 - a 2 , a 2 < 1 , m = 0 , 1 , 2 , \int_{0}^{\pi}\frac{\cos mx\ dx}{1-2a\cos x+a^{2}}=\frac{\pi a^{m}}{1-a^{2}}% \quad,a^{2}<1,\ m=0,1,2,\dots
  35. 0 sin a x 2 d x = 0 cos a x 2 = 1 2 π 2 a \int_{0}^{\infty}\sin ax^{2}\ dx=\int_{0}^{\infty}\cos ax^{2}=\frac{1}{2}\sqrt% {\frac{\pi}{2a}}
  36. 0 sin a x n = 1 n a 1 / n Γ ( 1 / n ) sin π 2 n , n > 1 \int_{0}^{\infty}\sin ax^{n}=\frac{1}{na^{1/n}}\Gamma(1/n)\sin\frac{\pi}{2n}% \quad,n>1
  37. 0 cos a x n = 1 n a 1 / n Γ ( 1 / n ) cos π 2 n , n > 1 \int_{0}^{\infty}\cos ax^{n}=\frac{1}{na^{1/n}}\Gamma(1/n)\cos\frac{\pi}{2n}% \quad,n>1
  38. 0 sin x x d x = 0 cos x x d x = π 2 \int_{0}^{\infty}\frac{\sin x}{\sqrt{x}}\ dx=\int_{0}^{\infty}\frac{\cos x}{% \sqrt{x}}\ dx=\sqrt{\frac{\pi}{2}}
  39. 0 sin x x p d x = π 2 Γ ( p ) sin ( p π / 2 ) , 0 < p < 1 \int_{0}^{\infty}\frac{\sin x}{x^{p}}\ dx=\frac{\pi}{2\Gamma(p)\sin(p\pi/2)},% \quad 0<p<1
  40. 0 cos x x p d x = π 2 Γ ( p ) cos ( p π / 2 ) , 0 < p < 1 \int_{0}^{\infty}\frac{\cos x}{x^{p}}\ dx=\frac{\pi}{2\Gamma(p)\cos(p\pi/2)},% \quad 0<p<1
  41. 0 sin a x 2 cos 2 b x d x = 1 2 π 2 a ( cos b 2 a - sin b 2 a ) \int_{0}^{\infty}\sin ax^{2}\cos 2bx\ dx=\frac{1}{2}\sqrt{\frac{\pi}{2a}}\left% (\cos\frac{b^{2}}{a}-\sin\frac{b^{2}}{a}\right)
  42. 0 cos a x 2 cos 2 b x d x = 1 2 π 2 a ( cos b 2 a + sin b 2 a ) \int_{0}^{\infty}\cos ax^{2}\cos 2bx\ dx=\frac{1}{2}\sqrt{\frac{\pi}{2a}}\left% (\cos\frac{b^{2}}{a}+\sin\frac{b^{2}}{a}\right)
  43. 0 e - a x cos b x d x = a a 2 + b 2 \int_{0}^{\infty}e^{-ax}\cos bx\,dx=\frac{a}{a^{2}+b^{2}}
  44. 0 e - a x sin b x d x = b a 2 + b 2 \int_{0}^{\infty}e^{-ax}\sin bx\,dx=\frac{b}{a^{2}+b^{2}}
  45. 0 e - a x sin b x x d x = tan - 1 b a \int_{0}^{\infty}\frac{{}e^{-ax}\sin bx}{x}\,dx=\tan^{-1}\frac{b}{a}
  46. 0 e - a x - e - b x x d x = ln b a \int_{0}^{\infty}\frac{e^{-ax}-e^{-bx}}{x}\,dx=\ln\frac{b}{a}
  47. 0 e - a x 2 d x = 1 2 π a \int_{0}^{\infty}{e^{-ax^{2}}}\,dx=\frac{1}{2}\sqrt{\frac{\pi}{a}}
  48. 0 e - a x 2 cos b x d x = 1 2 π a e - b 2 / 4 a \int_{0}^{\infty}{e^{-ax^{2}}}\cos bx\,dx=\frac{1}{2}\sqrt{\frac{\pi}{a}}e^{-b% ^{2}/4a}
  49. 0 e - ( a x 2 + b x + c ) d x = 1 2 π a e ( b 2 - 4 a c ) / 4 a erfc b 2 a , where erfc ( p ) = 2 π p e - x 2 d x \int_{0}^{\infty}e^{-(ax^{2}+bx+c)}\,dx=\frac{1}{2}\sqrt{\frac{\pi}{a}}e^{(b^{% 2}-4ac)/4a}\ \operatorname{erfc}\frac{b}{2\sqrt{a}},\,\text{ where }% \operatorname{erfc}(p)=\frac{2}{\sqrt{\pi}}\int_{p}^{\infty}e^{-x^{2}}\,dx
  50. - + e - ( a x 2 + b x + c ) d x = π a e ( b 2 - 4 a c ) / 4 a \int_{-\infty}^{+\infty}e^{-(ax^{2}+bx+c)}\ dx=\sqrt{\frac{\pi}{a}}e^{(b^{2}-4% ac)/4a}
  51. 0 x n e - a x d x = Γ ( n + 1 ) a n + 1 \int_{0}^{\infty}x^{n}e^{-ax}\ dx=\frac{\Gamma(n+1)}{a^{n+1}}
  52. 0 x m e - a x 2 d x = Γ [ ( m + 1 ) / 2 ] 2 a ( m + 1 ) / 2 \int_{0}^{\infty}x^{m}e^{-ax^{2}}\ dx=\frac{\Gamma[(m+1)/2]}{2a^{(m+1)/2}}
  53. 0 e - a x 2 - b / x 2 d x = 1 2 π a e - 2 a b \int_{0}^{\infty}e^{-ax^{2}-b/x^{2}}\ dx=\frac{1}{2}\sqrt{\frac{\pi}{a}}e^{-2% \sqrt{ab}}
  54. 0 x e x - 1 d x = ζ ( 2 ) = π 2 6 \int_{0}^{\infty}\frac{x}{e^{x}-1}\ dx=\zeta(2)=\frac{\pi^{2}}{6}
  55. 0 x n - 1 e x - 1 d x = Γ ( n ) ζ ( n ) \int_{0}^{\infty}\frac{x^{n-1}}{e^{x}-1}\ dx=\Gamma(n)\zeta(n)
  56. 0 x e x + 1 d x = 1 1 2 - 1 2 2 + 1 3 2 - 1 4 2 + = π 2 12 \int_{0}^{\infty}\frac{x}{e^{x}+1}\ dx=\frac{1}{1^{2}}-\frac{1}{2^{2}}+\frac{1% }{3^{2}}-\frac{1}{4^{2}}+\dots=\frac{\pi^{2}}{12}
  57. 0 sin m x e 2 π x - 1 d x = 1 4 coth m 2 - 1 2 m \int_{0}^{\infty}\frac{\sin mx}{e^{2\pi x}-1}\ dx=\frac{1}{4}\coth\frac{m}{2}-% \frac{1}{2m}
  58. 0 ( 1 1 + x - e - x ) d x x = γ \int_{0}^{\infty}\left(\frac{1}{1+x}-e^{-x}\right)\ \frac{dx}{x}=\gamma
  59. 0 e - x 2 - e - x x d x = γ 2 \int_{0}^{\infty}\frac{e^{-x^{2}}-e^{-x}}{x}\ dx=\frac{\gamma}{2}
  60. 0 ( 1 e x - 1 - e - x x ) d x = γ \int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{e^{-x}}{x}\right)\ dx=\gamma
  61. 0 e - a x - e - b x x sec p x d x = 1 2 ln b 2 + p 2 a 2 + p 2 \int_{0}^{\infty}\frac{e^{-ax}-e^{-bx}}{x\sec px}\ dx=\frac{1}{2}\ln\frac{b^{2% }+p^{2}}{a^{2}+p^{2}}
  62. 0 e - a x - e - b x x csc p x d x = tan - 1 b p - tan - 1 a p \int_{0}^{\infty}\frac{e^{-ax}-e^{-bx}}{x\csc px}\ dx=\tan^{-1}\frac{b}{p}-% \tan^{-1}\frac{a}{p}
  63. 0 e - a x ( 1 - cos x ) x 2 d x = cot - 1 a - a 2 ln | a 2 + 1 a 2 | \int_{0}^{\infty}\frac{e^{-ax}(1-\cos x)}{x^{2}}\ dx=\cot^{-1}a-\frac{a}{2}\ln% \left|\frac{a^{2}+1}{a^{2}}\right|
  64. - e - x 2 d x = π \int_{-\infty}^{\infty}e^{-x^{2}}\,dx=\sqrt{\pi}
  65. - x 2 ( n + 1 ) e - x 2 / 2 d x = ( 2 n + 1 ) ! 2 n n ! 2 π n = 0 , 1 , 2 , \int_{-\infty}^{\infty}x^{2(n+1)}e^{-x^{2}/2}\,dx=\frac{(2n+1)!}{2^{n}n!}\sqrt% {2\pi}\quad n=0,1,2,\ldots
  66. 0 1 x m ( ln x ) n d x = ( - 1 ) n n ! ( m + 1 ) n + 1 m > - 1 , n = 0 , 1 , 2 , \int_{0}^{1}x^{m}(\ln x)^{n}\,dx=\frac{(-1)^{n}n!}{(m+1)^{n+1}}\quad m>-1,n=0,% 1,2,\ldots
  67. 0 1 ln x 1 + x d x = - π 2 12 \int_{0}^{1}\frac{\ln x}{1+x}\,dx=-\frac{\pi^{2}}{12}
  68. 0 1 ln x 1 - x d x = - π 2 6 \int_{0}^{1}\frac{\ln x}{1-x}\,dx=-\frac{\pi^{2}}{6}
  69. 0 1 ln ( 1 + x ) x d x = π 2 12 \int_{0}^{1}\frac{\ln(1+x)}{x}\,dx=\frac{\pi^{2}}{12}
  70. 0 1 ln ( 1 - x ) x d x = - π 2 6 \int_{0}^{1}\frac{\ln(1-x)}{x}\,dx=-\frac{\pi^{2}}{6}
  71. 0 ln ( a 2 + x 2 ) b 2 + x 2 d x = π b ln ( a + b ) a , b > 0 \int_{0}^{\infty}\frac{\ln(a^{2}+x^{2})}{b^{2}+x^{2}}\ dx=\frac{\pi}{b}\ln(a+b% )\quad a,b>0
  72. 0 ln x x 2 + a 2 d x = π ln a 2 a a > 0 \int_{0}^{\infty}\frac{\ln x}{x^{2}+a^{2}}\ dx=\frac{\pi\ln a}{2a}\quad a>0
  73. 0 sin a x sinh b x d x = π 2 b tanh a π 2 b \int_{0}^{\infty}\frac{\sin ax}{\sinh bx}\ dx=\frac{\pi}{2b}\tanh\frac{a\pi}{2b}
  74. 0 cos a x cosh b x d x = π 2 b 1 cosh a π 2 b \int_{0}^{\infty}\frac{\cos ax}{\cosh bx}\ dx=\frac{\pi}{2b}\frac{1}{\cosh% \frac{a\pi}{2b}}
  75. 0 x sinh a x d x = π 2 4 a 2 \int_{0}^{\infty}\frac{x}{\sinh ax}\ dx=\frac{\pi^{2}}{4a^{2}}
  76. - 1 cosh x d x = π \int_{-\infty}^{\infty}\frac{1}{\cosh x}\ dx=\pi
  77. 0 f ( a x ) - f ( b x ) x d x = [ f ( 0 ) - f ( ) ] ln b a \int_{0}^{\infty}\frac{f(ax)-f(bx)}{x}\ dx=[{f(0)-f(\infty)}]\ln\frac{b}{a}

List_of_electromagnetism_equations.html

  1. q e = λ e d q_{e}=\int\lambda_{e}\mathrm{d}\ell
  2. q e = σ e d S q_{e}=\iint\sigma_{e}\mathrm{d}S
  3. q e = ρ e d V q_{e}=\iiint\rho_{e}\mathrm{d}V
  4. C = d q / d V C=\mathrm{d}q/\mathrm{d}V\,\!
  5. I = d q / d t I=\mathrm{d}q/\mathrm{d}t\,\!
  6. I = 𝐉 d 𝐒 I=\mathbf{J}\cdot\mathrm{d}\mathbf{S}
  7. 𝐉 d = ϵ 0 ( 𝐄 / t ) = 𝐃 / t \mathbf{J}_{\mathrm{d}}=\epsilon_{0}\left(\partial\mathbf{E}/\partial t\right)% =\partial\mathbf{D}/\partial t\,\!
  8. 𝐉 c = ρ 𝐯 \mathbf{J}_{\mathrm{c}}=\rho\mathbf{v}\,\!
  9. 𝐄 = 𝐅 / q \mathbf{E}=\mathbf{F}/q\,\!
  10. Φ E = S 𝐄 d 𝐀 \Phi_{E}=\int_{S}\mathbf{E}\cdot\mathrm{d}\mathbf{A}\,\!
  11. ϵ = ϵ r ϵ 0 \epsilon=\epsilon_{r}\epsilon_{0}\,\!
  12. 𝐩 = 2 q 𝐚 \mathbf{p}=2q\mathbf{a}\,\!
  13. 𝐏 = d 𝐩 / d V \mathbf{P}=\mathrm{d}\langle\mathbf{p}\rangle/\mathrm{d}V\,\!
  14. 𝐃 = ϵ 𝐄 = ϵ 0 𝐄 + 𝐏 \mathbf{D}=\epsilon\mathbf{E}=\epsilon_{0}\mathbf{E}+\mathbf{P}\,
  15. Φ D = S 𝐃 d 𝐀 \Phi_{D}=\int_{S}\mathbf{D}\cdot\mathrm{d}\mathbf{A}\,\!
  16. r 0 r_{0}\,\!
  17. r 0 = r_{0}=\infty\,\!
  18. r 0 = R earth r_{0}=R_{\mathrm{earth}}\,\!
  19. V = - W r q = - 1 q r 𝐅 d 𝐫 = - r 1 r 2 𝐄 d 𝐫 V=-\frac{W_{\infty r}}{q}=-\frac{1}{q}\int_{\infty}^{r}\mathbf{F}\cdot\mathrm{% d}\mathbf{r}=-\int_{r_{1}}^{r_{2}}\mathbf{E}\cdot\mathrm{d}\mathbf{r}\,\!
  20. Δ V = - Δ W q = - 1 q r 1 r 2 𝐅 d 𝐫 = - r 1 r 2 𝐄 d 𝐫 \Delta V=-\frac{\Delta W}{q}=-\frac{1}{q}\int_{r_{1}}^{r_{2}}\mathbf{F}\cdot% \mathrm{d}\mathbf{r}=-\int_{r_{1}}^{r_{2}}\mathbf{E}\cdot\mathrm{d}\mathbf{r}\,\!
  21. q m = λ m d q_{m}=\int\lambda_{m}\mathrm{d}\ell
  22. q m = σ m d S q_{m}=\iint\sigma_{m}\mathrm{d}S
  23. q m = ρ m d V q_{m}=\iiint\rho_{m}\mathrm{d}V
  24. I m = d q m / d t I_{m}=\mathrm{d}q_{m}/\mathrm{d}t\,\!
  25. I = 𝐉 m d 𝐀 I=\iint\mathbf{J}_{\mathrm{m}}\cdot\mathrm{d}\mathbf{A}
  26. 𝐅 = q e ( 𝐯 × 𝐁 ) \mathbf{F}=q_{e}\left(\mathbf{v}\times\mathbf{B}\right)\,\!
  27. 𝐁 = × 𝐀 \mathbf{B}=\nabla\times\mathbf{A}
  28. Φ B = S 𝐁 d 𝐀 \Phi_{B}=\int_{S}\mathbf{B}\cdot\mathrm{d}\mathbf{A}\,\!
  29. μ \mu\,\!
  30. μ = μ r μ 0 \mu\ =\mu_{r}\,\mu_{0}\,\!
  31. 𝐦 = q m 𝐚 \mathbf{m}=q_{m}\mathbf{a}\,\!
  32. 𝐦 = N I A 𝐧 ^ \mathbf{m}=NIA\mathbf{\hat{n}}\,\!
  33. 𝐌 = d 𝐦 / d V \mathbf{M}=\mathrm{d}\langle\mathbf{m}\rangle/\mathrm{d}V\,\!
  34. 𝐁 = μ 𝐇 = μ 0 ( 𝐇 + 𝐌 ) \mathbf{B}=\mu\mathbf{H}=\mu_{0}\left(\mathbf{H}+\mathbf{M}\right)\,
  35. 𝐇 = 𝐅 / q m \mathbf{H}=\mathbf{F}/q_{m}\,
  36. 𝐈 = μ 𝐌 \mathbf{I}=\mu\mathbf{M}\,\!
  37. L = N ( d Φ / d I ) L=N\left(\mathrm{d}\Phi/\mathrm{d}I\right)\,\!
  38. L ( d I / d t ) = - N V L\left(\mathrm{d}I/\mathrm{d}t\right)=-NV\,\!
  39. M 1 = N ( d Φ 2 / d I 1 ) M_{1}=N\left(\mathrm{d}\Phi_{2}/\mathrm{d}I_{1}\right)\,\!
  40. M ( d I 2 / d t ) = - N V 1 M\left(\mathrm{d}I_{2}/\mathrm{d}t\right)=-NV_{1}\,\!
  41. M 2 = N ( d Φ 1 / d I 2 ) M_{2}=N\left(\mathrm{d}\Phi_{1}/\mathrm{d}I_{2}\right)\,\!
  42. M ( d I 1 / d t ) = - N V 2 M\left(\mathrm{d}I_{1}/\mathrm{d}t\right)=-NV_{2}\,\!
  43. ω = γ B \omega=\gamma B\,\!
  44. R int = V ter / I R_{\mathrm{int}}=V_{\mathrm{ter}}/I\,\!
  45. R ext = V load / I R_{\mathrm{ext}}=V_{\mathrm{load}}/I\,\!
  46. = V ter + V load \mathcal{E}=V_{\mathrm{ter}}+V_{\mathrm{load}}\,\!
  47. V R = I R R V_{R}=I_{R}R\,\!
  48. V C = I C X C V_{C}=I_{C}X_{C}\,\!
  49. V L = I L X L V_{L}=I_{L}X_{L}\,\!
  50. X C = 1 ω d C X_{C}=\frac{1}{\omega_{\mathrm{d}}C}\,\!
  51. X L = ω d L X_{L}=\omega_{d}L\,\!
  52. V = I Z V=IZ\,\!
  53. Z = R 2 + ( X L - X C ) 2 Z=\sqrt{R^{2}+\left(X_{L}-X_{C}\right)^{2}}\,\!
  54. tan ϕ = X L - X C R \tan\phi=\frac{X_{L}-X_{C}}{R}\,\!
  55. I 0 = I rms 2 I_{0}=I_{\mathrm{rms}}\sqrt{2}\,\!
  56. I rms = 1 T 0 T [ I ( t ) ] 2 d t I_{\mathrm{rms}}=\sqrt{\frac{1}{T}\int_{0}^{T}\left[I\left(t\right)\right]^{2}% \mathrm{d}t}\,\!
  57. V 0 = V rms 2 V_{0}=V_{\mathrm{rms}}\sqrt{2}\,\!
  58. V rms = 1 T 0 T [ V ( t ) ] 2 d t V_{\mathrm{rms}}=\sqrt{\frac{1}{T}\int_{0}^{T}\left[V\left(t\right)\right]^{2}% \mathrm{d}t}\,\!
  59. rms , \mathcal{E}_{\mathrm{rms}},\sqrt{\langle\mathcal{E}\rangle}\,\!
  60. rms = m / 2 \mathcal{E}_{\mathrm{rms}}=\mathcal{E}_{\mathrm{m}}/\sqrt{2}\,\!
  61. P \langle P\rangle\,\!
  62. P = I rms cos ϕ \langle P\rangle=\mathcal{E}I_{\mathrm{rms}}\cos\phi\,\!
  63. τ C = R C \tau_{C}=RC\,\!
  64. τ L = L / R \tau_{L}=L/R\,\!
  65. , \mathcal{F},\mathcal{M}
  66. = N I \mathcal{M}=NI
  67. 𝐄 = - V \mathbf{E}=-\nabla V
  68. Δ V = - r 1 r 2 𝐄 d 𝐫 \Delta V=-\int_{r_{1}}^{r_{2}}\mathbf{E}\cdot d\mathbf{r}\,\!
  69. 𝐄 = q 4 π ϵ 0 | 𝐫 | 2 𝐫 ^ \mathbf{E}=\frac{q}{4\pi\epsilon_{0}\left|\mathbf{r}\right|^{2}}\mathbf{\hat{r% }}\,\!
  70. 𝐄 = 𝐄 i = 1 4 π ϵ 0 i q i | 𝐫 i - 𝐫 | 2 𝐫 ^ i \mathbf{E}=\sum\mathbf{E}_{i}=\frac{1}{4\pi\epsilon_{0}}\sum_{i}\frac{q_{i}}{% \left|\mathbf{r}_{i}-\mathbf{r}\right|^{2}}\mathbf{\hat{r}}_{i}\,\!
  71. 𝐄 = 1 4 π ϵ 0 V 𝐫 ρ d V | 𝐫 | 3 \mathbf{E}=\frac{1}{4\pi\epsilon_{0}}\int_{V}\frac{\mathbf{r}\rho\mathrm{d}V}{% \left|\mathbf{r}\right|^{3}}\,\!
  72. s y m b o l τ = V d 𝐩 × 𝐄 symbol{\tau}=\int_{V}\mathrm{d}\mathbf{p}\times\mathbf{E}
  73. U = V d 𝐩 𝐄 U=\int_{V}\mathrm{d}\mathbf{p}\cdot\mathbf{E}
  74. 𝐁 = × 𝐀 \mathbf{B}=\nabla\times\mathbf{A}
  75. 𝐀 = μ 0 4 π 𝐦 × 𝐫 | 𝐫 | 3 \mathbf{A}=\frac{\mu_{0}}{4\pi}\frac{\mathbf{m}\times\mathbf{r}}{\left|\mathbf% {r}\right|^{3}}
  76. 𝐁 ( 𝐫 ) = × 𝐀 = μ 0 4 π ( 3 𝐫 ( 𝐦 𝐫 ) | 𝐫 | 5 - 𝐦 | 𝐫 | 3 ) \mathbf{B}({\mathbf{r}})=\nabla\times{\mathbf{A}}=\frac{\mu_{0}}{4\pi}\left(% \frac{3\mathbf{r}(\mathbf{m}\cdot\mathbf{r})}{\left|\mathbf{r}\right|^{5}}-% \frac{{\mathbf{m}}}{\left|\mathbf{r}\right|^{3}}\right)
  77. 𝐦 = 1 2 V 𝐫 × 𝐉 d V \mathbf{m}=\frac{1}{2}\int_{V}\mathbf{r}\times\mathbf{J}\mathrm{d}V
  78. s y m b o l τ = V d 𝐦 × 𝐁 symbol{\tau}=\int_{V}\mathrm{d}\mathbf{m}\times\mathbf{B}
  79. U = V d 𝐦 𝐁 U=\int_{V}\mathrm{d}\mathbf{m}\cdot\mathbf{B}
  80. V s V p = N s N p = I p I s = η \frac{V_{s}}{V_{p}}=\frac{N_{s}}{N_{p}}=\frac{I_{p}}{I_{s}}=\eta\,\!
  81. R net = i = 1 N R i R_{\mathrm{net}}=\sum_{i=1}^{N}R_{i}\,\!
  82. 1 G net = i = 1 N 1 G i {1\over G_{\mathrm{net}}}=\sum_{i=1}^{N}{1\over G_{i}}\,\!
  83. 1 R net = i = 1 N 1 R i {1\over R_{\mathrm{net}}}=\sum_{i=1}^{N}{1\over R_{i}}\,\!
  84. G net = i = 1 N G i G_{\mathrm{net}}=\sum_{i=1}^{N}G_{i}\,\!
  85. q net = i = 1 N q i q_{\mathrm{net}}=\sum_{i=1}^{N}q_{i}\,\!
  86. 1 C net = i = 1 N 1 C i {1\over C_{\mathrm{net}}}=\sum_{i=1}^{N}{1\over C_{i}}\,\!
  87. I net = I i I_{\mathrm{net}}=I_{i}\,\!
  88. q net = i = 1 N q i q_{\mathrm{net}}=\sum_{i=1}^{N}q_{i}\,\!
  89. C net = i = 1 N C i C_{\mathrm{net}}=\sum_{i=1}^{N}C_{i}\,\!
  90. I net = i = 1 N I i I_{\mathrm{net}}=\sum_{i=1}^{N}I_{i}\,\!
  91. L net = i = 1 N L i L_{\mathrm{net}}=\sum_{i=1}^{N}L_{i}\,\!
  92. 1 L net = i = 1 N 1 L i {1\over L_{\mathrm{net}}}=\sum_{i=1}^{N}{1\over L_{i}}\,\!
  93. V i = j = 1 N L i j d I j d t V_{i}=\sum_{j=1}^{N}L_{ij}\frac{\mathrm{d}I_{j}}{\mathrm{d}t}\,\!
  94. R d q d t + q C = R\frac{\mathrm{d}q}{\mathrm{d}t}+\frac{q}{C}=\mathcal{E}\,\!
  95. q = C ( 1 - e - t / R C ) q=C\mathcal{E}\left(1-e^{-t/RC}\right)\,\!
  96. q = C e - t / R C q=C\mathcal{E}e^{-t/RC}\,\!
  97. L d I d t + R I = L\frac{\mathrm{d}I}{\mathrm{d}t}+RI=\mathcal{E}\,\!
  98. I = R ( 1 - e - R t / L ) I=\frac{\mathcal{E}}{R}\left(1-e^{-Rt/L}\right)\,\!
  99. I = R e - t / τ L = I 0 e - R t / L I=\frac{\mathcal{E}}{R}e^{-t/\tau_{L}}=I_{0}e^{-Rt/L}\,\!
  100. L d 2 q d t 2 + q / C = L\frac{\mathrm{d}^{2}q}{\mathrm{d}t^{2}}+q/C=\mathcal{E}\,\!
  101. L d 2 q d t 2 + q / C = sin ( ω 0 t + ϕ ) L\frac{\mathrm{d}^{2}q}{\mathrm{d}t^{2}}+q/C=\mathcal{E}\sin\left(\omega_{0}t+% \phi\right)\,\!
  102. ω res = 1 / L C \omega_{\mathrm{res}}=1/\sqrt{LC}\,\!
  103. q = q 0 cos ( ω t + ϕ ) q=q_{0}\cos(\omega t+\phi)\,\!
  104. I = - ω q 0 sin ( ω t + ϕ ) I=-\omega q_{0}\sin(\omega t+\phi)\,\!
  105. U E = q 2 / 2 C = Q 2 cos 2 ( ω t + ϕ ) / 2 C U_{E}=q^{2}/2C=Q^{2}\cos^{2}(\omega t+\phi)/2C\,\!
  106. U B = Q 2 sin 2 ( ω t + ϕ ) / 2 C U_{B}=Q^{2}\sin^{2}(\omega t+\phi)/2C\,\!
  107. L d 2 q d t 2 + R d q d t + q C = L\frac{\mathrm{d}^{2}q}{\mathrm{d}t^{2}}+R\frac{\mathrm{d}q}{\mathrm{d}t}+% \frac{q}{C}=\mathcal{E}\,\!
  108. L d 2 q d t 2 + R d q d t + q C = sin ( ω 0 t + ϕ ) L\frac{\mathrm{d}^{2}q}{\mathrm{d}t^{2}}+R\frac{\mathrm{d}q}{\mathrm{d}t}+% \frac{q}{C}=\mathcal{E}\sin\left(\omega_{0}t+\phi\right)\,\!
  109. q = q 0 e T - R t / 2 L cos ( ω t + ϕ ) q=q_{0}eT^{-Rt/2L}\cos(\omega^{\prime}t+\phi)\,\!

List_of_equations_in_fluid_mechanics.html

  1. 𝐭 ^ \mathbf{\hat{t}}\,\!
  2. 𝐮 = 𝐮 ( 𝐫 , t ) \mathbf{u}=\mathbf{u}\left(\mathbf{r},t\right)\,\!
  3. s y m b o l ω = × 𝐯 symbol{\omega}=\nabla\times\mathbf{v}
  4. ϕ V = S 𝐮 d 𝐀 \phi_{V}=\int_{S}\mathbf{u}\cdot\mathrm{d}\mathbf{A}\,\!
  5. s = d ρ / d t s=\mathrm{d}\rho/\mathrm{d}t\,\!
  6. I m = d m / d t I_{\mathrm{m}}=\mathrm{d}m/\mathrm{d}t\,\!
  7. I m = 𝐣 m d 𝐒 I_{\mathrm{m}}=\iint\mathbf{j}_{\mathrm{m}}\cdot\mathrm{d}\mathbf{S}\,\!
  8. I p = d | 𝐩 | / d t I_{\mathrm{p}}=\mathrm{d}\left|\mathbf{p}\right|/\mathrm{d}t\,\!
  9. I p = 𝐣 p d 𝐒 I_{\mathrm{p}}=\iint\mathbf{j}_{\mathrm{p}}\cdot\mathrm{d}\mathbf{S}
  10. P = ρ 𝐠 \nabla P=\rho\mathbf{g}\,\!
  11. 𝐅 b = - ρ f V imm 𝐠 = - 𝐅 g \mathbf{F}_{\mathrm{b}}=-\rho_{f}V_{\mathrm{imm}}\mathbf{g}=-\mathbf{F}_{% \mathrm{g}}\,\!
  12. 𝐖 app = 𝐖 - 𝐅 b \mathbf{W}_{\mathrm{app}}=\mathbf{W}-\mathbf{F}_{\mathrm{b}}\,\!
  13. p + ρ u 2 / 2 + ρ g y = p constant p+\rho u^{2}/2+\rho gy=p_{\mathrm{constant}}\,\!
  14. \otimes
  15. ρ t + ( ρ 𝐮 ) = 0 \frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\mathbf{u})=0\,\!
  16. ρ 𝐮 t + ( 𝐮 ( ρ 𝐮 ) ) + p = 0 \frac{\partial\rho{\mathbf{u}}}{\partial t}+\nabla\cdot\left(\mathbf{u}\otimes% \left(\rho\mathbf{u}\right)\right)+\nabla p=0\,\!
  17. E t + ( u ( E + p ) ) = 0 \frac{\partial E}{\partial t}+\nabla\cdot\left(u\left(E+p\right)\right)=0\,\!
  18. E = ρ ( U + 1 2 𝐮 2 ) E=\rho\left(U+\frac{1}{2}\mathbf{u}^{2}\right)\,\!
  19. 𝐚 = ( 𝐮 ) 𝐮 \mathbf{a}=\left(\mathbf{u}\cdot\nabla\right)\mathbf{u}
  20. 𝐟 \mathbf{f}
  21. \nabla
  22. ρ ( 𝐮 t + 𝐮 𝐮 ) = - p + 𝐓 D + 𝐟 \rho\left(\frac{\partial\mathbf{u}}{\partial t}+\mathbf{u}\cdot\nabla\mathbf{u% }\right)=-\nabla p+\nabla\cdot\mathbf{T}_{\mathrm{D}}+\mathbf{f}

List_of_equations_in_gravitation.html

  1. 𝐦 i = 𝐫 i m i \mathbf{m}_{i}=\mathbf{r}_{i}m_{i}\,\!
  2. 𝐫 cog = 1 M | 𝐠 ( 𝐫 i ) | i 𝐦 i | 𝐠 ( 𝐫 i ) | = 1 M | 𝐠 ( 𝐫 cog ) | i 𝐫 i m i | 𝐠 ( 𝐫 i ) | \begin{aligned}\displaystyle\mathbf{r}_{\mathrm{cog}}&\displaystyle=\frac{1}{M% \left|\mathbf{g}\left(\mathbf{r}_{i}\right)\right|}\sum_{i}\mathbf{m}_{i}\left% |\mathbf{g}\left(\mathbf{r}_{i}\right)\right|\\ &\displaystyle=\frac{1}{M\left|\mathbf{g}\left(\mathbf{r}_{\mathrm{cog}}\right% )\right|}\sum_{i}\mathbf{r}_{i}m_{i}\left|\mathbf{g}\left(\mathbf{r}_{i}\right% )\right|\end{aligned}\,\!
  3. 𝐫 cog = 1 M | 𝐠 ( 𝐫 cog ) | | 𝐠 ( 𝐫 ) | d 𝐦 = 1 M | 𝐠 ( 𝐫 cog ) | 𝐫 | 𝐠 ( 𝐫 ) | d n m = 1 M | 𝐠 ( 𝐫 cog ) | 𝐫 ρ n | 𝐠 ( 𝐫 ) | d n x \begin{aligned}\displaystyle\mathbf{r}_{\mathrm{cog}}&\displaystyle=\frac{1}{M% \left|\mathbf{g}\left(\mathbf{r}_{\mathrm{cog}}\right)\right|}\int\left|% \mathbf{g}\left(\mathbf{r}\right)\right|\mathrm{d}\mathbf{m}\\ &\displaystyle=\frac{1}{M\left|\mathbf{g}\left(\mathbf{r}_{\mathrm{cog}}\right% )\right|}\int\mathbf{r}\left|\mathbf{g}\left(\mathbf{r}\right)\right|\mathrm{d% }^{n}m\\ &\displaystyle=\frac{1}{M\left|\mathbf{g}\left(\mathbf{r}_{\mathrm{cog}}\right% )\right|}\int\mathbf{r}\rho_{n}\left|\mathbf{g}\left(\mathbf{r}\right)\right|% \mathrm{d}^{n}x\end{aligned}\,\!
  4. μ = G m \mu=Gm\,\!
  5. 𝐠 = 𝐅 / m \mathbf{g}=\mathbf{F}/m\,\!
  6. Φ G = S 𝐠 d 𝐀 \Phi_{G}=\int_{S}\mathbf{g}\cdot\mathrm{d}\mathbf{A}\,\!
  7. U = - W r m = - 1 m r 𝐅 d 𝐫 = - r 𝐠 d 𝐫 U=-\frac{W_{\infty r}}{m}=-\frac{1}{m}\int_{\infty}^{r}\mathbf{F}\cdot\mathrm{% d}\mathbf{r}=-\int_{\infty}^{r}\mathbf{g}\cdot\mathrm{d}\mathbf{r}\,\!
  8. Δ U = - W m = - 1 m r 1 r 2 𝐅 d 𝐫 = - r 1 r 2 𝐠 d 𝐫 \Delta U=-\frac{W}{m}=-\frac{1}{m}\int_{r_{1}}^{r_{2}}\mathbf{F}\cdot\mathrm{d% }\mathbf{r}=-\int_{r_{1}}^{r_{2}}\mathbf{g}\cdot\mathrm{d}\mathbf{r}\,\!
  9. E p = - W r E_{p}=-W_{\infty r}\,\!
  10. s y m b o l Ω = 2 s y m b o l ξ symbol{\Omega}=2symbol{\xi}\,\!
  11. Φ Ω = S s y m b o l Ω d 𝐀 \Phi_{\Omega}=\int_{S}symbol{\Omega}\cdot\mathrm{d}\mathbf{A}\,\!
  12. 𝐅 = m ( 𝐯 × 2 s y m b o l ξ ) \mathbf{F}=m\left(\mathbf{v}\times 2symbol{\xi}\right)\,\!
  13. Φ ξ = S s y m b o l ξ d 𝐀 \Phi_{\xi}=\int_{S}symbol{\xi}\cdot\mathrm{d}\mathbf{A}\,\!
  14. ξ = × 𝐡 \mathbf{\xi}=\nabla\times\mathbf{h}\,\!
  15. 𝐠 = - U \mathbf{g}=-\nabla U
  16. Δ U = - C 𝐠 d 𝐫 \Delta U=-\int_{C}\mathbf{g}\cdot d\mathbf{r}\,\!
  17. 𝐠 = G m | 𝐫 | 2 𝐫 ^ \mathbf{g}=\frac{Gm}{\left|\mathbf{r}\right|^{2}}\mathbf{\hat{r}}\,\!
  18. 𝐠 = i 𝐠 i = G i m i | 𝐫 i - 𝐫 | 2 𝐫 ^ i \mathbf{g}=\sum_{i}\mathbf{g}_{i}=G\sum_{i}\frac{m_{i}}{\left|\mathbf{r}_{i}-% \mathbf{r}\right|^{2}}\mathbf{\hat{r}}_{i}\,\!
  19. s y m b o l τ = V n d 𝐦 × 𝐠 symbol{\tau}=\int_{V_{n}}\mathrm{d}\mathbf{m}\times\mathbf{g}\,\!
  20. U = V n d 𝐦 𝐠 U=\int_{V_{n}}\mathrm{d}\mathbf{m}\cdot\mathbf{g}\,\!
  21. ϕ \phi
  22. 𝐚 ^ \mathbf{\hat{a}}\,\!
  23. 𝐠 = - G M | 𝐫 | 2 𝐫 ^ - ( | s y m b o l ω | 2 | 𝐫 | sin ϕ ) 𝐚 ^ \mathbf{g}=-\frac{GM}{\left|\mathbf{r}\right|^{2}}\mathbf{\hat{r}}-(\left|% symbol{\omega}\right|^{2}\left|\mathbf{r}\right|\sin\phi)\mathbf{\hat{a}}\,\!
  24. U = - G m 1 m 2 | 𝐫 | m | 𝐠 | y U=-\frac{Gm_{1}m_{2}}{\left|\mathbf{r}\right|}\approx m\left|\mathbf{g}\right|% y\,\!
  25. v = 2 G M r v=\sqrt{\frac{2GM}{r}}\,\!
  26. E = T + U = - G m M | 𝐫 | + 1 2 m | 𝐯 | 2 = m ( - G M | 𝐫 | + | s y m b o l ω × 𝐫 | 2 2 ) = - G m M 2 | 𝐫 | \begin{aligned}\displaystyle E&\displaystyle=T+U\\ &\displaystyle=-\frac{GmM}{\left|\mathbf{r}\right|}+\frac{1}{2}m\left|\mathbf{% v}\right|^{2}\\ &\displaystyle=m\left(-\frac{GM}{\left|\mathbf{r}\right|}+\frac{\left|symbol{% \omega}\times\mathbf{r}\right|^{2}}{2}\right)\\ &\displaystyle=-\frac{GmM}{2\left|\mathbf{r}\right|}\end{aligned}\,\!
  27. s y m b o l ξ = G 2 c 2 𝐋 - 3 ( 𝐋 𝐫 ^ ) 𝐫 ^ | 𝐫 | 3 symbol{\xi}=\frac{G}{2c^{2}}\frac{\mathbf{L}-3(\mathbf{L}\cdot\mathbf{\hat{r}}% )\mathbf{\hat{r}}}{\left|\mathbf{r}\right|^{3}}

List_of_equations_in_nuclear_and_particle_physics.html

  1. N 0 = N + N D N_{0}=N+N_{D}\,\!
  2. A = d N / d t A=\mathrm{d}N/\mathrm{d}t\,\!
  3. λ = A / N \lambda=A/N\,\!
  4. t t + T 1 / 2 t\rightarrow t+T_{1/2}\,\!
  5. N N / 2 N\rightarrow N/2\,\!
  6. n = t / T 1 / 2 n=t/T_{1/2}\,\!
  7. τ = 1 / λ \tau=1/\lambda\,\!
  8. H = D Q H=DQ\,\!
  9. E = j H j W j E=\sum_{j}H_{j}W_{j}\,\!
  10. j W j = 1 \sum_{j}W_{j}=1\,\!
  11. A = Z + N A=Z+N\,\!
  12. M Σ = Z m p + N m n M_{\Sigma}=Zm_{p}+Nm_{n}\,\!
  13. M Σ > M N M_{\Sigma}>M_{N}\,\!
  14. Δ M = M Σ - M nuc \Delta M=M_{\Sigma}-M_{\mathrm{nuc}}\,\!
  15. Δ E = Δ M c 2 \Delta E=\Delta Mc^{2}\,\!
  16. r = r 0 A 1 / 3 r=r_{0}A^{1/3}\,\!
  17. E B = a v A - a s A 2 / 3 - a c Z ( Z - 1 ) A - 1 / 3 - a a ( N - Z ) 2 A - 1 + 12 δ ( N , Z ) A - 1 / 2 \begin{aligned}\displaystyle E_{B}=&\displaystyle a_{v}A-a_{s}A^{2/3}-a_{c}Z(Z% -1)A^{-1/3}\\ &\displaystyle-a_{a}(N-Z)^{2}A^{-1}+12\delta(N,Z)A^{-1/2}\\ \end{aligned}
  18. d N d t = - λ N \frac{\mathrm{d}N}{\mathrm{d}t}=-\lambda N
  19. N = N 0 e - λ t N=N_{0}e^{-\lambda t}\,\!
  20. c i = j = 1 , i j D λ j λ j - λ i c_{i}=\prod_{j=1,i\neq j}^{D}\frac{\lambda_{j}}{\lambda_{j}-\lambda_{i}}
  21. N D = N 1 ( 0 ) λ D i = 1 D λ i c i e - λ i t N_{D}=\frac{N_{1}(0)}{\lambda_{D}}\sum_{i=1}^{D}\lambda_{i}c_{i}e^{-\lambda_{i% }t}
  22. I = I 0 e - μ x I=I_{0}e^{-\mu x}\,\!
  23. σ ( E ) = π g k 2 Γ a b Γ c ( E - E 0 ) 2 + Γ 2 / 4 \sigma(E)=\frac{\pi g}{k^{2}}\frac{\Gamma_{ab}\Gamma_{c}}{(E-E_{0})^{2}+\Gamma% ^{2}/4}
  24. g = 2 J + 1 ( 2 s a + 1 ) ( 2 s b + 1 ) g=\frac{2J+1}{(2s_{a}+1)(2s_{b}+1)}
  25. Γ = Γ a b + Γ c \Gamma=\Gamma_{ab}+\Gamma_{c}
  26. τ = / Γ \tau=\hbar/\Gamma
  27. d σ d Ω = | 2 μ 2 0 sin ( Δ k r ) Δ k r V ( r ) r 2 d r | 2 \frac{d\sigma}{d\Omega}=\left|\frac{2\mu}{\hbar^{2}}\int_{0}^{\infty}\frac{% \sin(\Delta kr)}{\Delta kr}V(r)r^{2}dr\right|^{2}
  28. d σ d Ω = ( α 4 E ) [ csc 4 χ 2 + sec 4 χ 2 + A cos ( α ν ln tan 2 χ 2 ) sin 2 χ 2 cos χ 2 ] 2 \frac{d\sigma}{d\Omega}=\left(\frac{\alpha}{4E}\right)\left[\csc^{4}\frac{\chi% }{2}+\sec^{4}\frac{\chi}{2}+\frac{A\cos\left(\frac{\alpha}{\hbar\nu}\ln\tan^{2% }\frac{\chi}{2}\right)}{\sin^{2}\frac{\chi}{2}\cos\frac{\chi}{2}}\right]^{2}
  29. V = - α / r V=-\alpha/r
  30. d σ d Ω = ( 1 n ) d N d Ω = ( α 4 E ) 2 csc 4 χ 2 \frac{d\sigma}{d\Omega}=\left(\frac{1}{n}\right)\frac{dN}{d\Omega}=\left(\frac% {\alpha}{4E}\right)^{2}\csc^{4}\frac{\chi}{2}
  31. QCD = ψ ¯ i ( i γ μ ( D μ ) i j - m δ i j ) ψ j - 1 4 G μ ν a G a μ ν = ψ ¯ i ( i γ μ μ - m ) ψ i - g G μ a ψ ¯ i γ μ T i j a ψ j - 1 4 G μ ν a G a μ ν , \begin{aligned}\displaystyle\mathcal{L}_{\mathrm{QCD}}&\displaystyle=\bar{\psi% }_{i}\left(i\gamma^{\mu}(D_{\mu})_{ij}-m\,\delta_{ij}\right)\psi_{j}-\frac{1}{% 4}G^{a}_{\mu\nu}G^{\mu\nu}_{a}\\ &\displaystyle=\bar{\psi}_{i}(i\gamma^{\mu}\partial_{\mu}-m)\psi_{i}-gG^{a}_{% \mu}\bar{\psi}_{i}\gamma^{\mu}T^{a}_{ij}\psi_{j}-\frac{1}{4}G^{a}_{\mu\nu}G^{% \mu\nu}_{a}\,,\\ \end{aligned}\,\!
  32. E W = g + f + h + y . \mathcal{L}_{EW}=\mathcal{L}_{g}+\mathcal{L}_{f}+\mathcal{L}_{h}+\mathcal{L}_{% y}.\,\!
  33. g = - 1 4 W a μ ν W μ ν a - 1 4 B μ ν B μ ν \mathcal{L}_{g}=-\frac{1}{4}W_{a}^{\mu\nu}W_{\mu\nu}^{a}-\frac{1}{4}B^{\mu\nu}% B_{\mu\nu}\,\!
  34. f = Q ¯ i i D / Q i + u ¯ i c i D / u i c + d ¯ i c i D / d i c + L ¯ i i D / L i + e ¯ i c i D / e i c \mathcal{L}_{f}=\overline{Q}_{i}iD\!\!\!\!/\;Q_{i}+\overline{u}_{i}^{c}iD\!\!% \!\!/\;u^{c}_{i}+\overline{d}_{i}^{c}iD\!\!\!\!/\;d^{c}_{i}+\overline{L}_{i}iD% \!\!\!\!/\;L_{i}+\overline{e}^{c}_{i}iD\!\!\!\!/\;e^{c}_{i}\,\!
  35. h = | D μ h | 2 - λ ( | h | 2 - v 2 2 ) 2 \mathcal{L}_{h}=|D_{\mu}h|^{2}-\lambda\left(|h|^{2}-\frac{v^{2}}{2}\right)^{2}\,\!
  36. y = - y u i j ϵ a b h b Q ¯ i a u j c - y d i j h Q ¯ i d j c - y e i j h L ¯ i e j c + h . c . \mathcal{L}_{y}=-y_{u\,ij}\epsilon^{ab}\,h_{b}^{\dagger}\,\overline{Q}_{ia}u_{% j}^{c}-y_{d\,ij}\,h\,\overline{Q}_{i}d^{c}_{j}-y_{e\,ij}\,h\,\overline{L}_{i}e% ^{c}_{j}+h.c.\,\!
  37. = ψ ¯ ( i γ μ D μ - m ) ψ - 1 4 F μ ν F μ ν , \mathcal{L}=\bar{\psi}(i\gamma^{\mu}D_{\mu}-m)\psi-\frac{1}{4}F_{\mu\nu}F^{\mu% \nu}\;,\,\!

List_of_equations_in_quantum_mechanics.html

  1. ρ = | Ψ | 2 = Ψ * Ψ \rho=\left|\Psi\right|^{2}=\Psi^{*}\Psi
  2. 𝐣 = - i 2 m ( Ψ * Ψ - Ψ Ψ * ) \mathbf{j}=\frac{-i\hbar}{2m}\left(\Psi^{*}\nabla\Psi-\Psi\nabla\Psi^{*}\right)
  3. = m Im ( Ψ * Ψ ) = Re ( Ψ * i m Ψ ) =\frac{\hbar}{m}\mathrm{Im}(\Psi^{*}\nabla\Psi)=\mathrm{Re}(\Psi^{*}\frac{% \hbar}{im}\nabla\Psi)
  4. Ψ = Ψ ( 𝐫 , 𝐬 𝐳 , t ) \Psi=\Psi\left(\mathbf{r},\mathbf{s_{z}},t\right)
  5. | Ψ = s z 1 s z 2 s z N V 1 V 2 V N d 𝐫 1 d 𝐫 2 d 𝐫 N Ψ | 𝐫 , 𝐬 𝐳 |\Psi\rangle=\sum_{s_{z1}}\sum_{s_{z2}}\cdots\sum_{s_{zN}}\int_{V_{1}}\int_{V_% {2}}\cdots\int_{V_{N}}\mathrm{d}\mathbf{r}_{1}\mathrm{d}\mathbf{r}_{2}\cdots% \mathrm{d}\mathbf{r}_{N}\Psi|\mathbf{r},\mathbf{s_{z}}\rangle
  6. Ψ = n = 1 N Ψ ( 𝐫 n , s z n , t ) \Psi=\prod_{n=1}^{N}\Psi\left(\mathbf{r}_{n},s_{zn},t\right)
  7. Φ ( 𝐩 , s z , t ) = 1 2 π all space e - i 𝐩 𝐫 / Ψ ( 𝐫 , s z , t ) d 3 𝐫 Ψ ( 𝐫 , s z , t ) = 1 2 π all space e + i 𝐩 𝐫 / Φ ( 𝐩 , s z , t ) d 3 𝐩 n \begin{aligned}\displaystyle\Phi(\mathbf{p},s_{z},t)&\displaystyle=\frac{1}{% \sqrt{2\pi\hbar}}\int\limits_{\mathrm{all\,space}}e^{-i\mathbf{p}\cdot\mathbf{% r}/\hbar}\Psi(\mathbf{r},s_{z},t)\mathrm{d}^{3}\mathbf{r}\\ &\displaystyle\upharpoonleft\downharpoonright\\ \displaystyle\Psi(\mathbf{r},s_{z},t)&\displaystyle=\frac{1}{\sqrt{2\pi\hbar}}% \int\limits_{\mathrm{all\,space}}e^{+i\mathbf{p}\cdot\mathbf{r}/\hbar}\Phi(% \mathbf{p},s_{z},t)\mathrm{d}^{3}\mathbf{p}_{n}\\ \end{aligned}
  8. P = s z N s z 2 s z 1 V N V 2 V 1 | Ψ | 2 d 3 𝐫 1 d 3 𝐫 2 d 3 𝐫 N P=\sum_{s_{zN}}\cdots\sum_{s_{z2}}\sum_{s_{z1}}\int_{V_{N}}\cdots\int_{V_{2}}% \int_{V_{1}}\left|\Psi\right|^{2}\mathrm{d}^{3}\mathbf{r}_{1}\mathrm{d}^{3}% \mathbf{r}_{2}\cdots\mathrm{d}^{3}\mathbf{r}_{N}\,\!
  9. P = s z N s z 2 s z 1 all space all space all space | Ψ | 2 d 3 𝐫 1 d 3 𝐫 2 d 3 𝐫 N = 1 P=\sum_{s_{zN}}\cdots\sum_{s_{z2}}\sum_{s_{z1}}\int\limits_{\mathrm{all\,space% }}\cdots\int\limits_{\mathrm{all\,space}}\int\limits_{\mathrm{all\,space}}% \left|\Psi\right|^{2}\mathrm{d}^{3}\mathbf{r}_{1}\mathrm{d}^{3}\mathbf{r}_{2}% \cdots\mathrm{d}^{3}\mathbf{r}_{N}=1\,\!
  10. 𝐏 = ( E / c , 𝐩 ) = ( ω / c , 𝐤 ) = 𝐊 \mathbf{P}=(E/c,\mathbf{p})=\hbar(\omega/c,\mathbf{k})=\hbar\mathbf{K}
  11. i t Ψ = H ^ Ψ i\hbar\frac{\partial}{\partial t}\Psi=\hat{H}\Psi
  12. H ^ Ψ = E Ψ \hat{H}\Psi=E\Psi
  13. \langle\,\rangle
  14. d d t A ^ ( t ) = i [ H ^ , A ^ ( t ) ] + A ^ ( t ) t , \frac{d}{dt}\hat{A}(t)=\frac{i}{\hbar}[\hat{H},\hat{A}(t)]+\frac{\partial\hat{% A}(t)}{\partial t},
  15. d d t A ^ = 1 i [ A ^ , H ^ ] + A ^ t \frac{d}{dt}\langle\hat{A}\rangle=\frac{1}{i\hbar}\langle[\hat{A},\hat{H}]% \rangle+\left\langle\frac{\partial\hat{A}}{\partial t}\right\rangle
  16. m d d t 𝐫 = 𝐩 m\frac{d}{dt}\langle\mathbf{r}\rangle=\langle\mathbf{p}\rangle
  17. d d t 𝐩 = - V \frac{d}{dt}\langle\mathbf{p}\rangle=-\langle\nabla V\rangle
  18. H ^ = p ^ 2 2 m + V ( x ) = - 2 2 m d 2 d x 2 + V ( x ) \hat{H}=\frac{\hat{p}^{2}}{2m}+V(x)=-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}+% V(x)
  19. H ^ = n = 1 N p ^ n 2 2 m n + V ( x 1 , x 2 , x N ) = - 2 2 n = 1 N 1 m n 2 x n 2 + V ( x 1 , x 2 , x N ) \begin{aligned}\displaystyle\hat{H}&\displaystyle=\sum_{n=1}^{N}\frac{\hat{p}_% {n}^{2}}{2m_{n}}+V(x_{1},x_{2},\cdots x_{N})\\ &\displaystyle=-\frac{\hbar^{2}}{2}\sum_{n=1}^{N}\frac{1}{m_{n}}\frac{\partial% ^{2}}{\partial x_{n}^{2}}+V(x_{1},x_{2},\cdots x_{N})\end{aligned}
  20. E Ψ = - 2 2 m d 2 d x 2 Ψ + V Ψ E\Psi=-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}\Psi+V\Psi
  21. E Ψ = - 2 2 n = 1 N 1 m n 2 x n 2 Ψ + V Ψ . E\Psi=-\frac{\hbar^{2}}{2}\sum_{n=1}^{N}\frac{1}{m_{n}}\frac{\partial^{2}}{% \partial x_{n}^{2}}\Psi+V\Psi\,.
  22. Ψ ( x , t ) = ψ ( x ) e - i E t / . \Psi(x,t)=\psi(x)e^{-iEt/\hbar}\,.
  23. ψ 2 = | ψ ( x ) | 2 d x . \|\psi\|^{2}=\int|\psi(x)|^{2}\,dx.\,
  24. Ψ = e - i E t / ψ ( x 1 , x 2 x N ) \Psi=e^{-iEt/\hbar}\psi(x_{1},x_{2}\cdots x_{N})
  25. Ψ = e - i E t / n = 1 N ψ ( x n ) , V ( x 1 , x 2 , x N ) = n = 1 N V ( x n ) . \Psi=e^{-i{Et/\hbar}}\prod_{n=1}^{N}\psi(x_{n})\,,\quad V(x_{1},x_{2},\cdots x% _{N})=\sum_{n=1}^{N}V(x_{n})\,.
  26. H ^ = 𝐩 ^ 𝐩 ^ 2 m + V ( 𝐫 ) = - 2 2 m 2 + V ( 𝐫 ) \begin{aligned}\displaystyle\hat{H}&\displaystyle=\frac{\hat{\mathbf{p}}\cdot% \hat{\mathbf{p}}}{2m}+V(\mathbf{r})\\ &\displaystyle=-\frac{\hbar^{2}}{2m}\nabla^{2}+V(\mathbf{r})\end{aligned}
  27. H ^ = n = 1 N 𝐩 ^ n 𝐩 ^ n 2 m n + V ( 𝐫 1 , 𝐫 2 , 𝐫 N ) = - 2 2 n = 1 N 1 m n n 2 + V ( 𝐫 1 , 𝐫 2 , 𝐫 N ) \begin{aligned}\displaystyle\hat{H}&\displaystyle=\sum_{n=1}^{N}\frac{\hat{% \mathbf{p}}_{n}\cdot\hat{\mathbf{p}}_{n}}{2m_{n}}+V(\mathbf{r}_{1},\mathbf{r}_% {2},\cdots\mathbf{r}_{N})\\ &\displaystyle=-\frac{\hbar^{2}}{2}\sum_{n=1}^{N}\frac{1}{m_{n}}\nabla_{n}^{2}% +V(\mathbf{r}_{1},\mathbf{r}_{2},\cdots\mathbf{r}_{N})\end{aligned}
  28. n 2 = 2 x n 2 + 2 y n 2 + 2 z n 2 \nabla_{n}^{2}=\frac{\partial^{2}}{{\partial x_{n}}^{2}}+\frac{\partial^{2}}{{% \partial y_{n}}^{2}}+\frac{\partial^{2}}{{\partial z_{n}}^{2}}
  29. E Ψ = - 2 2 m 2 Ψ + V Ψ E\Psi=-\frac{\hbar^{2}}{2m}\nabla^{2}\Psi+V\Psi
  30. E Ψ = - 2 2 n = 1 N 1 m n n 2 Ψ + V Ψ E\Psi=-\frac{\hbar^{2}}{2}\sum_{n=1}^{N}\frac{1}{m_{n}}\nabla_{n}^{2}\Psi+V\Psi
  31. Ψ = ψ ( 𝐫 ) e - i E t / \Psi=\psi(\mathbf{r})e^{-iEt/\hbar}
  32. Ψ = e - i E t / ψ ( 𝐫 1 , 𝐫 2 𝐫 N ) \Psi=e^{-iEt/\hbar}\psi(\mathbf{r}_{1},\mathbf{r}_{2}\cdots\mathbf{r}_{N})
  33. Ψ = e - i E t / n = 1 N ψ ( 𝐫 n ) , V ( 𝐫 1 , 𝐫 2 , 𝐫 N ) = n = 1 N V ( 𝐫 n ) \Psi=e^{-i{Et/\hbar}}\prod_{n=1}^{N}\psi(\mathbf{r}_{n})\,,\quad V(\mathbf{r}_% {1},\mathbf{r}_{2},\cdots\mathbf{r}_{N})=\sum_{n=1}^{N}V(\mathbf{r}_{n})
  34. H ^ = p ^ 2 2 m + V ( x , t ) = - 2 2 m 2 x 2 + V ( x , t ) \hat{H}=\frac{\hat{p}^{2}}{2m}+V(x,t)=-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}% {\partial x^{2}}+V(x,t)
  35. H ^ = n = 1 N p ^ n 2 2 m n + V ( x 1 , x 2 , x N , t ) = - 2 2 n = 1 N 1 m n 2 x n 2 + V ( x 1 , x 2 , x N , t ) \hat{H}=\sum_{n=1}^{N}\frac{\hat{p}_{n}^{2}}{2m_{n}}+V(x_{1},x_{2},\cdots x_{N% },t)=-\frac{\hbar^{2}}{2}\sum_{n=1}^{N}\frac{1}{m_{n}}\frac{\partial^{2}}{% \partial x_{n}^{2}}+V(x_{1},x_{2},\cdots x_{N},t)
  36. i t Ψ = - 2 2 m 2 x 2 Ψ + V Ψ i\hbar\frac{\partial}{\partial t}\Psi=-\frac{\hbar^{2}}{2m}\frac{\partial^{2}}% {\partial x^{2}}\Psi+V\Psi
  37. i t Ψ = - 2 2 n = 1 N 1 m n 2 x n 2 Ψ + V Ψ . i\hbar\frac{\partial}{\partial t}\Psi=-\frac{\hbar^{2}}{2}\sum_{n=1}^{N}\frac{% 1}{m_{n}}\frac{\partial^{2}}{\partial x_{n}^{2}}\Psi+V\Psi\,.
  38. Ψ = Ψ ( x , t ) \Psi=\Psi(x,t)
  39. Ψ = Ψ ( x 1 , x 2 x N , t ) \Psi=\Psi(x_{1},x_{2}\cdots x_{N},t)
  40. H ^ = 𝐩 ^ 𝐩 ^ 2 m + V ( 𝐫 , t ) = - 2 2 m 2 + V ( 𝐫 , t ) \begin{aligned}\displaystyle\hat{H}&\displaystyle=\frac{\hat{\mathbf{p}}\cdot% \hat{\mathbf{p}}}{2m}+V(\mathbf{r},t)\\ &\displaystyle=-\frac{\hbar^{2}}{2m}\nabla^{2}+V(\mathbf{r},t)\\ \end{aligned}
  41. H ^ = n = 1 N 𝐩 ^ n 𝐩 ^ n 2 m n + V ( 𝐫 1 , 𝐫 2 , 𝐫 N , t ) = - 2 2 n = 1 N 1 m n n 2 + V ( 𝐫 1 , 𝐫 2 , 𝐫 N , t ) \begin{aligned}\displaystyle\hat{H}&\displaystyle=\sum_{n=1}^{N}\frac{\hat{% \mathbf{p}}_{n}\cdot\hat{\mathbf{p}}_{n}}{2m_{n}}+V(\mathbf{r}_{1},\mathbf{r}_% {2},\cdots\mathbf{r}_{N},t)\\ &\displaystyle=-\frac{\hbar^{2}}{2}\sum_{n=1}^{N}\frac{1}{m_{n}}\nabla_{n}^{2}% +V(\mathbf{r}_{1},\mathbf{r}_{2},\cdots\mathbf{r}_{N},t)\end{aligned}
  42. i t Ψ = - 2 2 m 2 Ψ + V Ψ i\hbar\frac{\partial}{\partial t}\Psi=-\frac{\hbar^{2}}{2m}\nabla^{2}\Psi+V\Psi
  43. i t Ψ = - 2 2 n = 1 N 1 m n n 2 Ψ + V Ψ i\hbar\frac{\partial}{\partial t}\Psi=-\frac{\hbar^{2}}{2}\sum_{n=1}^{N}\frac{% 1}{m_{n}}\nabla_{n}^{2}\Psi+V\Psi
  44. Ψ = Ψ ( 𝐫 , t ) \Psi=\Psi(\mathbf{r},t)
  45. Ψ = Ψ ( 𝐫 1 , 𝐫 2 , 𝐫 N , t ) \Psi=\Psi(\mathbf{r}_{1},\mathbf{r}_{2},\cdots\mathbf{r}_{N},t)
  46. K max = h f - Φ K_{\mathrm{max}}=hf-\Phi\,\!
  47. ϕ = h f 0 \phi=hf_{0}\,\!
  48. p = h f / c = h / λ p=hf/c=h/\lambda\,\!
  49. σ ( x ) σ ( p ) 2 \sigma(x)\sigma(p)\geq\frac{\hbar}{2}\,\!
  50. σ ( E ) σ ( t ) 2 \sigma(E)\sigma(t)\geq\frac{\hbar}{2}\,\!
  51. σ ( n ) σ ( ϕ ) 2 \sigma(n)\sigma(\phi)\geq\frac{\hbar}{2}\,\!
  52. σ ( A ) 2 = ( A - A ) 2 = A 2 - A 2 \begin{aligned}\displaystyle\sigma(A)^{2}&\displaystyle=\langle(A-\langle A% \rangle)^{2}\rangle\\ &\displaystyle=\langle A^{2}\rangle-\langle A\rangle^{2}\end{aligned}
  53. σ ( A ) σ ( B ) 1 2 i [ A ^ , B ^ ] \sigma(A)\sigma(B)\geq\frac{1}{2}\langle i[\hat{A},\hat{B}]\rangle
  54. N ( E ) = 8 2 π m 3 / 2 E 1 / 2 / h 3 N(E)=8\sqrt{2}\pi m^{3/2}E^{1/2}/h^{3}\,\!
  55. P ( E i ) = g ( E i ) / ( e ( E - μ ) / k T + 1 ) P(E_{i})=g(E_{i})/(e^{(E-\mu)/kT}+1)\,\!
  56. P ( E i ) = g ( E i ) / ( e ( E i - μ ) / k T - 1 ) P(E_{i})=g(E_{i})/(e^{(E_{i}-\mu)/kT}-1)\,\!
  57. m s { - s , - s + 1 s - 1 , s } m_{s}\in\{-s,-s+1\cdots s-1,s\}\,\!
  58. m { - , - + 1 - 1 , } m_{\ell}\in\{-\ell,-\ell+1\cdots\ell-1,\ell\}\,\!
  59. m { 0 n - 1 } m_{\ell}\in\{0\cdots n-1\}\,\!
  60. j = + s j { | - s | , | - s | + 1 | + s | - 1 , | + s | } \begin{aligned}&\displaystyle j=\ell+s\\ &\displaystyle j\in\{|\ell-s|,|\ell-s|+1\cdots|\ell+s|-1,|\ell+s|\}\\ \end{aligned}\,\!
  61. | 𝐒 | = s ( s + 1 ) |\mathbf{S}|=\hbar\sqrt{s(s+1)}\,\!
  62. | 𝐋 | = ( + 1 ) |\mathbf{L}|=\hbar\sqrt{\ell(\ell+1)}\,\!
  63. 𝐉 = 𝐋 + 𝐒 \mathbf{J}=\mathbf{L}+\mathbf{S}\,\!
  64. | 𝐉 | = j ( j + 1 ) |\mathbf{J}|=\hbar\sqrt{j(j+1)}\,\!
  65. S z = m s S_{z}=m_{s}\hbar\,\!
  66. L z = m L_{z}=m_{\ell}\hbar\,\!
  67. s y m b o l μ = - e 𝐋 / 2 m e = g μ B 𝐋 symbol{\mu}_{\ell}=-e\mathbf{L}/2m_{e}=g_{\ell}\frac{\mu_{B}}{\hbar}\mathbf{L}\,\!
  68. μ , z = - m μ B \mu_{\ell,z}=-m_{\ell}\mu_{B}\,\!
  69. s y m b o l μ s = - e 𝐒 / m e = g s μ B 𝐒 symbol{\mu}_{s}=-e\mathbf{S}/m_{e}=g_{s}\frac{\mu_{B}}{\hbar}\mathbf{S}\,\!
  70. μ s , z = - e S z / m e = g s e S z / 2 m e \mu_{s,z}=-eS_{z}/m_{e}=g_{s}eS_{z}/2m_{e}\,\!
  71. U = - s y m b o l μ 𝐁 = - μ z B U=-symbol{\mu}\cdot\mathbf{B}=-\mu_{z}B\,\!
  72. E n = - m e 4 / 8 ϵ 0 2 h 2 n 2 = 13.61 e V / n 2 E_{n}=-me^{4}/8\epsilon_{0}^{2}h^{2}n^{2}=13.61eV/n^{2}\,\!
  73. 1 λ = R ( 1 n j 2 - 1 n i 2 ) , n j < n i \frac{1}{\lambda}=R\left(\frac{1}{n_{j}^{2}}-\frac{1}{n_{i}^{2}}\right),\,n_{j% }<n_{i}\,\!

List_of_equations_in_wave_theory.html

  1. 𝐀 = A 𝐞 ^ \mathbf{A}=A\mathbf{\hat{e}}_{\parallel}\,\!
  2. 𝐀 = A 𝐞 ^ \mathbf{A}=A\mathbf{\hat{e}}_{\bot}\,\!
  3. 𝐫 r 𝐞 ^ 𝐝 - 𝐫 0 \mathbf{r}\equiv r\mathbf{\hat{e}}_{\parallel}\equiv\mathbf{d}-\mathbf{r}_{0}\,\!
  4. λ = d r / d N \lambda=\mathrm{d}r/\mathrm{d}N\,\!
  5. λ = Δ r / Δ N \lambda=\Delta r/\Delta N\,\!
  6. 𝐤 = ( 2 π / λ ) 𝐞 ^ \mathbf{k}=\left(2\pi/\lambda\right)\mathbf{\hat{e}}_{\angle}\,\!
  7. 𝐤 = ( 1 / λ ) 𝐞 ^ \mathbf{k}=\left(1/\lambda\right)\mathbf{\hat{e}}_{\angle}\,\!
  8. f = d N / d t f=\mathrm{d}N/\mathrm{d}t\,\!
  9. f = Δ N / Δ t f=\Delta N/\Delta t\,\!
  10. f = 1 / T f=1/T\,\!
  11. ω = 2 π f = 2 π / T \omega=2\pi f=2\pi/T\,\!
  12. 𝐯 = 𝐞 ^ ( A / t ) \mathbf{v}=\mathbf{\hat{e}}_{\parallel}\left(\partial A/\partial t\right)\,\!
  13. 𝐯 = 𝐞 ^ ( A / t ) \mathbf{v}=\mathbf{\hat{e}}_{\bot}\left(\partial A/\partial t\right)\,\!
  14. 𝐚 = 𝐞 ^ ( 2 A / t 2 ) \mathbf{a}=\mathbf{\hat{e}}_{\parallel}\left(\partial^{2}A/\partial t^{2}% \right)\,\!
  15. 𝐚 = 𝐞 ^ ( 2 A / t 2 ) \mathbf{a}=\mathbf{\hat{e}}_{\bot}\left(\partial^{2}A/\partial t^{2}\right)\,\!
  16. 𝐫 = 𝐫 2 - 𝐫 1 \mathbf{r}=\mathbf{r}_{2}-\mathbf{r}_{1}\,\!
  17. 𝐯 p = 𝐞 ^ ( Δ r / Δ t ) \mathbf{v}_{\mathrm{p}}=\mathbf{\hat{e}}_{\parallel}\left(\Delta r/\Delta t% \right)\,\!
  18. 𝐯 p = λ f 𝐞 ^ = ( ω / k ) 𝐞 ^ \mathbf{v}_{\mathrm{p}}=\lambda f\mathbf{\hat{e}}_{\parallel}=\left(\omega/k% \right)\mathbf{\hat{e}}_{\parallel}\,\!
  19. 𝐯 g = 𝐞 ^ ( ω / k ) \mathbf{v}_{\mathrm{g}}=\mathbf{\hat{e}}_{\parallel}\left(\partial\omega/% \partial k\right)\,\!
  20. Δ t = t 2 - t 1 \Delta t=t_{2}-t_{1}\,\!
  21. Δ ϕ = ϕ 2 - ϕ 1 \Delta\phi=\phi_{2}-\phi_{1}\,\!
  22. 𝐤 𝐫 ω t + ϕ = 2 π N \mathbf{k}\cdot\mathbf{r}\mp\omega t+\phi=2\pi N\,\!
  23. Δ r λ = Δ t T = ϕ 2 π \frac{\Delta r}{\lambda}=\frac{\Delta t}{T}=\frac{\phi}{2\pi}\,\!
  24. h A M = A / A m h_{AM}=A/A_{m}\,\!
  25. h F M = Δ f / f m h_{FM}=\Delta f/f_{m}\,\!
  26. h P M = Δ ϕ h_{PM}=\Delta\phi\,\!
  27. Z = ρ v Z=\rho v\,\!
  28. z = Z S z=ZS\,\!
  29. β = ( dB ) 10 log | I I 0 | \beta=\left(\mathrm{dB}\right)10\log\left|\frac{I}{I_{0}}\right|\,\!
  30. n , m 𝐙 n,m\in\mathbf{Z}\,\!
  31. f n = v λ n = n v 2 L = n f 1 f_{n}=\frac{v}{\lambda_{n}}=\frac{nv}{2L}=nf_{1}\,\!
  32. P = μ v ω 2 x m 2 / 2 \langle P\rangle=\mu v\omega^{2}x_{m}^{2}/2\,\!
  33. I = P 0 / ( Ω r 2 ) I=P_{0}/(\Omega r^{2})\,\!
  34. I = P / A = ρ v ω 2 s m 2 / 2 I=P/A=\rho v\omega^{2}s^{2}_{m}/2\,\!
  35. f beat = | f 2 - f 1 | f_{\mathrm{beat}}=\left|f_{2}-f_{1}\right|\,\!
  36. f r = f 0 V ± v r v v 0 f_{r}=f_{0}\frac{V\pm v_{r}}{v\mp v_{0}}\,\!
  37. sin θ = v v s \sin\theta=\frac{v}{v_{s}}\,\!
  38. p 0 = ( v ρ ω ) s 0 p_{0}=\left(v\rho\omega\right)s_{0}\,\!
  39. s = [ 2 s 0 cos ( ω t ) ] cos ( ω t ) s=\left[2s_{0}\cos\left(\omega^{\prime}t\right)\right]\cos\left(\omega t\right% )\,\!
  40. s = s 0 cos ( k r - ω t ) s=s_{0}\cos(kr-\omega t)\,\!
  41. p = p 0 sin ( k r - ω t ) p=p_{0}\sin(kr-\omega t)\,\!
  42. P = d E d t = - 32 5 G 4 c 5 ( m 1 m 2 ) 2 ( m 1 + m 2 ) r 5 P=\frac{\mathrm{d}E}{\mathrm{d}t}=-\frac{32}{5}\,\frac{G^{4}}{c^{5}}\,\frac{(m% _{1}m_{2})^{2}(m_{1}+m_{2})}{r^{5}}
  43. d r d t = - 64 5 G 3 c 5 ( m 1 m 2 ) ( m 1 + m 2 ) r 3 \frac{\mathrm{d}r}{\mathrm{d}t}=-\frac{64}{5}\frac{G^{3}}{c^{5}}\frac{(m_{1}m_% {2})(m_{1}+m_{2})}{r^{3}}
  44. t = 5 256 c 5 G 3 r 0 4 ( m 1 m 2 ) ( m 1 + m 2 ) t=\frac{5}{256}\frac{c^{5}}{G^{3}}\frac{r_{0}^{4}}{(m_{1}m_{2})(m_{1}+m_{2})}
  45. y net = i = 1 N y i y_{\mathrm{net}}=\sum_{i=1}^{N}y_{i}\,\!
  46. ω d = ω nat \omega_{d}=\omega_{\mathrm{nat}}\,\!
  47. Δ r λ = Δ t T = ϕ 2 π \frac{\Delta r}{\lambda}=\frac{\Delta t}{T}=\frac{\phi}{2\pi}\,\!
  48. n = λ Δ x n=\frac{\lambda}{\Delta x}\,\!
  49. n + 1 2 = λ Δ x n+\frac{1}{2}=\frac{\lambda}{\Delta x}\,\!
  50. v = p ρ = F μ v=\sqrt{\frac{p}{\rho}}=\sqrt{\frac{F}{\mu}}\,\!
  51. D ( ω , k ) = 0 D\left(\omega,k\right)=0
  52. ω = ω ( k ) \omega=\omega\left(k\right)
  53. A = A ( t ) A=A\left(t\right)
  54. f = f ( t ) f=f\left(t\right)
  55. 2 A = 1 v 2 2 A t 2 \nabla^{2}A=\frac{1}{v_{\parallel}^{2}}\frac{\partial^{2}A}{\partial t^{2}}\,\!
  56. A ( 𝐫 , t ) = A ( x - v t ) A\left(\mathbf{r},t\right)=A\left(x-v_{\parallel}t\right)\,\!
  57. A = A 0 e - b t sin ( k x - ω t + ϕ ) A=A_{0}e^{-bt}\sin\left(kx-\omega t+\phi\right)\,\!
  58. y t + α y y x + 3 y x 3 = 0 \frac{\partial y}{\partial t}+\alpha y\frac{\partial y}{\partial x}+\frac{% \partial^{3}y}{\partial x^{3}}=0\,\!
  59. A ( x , t ) = 3 v α sech 2 [ v 2 ( x - v t ) ] A(x,t)=\frac{3v_{\parallel}}{\alpha}\mathrm{sech}^{2}\left[\frac{\sqrt{v_{% \parallel}}}{2}\left(x-v_{\parallel}t\right)\right]\,\!
  60. A n = | A n | e i ( 𝐤 n 𝐫 - ω n t + ϕ n ) A_{n}=\left|A_{n}\right|e^{i\left(\mathbf{k}_{\mathrm{n}}\cdot\mathbf{r}-% \omega_{n}t+\phi_{n}\right)}\,\!
  61. A = n = 1 N A n A=\sum_{n=1}^{N}A_{n}\,\!
  62. A = A A * = n = 1 N m = 1 N | A n | | A m | cos [ ( 𝐤 n - 𝐤 m ) 𝐫 + ( ω n - ω m ) t + ( ϕ n - ϕ m ) ] A=\sqrt{AA^{*}}=\sqrt{\sum_{n=1}^{N}\sum_{m=1}^{N}\left|A_{n}\right|\left|A_{m% }\right|\cos\left[\left(\mathbf{k}_{n}-\mathbf{k}_{m}\right)\cdot\mathbf{r}+% \left(\omega_{n}-\omega_{m}\right)t+\left(\phi_{n}-\phi_{m}\right)\right]}\,\!
  63. y 1 + y 2 = A sin ( k x - ω t ) + A sin ( k x + ω t ) \begin{aligned}\displaystyle y_{1}+y_{2}&\displaystyle=A\sin\left(kx-\omega t% \right)\\ &\displaystyle+A\sin\left(kx+\omega t\right)\end{aligned}\,\!
  64. y = 2 A sin ( k x ) cos ( ω t ) y=2A\sin\left(kx\right)\cos\left(\omega t\right)\,\!
  65. ω = ω 1 + ω 2 2 \langle\omega\rangle=\frac{\omega_{1}+\omega_{2}}{2}\,\!
  66. k = k 1 + k 2 2 \langle k\rangle=\frac{k_{1}+k_{2}}{2}\,\!
  67. Δ ω = ω 1 - ω 2 \Delta\omega=\omega_{1}-\omega_{2}\,\!
  68. Δ k = k 1 - k 2 \Delta k=k_{1}-k_{2}\,\!
  69. y 1 + y 2 = A sin ( k 1 x - ω 1 t ) + A sin ( k 2 x + ω 2 t ) \begin{aligned}\displaystyle y_{1}+y_{2}&\displaystyle=A\sin\left(k_{1}x-% \omega_{1}t\right)\\ &\displaystyle+A\sin\left(k_{2}x+\omega_{2}t\right)\end{aligned}\,\!
  70. y = 2 A sin ( k x - ω t ) cos ( Δ k 2 x - Δ ω 2 t ) y=2A\sin\left(\langle k\rangle x-\langle\omega\rangle t\right)\cos\left(\frac{% \Delta k}{2}x-\frac{\Delta\omega}{2}t\right)\,\!
  71. y 1 + y 2 = 2 A sin ( k x - ω t ) + A sin ( k x + ω t + ϕ ) \begin{aligned}\displaystyle y_{1}+y_{2}&\displaystyle=2A\sin\left(kx-\omega t% \right)\\ &\displaystyle+A\sin\left(kx+\omega t+\phi\right)\end{aligned}\,\!
  72. y = 2 A cos ( ϕ 2 ) sin ( k x - ω t + ϕ 2 ) y=2A\cos\left(\frac{\phi}{2}\right)\sin\left(kx-\omega t+\frac{\phi}{2}\right)\,\!

List_of_Kings_XI_Punjab_cricketers.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_OEIS_sequences.html

  1. φ ( n ) φ(n)
  2. φ ( n ) φ(n)
  3. C n = 1 n + 1 ( 2 n n ) = ( 2 n ) ! ( n + 1 ) ! n ! = k = 2 n n + k k C_{n}=\frac{1}{n+1}{2n\choose n}=\frac{(2n)!}{(n+1)!\,n!}=\prod\limits_{k=2}^{% n}\frac{n+k}{k}\qquad
  4. γ = lim n ( k = 1 n 1 k - ln ( n ) ) = lim b 1 b ( 1 x - 1 x ) d x . \gamma=\lim_{n\rightarrow\infty}\left(\sum_{k=1}^{n}\frac{1}{k}-\ln(n)\right)=% \lim_{b\rightarrow\infty}\int_{1}^{b}\left({1\over\lfloor x\rfloor}-{1\over x}% \right)\,dx.
  5. φ = 1 + 5 2 = 1.61803 39887 . \varphi=\frac{1+\sqrt{5}}{2}=1.61803\,39887\ldots.
  6. τ = i = 0 t i 2 i + 1 \tau=\sum_{i=0}^{\infty}\frac{t_{i}}{2^{i+1}}
  7. σ 2 ( n ) = σ ( σ ( n ) ) = 2 n , \sigma^{2}(n)=\sigma(\sigma(n))=2n\,,
  8. { k 2 n + 1 : n } \left\{\,k2^{n}+1:n\in\mathbb{N}\,\right\}
  9. { k 2 n - 1 : n } \left\{\,k2^{n}-1:n\in\mathbb{N}\,\right\}
  10. π ( x ) - π ( x / 2 ) \pi(x)-\pi(x/2)
  11. 1 cosh t = 2 e t + e - t = n = 0 E n n ! t n \frac{1}{\cosh t}=\frac{2}{e^{t}+e^{-t}}=\sum_{n=0}^{\infty}\frac{E_{n}}{n!}% \cdot t^{n}\!

List_of_photonics_equations.html

  1. P = 1 / f P=1/f\,\!
  2. m = - x 2 / x 1 = y 2 / y 1 m=-x_{2}/x_{1}=y_{2}/y_{1}\,\!
  3. m = θ 2 / θ 1 m=\theta_{2}/\theta_{1}\,\!
  4. 𝐍 = 1 μ 0 𝐄 × 𝐁 = 𝐄 × 𝐇 \mathbf{N}=\frac{1}{\mu_{0}}\mathbf{E}\times\mathbf{B}=\mathbf{E}\times\mathbf% {H}\,\!
  5. Φ N = S 𝐍 d 𝐒 \Phi_{N}=\int_{S}\mathbf{N}\cdot\mathrm{d}\mathbf{S}\,\!
  6. E rms = E 2 = E / 2 E_{\mathrm{rms}}=\sqrt{\langle E^{2}\rangle}=E/\sqrt{2}\,\!
  7. p E M = U / c p_{EM}=U/c\,\!
  8. P E M = I / c = p E M / A t P_{EM}=I/c=p_{EM}/At\,\!
  9. 𝐧 ^ \mathbf{\hat{n}}\,\!
  10. d 𝐀 = 𝐧 ^ d A \mathrm{d}\mathbf{A}=\mathbf{\hat{n}}\mathrm{d}A\,\!
  11. 𝐞 ^ \mathbf{\hat{e}}_{\angle}\,\!
  12. 𝐧 ^ 𝐞 ^ d A = 𝐞 ^ d 𝐀 = cos θ d A \mathbf{\hat{n}}\cdot\mathbf{\hat{e}}_{\angle}\mathrm{d}A=\mathbf{\hat{e}}_{% \angle}\cdot\mathrm{d}\mathbf{A}=\cos\theta\mathrm{d}A\,\!
  13. H e = d Q / ( 𝐞 ^ d 𝐀 ) H_{e}=\mathrm{d}Q/\left(\mathbf{\hat{e}}_{\angle}\cdot\mathrm{d}\mathbf{A}% \right)\,\!
  14. ω e = d Q / d V \omega_{e}=\mathrm{d}Q/\mathrm{d}V\,\!
  15. Q = Φ d t Q=\int\Phi\mathrm{d}t
  16. Φ = I d Ω \Phi=I\mathrm{d}\Omega\,\!
  17. Φ = L ( 𝐞 ^ d 𝐀 ) d Ω \Phi=\iint L\left(\mathbf{\hat{e}}_{\angle}\cdot\mathrm{d}\mathbf{A}\right)% \mathrm{d}\Omega
  18. Φ = E ( 𝐞 ^ d 𝐀 ) \Phi=\int E\left(\mathbf{\hat{e}}_{\angle}\cdot\mathrm{d}\mathbf{A}\right)
  19. Φ = M ( 𝐞 ^ d 𝐀 ) \Phi=\int M\left(\mathbf{\hat{e}}_{\angle}\cdot\mathrm{d}\mathbf{A}\right)
  20. J = E + M J=E+M\,\!
  21. Q = Φ λ d λ d t Q=\iint\Phi_{\lambda}{\mathrm{d}\lambda\mathrm{d}t}
  22. Q = Φ ν d ν d t Q=\iint\Phi_{\nu}\mathrm{d}\nu\mathrm{d}t
  23. Φ = I λ d λ d Ω \Phi=\iint I_{\lambda}\mathrm{d}\lambda\mathrm{d}\Omega
  24. Φ = I ν d ν d Ω \Phi=\iint I_{\nu}\mathrm{d}\nu\mathrm{d}\Omega
  25. Φ = L λ d λ ( 𝐞 ^ d 𝐀 ) d Ω \Phi=\iiint L_{\lambda}\mathrm{d}\lambda\left(\mathbf{\hat{e}}_{\angle}\cdot% \mathrm{d}\mathbf{A}\right)\mathrm{d}\Omega
  26. Φ = L ν d ν ( 𝐞 ^ d 𝐀 ) d Ω \Phi=\iiint L_{\nu}\mathrm{d}\nu\left(\mathbf{\hat{e}}_{\angle}\cdot\mathrm{d}% \mathbf{A}\right)\mathrm{d}\Omega\,\!
  27. Φ = E λ d λ ( 𝐞 ^ d 𝐀 ) \Phi=\iint E_{\lambda}\mathrm{d}\lambda\left(\mathbf{\hat{e}}_{\angle}\cdot% \mathrm{d}\mathbf{A}\right)
  28. Φ = E ν d ν ( 𝐞 ^ d 𝐀 ) \Phi=\iint E_{\nu}\mathrm{d}\nu\left(\mathbf{\hat{e}}_{\angle}\cdot\mathrm{d}% \mathbf{A}\right)
  29. u \langle u\rangle\,\!
  30. u = 1 2 ( ϵ 𝐄 2 + μ 𝐁 2 ) \langle u\rangle=\frac{1}{2}\left(\epsilon\mathbf{E}^{2}+\mu\mathbf{B}^{2}% \right)\,\!
  31. 𝐩 p = q 𝐀 \mathbf{p}_{\mathrm{p}}=q\mathbf{A}\,\!
  32. 𝐩 k = m 𝐯 \mathbf{p}_{\mathrm{k}}=m\mathbf{v}\,\!
  33. 𝐩 = m 𝐯 + q 𝐀 \mathbf{p}=m\mathbf{v}+q\mathbf{A}\,\!
  34. 𝐒 \langle\mathbf{S}\rangle\,\!
  35. I = 𝐒 = E rms 2 / c μ 0 I=\langle\mathbf{S}\rangle=E^{2}_{\mathrm{rms}}/c\mu_{0}\,\!
  36. I = P 0 Ω | r | 2 I=\frac{P_{0}}{\Omega\left|r\right|^{2}}\,\!
  37. λ = λ 0 c - v c + v \lambda=\lambda_{0}\sqrt{\frac{c-v}{c+v}}\,\!
  38. v = | Δ λ | c / λ 0 v=|\Delta\lambda|c/\lambda_{0}\,\!
  39. cos θ = c n v = 1 v ϵ μ \cos\theta=\frac{c}{nv}=\frac{1}{v\sqrt{\epsilon\mu}}\,\!
  40. | 𝐄 | = ϵ μ | 𝐇 | \left|\mathbf{E}\right|=\sqrt{\frac{\epsilon}{\mu}}\left|\mathbf{H}\right|\,\!
  41. 𝐄 = 𝐄 0 sin ( k x - ω t ) \mathbf{E}=\mathbf{E}_{0}\sin(kx-\omega t)\,\!
  42. 𝐁 = 𝐁 0 sin ( k x - ω t ) \mathbf{B}=\mathbf{B}_{0}\sin(kx-\omega t)\,\!
  43. sin θ c = n 2 n 1 \sin\theta_{c}=\frac{n_{2}}{n_{1}}\,\!
  44. 1 x 1 + 1 x 2 = 1 f \frac{1}{x_{1}}+\frac{1}{x_{2}}=\frac{1}{f}\,\!
  45. 1 f = ( n lens n med - 1 ) ( 1 r 1 - 1 r 2 ) \frac{1}{f}=\left(\frac{{n}_{\mathrm{lens}}}{n_{\mathrm{med}}-1}\right)\left(% \frac{1}{r_{1}}-\frac{1}{r_{2}}\right)\,\!
  46. x 2 = - x 1 x_{2}=-x_{1}\,\!
  47. 1 x 1 + 1 x 2 = 1 f = 2 r \frac{1}{x_{1}}+\frac{1}{x_{2}}=\frac{1}{f}=\frac{2}{r}\,\!
  48. n 1 x 1 + n 2 x 2 = ( n 2 - n 1 ) r \frac{n_{1}}{x_{1}}+\frac{n_{2}}{x_{2}}=\frac{\left(n_{2}-n_{1}\right)}{r}\,\!
  49. n 1 n 2 = v 2 v 1 = λ 2 λ 1 = ϵ 1 μ 1 ϵ 2 μ 2 \frac{n_{1}}{n_{2}}=\frac{v_{2}}{v_{1}}=\frac{\lambda_{2}}{\lambda_{1}}=\sqrt{% \frac{\epsilon_{1}\mu_{1}}{\epsilon_{2}\mu_{2}}}\,\!
  50. tan θ B = n 2 / n 1 \tan\theta_{B}=n_{2}/n_{1}\,\!
  51. I = I 0 cos 2 θ I=I_{0}\cos^{2}\theta\,\!
  52. N λ / n 2 N\lambda/n_{2}\,\!
  53. 2 L = ( N + 1 / 2 ) λ / n 2 2L=(N+1/2)\lambda/n_{2}\,\!
  54. δ 2 π λ = a ( sin θ + sin α ) \frac{\delta}{2\pi}\lambda=a\left(\sin\theta+\sin\alpha\right)\,\!
  55. θ R = 1.22 λ / d \theta_{R}=1.22\lambda/\,\!d
  56. δ 2 π λ = 2 d sin θ \frac{\delta}{2\pi}\lambda=2d\sin\theta\,\!
  57. δ / 2 π = n \delta/2\pi=n\,\!
  58. δ / 2 π = n / 2 \delta/2\pi=n/2\,\!
  59. n 𝐍 n\in\mathbf{N}\,\!
  60. ϕ = 2 π a λ sin θ \phi=\frac{2\pi a}{\lambda}\sin\theta\,\!
  61. I = I 0 [ sin ( ϕ / 2 ) ( ϕ / 2 ) ] 2 I=I_{0}\left[\frac{\sin\left(\phi/2\right)}{\left(\phi/2\right)}\right]^{2}\,\!
  62. δ = 2 π d λ sin θ \delta=\frac{2\pi d}{\lambda}\sin\theta\,\!
  63. I = I 0 [ sin ( N δ / 2 ) sin ( δ / 2 ) ] 2 I=I_{0}\left[\frac{\sin\left(N\delta/2\right)}{\sin\left(\delta/2\right)}% \right]^{2}\,\!
  64. I = I 0 [ sin ( ϕ / 2 ) ( ϕ / 2 ) sin ( N δ / 2 ) sin ( δ / 2 ) ] 2 I=I_{0}\left[\frac{\sin\left(\phi/2\right)}{\left(\phi/2\right)}\frac{\sin% \left(N\delta/2\right)}{\sin\left(\delta/2\right)}\right]^{2}\,\!
  65. I = I 0 ( 2 J 1 ( k a sin θ ) k a sin θ ) 2 I=I_{0}\left(\frac{2J_{1}(ka\sin\theta)}{ka\sin\theta}\right)^{2}
  66. A ( 𝐫 ) aperture E inc ( 𝐫 ) e i k | 𝐫 - 𝐫 | 4 π | 𝐫 - 𝐫 | d x d y A\left(\mathbf{r}\right)\propto\iint_{\mathrm{aperture}}E_{\mathrm{inc}}\left(% \mathbf{r}^{\prime}\right)~{}\frac{e^{ik\left|\mathbf{r}-\mathbf{r}^{\prime}% \right|}}{4\pi\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}\mathrm{d}x^{\prime}% \mathrm{d}y^{\prime}
  67. A ( 𝐫 ) e i k r 4 π r aperture E inc ( 𝐫 ) e - i k [ sin θ ( cos ϕ x + sin ϕ y ) ] d x d y A\left(\mathbf{r}\right)\propto\frac{e^{ikr}}{4\pi r}\iint_{\mathrm{aperture}}% E_{\mathrm{inc}}\left(\mathbf{r}^{\prime}\right)e^{-ik\left[\sin\theta\left(% \cos\phi x^{\prime}+\sin\phi y^{\prime}\right)\right]}\mathrm{d}x^{\prime}% \mathrm{d}y^{\prime}
  68. 𝐧 ^ \mathbf{\hat{n}}\,\!
  69. 𝐫 0 𝐧 ^ = | 𝐫 0 | cos α 0 \mathbf{r}_{0}\cdot\mathbf{\hat{n}}=\left|\mathbf{r}_{0}\right|\cos\alpha_{0}\,\!
  70. 𝐫 𝐧 ^ = | 𝐫 | cos α \mathbf{r}\cdot\mathbf{\hat{n}}=\left|\mathbf{r}\right|\cos\alpha\,\!
  71. | 𝐫 | | 𝐫 0 | λ \left|\mathbf{r}\right|\left|\mathbf{r}_{0}\right|\ll\lambda\,\!
  72. A ( 𝐫 ) = - i 2 λ aperture e i 𝐤 ( 𝐫 + 𝐫 0 ) | 𝐫 | | 𝐫 0 | [ cos α 0 - cos α ] d S A\mathbf{(}\mathbf{r})=\frac{-i}{2\lambda}\iint_{\mathrm{aperture}}\frac{e^{i% \mathbf{k}\cdot\left(\mathbf{r}+\mathbf{r}_{0}\right)}}{\left|\mathbf{r}\right% |\left|\mathbf{r}_{0}\right|}\left[\cos\alpha_{0}-\cos\alpha\right]\mathrm{d}S\,\!
  73. A ( 𝐫 ) = - 1 4 π aperture e i 𝐤 𝐫 0 | 𝐫 0 | [ i | 𝐤 | U 0 ( 𝐫 0 ) cos α + A 0 ( 𝐫 0 ) n ] d S A\left(\mathbf{r}\right)=-\frac{1}{4\pi}\iint_{\mathrm{aperture}}\frac{e^{i% \mathbf{k}\cdot\mathbf{r}_{0}}}{\left|\mathbf{r}_{0}\right|}\left[i\left|% \mathbf{k}\right|U_{0}\left(\mathbf{r}_{0}\right)\cos{\alpha}+\frac{\partial A% _{0}\left(\mathbf{r}_{0}\right)}{\partial n}\right]\mathrm{d}S
  74. D L = L 4 π F D_{L}=\sqrt{\frac{L}{4\pi F}}\,
  75. m j = - 5 2 log 10 | F j F j 0 | m_{j}=-\frac{5}{2}\log_{10}\left|\frac{F_{j}}{F_{j}^{0}}\right|\,
  76. M = m - 5 [ ( log 10 D L ) - 1 ] M=m-5\left[\left(\log_{10}{D_{L}}\right)-1\right]\!\,
  77. μ = m - M \mu=m-M\!\,
  78. U - B = M U - M B U-B=M_{U}-M_{B}\!\,
  79. B - V = M B - M V B-V=M_{B}-M_{V}\!\,
  80. C bol = m bol - V = M bol - M V \begin{aligned}\displaystyle C_{\mathrm{bol}}&\displaystyle=m_{\mathrm{bol}}-V% \\ &\displaystyle=M_{\mathrm{bol}}-M_{V}\end{aligned}\!\,

List_of_Rajasthan_Royals_cricketers.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_Taylor_polyhedra.html

  1. { 4 4 / 2 } \left\{{4\atop 4/2}\right\}
  2. { 4 4 / 2 } \left\{{4\atop 4/2}\right\}
  3. { 4 4 / 2 } \left\{{4\atop 4/2}\right\}
  4. { 4 4 / 2 } \left\{{4\atop 4/2}\right\}
  5. { 4 4 / 2 } \left\{{4^{\prime}\atop 4/2}\right\}

Local_language_(formal_language).html

  1. ( R A * A * S ) A * F A * . (RA^{*}\cap A^{*}S)\setminus A^{*}FA^{*}\ .
  2. a a * , { a b } . aa^{*},\ \{ab\}\ .

Local_ternary_patterns.html

  1. { 1 , if p > c + k 0 , if p > c - k and p < c + k - 1 if p < c - k \begin{cases}1,&\,\text{if }p>c+k\\ 0,&\,\text{if }p>c-k\,\text{ and }p<c+k\\ -1&\,\text{if }p<c-k\\ \end{cases}

Locally_acyclic_morphism.html

  1. f : X S f:X\to S

Locally_finite_operator.html

  1. f : V V f:V\to V
  2. V V
  3. f f
  4. { V i | i I } \{V_{i}|i\in I\}
  5. V V
  6. i I V i = V \bigcup_{i\in I}V_{i}=V
  7. ( i I ) f [ V i ] V i (\forall i\in I)f[V_{i}]\subseteq V_{i}
  8. V i V_{i}
  9. V V
  10. f f
  11. f f

Log-linear_analysis.html

  1. \Chi 2 \Chi^{2}
  2. \Chi 2 = 2 O i j ln O i j E i j , \Chi^{2}=2\sum O_{ij}\ln\frac{O_{ij}}{E_{ij}},
  3. ln = \ln=
  4. O i j = O_{ij}=
  5. E i j = E_{ij}=
  6. \Chi 2 = \Chi^{2}=
  7. O i j E i j = 1 \frac{O_{ij}}{E_{ij}}=1
  8. ln ( 1 ) = 0 \ln(1)=0
  9. ln ( F i j k ) = λ + λ A + λ B + λ C + λ A B + λ A C + λ B C + λ A B C , \ln(F_{ijk})=\lambda+\lambda^{A}+\lambda^{B}+\lambda^{C}+\lambda^{AB}+\lambda^% {AC}+\lambda^{BC}+\lambda^{ABC},\,
  10. F i j k = F_{ijk}=
  11. λ = \lambda=

Log5.html

  1. p i p_{i}
  2. i i
  3. q i = 1 - p i q_{i}=1-p_{i}
  4. i i
  5. p A , B = p A - p A × p B p A + p B - 2 × p A × p B . p_{A,B}=\frac{p_{A}-p_{A}\times p_{B}}{p_{A}+p_{B}-2\times p_{A}\times p_{B}}.
  6. p A = 1 p_{A}=1
  7. p A = 0 p_{A}=0
  8. p A = p B p_{A}=p_{B}
  9. p A = 1 / 2 p_{A}=1/2
  10. 1 - p B 1-p_{B}
  11. p A , B q A , B = p A q A × q B p B . \frac{p_{A,B}}{q_{A,B}}=\frac{p_{A}}{q_{A}}\times\frac{q_{B}}{p_{B}}.
  12. q A , B = 1 - p A , B q_{A,B}=1-p_{A,B}

Log_structure.html

  1. 𝒪 X \mathcal{O}_{X}
  2. 𝒪 X \mathcal{O}_{X}
  3. j : U X j\colon U\to X
  4. D = X - U D=X-U
  5. \mathcal{M}
  6. α : 𝒪 X \alpha\colon\mathcal{M}\to\mathcal{O}_{X}
  7. 𝒪 X \mathcal{O}_{X}
  8. ( , α ) (\mathcal{M},\alpha)
  9. α \alpha
  10. α : α - 1 ( 𝒪 X × ) 𝒪 X × \alpha\colon\alpha^{-1}(\mathcal{O}_{X}^{\times})\to\mathcal{O}_{X}^{\times}
  11. 𝒪 X \mathcal{O}_{X}
  12. = 𝒪 X × \mathcal{M}=\mathcal{O}_{X}^{\times}
  13. α \alpha
  14. j : U X j\colon U\to X
  15. D = X - U D=X-U
  16. = 𝒪 X j * 𝒪 U × \mathcal{M}=\mathcal{O}_{X}\cap j_{*}\mathcal{O}_{U}^{\times}
  17. α \alpha
  18. 𝒪 X \mathcal{O}_{X}
  19. Spec ( R ) \mathrm{Spec}(R)
  20. R { 0 } R\setminus\{0\}
  21. R × R^{\times}
  22. R R
  23. j : Spec ( K ) Spec ( R ) j\colon\mathrm{Spec}(K)\to\mathrm{Spec}(R)
  24. Spec ( R ) \mathrm{Spec}(R)

Logarithmic_resistor_ladder.html

  1. CodeValue = i = 1 N s i 2 i - 1 \mathrm{CodeValue}=\sum_{i=1}^{N}s_{i}\cdot 2^{i-1}
  2. s i s_{i}
  3. V o u t = a ( CodeValue + b ) V i n V_{out}=a\cdot(\mathrm{CodeValue}+b)\cdot V_{in}
  4. a a
  5. b b
  6. V i n V_{in}
  7. log ( V o u t / V i n ) = a ( CodeValue + b ) \log(V_{out}/V_{in})=a\cdot(\mathrm{CodeValue}+b)
  8. V i n V_{in}
  9. R a t i o i = if s w i then α 2 i - 1 else 1 Ratio_{i}=\,\text{if}\;sw_{i}\;\,\text{then}\;\alpha^{2^{i-1}}\;\,\text{else}\;1
  10. d B ( R a t i o i ) = 20 log 10 α 2 i - 1 = 2 i - 1 20 log 10 α dB(Ratio_{i})=20\log_{10}\alpha^{2^{i-1}}=2^{i-1}\cdot 20\cdot\log_{10}\alpha
  11. i = 1.. N i=1..N
  12. s w i = 1 sw_{i}=1
  13. d B ( R a t i o i + 1 ) = 2 d B ( R a t i o i ) dB(Ratio_{i+1})=2\cdot dB(Ratio_{i})
  14. R a t i o i Ratio_{i}
  15. log ( V o u t / V i n ) = log ( i = 1 N R a t i o i ) = i = 1 N log ( R a t i o i ) = a ( C o d e V a l u e + b ) \log(V_{out}/V_{in})=\log(\prod_{i=1}^{N}Ratio_{i})=\sum_{i=1}^{N}\log(Ratio_{% i})=a\cdot(CodeValue+b)
  16. b = 0 b=0
  17. a = log ( α ) a=\log(\alpha)
  18. { R i , p a r r = ( R i , b R l o a d ) / ( R i , b + R l o a d ) R i , a + R i , p a r r = R l o a d R i , p a r r / ( R i , a + R i , p a r r ) = R a t i o i \begin{cases}R_{i,parr}=(R_{i,b}\cdot R_{load})/(R_{i,b}+R_{load})\\ R_{i,a}+R_{i,parr}=R_{load}\\ R_{i,parr}/(R_{i,a}+R_{i,parr})=Ratio_{i}\end{cases}
  19. α \alpha
  20. { R i , s e r = R i , a + R s o u r c e R i , s e r R i , b / ( R i , s e r + R i , b ) = R s o u r c e R i , b / ( R i , s e r + R i , b ) = R a t i o i \begin{cases}R_{i,ser}=R_{i,a}+R_{source}\\ R_{i,ser}\cdot R_{i,b}/(R_{i,ser}+R_{i,b})=R_{source}\\ R_{i,b}/(R_{i,ser}+R_{i,b})=Ratio_{i}\end{cases}
  21. α \alpha

Logical_consequence.html

  1. \mathcal{L}
  2. \mathcal{L}
  3. \mathcal{L}
  4. A A
  5. B B
  6. C C
  7. A A
  8. C C
  9. B B
  10. P P
  11. Q Q
  12. P P
  13. Q Q
  14. Q Q
  15. P P
  16. P P
  17. Q Q
  18. Q Q
  19. P P
  20. A A
  21. 𝒮 \mathcal{FS}
  22. Γ \Gamma
  23. 𝒮 \mathcal{FS}
  24. A A
  25. Γ \Gamma
  26. Γ 𝒮 A \Gamma\vdash_{\mathcal{FS}}A
  27. A A
  28. 𝒮 \mathcal{FS}
  29. Γ \Gamma
  30. Γ 𝒮 A , \Gamma\models_{\mathcal{FS}}A,
  31. \mathcal{I}
  32. Γ \Gamma
  33. A A
  34. Γ \Gamma
  35. A A
  36. Γ \Gamma
  37. \vdash
  38. A A
  39. Γ \Gamma
  40. A A
  41. Γ \Gamma
  42. \vdash
  43. A A
  44. Γ \Gamma
  45. A A
  46. Γ \Gamma
  47. \vdash
  48. A A
  49. Γ \Gamma
  50. A A
  51. Γ \Gamma
  52. \vdash
  53. A A
  54. Γ \Gamma
  55. A A
  56. A A
  57. Γ \Gamma
  58. A A
  59. Γ \Gamma

Long-chain_acyl-(acyl-carrier-protein)_reductase.html

  1. \rightleftharpoons

Long-chain_acyl-CoA_dehydrogenase.html

  1. \rightleftharpoons

Longest_alternating_subsequence.html

  1. 𝐱 = { x 1 , x 2 , , x n } \mathbf{x}=\{x_{1},x_{2},\ldots,x_{n}\}
  2. { x i 1 , x i 2 , , x i k } \{x_{i_{1}},x_{i_{2}},\ldots,x_{i_{k}}\}
  3. x i 1 > x i 2 < x i 3 > x i k and 1 i 1 < i 2 < < i k n . x_{i_{1}}>x_{i_{2}}<x_{i_{3}}>\cdots x_{i_{k}}\qquad\,\text{and}\qquad 1\leq i% _{1}<i_{2}<\cdots<i_{k}\leq n.
  4. 𝐱 \mathbf{x}
  5. x i 1 < x i 2 > x i 3 < x i k and 1 i 1 < i 2 < < i k n . x_{i_{1}}<x_{i_{2}}>x_{i_{3}}<\cdots x_{i_{k}}\qquad\,\text{and}\qquad 1\leq i% _{1}<i_{2}<\cdots<i_{k}\leq n.
  6. as n ( 𝐱 ) {\rm as}_{n}(\mathbf{x})
  7. 𝐱 \mathbf{x}
  8. as 5 ( 1 , 2 , 3 , 4 , 5 ) = 2 {\rm as}_{5}(1,2,3,4,5)=2
  9. as 5 ( 1 , 5 , 3 , 2 , 4 ) = 4 , {\rm as}_{5}(1,5,3,2,4)=4,
  10. as 5 ( 5 , 3 , 4 , 1 , 2 ) = 5 , {\rm as}_{5}(5,3,4,1,2)=5,
  11. O ( n ) O(n)
  12. n n
  13. 𝐱 \mathbf{x}
  14. 1 , 2 , , n 1,2,\ldots,n
  15. A n as n ( 𝐱 ) A_{n}\equiv{\rm as}_{n}(\mathbf{x})
  16. E [ A n ] = 2 n 3 + 1 6 and Var [ A n ] = 8 n 45 - 13 180 . E[A_{n}]=\frac{2n}{3}+\frac{1}{6}\qquad\,\text{and}\qquad\operatorname{Var}[A_% {n}]=\frac{8n}{45}-\frac{13}{180}.
  17. n n\rightarrow\infty
  18. A n A_{n}
  19. 𝐱 \mathbf{x}
  20. X 1 , X 2 , , X n X_{1},X_{2},\ldots,X_{n}
  21. F F
  22. ( 2 - 2 ) n + O ( 1 ) (2-\sqrt{2})n+O(1)
  23. n n\rightarrow\infty
  24. n n

LOOP_(programming_language).html

  1. P : = x i := x j + c | x i := x j - c | P ; P | LOOP x i DO P END \begin{array}[]{lrl}P&:=&x_{i}:=x_{j}+c\\ &|&x_{i}:=x_{j}-c\\ &|&P;P\\ &|&\mathrm{LOOP}\,x_{i}\,\mathrm{DO}\,P\,\mathrm{END}\end{array}
  2. V a r := { x 0 , x 1 , } Var:=\{x_{0},x_{1},...\}
  3. c c\in\mathbb{N}
  4. f : k f:\mathbb{N}^{k}\rightarrow\mathbb{N}
  5. x 1 x_{1}
  6. x k x_{k}
  7. f f
  8. x 0 x_{0}
  9. f f
  10. x 1 x_{1}
  11. x k x_{k}
  12. c c
  13. x 1 x_{1}
  14. x 0 x_{0}
  15. c c
  16. c c
  17. x 1 x_{1}
  18. x 0 x_{0}
  19. c c
  20. x 1 x_{1}
  21. P 1 P_{1}
  22. P 2 P_{2}
  23. P P
  24. x x
  25. x x
  26. P P
  27. x x
  28. P P
  29. x x
  30. P P
  31. x 0 x_{0}
  32. x 1 x_{1}
  33. x 2 x_{2}
  34. x 0 x_{0}
  35. x 1 x_{1}
  36. x 0 x_{0}
  37. x 2 x_{2}
  38. x 0 x_{0}
  39. x 1 x_{1}
  40. x 2 x_{2}
  41. x 2 x_{2}
  42. x 1 x_{1}
  43. x 0 x_{0}

Lorden's_inequality.html

  1. E ( R b ) E ( X 2 ) E ( X ) . \operatorname{E}(R_{b})\leq\frac{\operatorname{E}(X^{2})}{\operatorname{E}(X)}.

Lorentz_force_velocimetry.html

  1. f = j × B \vec{f}=\vec{j}\times\vec{B}
  2. f σ v B 2 f\sim\sigma vB^{2}
  3. σ \sigma
  4. v v
  5. B B
  6. B ( r ) \vec{B}\left(\vec{r}\right)
  7. J ( r ) \vec{J}\left(\vec{r}\right)
  8. j ( r ) \vec{j}\left(\vec{r}\right)
  9. F f = f j × B d 3 r \vec{F}_{f}=\int_{f}\vec{j}\times\vec{B}d^{3}\vec{r}
  10. b ( r ) \vec{b}\left(\vec{r}\right)
  11. F m = m J × b d 3 r \vec{F}_{m}=\int_{m}\vec{J}\times\vec{b}d^{3}\vec{r}
  12. F m = - F f \vec{F}_{m}=-\vec{F}_{f}
  13. m m
  14. L L
  15. v v
  16. m = m e ^ z \vec{m}=m\hat{e}_{z}
  17. B ( R ) = μ 0 4 π { 3 ( m R ) R R 5 - m R 3 } \vec{B}\left(\vec{R}\right)=\frac{\mu_{0}}{4\pi}\left\{3\frac{\left(\vec{m}% \cdot\vec{R}\right)\vec{R}}{R^{5}}-\frac{\vec{m}}{R^{3}}\right\}
  18. R = r - L e ^ z \vec{R}=\vec{r}-L\hat{e}_{z}
  19. R = R R=\mid\vec{R}\mid
  20. v = v e ^ x \vec{v}=v\hat{e}_{x}
  21. z < 0 z<0
  22. J = σ ( - ϕ + v × B ) \vec{J}=\sigma\left(-\nabla\phi+\vec{v}\times\vec{B}\right)
  23. J z = 0 J_{z}=0
  24. z = 0 z=0
  25. J z 0 J_{z}\to 0
  26. z 1 z\to 1
  27. ϕ ( r ) = - μ 0 v m 4 π x R 3 \phi\left(\vec{r}\right)=-\frac{\mu_{0}vm}{4\pi}\frac{x}{R^{3}}
  28. b ( r ) \vec{b}\left(\vec{r}\right)
  29. F = ( m ) b \vec{F}=\left(\vec{m}\cdot\nabla\right)\vec{b}
  30. b \vec{b}
  31. F = μ 0 2 σ v m 2 128 π L 3 e ^ z F=\frac{\mu_{0}^{2}\sigma vm^{2}}{128\pi L^{3}}\hat{e}_{z}
  32. F μ 0 2 σ v m 2 L - 3 F\sim\mu_{0}^{2}\sigma vm^{2}L^{-3}
  33. 10 6 S / m 10^{6}~{}S/m
  34. 1 S / m \sim~{}1~{}S/m
  35. 3.2 10 4 3.2\cdot 10^{4}
  36. 1.3 10 5 1.3\cdot 10^{5}
  37. F G F_{G}
  38. F G = m g F_{G}=m\cdot g
  39. F / F G = 10 - 7 F/F_{G}=10^{-7}
  40. F M F_{M}
  41. F G F_{G}
  42. F C F_{C}
  43. c s c_{s}
  44. l p l_{p}
  45. a a b a_{ab}
  46. α \alpha
  47. m 0 m_{0}
  48. g g
  49. m ˙ \dot{m}
  50. F F
  51. m ˙ ( t ) = K Σ F ( t ) \dot{m}\left(t\right)=\frac{K}{\Sigma}F\left(t\right)\quad
  52. Σ = σ ρ \Sigma=\frac{\sigma}{\rho}
  53. σ \sigma
  54. ρ \rho
  55. K K
  56. M = t 1 t 2 m ˙ ( t ) d t = K Σ t 1 t 2 F ( t ) d t = K Σ F ~ , M=\int_{t1}^{t2}\dot{m}\left(t\right)dt=\frac{K}{\Sigma}\int_{t1}^{t2}F\left(t% \right)dt=\frac{K}{\Sigma}\tilde{F}\quad,
  57. F ~ \tilde{F}
  58. σ = ρ K F ~ M . \sigma=\rho K\frac{\tilde{F}}{M}\quad.
  59. Q f l o w = k D τ Q_{flow}=k\frac{D}{\tau}
  60. D D
  61. τ \tau
  62. k k

Loss_network.html

  1. π ( n ) = G ( C ) - 1 r R v r n r n r ! for n S ( C ) \pi(n)=G(C)^{-1}\prod_{r\in R}\frac{v_{r}^{n_{r}}}{n_{r}!}\text{ for }n\in S(C)
  2. S ( C ) = { n + R : A n C } S(C)=\{n\in\mathbb{Z}_{+}^{R}:An\leq C\}
  3. G ( C ) = ( n S ( C ) r R v r n r n r ! ) . G(C)=\left(\sum_{n\in S(C)}\prod_{r\in R}\frac{v_{r}^{n_{r}}}{n_{r}!}\right).

Low-specificity_L-threonine_aldolase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Luminosity_(scattering_theory).html

  1. L = 1 σ d N d t . L=\frac{1}{\sigma}\frac{dN}{dt}.
  2. L int = L d t . L_{\mathrm{int}}=\int L\ dt.
  3. p ¯ \overline{p}
  4. p ¯ \overline{p}

Lupanine_17-hydroxylase_(cytochrome_c).html

  1. \rightleftharpoons

Lyapunov_optimization.html

  1. Q ( t ) = ( Q 1 ( t ) , Q 2 ( t ) , , Q N ( t ) ) Q(t)=(Q_{1}(t),Q_{2}(t),...,Q_{N}(t))
  2. L ( t ) = 1 2 i = 1 N Q i ( t ) 2 L(t)=\frac{1}{2}\sum_{i=1}^{N}Q_{i}(t)^{2}
  3. Δ L ( t ) = L ( t + 1 ) - L ( t ) \Delta L(t)=L(t+1)-L(t)
  4. Q i ( t + 1 ) = max [ Q i ( t ) + a i ( t ) - b i ( t ) , 0 ] Q_{i}(t+1)=\max[Q_{i}(t)+a_{i}(t)-b_{i}(t),0]
  5. a i ( t ) a_{i}(t)
  6. b i ( t ) b_{i}(t)
  7. Q i ( t + 1 ) 2 = ( max [ Q i ( t ) + a i ( t ) - b i ( t ) , 0 ] ) 2 ( Q i ( t ) + a i ( t ) - b i ( t ) ) 2 Q_{i}(t+1)^{2}=(\max[Q_{i}(t)+a_{i}(t)-b_{i}(t),0])^{2}\leq(Q_{i}(t)+a_{i}(t)-% b_{i}(t))^{2}
  8. ( E q .1 ) Δ L ( t ) B ( t ) + i = 1 N Q i ( t ) ( a i ( t ) - b i ( t ) ) (Eq.1)\,\text{ }\Delta L(t)\leq B(t)+\sum_{i=1}^{N}Q_{i}(t)(a_{i}(t)-b_{i}(t))
  9. B ( t ) = 1 2 i = 1 N [ a i ( t ) 2 + b i ( t ) 2 - 2 a i ( t ) b i ( t ) ] B(t)=\frac{1}{2}\sum_{i=1}^{N}[a_{i}(t)^{2}+b_{i}(t)^{2}-2a_{i}(t)b_{i}(t)]
  10. E [ B ( t ) | Q ( t ) ] B E[B(t)|Q(t)]\leq B
  11. ( E q .2 ) E [ Δ L ( t ) | Q ( t ) ] B + i = 1 N Q i ( t ) E [ a i ( t ) - b i ( t ) | Q ( t ) ] (Eq.2)\,\text{ }E[\Delta L(t)|Q(t)]\leq B+\sum_{i=1}^{N}Q_{i}(t)E[a_{i}(t)-b_{% i}(t)|Q(t)]
  12. ϵ > 0 \epsilon>0
  13. E [ a i ( t ) - b i ( t ) | Q ( t ) ] - ϵ E[a_{i}(t)-b_{i}(t)|Q(t)]\leq-\epsilon
  14. Theorem (Lyapunov Drift): \,\text{ Theorem (Lyapunov Drift):}
  15. B 0 , ϵ > 0 B\geq 0,\epsilon>0
  16. E [ Δ L ( t ) | Q ( t ) ] B - ϵ i = 1 N Q i ( t ) E[\Delta L(t)|Q(t)]\leq B-\epsilon\sum_{i=1}^{N}Q_{i}(t)
  17. 1 t τ = 0 t - 1 i = 1 N E [ Q i ( τ ) ] B ϵ + E [ L ( 0 ) ] ϵ t \frac{1}{t}\sum_{\tau=0}^{t-1}\sum_{i=1}^{N}E[Q_{i}(\tau)]\leq\frac{B}{% \epsilon}+\frac{E[L(0)]}{\epsilon t}
  18. Proof: \,\text{ Proof:}
  19. E [ Δ L ( t ) ] B - ϵ i = 1 N E [ Q i ( t ) ] E[\Delta L(t)]\leq B-\epsilon\sum_{i=1}^{N}E[Q_{i}(t)]
  20. τ { 0 , 1 , , t - 1 } \tau\in\{0,1,\ldots,t-1\}
  21. E [ L ( t ) ] - E [ L ( 0 ) ] B t - ϵ τ = 0 t - 1 i = 1 N E [ Q i ( τ ) ] E[L(t)]-E[L(0)]\leq Bt-\epsilon\sum_{\tau=0}^{t-1}\sum_{i=1}^{N}E[Q_{i}(\tau)]
  22. Δ L ( t ) + V p ( t ) \Delta L(t)+Vp(t)
  23. p ( t ) p m i n t { 0 , 1 , 2 , } p(t)\geq p_{min}\,\text{ }\forall t\in\{0,1,2,...\}
  24. Theorem (Lyapunov Optimization): \,\text{ Theorem (Lyapunov Optimization):}
  25. B 0 , ϵ > 0 , V 0 , p * B\geq 0,\epsilon>0,V\geq 0,p^{*}
  26. ( E q .3 ) E [ Δ L ( t ) + V p ( t ) | Q ( t ) ] B + V p * - ϵ i = 1 N Q i ( t ) (Eq.3)\,\text{ }E[\Delta L(t)+Vp(t)|Q(t)]\leq B+Vp^{*}-\epsilon\sum_{i=1}^{N}Q% _{i}(t)
  27. 1 t τ = 0 t - 1 E [ p ( τ ) ] p * + B V + E [ L ( 0 ) ] V t \,\text{ }\frac{1}{t}\sum_{\tau=0}^{t-1}E[p(\tau)]\leq p^{*}+\frac{B}{V}+\frac% {E[L(0)]}{Vt}
  28. 1 t τ = 0 t - 1 i = 1 N E [ Q i ( τ ) ] B + V ( p * - p m i n ) ϵ + E [ L ( 0 ) ] ϵ t \,\text{ }\frac{1}{t}\sum_{\tau=0}^{t-1}\sum_{i=1}^{N}E[Q_{i}(\tau)]\leq\frac{% B+V(p^{*}-p_{min})}{\epsilon}+\frac{E[L(0)]}{\epsilon t}
  29. Proof: \,\text{ Proof:}
  30. E [ Δ L ( t ) ] + V E [ p ( t ) ] B + V p * - ϵ i = 1 N E [ Q i ( t ) ] E[\Delta L(t)]+VE[p(t)]\leq B+Vp^{*}-\epsilon\sum_{i=1}^{N}E[Q_{i}(t)]
  31. E [ L ( t ) ] - E [ L ( 0 ) ] + V τ = 0 t - 1 E [ p ( τ ) ] ( B + V p * ) t - ϵ τ = 0 t - 1 i = 1 N E [ Q i ( τ ) ] E[L(t)]-E[L(0)]+V\sum_{\tau=0}^{t-1}E[p(\tau)]\leq(B+Vp^{*})t-\epsilon\sum_{% \tau=0}^{t-1}\sum_{i=1}^{N}E[Q_{i}(\tau)]
  32. - E [ L ( 0 ) ] + V τ = 0 t - 1 E [ p ( τ ) ] ( B + V p * ) t -E[L(0)]+V\sum_{\tau=0}^{t-1}E[p(\tau)]\leq(B+Vp^{*})t

Lyddane–Sachs–Teller_relation.html

  1. ω L \omega_{L}
  2. ω T \omega_{T}
  3. ε s t \varepsilon_{st}
  4. ε ( 0 ) \varepsilon(0)
  5. ε \varepsilon_{\infty}
  6. ε ( ω s ) \varepsilon(\omega_{s})
  7. ω L 2 ω T 2 = ε s t ε \frac{\omega_{L}^{2}}{\omega_{T}^{2}}=\frac{\varepsilon_{st}}{\varepsilon_{% \infty}}

Lysine_6-dehydrogenase.html

  1. \rightleftharpoons
  2. \rightleftharpoons
  3. \rightleftharpoons

M::D::1_queue.html

  1. π 0 = 1 - λ π 1 = ( 1 - λ ) ( e λ - 1 ) π n = ( 1 - λ ) ( e n λ + k = 1 n - 1 e k λ ( - 1 ) n - k [ ( k λ ) n - k ( n - k ) ! + ( k λ ) n - k - 1 ( n - k - 1 ) ! ] ) ( n > 2 ) . \begin{aligned}\displaystyle\pi_{0}&\displaystyle=1-\lambda\\ \displaystyle\pi_{1}&\displaystyle=(1-\lambda)(e^{\lambda}-1)\\ \displaystyle\pi_{n}&\displaystyle=(1-\lambda)\left(e^{n\lambda}+\sum_{k=1}^{n% -1}e^{k\lambda}(-1)^{n-k}\left[\frac{(k\lambda)^{n-k}}{(n-k)!}+\frac{(k\lambda% )^{n-k-1}}{(n-k-1)!}\right]\right)\quad(n>2).\end{aligned}
  2. 1 2 μ 2 - ρ 1 - ρ . \frac{1}{2\mu}\cdot\frac{2-\rho}{1-\rho}.
  3. 1 2 μ ρ 1 - ρ . \frac{1}{2\mu}\cdot\frac{\rho}{1-\rho}.

M::M::∞_queue.html

  1. Q = ( - λ λ μ - ( μ + λ ) λ 2 μ - ( 2 μ + λ ) λ 3 μ - ( 3 μ + λ ) λ ) . Q=\begin{pmatrix}-\lambda&\lambda\\ \mu&-(\mu+\lambda)&\lambda\\ &2\mu&-(2\mu+\lambda)&\lambda\\ &&3\mu&-(3\mu+\lambda)&\lambda\\ &&&&\ddots\end{pmatrix}.
  2. p 0 j ( t ) = exp ( - λ μ ( 1 - e - μ t ) ) ( λ μ ( 1 - e - μ t ) ) j j ! for j 0 p_{0j}(t)=\exp\left(-\frac{\lambda}{\mu}(1-e^{-\mu t})\right)\frac{\left(\frac% {\lambda}{\mu}(1-e^{-\mu t})\right)^{j}}{j!}\,\text{ for }j\geq 0
  3. 𝔼 ( N ( t ) | N ( 0 ) = 0 ) = λ μ ( 1 - e - μ t ) for t 0. \mathbb{E}(N(t)|N(0)=0)=\frac{\lambda}{\mu}(1-e^{-\mu t})\,\text{ for }t\geq 0.
  4. 1 λ i > c c ! i ! ( λ μ ) i - c \frac{1}{\lambda}\sum_{i>c}\frac{c!}{i!}\left(\frac{\lambda}{\mu}\right)^{i-c}
  5. π k = ( λ / μ ) k e - λ / μ k ! k 0 \pi_{k}=\frac{(\lambda/\mu)^{k}e^{-\lambda/\mu}}{k!}\quad k\geq 0
  6. X t = N t - λ / μ λ / μ X_{t}=\frac{N_{t}-\lambda/\mu}{\sqrt{\lambda/\mu}}
  7. d X t = - X d t + 2 d W t dX_{t}=-Xdt+\sqrt{2}dW_{t}

M7GpppN-mRNA_hydrolase.html

  1. \rightleftharpoons

M7GpppX_diphosphatase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Macroemulsion.html

  1. Δ G em = 3 γ V R f \Delta G_{\rm em}=3{\gamma V\over\ R_{\rm f}}
  2. Δ G em \Delta G_{\rm em}
  3. γ \gamma
  4. V V
  5. R f R_{\rm f}
  6. P = 1 Δ R 2 π exp [ ( ln R - ln R ¯ ) 2 2 Δ R 2 ] \ P={{\ 1\over\Delta R\sqrt{2\pi\,}\ }\exp[{\ (\ln{R}\ -\ln{\bar{R}}\ )^{2}% \over\ 2\Delta R^{2}\ \ }\ }]
  7. R ¯ \bar{R}
  8. Δ R \Delta R
  9. d c d t = - 8 π D R c 2 \frac{dc}{dt}=-8\pi\,DRc^{2}
  10. D = k B T 6 π η R D=\frac{k_{B}T}{6\pi\,\eta\,R}
  11. d c d t = - k f c 2 \frac{dc}{dt}=-k_{f}c^{2}
  12. k f k_{f}
  13. ( k f = 4 k B T 3 η ) (k_{f}=\frac{4k_{B}T}{3\eta\,})
  14. 8 k B T 3 η \frac{8k_{B}T}{3\eta\,}