wpmath0000003_8

Integral_test_for_convergence.html

  1. N N
  2. f f
  3. [ N , ) [N,∞)
  4. n = N f ( n ) \sum_{n=N}^{\infty}f(n)
  5. N f ( x ) d x \int_{N}^{\infty}f(x)\,dx
  6. N f ( x ) d x n = N f ( n ) f ( N ) + N f ( x ) d x \int_{N}^{\infty}f(x)\,dx\leq\sum_{n=N}^{\infty}f(n)\leq f(N)+\int_{N}^{\infty% }f(x)\,dx
  7. f ( n ) f(n)
  8. f f
  9. [ n 1 , n ) [n−1,n)
  10. [ n , n + 1 ) [n,n+1)
  11. f f
  12. f ( x ) f ( n ) for all x [ n , ) f(x)\leq f(n)\quad\,\text{for all }x\in[n,\infty)
  13. f ( n ) f ( x ) for all x [ N , n ] . f(n)\leq f(x)\quad\,\text{for all }x\in[N,n].
  14. n N n≥N
  15. n n + 1 f ( x ) d x n n + 1 f ( n ) d x = f ( n ) \int_{n}^{n+1}f(x)\,dx\leq\int_{n}^{n+1}f(n)\,dx=f(n)
  16. n N + 1 n≥N+1
  17. f ( n ) = n - 1 n f ( n ) d x n - 1 n f ( x ) d x . f(n)=\int_{n-1}^{n}f(n)\,dx\leq\int_{n-1}^{n}f(x)\,dx.
  18. n n
  19. N N
  20. M M
  21. N M + 1 f ( x ) d x = n = N M n n + 1 f ( x ) d x f ( n ) n = N M f ( n ) \int_{N}^{M+1}f(x)\,dx=\sum_{n=N}^{M}\underbrace{\int_{n}^{n+1}f(x)\,dx}_{\leq% \,f(n)}\leq\sum_{n=N}^{M}f(n)
  22. n = N M f ( n ) f ( N ) + n = N + 1 M n - 1 n f ( x ) d x f ( n ) = f ( N ) + N M f ( x ) d x . \sum_{n=N}^{M}f(n)\leq f(N)+\sum_{n=N+1}^{M}\underbrace{\int_{n-1}^{n}f(x)\,dx% }_{\geq\,f(n)}=f(N)+\int_{N}^{M}f(x)\,dx.
  23. N M + 1 f ( x ) d x n = N M f ( n ) f ( N ) + N M f ( x ) d x . \int_{N}^{M+1}f(x)\,dx\leq\sum_{n=N}^{M}f(n)\leq f(N)+\int_{N}^{M}f(x)\,dx.
  24. M M
  25. n = 1 1 n \sum_{n=1}^{\infty}\frac{1}{n}
  26. 1 M 1 x d x = ln x | 1 M = ln M for M . \int_{1}^{M}\frac{1}{x}\,dx=\ln x\Bigr|_{1}^{M}=\ln M\to\infty\quad\,\text{for% }M\to\infty.
  27. ζ ( 1 + ε ) = n = 1 1 n 1 + ε \zeta(1+\varepsilon)=\sum_{n=1}^{\infty}\frac{1}{n^{1+\varepsilon}}
  28. ε > 0 ε>0
  29. 1 M 1 x 1 + ε d x = - 1 ε x ε | 1 M = 1 ε ( 1 - 1 M ε ) 1 ε < for all M 1. \int_{1}^{M}\frac{1}{x^{1+\varepsilon}}\,dx=-\frac{1}{\varepsilon x^{% \varepsilon}}\biggr|_{1}^{M}=\frac{1}{\varepsilon}\Bigl(1-\frac{1}{M^{% \varepsilon}}\Bigr)\leq\frac{1}{\varepsilon}<\infty\quad\,\text{for all }M\geq 1.
  30. ζ ( 1 + ε ) = n = 1 1 n 1 + ε 1 + ε ε , \zeta(1+\varepsilon)=\sum_{n=1}^{\infty}\frac{1}{n^{1+\varepsilon}}\leq\frac{1% +\varepsilon}{\varepsilon},
  31. f ( n ) f(n)
  32. 1 / n 1/n
  33. lim n f ( n ) 1 / n = 0 and lim n f ( n ) 1 / n 1 + ε = \lim_{n\to\infty}\frac{f(n)}{1/n}=0\quad\,\text{and}\quad\lim_{n\to\infty}% \frac{f(n)}{1/n^{1+\varepsilon}}=\infty
  34. ε > 0 ε>0
  35. f ( n ) f(n)
  36. f ( n ) f(n)
  37. 1 / n 1/n
  38. k k
  39. n = N k 1 n ln ( n ) ln 2 ( n ) ln k - 1 ( n ) ln k ( n ) \sum_{n=N_{k}}^{\infty}\frac{1}{n\ln(n)\ln_{2}(n)\cdots\ln_{k-1}(n)\ln_{k}(n)}
  40. k = 1 k=1
  41. n = N k 1 n ln ( n ) ln 2 ( n ) ln k - 1 ( n ) ( ln k ( n ) ) 1 + ε \sum_{n=N_{k}}^{\infty}\frac{1}{n\ln(n)\ln_{2}(n)\cdots\ln_{k-1}(n)(\ln_{k}(n)% )^{1+\varepsilon}}
  42. ε > 0 ε>0
  43. k k
  44. ln k ( x ) = { ln ( x ) for k = 1 , ln ( ln k - 1 ( x ) ) for k 2. \ln_{k}(x)=\begin{cases}\ln(x)&\,\text{for }k=1,\\ \ln(\ln_{k-1}(x))&\,\text{for }k\geq 2.\end{cases}
  45. k k
  46. N k e e e k e s = e k N_{k}\geq\underbrace{e^{e^{\cdot^{\cdot^{e}}}}}_{k\ e^{\prime}\,\text{s}}=e% \uparrow\uparrow k
  47. d d x ln k + 1 ( x ) = d d x ln ( ln k ( x ) ) = 1 ln k ( x ) d d x ln k ( x ) = = 1 x ln ( x ) ln k ( x ) , \frac{d}{dx}\ln_{k+1}(x)=\frac{d}{dx}\ln(\ln_{k}(x))=\frac{1}{\ln_{k}(x)}\frac% {d}{dx}\ln_{k}(x)=\cdots=\frac{1}{x\ln(x)\cdots\ln_{k}(x)},
  48. N k d x x ln ( x ) ln k ( x ) = ln k + 1 ( x ) | N k = . \int_{N_{k}}^{\infty}\frac{dx}{x\ln(x)\cdots\ln_{k}(x)}=\ln_{k+1}(x)\bigr|_{N_% {k}}^{\infty}=\infty.
  49. - d d x 1 ε ( ln k ( x ) ) ε = 1 ( ln k ( x ) ) 1 + ε d d x ln k ( x ) = = 1 x ln ( x ) ln k - 1 ( x ) ( ln k ( x ) ) 1 + ε , -\frac{d}{dx}\frac{1}{\varepsilon(\ln_{k}(x))^{\varepsilon}}=\frac{1}{(\ln_{k}% (x))^{1+\varepsilon}}\frac{d}{dx}\ln_{k}(x)=\cdots=\frac{1}{x\ln(x)\cdots\ln_{% k-1}(x)(\ln_{k}(x))^{1+\varepsilon}},
  50. N k d x x ln ( x ) ln k - 1 ( x ) ( ln k ( x ) ) 1 + ε = - 1 ε ( ln k ( x ) ) ε | N k < \int_{N_{k}}^{\infty}\frac{dx}{x\ln(x)\cdots\ln_{k-1}(x)(\ln_{k}(x))^{1+% \varepsilon}}=-\frac{1}{\varepsilon(\ln_{k}(x))^{\varepsilon}}\biggr|_{N_{k}}^% {\infty}<\infty

Integral_transform.html

  1. ( T f ) ( u ) = t 1 t 2 K ( t , u ) f ( t ) d t (Tf)(u)=\int\limits_{t_{1}}^{t_{2}}K(t,u)\,f(t)\,dt
  2. f ( t ) = u 1 u 2 K - 1 ( u , t ) ( T f ) ( u ) d u f(t)=\int\limits_{u_{1}}^{u_{2}}K^{-1}(u,t)\,(Tf)(u)\,du
  3. ψ ( x , t ) = - ψ ( x , t ) K ( x , t ; x , t ) d x . \psi(x,t)=\int_{-\infty}^{\infty}\psi(x^{\prime},t^{\prime})K(x,t;x^{\prime},t% ^{\prime})dx^{\prime}.
  4. ( x , t ) (x,t)
  5. ψ ( x , t ) \psi(x,t)
  6. x x^{\prime}
  7. ( x , t ) (x^{\prime},t^{\prime})
  8. ψ ( x , t ) \psi(x^{\prime},t^{\prime})
  9. K ( x , t ; x , t ) K(x,t;x^{\prime},t^{\prime})
  10. \mathcal{F}
  11. e - i u t 2 π \frac{e^{-iut}}{\sqrt{2\pi}}
  12. L 1 L_{1}
  13. e i u t 2 π \frac{e^{iut}}{\sqrt{2\pi}}
  14. s \mathcal{F}_{s}
  15. 2 π sin ( u t ) \sqrt{\frac{2}{\pi}}\sin(ut)
  16. [ 0 , ) [0,\infty)
  17. 2 π sin ( u t ) \sqrt{\frac{2}{\pi}}\sin(ut)
  18. c \mathcal{F}_{c}
  19. 2 π cos ( u t ) \sqrt{\frac{2}{\pi}}\cos(ut)
  20. [ 0 , ) [0,\infty)
  21. 2 π cos ( u t ) \sqrt{\frac{2}{\pi}}\cos(ut)
  22. \mathcal{H}
  23. cos ( u t ) + sin ( u t ) 2 π \frac{\cos(ut)+\sin(ut)}{\sqrt{2\pi}}
  24. cos ( u t ) + sin ( u t ) 2 π \frac{\cos(ut)+\sin(ut)}{\sqrt{2\pi}}
  25. \mathcal{M}
  26. t - u 2 π i \frac{t^{-u}}{2\pi i}\,
  27. c - i c\!-\!i\infty
  28. c + i c\!+\!i\infty
  29. \mathcal{B}
  30. e u t 2 π i \frac{e^{ut}}{2\pi i}
  31. c - i c\!-\!i\infty
  32. c + i c\!+\!i\infty
  33. \mathcal{L}
  34. e u t 2 π i \frac{e^{ut}}{2\pi i}
  35. c - i c\!-\!i\infty
  36. c + i c\!+\!i\infty
  37. 𝒲 \mathcal{W}
  38. e - ( u - t ) 2 4 4 π \frac{e^{-\frac{(u-t)^{2}}{4}}}{\sqrt{4\pi}}\,
  39. e ( u - t ) 2 4 i 4 π \frac{e^{\frac{(u-t)^{2}}{4}}}{i\sqrt{4\pi}}
  40. c - i c\!-\!i\infty
  41. c + i c\!+\!i\infty
  42. t J ν ( u t ) t\,J_{\nu}(ut)
  43. u J ν ( u t ) u\,J_{\nu}(ut)
  44. 2 t t 2 - u 2 \frac{2t}{\sqrt{t^{2}-u^{2}}}
  45. - 1 π u 2 - t 2 d d u \frac{-1}{\pi\sqrt{u^{2}\!-\!t^{2}}}\frac{d}{du}
  46. i l \mathcal{H}il
  47. 1 π 1 u - t \frac{1}{\pi}\frac{1}{u-t}
  48. 1 π 1 u - t \frac{1}{\pi}\frac{1}{u-t}
  49. 1 - r 2 1 - 2 r cos θ + r 2 \frac{1-r^{2}}{1-2r\cos\theta+r^{2}}
  50. C n C_{n}
  51. 𝐙 / n 𝐙 \mathbf{Z}/n\mathbf{Z}

Interaction_(statistics).html

  1. Y = c + a x 1 + b x 2 + error Y=c+ax_{1}+bx_{2}+\,\text{error}\,
  2. Y = c + a x 1 + b x 2 + d ( x 1 × x 2 ) + error Y=c+ax_{1}+bx_{2}+d(x_{1}\times x_{2})+\,\text{error}\,
  3. d ( x 1 × x 2 ) d(x_{1}\times x_{2})
  4. ( n 0 ) + ( n 1 ) + ( n 2 ) + + ( n n ) = 2 n {\textstyle\left({{n}\atop{0}}\right)}+{\textstyle\left({{n}\atop{1}}\right)}+% {\textstyle\left({{n}\atop{2}}\right)}+\cdots+{\textstyle\left({{n}\atop{n}}% \right)}=2^{n}
  5. ( n 0 ) + ( n 1 ) + ( n 2 ) = 1 + 1 2 n + 1 2 n 2 {\textstyle\left({{n}\atop{0}}\right)}+{\textstyle\left({{n}\atop{1}}\right)}+% {\textstyle\left({{n}\atop{2}}\right)}=1+\tfrac{1}{2}n+\tfrac{1}{2}n^{2}

Intermodulation.html

  1. f a , ~{}f_{a},
  2. f a , 2 f a , 3 f a , 4 f a , ~{}f_{a},2f_{a},3f_{a},4f_{a},\ldots
  3. f a ~{}f_{a}
  4. f b ~{}f_{b}
  5. f c ~{}f_{c}
  6. x ( t ) = M a sin ( 2 π f a t + ϕ a ) + M b sin ( 2 π f b t + ϕ b ) + M c sin ( 2 π f c t + ϕ c ) \ x(t)=M_{a}\sin(2\pi f_{a}t+\phi_{a})+M_{b}\sin(2\pi f_{b}t+\phi_{b})+M_{c}% \sin(2\pi f_{c}t+\phi_{c})
  7. M \ M
  8. ϕ \ \phi
  9. y ( t ) \ y(t)
  10. G G
  11. y ( t ) = G ( x ( t ) ) \ y(t)=G\left(x(t)\right)\,
  12. y ( t ) \ y(t)
  13. f a ~{}f_{a}
  14. f b ~{}f_{b}
  15. f c ~{}f_{c}
  16. k a f a + k b f b + k c f c \ k_{a}f_{a}+k_{b}f_{b}+k_{c}f_{c}
  17. k a ~{}k_{a}
  18. k b ~{}k_{b}
  19. k c ~{}k_{c}
  20. N N
  21. f a , f b , , f N f_{a},f_{b},\ldots,f_{N}
  22. k a f a + k b f b + + k N f N , k_{a}f_{a}+k_{b}f_{b}+\cdots+k_{N}f_{N},\,
  23. k a , k b , , k N k_{a},k_{b},\ldots,k_{N}
  24. O \ O
  25. O = | k a | + | k b | + + | k N | , \ O=\left|k_{a}\right|+\left|k_{b}\right|+\cdots+\left|k_{N}\right|,
  26. | k a | + | k b | + | k c | = 3 \ |k_{a}|+|k_{b}|+|k_{c}|=3
  27. ( f a + f b - f c ) , ( f a + f c - f b ) , ( f b + f c - f a ) \ (f_{a}+f_{b}-f_{c}),(f_{a}+f_{c}-f_{b}),(f_{b}+f_{c}-f_{a})
  28. ( 2 f a - f b ) , ( 2 f a - f c ) , ( 2 f b - f a ) , ( 2 f b - f c ) , ( 2 f c - f a ) , ( 2 f c - f b ) \ (2f_{a}-f_{b}),(2f_{a}-f_{c}),(2f_{b}-f_{a}),(2f_{b}-f_{c}),(2f_{c}-f_{a}),(% 2f_{c}-f_{b})

Internal_conversion.html

  1. α = e / γ \alpha=e/{\gamma}
  2. e e
  3. γ \gamma
  4. α = 93 / 7 = 13.3 \alpha=93/7=13.3
  5. E = ( E i - E f ) - E B E=(E_{i}-E_{f})-E_{B}
  6. E i E_{i}
  7. E f E_{f}
  8. E B E_{B}

Internal_energy.html

  1. S S
  2. V V
  3. S S
  4. U U
  5. U U
  6. Δ U = i E i \Delta U=\sum_{i}E_{i}\,
  7. Δ U ΔU
  8. U = U micro pot + U micro kin U=U_{\mathrm{micro\,pot}}+U_{\mathrm{micro\,kin}}
  9. U = i = 1 N p i E i . U=\sum_{i=1}^{N}p_{i}\,E_{i}\ .
  10. Δ U ΔU
  11. Q Q
  12. Δ U ΔU
  13. Δ U = Q + W pressure - volume + W isochoric ( closed system , no transfer of matter ) . \Delta U=Q+W_{\mathrm{pressure-volume}}+W_{\mathrm{isochoric}}\,\,\,\,\,\,\,\,% \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathrm{(closed\,\,% system,\,\,no\,\,transfer\,\,of\,\,matter)}.
  14. Δ U = Q + W pressure - volume + W isochoric + Δ U matter \Delta U=Q+W_{\mathrm{pressure-volume}}+W_{\mathrm{isochoric}}+\Delta U_{% \mathrm{matter}}
  15. ( separate pathway for matter transfer from heat and work transfer pathways ) . \mathrm{(separate\,\,pathway\,\,for\,\,matter\,\,transfer\,\,from\,\,heat\,\,% and\,\,work\,\,transfer\,\,pathways)}.
  16. U = c N T , U=cNT,
  17. U ( S , V , N ) = c o n s t e S c N V - R c N R + c c , U(S,V,N)=const\cdot e^{\frac{S}{cN}}V^{\frac{-R}{c}}N^{\frac{R+c}{c}},
  18. T = U S , T=\frac{\partial U}{\partial S},
  19. p = - U V , p=-\frac{\partial U}{\partial V},
  20. p V = R N T pV=RNT
  21. d U = δ Q + δ W dU=\delta Q+\delta W\,
  22. δ W = - p d V \delta W=-p\mathrm{d}V\,
  23. δ Q = T d S \delta Q=T\mathrm{d}S\,
  24. T T
  25. S S
  26. d U = T d S - p d V \mathrm{d}U=T\mathrm{d}S-p\mathrm{d}V\!
  27. d U = C V d T + [ T ( p T ) V - p ] d V (1) . dU=C_{V}dT+\left[T\left(\frac{\partial p}{\partial T}\right)_{V}-p\right]dV\,% \,\,\text{ (1)}.\,
  28. d U = C v d T dU=C_{v}dT
  29. d U = C V d T + [ T ( p T ) V - p ] d V . dU=C_{V}dT+\left[T\left(\frac{\partial p}{\partial T}\right)_{V}-p\right]dV.\,
  30. p V = n R T . pV=nRT.\,
  31. p = n R T V . p=\frac{nRT}{V}.
  32. d U = C V d T + [ T ( p T ) V - n R T V ] d V . dU=C_{V}dT+\left[T\left(\frac{\partial p}{\partial T}\right)_{V}-\frac{nRT}{V}% \right]dV.\,
  33. ( p T ) V = n R V . \left(\frac{\partial p}{\partial T}\right)_{V}=\frac{nR}{V}.
  34. d U = C V d T + [ n R T V - n R T V ] d V . dU=C_{V}dT+\left[\frac{nRT}{V}-\frac{nRT}{V}\right]dV.
  35. d U = C V d T . dU=C_{V}dT.\,
  36. d S = ( S T ) V d T + ( S V ) T d V dS=\left(\frac{\partial S}{\partial T}\right)_{V}dT+\left(\frac{\partial S}{% \partial V}\right)_{T}dV\,
  37. d U = T d S - p d V . dU=TdS-pdV.\,
  38. d U = T ( S T ) V d T + [ T ( S V ) T - p ] d V . dU=T\left(\frac{\partial S}{\partial T}\right)_{V}dT+\left[T\left(\frac{% \partial S}{\partial V}\right)_{T}-p\right]dV.\,
  39. T ( S T ) V T\left(\frac{\partial S}{\partial T}\right)_{V}
  40. C V . C_{V}.
  41. d A = - S d T - p d V . dA=-SdT-pdV.\,
  42. ( S V ) T = ( p T ) V . \left(\frac{\partial S}{\partial V}\right)_{T}=\left(\frac{\partial p}{% \partial T}\right)_{V}.\,
  43. d U = ( C p - α p V ) d T + ( β T p - α T ) V d p dU=\left(C_{p}-\alpha pV\right)dT+\left(\beta_{T}p-\alpha T\right)Vdp\,
  44. C p = C V + V T α 2 β T C_{p}=C_{V}+VT\frac{\alpha^{2}}{\beta_{T}}\,
  45. α 1 V ( V T ) p \alpha\equiv\frac{1}{V}\left(\frac{\partial V}{\partial T}\right)_{p}\,
  46. β T - 1 V ( V p ) T \beta_{T}\equiv-\frac{1}{V}\left(\frac{\partial V}{\partial p}\right)_{T}\,
  47. d V = ( V p ) T d p + ( V T ) p d T = V ( α d T - β T d p ) (2) dV=\left(\frac{\partial V}{\partial p}\right)_{T}dp+\left(\frac{\partial V}{% \partial T}\right)_{p}dT=V\left(\alpha dT-\beta_{T}dp\right)\,\,\,\text{ (2)}\,
  48. ( p T ) V = - ( V T ) p ( V p ) T = α β T (3) \left(\frac{\partial p}{\partial T}\right)_{V}=-\frac{\left(\frac{\partial V}{% \partial T}\right)_{p}}{\left(\frac{\partial V}{\partial p}\right)_{T}}=\frac{% \alpha}{\beta_{T}}\,\,\,\text{ (3)}\,
  49. π T = ( U V ) T \pi_{T}=\left(\frac{\partial U}{\partial V}\right)_{T}
  50. U = U ( S , V , N 1 , , N n ) U=U(S,V,N_{1},\ldots,N_{n})\,
  51. U ( α S , α V , α N 1 , α N 2 , ) = α U ( S , V , N 1 , N 2 , ) U(\alpha S,\alpha V,\alpha N_{1},\alpha N_{2},\ldots)=\alpha U(S,V,N_{1},N_{2}% ,\ldots)\,
  52. d U = U S d S + U V d V + i U N i d N i = T d S - p d V + i μ i d N i \mathrm{d}U=\frac{\partial U}{\partial S}\mathrm{d}S+\frac{\partial U}{% \partial V}\mathrm{d}V+\sum_{i}\ \frac{\partial U}{\partial N_{i}}\mathrm{d}N_% {i}\ =T\,\mathrm{d}S-p\,\mathrm{d}V+\sum_{i}\mu_{i}\mathrm{d}N_{i}\,
  53. T = U S , T=\frac{\partial U}{\partial S},
  54. p = - U V , p=-\frac{\partial U}{\partial V},
  55. μ i \mu_{i}
  56. μ i = ( U N i ) S , V , N j i \mu_{i}=\left(\frac{\partial U}{\partial N_{i}}\right)_{S,V,N_{j\neq i}}
  57. { N j } \{N_{j}\}
  58. U = T S - p V + i μ i N i U=TS-pV+\sum_{i}\mu_{i}N_{i}\,
  59. G = i μ i N i G=\sum_{i}\mu_{i}N_{i}\,
  60. { N j } \{N_{j}\}
  61. σ i j \sigma_{ij}
  62. ε i j \varepsilon_{ij}
  63. d U = T d S + V σ i j d ε i j \mathrm{d}U=T\mathrm{d}S+V\sigma_{ij}\mathrm{d}\varepsilon_{ij}
  64. U = T S + 1 2 σ i j ε i j U=TS+\frac{1}{2}\sigma_{ij}\varepsilon_{ij}
  65. σ i j = C i j k l ε k l \sigma_{ij}=C_{ijkl}\varepsilon_{kl}

Interpretability_logic.html

  1. \Box
  2. \triangleright
  3. p \Diamond p
  4. ¬ ¬ p \neg\Box\neg p
  5. p \Box p
  6. p p
  7. p q p\triangleright q
  8. P A + q PA+q
  9. P A + p PA+p
  10. ( p q ) ( p q ) \Box(p\rightarrow q)\rightarrow(\Box p\rightarrow\Box q)
  11. ( p p ) p \Box(\Box p\rightarrow p)\rightarrow\Box p
  12. ( p q ) ( p q ) \Box(p\rightarrow q)\rightarrow(p\triangleright q)
  13. ( p q ) ( q r ) ( p r ) (p\triangleright q)\wedge(q\triangleright r)\rightarrow(p\triangleright r)
  14. ( p r ) ( q r ) ( ( p q ) r ) (p\triangleright r)\wedge(q\triangleright r)\rightarrow((p\vee q)% \triangleright r)
  15. ( p q ) ( p q ) (p\triangleright q)\rightarrow(\Diamond p\triangleright\Diamond q)
  16. p p \Diamond p\triangleright p
  17. ( p q ) ( ( p r ) ( q r ) ) (p\triangleright q)\rightarrow((p\wedge\Box r)\triangleright(q\wedge\Box r))
  18. p p
  19. p q p\rightarrow q
  20. q q
  21. p p
  22. p \Box p
  23. \Diamond
  24. ( p 1 , , p n ) \Diamond(p_{1},\ldots,p_{n})
  25. ( P A + p 1 , , P A + p n ) (PA+p_{1},\ldots,PA+p_{n})
  26. p , q p,q
  27. r , s \vec{r},\vec{s}
  28. ( ) \Diamond()
  29. ( r , p , s ) ( r , p ¬ q , s ) ( r , q , s ) \Diamond(\vec{r},p,\vec{s})\rightarrow\Diamond(\vec{r},p\wedge\neg q,\vec{s})% \vee\Diamond(\vec{r},q,\vec{s})
  30. ( p ) ( p ¬ ( p ) ) \Diamond(p)\rightarrow\Diamond(p\wedge\neg\Diamond(p))
  31. ( r , p , s ) ( r , s ) \Diamond(\vec{r},p,\vec{s})\rightarrow\Diamond(\vec{r},\vec{s})
  32. ( r , p , s ) ( r , p , p , s ) \Diamond(\vec{r},p,\vec{s})\rightarrow\Diamond(\vec{r},p,p,\vec{s})
  33. ( p , ( r ) ) ( p ( r ) ) \Diamond(p,\Diamond(\vec{r}))\rightarrow\Diamond(p\wedge\Diamond(\vec{r}))
  34. ( r , ( s ) ) ( r , s ) \Diamond(\vec{r},\Diamond(\vec{s}))\rightarrow\Diamond(\vec{r},\vec{s})
  35. p p
  36. p q p\rightarrow q
  37. q q
  38. ¬ p \neg p
  39. ¬ ( p ) \neg\Diamond(p)

Intersection_number.html

  1. c : S 1 X c:S^{1}\to X
  2. η c \eta_{c}
  3. c α = - X α η c = ( α , * η c ) \int_{c}\alpha=-\int\int_{X}\alpha\wedge\eta_{c}=(\alpha,*\eta_{c})
  4. α \alpha
  5. \wedge
  6. * *
  7. a b := X η a η b = ( η a , - * η b ) = - b η a a\cdot b:=\int\int_{X}\eta_{a}\wedge\eta_{b}=(\eta_{a},-*\eta_{b})=-\int_{b}% \eta_{a}
  8. η c \eta_{c}
  9. Ω \Omega
  10. Ω c \Omega\setminus c
  11. Ω + \Omega^{+}
  12. Ω - \Omega^{-}
  13. Ω 0 \Omega_{0}
  14. Ω 0 - \Omega_{0}^{-}
  15. Ω 0 + \Omega_{0}^{+}
  16. f c ( x ) = { 1 , x Ω 0 - 0 , x X Ω - smooth interpolation , x Ω - Ω 0 - f_{c}(x)=\begin{cases}1,&x\in\Omega_{0}^{-}\\ 0,&x\in X\setminus\Omega^{-}\\ \mbox{smooth interpolation}~{},&x\in\Omega^{-}\setminus\Omega_{0}^{-}\end{cases}
  17. i = 1 N k i c i \sum_{i=1}^{N}k_{i}c_{i}
  18. c ω = i k i c i ω = i = 1 N k i c i ω \int_{c}\omega=\int_{\sum_{i}k_{i}c_{i}}\omega=\sum_{i=1}^{N}k_{i}\int_{c_{i}}\omega
  19. ω \omega
  20. η c \eta_{c}
  21. η c = i = 1 N k i η c i \eta_{c}=\sum_{i=1}^{N}k_{i}\eta_{c_{i}}
  22. n = dim k X n=\dim_{k}X
  23. f i ( x ) = 0 f_{i}(x)=0
  24. dim x i = 1 n Z i = 0 \dim_{x}\cap_{i=1}^{n}Z_{i}=0
  25. f i f_{i}
  26. ( Z 1 Z n ) x := dim k 𝒪 X , x / ( f 1 , , f n ) (Z_{1}\cdots Z_{n})_{x}:=\dim_{k}\mathcal{O}_{X,x}/(f_{1},\dots,f_{n})
  27. 𝒪 X , x \mathcal{O}_{X,x}
  28. k [ U ] 𝔪 x k[U]_{\mathfrak{m}_{x}}
  29. 𝔪 x \mathfrak{m}_{x}
  30. ( Z 1 , Z n ) = x i Z i ( Z 1 Z n ) x (Z_{1}\cdots,Z_{n})=\sum_{x\in\cap_{i}Z_{i}}(Z_{1}\cdots Z_{n})_{x}
  31. ( n Z 1 Z n ) = n ( Z 1 Z n ) (nZ_{1}\cdots Z_{n})=n(Z_{1}\cdots Z_{n})
  32. ( ( Y 1 + Z 1 ) Z 2 Z n ) = ( Y 1 Z 2 Z n ) + ( Z 1 Z 2 Z n ) ((Y_{1}+Z_{1})Z_{2}\cdots Z_{n})=(Y_{1}Z_{2}\cdots Z_{n})+(Z_{1}Z_{2}\cdots Z_% {n})
  33. ( ( P 1 - N 1 ) P 2 P n ) = ( P 1 P 2 P n ) - ( N 1 P 2 P n ) ((P_{1}-N_{1})P_{2}\cdots P_{n})=(P_{1}P_{2}\cdots P_{n})-(N_{1}P_{2}\cdots P_% {n})
  34. D M X D M Y D_{M}X\smile D_{M}Y
  35. I p ( P , Q ) = I p ( Q , P ) . I_{p}(P,Q)=I_{p}(Q,P).\,
  36. I p ( P , Q ) I_{p}(P,Q)
  37. I p ( P , Q ) I_{p}(P,Q)
  38. I p ( x , y ) = 1 I_{p}(x,y)=1
  39. I p ( P , Q 1 Q 2 ) = I p ( P , Q 1 ) + I p ( P , Q 2 ) I_{p}(P,Q_{1}Q_{2})=I_{p}(P,Q_{1})+I_{p}(P,Q_{2})
  40. I p ( P + Q R , Q ) = I p ( P , Q ) I_{p}(P+QR,Q)=I_{p}(P,Q)
  41. y = x 2 . y=x^{2}.
  42. P = y , P=y,
  43. Q = y - x 2 , Q=y-x^{2},
  44. I p ( P , Q ) = I p ( y , y - x 2 ) = I p ( y , x 2 ) = I p ( y , x ) + I p ( y , x ) = 1 + 1 = 2. I_{p}(P,Q)=I_{p}(y,y-x^{2})=I_{p}(y,x^{2})=I_{p}(y,x)+I_{p}(y,x)=1+1=2.\,

Interval_class.html

  1. i ( a , b ) = the smaller of i a , b and i b , a , i(a,b)=\,\text{ the smaller of }i\langle a,b\rangle\,\text{ and }i\langle b,a\rangle,

Intransitivity.html

  1. R R
  2. ¬ ( a , b , c : a R b b R c a R c ) . \lnot\left(\forall a,b,c:aRb\land bRc\Rightarrow aRc\right).
  3. a , b , c : a R b b R c ¬ ( a R c ) \exists a,b,c:aRb\land bRc\land\lnot(aRc)
  4. a , b , c : a R b b R c ¬ a R c \forall a,b,c:aRb\wedge bRc\Rightarrow\neg aRc

Intrinsic_metric.html

  1. ( M , d ) (M,d)
  2. d I d\text{I}
  3. M M
  4. d I ( x , y ) d\text{I}(x,y)
  5. x x
  6. y y
  7. x x
  8. y y
  9. γ : [ 0 , 1 ] M \gamma\colon[0,1]\rightarrow M
  10. γ ( 0 ) = x \gamma(0)=x
  11. γ ( 1 ) = y \gamma(1)=y
  12. d I ( x , y ) = d\text{I}(x,y)=\infty
  13. x x
  14. y y
  15. d I ( x , y ) = d ( x , y ) d\text{I}(x,y)=d(x,y)
  16. x x
  17. y y
  18. M M
  19. ( M , d ) (M,d)
  20. d d
  21. d d
  22. ε > 0 \varepsilon>0
  23. x x
  24. y y
  25. M M
  26. c c
  27. M M
  28. d ( x , c ) d(x,c)
  29. d ( c , y ) d(c,y)
  30. d ( x , y ) / 2 + ε {d(x,y)}/{2}+\varepsilon
  31. \R n \R^{n}
  32. \R n - { 0 } \R^{n}-\{0\}
  33. S 1 S^{1}
  34. \R 2 \R^{2}
  35. S 1 S^{1}
  36. d d I d\leq d\text{I}
  37. d I d\text{I}
  38. d d
  39. ( M , d I ) (M,d\text{I})
  40. d I d\text{I}
  41. ( M , d ) (M,d)
  42. M M
  43. M M

Intrinsic_parity.html

  1. P 2 ψ = e i ϕ ψ P^{2}\psi=e^{i\phi}\psi
  2. P ψ = ± e i ϕ / 2 ψ P\psi=\pm e^{i\phi/2}\psi
  3. P ( | 1 | 2 ) = ( P | 1 ) ( P | 2 ) P(|1\rangle|2\rangle)=(P|1\rangle)(P|2\rangle)

Intuitionistic_type_theory.html

  1. Vec ( , n ) \operatorname{Vec}({\mathbb{R}},n)
  2. \mathbb{N}
  3. : n , Vec ( , n ) \prod_{n\mathbin{:}{\mathbb{N}}}\operatorname{Vec}({\mathbb{R}},n)
  4. : n , \prod_{n\mathbin{:}{\mathbb{N}}}{\mathbb{R}}
  5. {\mathbb{N}}\to{\mathbb{R}}
  6. m , : n , ( m + n = n + m ) \prod_{m,n\mathbin{:}{\mathbb{N}}}(m+n=n+m)
  7. x = y x=y
  8. : n , Vec ( , n ) \sum_{n\mathbin{:}{\mathbb{N}}}\operatorname{Vec}({\mathbb{R}},n)
  9. n n
  10. n n
  11. : n , \sum_{n\mathbin{:}{\mathbb{N}}}{\mathbb{R}}
  12. × {\mathbb{N}}\times{\mathbb{R}}
  13. ¬ A A . \neg A\equiv A\to\bot.
  14. a , : b , A a,b\mathbin{:}A
  15. a = b a=b
  16. a a
  17. b b
  18. a = b a=b
  19. a a
  20. b b
  21. a = a a=a
  22. refl : : a , A ( a = a ) . \operatorname{refl}\mathbin{:}\prod_{a\mathbin{:}A}(a=a).
  23. \mathbb{N}
  24. : 0 , 0\mathbin{:}{\mathbb{N}}
  25. : succ , \operatorname{succ}\mathbin{:}{\mathbb{N}}\to{\mathbb{N}}
  26. - elim : P ( 0 ) ( : n , P ( n ) P ( succ ( n ) ) ) : n , P ( n ) {\operatorname{{\mathbb{N}}-elim}}\,\mathbin{:}P(0)\,\to\left(\prod_{n\mathbin% {:}{\mathbb{N}}}P(n)\to P(\operatorname{succ}(n))\right)\to\prod_{n\mathbin{:}% {\mathbb{N}}}P(n)
  27. P ( n ) P(n)
  28. : n , n\mathbin{:}{\mathbb{N}}
  29. Vec ( A , n ) \operatorname{Vec}(A,n)
  30. : vnil , Vec ( A , 0 ) \operatorname{vnil}\mathbin{:}\operatorname{Vec}(A,0)
  31. : vcons , A : n , Vec ( A , n ) Vec ( A , succ ( n ) ) . \operatorname{vcons}\,\mathbin{:}\,A\to\prod_{n\mathbin{:}{\mathbb{N}}}% \operatorname{Vec}(A,n)\to\operatorname{Vec}(A,\operatorname{succ}(n)).
  32. 𝒰 0 \mathcal{U}_{0}
  33. : a , 𝒰 0 a\mathbin{:}\mathcal{U}_{0}
  34. El ( a ) \operatorname{El}(a)
  35. 𝒰 n \mathcal{U}_{n}
  36. : n , n\mathbin{:}{\mathbb{N}}
  37. 𝒰 n + 1 \mathcal{U}_{n+1}
  38. : u n , 𝒰 n + 1 u_{n}\mathbin{:}\mathcal{U}_{n+1}
  39. El ( u n ) 𝒰 n \operatorname{El}(u_{n})\equiv\mathcal{U}_{n}
  40. A 𝖳𝗒𝗉𝖾 A\ \mathsf{Type}
  41. : a , A a\mathbin{:}A
  42. a = b a=b
  43. A = B A=B
  44. ( : x , A ) B (x\mathbin{:}A)B
  45. B [ x / a ] B[x/a]
  46. x x
  47. a a
  48. B B
  49. [ x ] b [x]b
  50. b [ x / a ] b[x/a]
  51. x x
  52. a a
  53. b b
  54. : 0 , 0\mathbin{:}\mathbb{N}
  55. : succ , \operatorname{succ}\mathbin{:}\mathbb{N}\to\mathbb{N}
  56. succ \operatorname{succ}
  57. succ \operatorname{succ}
  58. succ \operatorname{succ}
  59. add \displaystyle\operatorname{add}
  60. succ \operatorname{succ}
  61. Γ σ 𝖳𝗒𝗉𝖾 \Gamma\vdash\sigma\ \mathsf{Type}
  62. Γ : t , σ \Gamma\vdash t\mathbin{:}\sigma
  63. Γ σ τ \Gamma\vdash\sigma\equiv\tau
  64. Γ t : u , σ \Gamma\vdash t\equiv u\mathbin{:}\sigma
  65. Γ 𝖢𝗈𝗇𝗍𝖾𝗑𝗍 \vdash\Gamma\ \mathsf{Context}
  66. 𝒰 \mathcal{U}
  67. Set \operatorname{Set}
  68. 𝒰 \mathcal{U}
  69. El \operatorname{El}
  70. El \operatorname{El}
  71. A A
  72. 𝒰 \mathcal{U}
  73. A × B A\times B
  74. A A
  75. B B
  76. A A
  77. B B
  78. A A
  79. B B
  80. A × B A\times B

Invariant_subspace.html

  1. T T
  2. T : n n T:\mathbb{R}^{n}\to\mathbb{R}^{n}
  3. W W
  4. T T
  5. v W v\in W
  6. T T
  7. W W
  8. v W T ( v ) W v\in W\Rightarrow T(v)\in W
  9. n \mathbb{R}^{n}
  10. T T
  11. n \mathbb{R}^{n}
  12. n \mathbb{R}^{n}
  13. { 0 } \{0\}
  14. 0 0 0\to 0
  15. v v
  16. x U x\in U
  17. λ v \lambda v
  18. λ \lambda
  19. T T
  20. A A
  21. U U
  22. x U α : A x = α v \forall_{x\in U}\;\exists_{\alpha\in\mathbb{R}}:Ax=\alpha v
  23. x U x = β v x\in U\Rightarrow x=\beta v
  24. β \beta\in\mathbb{R}
  25. A v = λ v Av=\lambda v
  26. A A
  27. T T
  28. Lat ( Σ ) = T Σ Lat ( T ) . \mathrm{Lat}(\Sigma)=\bigcap_{T\in\Sigma}\mathrm{Lat}(T)\;.
  29. T = [ T 11 T 12 0 T 22 ] T=\begin{bmatrix}T_{11}&T_{12}\\ 0&T_{22}\end{bmatrix}
  30. V = W W . V=W\oplus W^{\prime}.
  31. T = [ T 11 T 12 T 21 T 22 ] : W W W W , T=\begin{bmatrix}T_{11}&T_{12}\\ T_{21}&T_{22}\end{bmatrix}:\begin{matrix}W\\ \oplus\\ W^{\prime}\end{matrix}\rightarrow\begin{matrix}W\\ \oplus\\ W^{\prime}\end{matrix},
  32. P = [ 1 0 0 0 ] : W W W W . P=\begin{bmatrix}1&0\\ 0&0\end{bmatrix}:\begin{matrix}W\\ \oplus\\ W^{\prime}\end{matrix}\rightarrow\begin{matrix}W\\ \oplus\\ W^{\prime}\end{matrix}.
  33. T = [ T 11 0 0 T 22 ] : Ran P Ran ( 1 - P ) Ran P Ran ( 1 - P ) . T=\begin{bmatrix}T_{11}&0\\ 0&T_{22}\end{bmatrix}:\begin{matrix}\mbox{Ran}~{}P\\ \oplus\\ \mbox{Ran}~{}(1-P)\end{matrix}\rightarrow\begin{matrix}\mbox{Ran}~{}P\\ \oplus\\ \mbox{Ran}~{}(1-P)\end{matrix}\;.
  34. W Σ W = W Σ W \bigwedge_{W\in\Sigma^{\prime}}W=\bigcap_{W\in\Sigma^{\prime}}W
  35. W Σ W = span W Σ W . \bigvee_{W\in\Sigma^{\prime}}W=\mbox{span}~{}\bigcup_{W\in\Sigma^{\prime}}W\;.
  36. span { e 1 , , e k } Lat ( Σ ) k 1 . \mbox{span}~{}\{e_{1},\cdots,e_{k}\}\in\mbox{Lat}~{}(\Sigma)\quad\forall k\geq 1\;.

Invariant_theory.html

  1. π : G G L ( V ) \pi:G\to GL(V)
  2. ( g f ) ( x ) := f ( g - 1 x ) x V , g G , f k [ V ] . (g\cdot f)(x):=f(g^{-1}x)\qquad\forall x\in V,g\in G,f\in k[V].
  3. 𝐂 2 \mathbf{C}^{2}

Inverse_iteration.html

  1. μ \mu
  2. b k + 1 = ( A - μ I ) - 1 b k C k , b_{k+1}=\frac{(A-\mu I)^{-1}b_{k}}{C_{k}},
  3. C k = ( A - μ I ) - 1 b k . C_{k}=\|(A-\mu I)^{-1}b_{k}\|.
  4. C k C_{k}
  5. ( A - μ I ) (A-\mu I)
  6. ( A - μ I ) - 1 . (A-\mu I)^{-1}.
  7. μ \mu
  8. μ \mu
  9. A b , A 2 b , A 3 b , Ab,A^{2}b,A^{3}b,...
  10. ( A - μ I ) - 1 (A-\mu I)^{-1}
  11. ( A - μ I ) - 1 (A-\mu I)^{-1}
  12. ( λ 1 - μ ) - 1 , , ( λ n - μ ) - 1 , (\lambda_{1}-\mu)^{-1},...,(\lambda_{n}-\mu)^{-1},
  13. λ i \lambda_{i}
  14. ( λ 1 - μ ) , , ( λ n - μ ) . (\lambda_{1}-\mu),...,(\lambda_{n}-\mu).
  15. ( A - μ I ) - 1 (A-\mu I)^{-1}
  16. μ . \mu.
  17. μ = 0 \mu=0
  18. ( A ) - k b (A)^{-k}b
  19. Distance ( b ideal , b Power Method k ) = O ( | λ subdominant λ dominant | k ) , \mathrm{Distance}(b^{\mathrm{ideal}},b^{k}_{\mathrm{Power~{}Method}})=O\left(% \left|\frac{\lambda_{\mathrm{subdominant}}}{\lambda_{\mathrm{dominant}}}\right% |^{k}\right),
  20. Distance ( b ideal , b Inverse iteration k ) = O ( | μ - λ closest to μ μ - λ second closest to μ | k ) . \mathrm{Distance}(b^{\mathrm{ideal}},b^{k}_{\mathrm{Inverse~{}iteration}})=O% \left(\left|\frac{\mu-\lambda_{\mathrm{closest~{}to~{}}\mu}}{\mu-\lambda_{% \mathrm{second~{}closest~{}to~{}}\mu}}\right|^{k}\right).
  21. μ \mu
  22. λ \lambda
  23. μ - λ = ϵ \mu-\lambda=\epsilon
  24. | ϵ | / | λ + ϵ - λ closest to λ | |\epsilon|/|\lambda+\epsilon-\lambda_{\mathrm{closest~{}to~{}}\lambda}|
  25. ϵ \epsilon
  26. μ \mu
  27. λ \lambda
  28. | ϵ | |\epsilon|
  29. | ϵ | / | λ - λ closest to λ | |\epsilon|/|\lambda-\lambda_{\mathrm{closest~{}to~{}}\lambda}|
  30. μ \mu
  31. ϵ \epsilon
  32. O ( n 3 ) O(n^{3})
  33. b k + 1 = ( A - μ I ) - 1 b k C k , b_{k+1}=\frac{(A-\mu I)^{-1}b_{k}}{C_{k}},
  34. ( A - μ I ) b k + 1 = b k C k , (A-\mu I)b_{k+1}=\frac{b_{k}}{C_{k}},
  35. b k + 1 b_{k+1}
  36. ( A - μ I ) - 1 (A-\mu I)^{-1}
  37. μ \mu
  38. ( A - μ I ) (A-\mu I)
  39. 10 3 n 3 + O ( n 2 ) \begin{matrix}\frac{10}{3}\end{matrix}n^{3}+O(n^{2})
  40. 4 3 n 3 + O ( n 2 ) \begin{matrix}\frac{4}{3}\end{matrix}n^{3}+O(n^{2})
  41. A | λ | , \left\|A\right\|\geq|\lambda|,
  42. λ \lambda

Inverse_kinematics.html

  1. m m
  2. p ( x ) : m 3 p(x):\Re^{m}\rightarrow\Re^{3}
  3. p 0 = p ( x 0 ) p_{0}=p(x_{0})
  4. p 1 = p ( x 0 + Δ x ) p_{1}=p(x_{0}+\Delta x)
  5. Δ x \Delta x
  6. || p ( x 0 + Δ x ) - p 1 || ||p(x_{0}+\Delta x)-p_{1}||
  7. Δ x \Delta x
  8. p ( x 1 ) p ( x 0 ) + J p ( x ^ 0 ) Δ x p(x_{1})\approx p(x_{0})+J_{p}(\hat{x}_{0})\Delta x
  9. J p ( x 0 ) J_{p}(x_{0})
  10. x 0 x_{0}
  11. p i x k p i ( x 0 , k + h ) - p i ( x 0 ) h \frac{\partial p_{i}}{\partial x_{k}}\approx\frac{p_{i}(x_{0,k}+h)-p_{i}(x_{0}% )}{h}
  12. p i ( x ) p_{i}(x)
  13. x 0 , k + h x_{0,k}+h
  14. x 0 x_{0}
  15. h h
  16. Δ x J p + ( x 0 ) Δ p \Delta x\approx J_{p}^{+}(x_{0})\Delta p
  17. Δ p = p ( x 0 + Δ x ) - p ( x 0 ) \Delta p=p(x_{0}+\Delta x)-p(x_{0})
  18. Δ x \Delta x
  19. Δ x \Delta x
  20. Δ x \Delta x
  21. Δ x k + 1 = J p + ( x k ) Δ p k \Delta x_{k+1}=J_{p}^{+}(x_{k})\Delta p_{k}
  22. Δ x \Delta x
  23. Δ x \Delta x

Inverse_trigonometric_functions.html

  1. c o s ( x ) < s u p > 1 cos(x)<sup>−1
  2. θ \theta
  3. sin θ \sin\theta
  4. cos θ \cos\theta
  5. tan θ \tan\theta
  6. arcsin x \arcsin x
  7. sin ( arcsin x ) = x \sin(\arcsin x)=x
  8. cos ( arcsin x ) = 1 - x 2 \cos(\arcsin x)=\sqrt{1-x^{2}}
  9. tan ( arcsin x ) = x 1 - x 2 \tan(\arcsin x)=\frac{x}{\sqrt{1-x^{2}}}
  10. arccos x \arccos x
  11. sin ( arccos x ) = 1 - x 2 \sin(\arccos x)=\sqrt{1-x^{2}}
  12. cos ( arccos x ) = x \cos(\arccos x)=x
  13. tan ( arccos x ) = 1 - x 2 x \tan(\arccos x)=\frac{\sqrt{1-x^{2}}}{x}
  14. arctan x \arctan x
  15. sin ( arctan x ) = x 1 + x 2 \sin(\arctan x)=\frac{x}{\sqrt{1+x^{2}}}
  16. cos ( arctan x ) = 1 1 + x 2 \cos(\arctan x)=\frac{1}{\sqrt{1+x^{2}}}
  17. tan ( arctan x ) = x \tan(\arctan x)=x
  18. \arccot x \arccot x
  19. sin ( \arccot x ) = 1 1 + x 2 \sin(\arccot x)=\frac{1}{\sqrt{1+x^{2}}}
  20. cos ( \arccot x ) = x 1 + x 2 \cos(\arccot x)=\frac{x}{\sqrt{1+x^{2}}}
  21. tan ( \arccot x ) = 1 x \tan(\arccot x)=\frac{1}{x}
  22. \arcsec x \arcsec x
  23. sin ( \arcsec x ) = x 2 - 1 x \sin(\arcsec x)=\frac{\sqrt{x^{2}-1}}{x}
  24. cos ( \arcsec x ) = 1 x \cos(\arcsec x)=\frac{1}{x}
  25. tan ( \arcsec x ) = x 2 - 1 \tan(\arcsec x)=\sqrt{x^{2}-1}
  26. \arccsc x \arccsc x
  27. sin ( \arccsc x ) = 1 x \sin(\arccsc x)=\frac{1}{x}
  28. cos ( \arccsc x ) = x 2 - 1 x \cos(\arccsc x)=\frac{\sqrt{x^{2}-1}}{x}
  29. tan ( \arccsc x ) = 1 x 2 - 1 \tan(\arccsc x)=\frac{1}{\sqrt{x^{2}-1}}
  30. arccos x \displaystyle\arccos x
  31. arcsin ( - x ) \displaystyle\arcsin(-x)
  32. arccos 1 x \displaystyle\arccos\tfrac{1}{x}
  33. arccos x \displaystyle\arccos x
  34. tan θ 2 = sin θ 1 + cos θ \tan\tfrac{\theta}{2}=\tfrac{\sin\theta}{1+\cos\theta}
  35. arcsin x \displaystyle\arcsin x
  36. arctan u + arctan v = arctan ( u + v 1 - u v ) ( mod π ) , u v 1 . \arctan u+\arctan v=\arctan\left(\frac{u+v}{1-uv}\right)\;\;(\mathop{{\rm mod}% }\pi)\,,\quad uv\neq 1\,.
  37. tan ( α + β ) = tan α + tan β 1 - tan α tan β , \tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}\,,
  38. α = arctan u , β = arctan v . \alpha=\arctan u\,,\quad\beta=\arctan v\,.
  39. d d z arcsin z \displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\arcsin z
  40. d d x \arcsec x \displaystyle\frac{\mathrm{d}}{\mathrm{d}x}\arcsec x
  41. θ = arcsin x \theta=\arcsin x\!
  42. d arcsin x d x = d θ d sin θ = d θ cos θ d θ = 1 cos θ = 1 1 - sin 2 θ = 1 1 - x 2 \frac{\mathrm{d}\arcsin x}{\mathrm{d}x}=\frac{\mathrm{d}\theta}{\mathrm{d}\sin% \theta}=\frac{\mathrm{d}\theta}{\cos\theta\mathrm{d}\theta}=\frac{1}{\cos% \theta}=\frac{1}{\sqrt{1-\sin^{2}\theta}}=\frac{1}{\sqrt{1-x^{2}}}
  43. arcsin x \displaystyle\arcsin x
  44. 1 1 - z 2 \frac{1}{\sqrt{1-z^{2}}}
  45. 1 1 + z 2 \frac{1}{1+z^{2}}
  46. arcsin z = z + ( 1 2 ) z 3 3 + ( 1 3 2 4 ) z 5 5 + ( 1 3 5 2 4 6 ) z 7 7 + = n = 0 ( 2 n n ) z 2 n + 1 4 n ( 2 n + 1 ) ; | z | 1 \arcsin z=z+\left(\frac{1}{2}\right)\frac{z^{3}}{3}+\left(\frac{1\cdot 3}{2% \cdot 4}\right)\frac{z^{5}}{5}+\left(\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}% \right)\frac{z^{7}}{7}+\cdots=\sum_{n=0}^{\infty}\frac{{\left({{2n}\atop{n}}% \right)}z^{2n+1}}{4^{n}(2n+1)}\,;\qquad|z|\leq 1
  47. arccos z = π 2 - arcsin z = π 2 - ( z + ( 1 2 ) z 3 3 + ( 1 3 2 4 ) z 5 5 + ) = π 2 - n = 0 ( 2 n n ) z 2 n + 1 4 n ( 2 n + 1 ) ; | z | 1 \arccos z=\frac{\pi}{2}-\arcsin z=\frac{\pi}{2}-\left(z+\left(\frac{1}{2}% \right)\frac{z^{3}}{3}+\left(\frac{1\cdot 3}{2\cdot 4}\right)\frac{z^{5}}{5}+% \cdots\right)=\frac{\pi}{2}-\sum_{n=0}^{\infty}\frac{{\left({{2n}\atop{n}}% \right)}z^{2n+1}}{4^{n}(2n+1)}\,;\qquad|z|\leq 1
  48. arctan z = z - z 3 3 + z 5 5 - z 7 7 + = n = 0 ( - 1 ) n z 2 n + 1 2 n + 1 ; | z | 1 z i , - i \arctan z=z-\frac{z^{3}}{3}+\frac{z^{5}}{5}-\frac{z^{7}}{7}+\cdots=\sum_{n=0}^% {\infty}\frac{(-1)^{n}z^{2n+1}}{2n+1}\,;\qquad|z|\leq 1\qquad z\neq\mathrm{i},% -\mathrm{i}
  49. \arccot z = π 2 - arctan z = π 2 - ( z - z 3 3 + z 5 5 - z 7 7 + ) = π 2 - n = 0 ( - 1 ) n z 2 n + 1 2 n + 1 ; | z | 1 z i , - i \arccot z=\frac{\pi}{2}-\arctan z=\frac{\pi}{2}-\left(z-\frac{z^{3}}{3}+\frac{% z^{5}}{5}-\frac{z^{7}}{7}+\cdots\right)=\frac{\pi}{2}-\sum_{n=0}^{\infty}\frac% {(-1)^{n}z^{2n+1}}{2n+1}\,;\qquad|z|\leq 1\qquad z\neq\mathrm{i},-\mathrm{i}
  50. \arcsec z = arccos ( 1 / z ) = π 2 - ( z - 1 + ( 1 2 ) z - 3 3 + ( 1 3 2 4 ) z - 5 5 + ) = π 2 - n = 0 ( 2 n n ) z - ( 2 n + 1 ) 4 n ( 2 n + 1 ) ; | z | 1 \arcsec z=\arccos(1/z)=\frac{\pi}{2}-\left(z^{-1}+\left(\frac{1}{2}\right)% \frac{z^{-3}}{3}+\left(\frac{1\cdot 3}{2\cdot 4}\right)\frac{z^{-5}}{5}+\cdots% \right)=\frac{\pi}{2}-\sum_{n=0}^{\infty}\frac{{\left({{2n}\atop{n}}\right)}z^% {-(2n+1)}}{4^{n}(2n+1)}\,;\qquad|z|\geq 1
  51. \arccsc z = arcsin ( 1 / z ) = z - 1 + ( 1 2 ) z - 3 3 + ( 1 3 2 4 ) z - 5 5 + = n = 0 ( 2 n n ) z - ( 2 n + 1 ) 4 n ( 2 n + 1 ) ; | z | 1 \arccsc z=\arcsin(1/z)=z^{-1}+\left(\frac{1}{2}\right)\frac{z^{-3}}{3}+\left(% \frac{1\cdot 3}{2\cdot 4}\right)\frac{z^{-5}}{5}+\cdots=\sum_{n=0}^{\infty}% \frac{{\left({{2n}\atop{n}}\right)}z^{-(2n+1)}}{4^{n}(2n+1)}\,;\qquad|z|\geq 1
  52. arctan z = z 1 + z 2 n = 0 k = 1 n 2 k z 2 ( 2 k + 1 ) ( 1 + z 2 ) . \arctan z=\frac{z}{1+z^{2}}\sum_{n=0}^{\infty}\prod_{k=1}^{n}\frac{2kz^{2}}{(2% k+1)(1+z^{2})}\,.
  53. arctan z = n = 0 2 2 n ( n ! ) 2 ( 2 n + 1 ) ! z 2 n + 1 ( 1 + z 2 ) n + 1 \arctan z=\sum_{n=0}^{\infty}\frac{2^{2n}(n!)^{2}}{(2n+1)!}\;\frac{z^{2n+1}}{(% 1+z^{2})^{n+1}}
  54. arctan z = z 1 + ( 1 z ) 2 3 - 1 z 2 + ( 3 z ) 2 5 - 3 z 2 + ( 5 z ) 2 7 - 5 z 2 + ( 7 z ) 2 9 - 7 z 2 + = z 1 + ( 1 z ) 2 3 + ( 2 z ) 2 5 + ( 3 z ) 2 7 + ( 4 z ) 2 9 + \arctan z=\frac{z}{1+\cfrac{(1z)^{2}}{3-1z^{2}+\cfrac{(3z)^{2}}{5-3z^{2}+% \cfrac{(5z)^{2}}{7-5z^{2}+\cfrac{(7z)^{2}}{9-7z^{2}+\ddots}}}}}=\frac{z}{1+% \cfrac{(1z)^{2}}{3+\cfrac{(2z)^{2}}{5+\cfrac{(3z)^{2}}{7+\cfrac{(4z)^{2}}{9+% \ddots}}}}}
  55. arcsin x d x \displaystyle\int\arcsin x\,\mathrm{d}x
  56. \arcsec x d x \displaystyle\int\arcsec x\,\mathrm{d}x
  57. u d v = u v - v d u \int u\,\mathrm{d}v=uv-\int v\,\mathrm{d}u
  58. u = arcsin x d v = d x d u = d x 1 - x 2 v = x \begin{aligned}\displaystyle u&\displaystyle{}=&\displaystyle\arcsin x&% \displaystyle\quad\quad\mathrm{d}v=\mathrm{d}x\\ \displaystyle\mathrm{d}u&\displaystyle{}=&\displaystyle\frac{\mathrm{d}x}{% \sqrt{1-x^{2}}}&\displaystyle\quad\quad{}v=x\end{aligned}
  59. arcsin ( x ) d x = x arcsin x - x 1 - x 2 d x \int\arcsin(x)\,\mathrm{d}x=x\arcsin x-\int\frac{x}{\sqrt{1-x^{2}}}\,\mathrm{d}x
  60. k = 1 - x 2 . k=1-x^{2}\,.
  61. d k = - 2 x d x \mathrm{d}k=-2x\,\mathrm{d}x
  62. x 1 - x 2 d x = - 1 2 d k k = - k \int\frac{x}{\sqrt{1-x^{2}}}\,\mathrm{d}x=-\frac{1}{2}\int\frac{\mathrm{d}k}{% \sqrt{k}}=-\sqrt{k}
  63. arcsin ( x ) d x = x arcsin x + 1 - x 2 + C \int\arcsin(x)\,\mathrm{d}x=x\arcsin x+\sqrt{1-x^{2}}+C
  64. arctan z = 0 z d x 1 + x 2 z - i , + i \arctan z=\int_{0}^{z}\frac{\mathrm{d}x}{1+x^{2}}\quad z\neq-\mathrm{i},+% \mathrm{i}\,
  65. arcsin z = arctan z 1 - z 2 z - 1 , + 1 \arcsin z=\arctan\frac{z}{\sqrt{1-z^{2}}}\quad z\neq-1,+1\,
  66. arccos z = π 2 - arcsin z z - 1 , + 1 \arccos z=\frac{\mathrm{\pi}}{2}-\arcsin z\quad z\neq-1,+1\,
  67. \arccot z = π 2 - arctan z z - i , + i \arccot z=\frac{\mathrm{\pi}}{2}-\arctan z\quad z\neq-\mathrm{i},+\mathrm{i}\,
  68. \arcsec z = arccos 1 z z - 1 , 0 , + 1 \arcsec z=\arccos\frac{1}{z}\quad z\neq-1,0,+1\,
  69. \arccsc z = arcsin 1 z z - 1 , 0 , + 1 \arccsc z=\arcsin\frac{1}{z}\quad z\neq-1,0,+1\,
  70. arcsin x = - i ln ( i x + 1 - x 2 ) = \arccsc 1 x arccos x = i ln ( x - i 1 - x 2 ) = π 2 + i ln ( i x + 1 - x 2 ) = π 2 - arcsin x = \arcsec 1 x arctan x = 1 2 i ( ln ( 1 - i x ) - ln ( 1 + i x ) ) = \arccot 1 x \arccot x = 1 2 i ( ln ( 1 - i x ) - ln ( 1 + i x ) ) = arctan 1 x \arcsec x = - i ln ( i 1 - 1 x 2 + 1 x ) = i ln ( 1 - 1 x 2 + i x ) + π 2 = π 2 - \arccsc x = arccos 1 x \arccsc x = - i ln ( 1 - 1 x 2 + i x ) = arcsin 1 x \begin{aligned}\displaystyle\arcsin x&\displaystyle{}=-\mathrm{i}\,\ln\left(% \mathrm{i}\,x+\sqrt{1-x^{2}}\right)&\displaystyle{}=\arccsc\frac{1}{x}\\ \displaystyle\arccos x&\displaystyle{}=\mathrm{i}\,\ln\left(x-\mathrm{i}\,% \sqrt{1-x^{2}}\right)=\frac{\mathrm{\pi}}{2}\,+\mathrm{i}\ln\left(\mathrm{i}\,% x+\sqrt{1-x^{2}}\right)=\frac{\mathrm{\pi}}{2}-\arcsin x&\displaystyle{}=% \arcsec\frac{1}{x}\\ \displaystyle\arctan x&\displaystyle{}=\tfrac{1}{2}\mathrm{i}\,\left(\ln\left(% 1-\mathrm{i}\,x\right)-\ln\left(1+\mathrm{i}\,x\right)\right)&\displaystyle{}=% \arccot\frac{1}{x}\\ \displaystyle\arccot x&\displaystyle{}=\tfrac{1}{2}\mathrm{i}\,\left(\ln\left(% 1-\frac{\mathrm{i}}{x}\right)-\ln\left(1+\frac{\mathrm{i}}{x}\right)\right)&% \displaystyle{}=\arctan\frac{1}{x}\\ \displaystyle\arcsec x&\displaystyle{}=-\mathrm{i}\,\ln\left(\mathrm{i}\,\sqrt% {1-\frac{1}{x^{2}}}+\frac{1}{x}\right)=\mathrm{i}\,\ln\left(\sqrt{1-\frac{1}{x% ^{2}}}+\frac{\mathrm{i}}{x}\right)+\frac{\mathrm{\pi}}{2}=\frac{\mathrm{\pi}}{% 2}-\arccsc x&\displaystyle{}=\arccos\frac{1}{x}\\ \displaystyle\arccsc x&\displaystyle{}=-\mathrm{i}\,\ln\left(\sqrt{1-\frac{1}{% x^{2}}}+\frac{\mathrm{i}}{x}\right)&\displaystyle{}=\arcsin\frac{1}{x}\end{aligned}
  71. θ = arcsin x \theta=\arcsin x
  72. sin ( θ ) = sin ( arcsin x ) \sin(\theta)=\sin(\arcsin x)
  73. sin ( θ ) = x \sin(\theta)=x
  74. e i ϕ - e - i ϕ 2 i = sin ( ϕ ) \frac{\mathrm{e}^{\mathrm{i}\phi}-\mathrm{e}^{-\mathrm{i}\phi}}{2\mathrm{i}}=% \sin(\phi)
  75. e i θ - e - i θ 2 i = x \frac{\mathrm{e}^{\mathrm{i}\theta}-\mathrm{e}^{-\mathrm{i}\theta}}{2\mathrm{i% }}=x
  76. k = e i θ k=\mathrm{e}^{\mathrm{i}\,\theta}\,
  77. k - 1 k 2 i = x \frac{k-\frac{1}{k}}{2\mathrm{i}}=x
  78. k - 1 k = 2 i x {k-\frac{1}{k}}=2\mathrm{i}x
  79. k - 2 i x - 1 k = 0 {k-2\mathrm{i}x-\frac{1}{k}}=0
  80. k 2 - 2 i k x - 1 = 0 k^{2}-2\,\mathrm{i}\,k\,x-1\,=\,0
  81. k = i x ± 1 - x 2 k=\mathrm{i}x\pm\sqrt{1-x^{2}}
  82. e i θ = i x ± 1 - x 2 \mathrm{e}^{\mathrm{i}\theta}=\mathrm{i}x\pm\sqrt{1-x^{2}}
  83. i θ = ln ( i x ± 1 - x 2 ) \mathrm{i}\theta=\ln\left(\mathrm{i}x\pm\sqrt{1-x^{2}}\right)
  84. θ = - i ln ( i x ± 1 - x 2 ) \theta=-\mathrm{i}\ln\left(\mathrm{i}x\pm\sqrt{1-x^{2}}\right)
  85. θ = arcsin x = - i ln ( i x + 1 - x 2 ) \theta=\arcsin x=-\mathrm{i}\ln\left(\mathrm{i}x+\sqrt{1-x^{2}}\right)
  86. θ = arcsin x \theta=\arcsin x
  87. e i θ = cos ( θ ) + i sin ( θ ) \mathrm{e}^{\mathrm{i}\theta}=\cos(\theta)+\mathrm{i}\sin(\theta)
  88. arcsin x = - i ln ( cos ( arcsin x ) + i sin ( arcsin x ) ) \arcsin x=-\mathrm{i}\ln(\cos(\arcsin x)+\mathrm{i}\sin(\arcsin x))
  89. arcsin x = - i ln ( 1 - x 2 + i x ) \arcsin x=-\mathrm{i}\ln(\sqrt{1-x^{2}}+\mathrm{i}x)
  90. arcsin ( z ) \arcsin(z)
  91. arccos ( z ) \arccos(z)
  92. arctan ( z ) \arctan(z)
  93. \arccot ( z ) \arccot(z)
  94. \arcsec ( z ) \arcsec(z)
  95. \arccsc ( z ) \arccsc(z)
  96. π \pi
  97. π \pi
  98. π \pi
  99. π \pi
  100. π \pi
  101. π \pi
  102. π \pi
  103. π \pi
  104. π \pi
  105. π \pi
  106. π \pi
  107. π \pi
  108. π \pi
  109. π \pi
  110. π \pi
  111. π \pi
  112. π \pi
  113. π \pi
  114. π \pi
  115. π \pi
  116. π \pi
  117. π \pi
  118. π \pi
  119. π \pi
  120. π \pi
  121. π \pi
  122. π \pi
  123. π \pi
  124. π \pi
  125. π \pi
  126. π \pi
  127. sin ( y ) = x y = arcsin ( x ) + 2 π k or y = π - arcsin ( x ) + 2 π k \sin(y)=x\;\Leftrightarrow\;y=\arcsin(x)+2\mathrm{\pi}k\;\,\text{ or }\;y=% \mathrm{\pi}-\arcsin(x)+2\mathrm{\pi}k
  128. sin ( y ) = x y = ( - 1 ) k arcsin ( x ) + π k \sin(y)=x\;\Leftrightarrow\;y=(-1)^{k}\arcsin(x)+\mathrm{\pi}k
  129. cos ( y ) = x y = arccos ( x ) + 2 π k or y = 2 π - arccos ( x ) + 2 π k \cos(y)=x\;\Leftrightarrow\;y=\arccos(x)+2\mathrm{\pi}k\;\,\text{ or }\;y=2% \mathrm{\pi}-\arccos(x)+2\mathrm{\pi}k
  130. cos ( y ) = x y = ± arccos ( x ) + 2 π k \cos(y)=x\;\Leftrightarrow\;y=\pm\arccos(x)+2\mathrm{\pi}k
  131. tan ( y ) = x y = arctan ( x ) + π k \tan(y)=x\;\Leftrightarrow\;y=\arctan(x)+\mathrm{\pi}k
  132. cot ( y ) = x y = \arccot ( x ) + π k \cot(y)=x\;\Leftrightarrow\;y=\arccot(x)+\mathrm{\pi}k
  133. sec ( y ) = x y = \arcsec ( x ) + 2 π k or y = 2 π - \arcsec ( x ) + 2 π k \sec(y)=x\;\Leftrightarrow\;y=\arcsec(x)+2\mathrm{\pi}k\,\text{ or }y=2\mathrm% {\pi}-\arcsec(x)+2\mathrm{\pi}k
  134. csc ( y ) = x y = \arccsc ( x ) + 2 π k or y = π - \arccsc ( x ) + 2 π k \csc(y)=x\;\Leftrightarrow\;y=\arccsc(x)+2\mathrm{\pi}k\,\text{ or }y=\mathrm{% \pi}-\arccsc(x)+2\mathrm{\pi}k
  135. θ = arcsin ( opposite hypotenuse ) . \theta=\arcsin\left(\frac{\,\text{opposite}}{\,\text{hypotenuse}}\right)\,.
  136. a 2 + b 2 = h 2 a^{2}+b^{2}=h^{2}
  137. h h
  138. θ = arctan ( opposite adjacent ) . \theta=\arctan\left(\frac{\,\text{opposite}}{\,\text{adjacent}}\right)\,.
  139. θ = arctan ( opposite adjacent ) = arctan ( rise run ) = arctan ( 8 20 ) 21.8 . \theta=\arctan\left(\frac{\,\text{opposite}}{\,\text{adjacent}}\right)=\arctan% \left(\frac{\,\text{rise}}{\,\text{run}}\right)=\arctan\left(\frac{8}{20}% \right)\approx 21.8^{\circ}\,.
  140. π \pi
  141. π \pi
  142. π \pi
  143. atan2 ( y , x ) = { arctan ( y x ) x > 0 arctan ( y x ) + π y 0 , x < 0 arctan ( y x ) - π y < 0 , x < 0 π 2 y > 0 , x = 0 - π 2 y < 0 , x = 0 undefined y = 0 , x = 0 \operatorname{atan2}(y,x)=\begin{cases}\arctan(\frac{y}{x})&\quad x>0\\ \arctan(\frac{y}{x})+\mathrm{\pi}&\quad y\geq 0\;,\;x<0\\ \arctan(\frac{y}{x})-\mathrm{\pi}&\quad y<0\;,\;x<0\\ \frac{\mathrm{\pi}}{2}&\quad y>0\;,\;x=0\\ -\frac{\mathrm{\pi}}{2}&\quad y<0\;,\;x=0\\ \,\text{undefined}&\quad y=0\;,\;x=0\end{cases}
  144. atan2 ( y , x ) = 2 arctan y x 2 + y 2 + x \operatorname{atan2}(y,x)=2\arctan\frac{y}{\sqrt{x^{2}+y^{2}}+x}
  145. y y
  146. x = tan y x=\tan y
  147. - < η < -\infty<\eta<\infty
  148. y = arctan η x := arctan x + π rni η - arctan x π . y=\arctan_{\eta}x:=\arctan x+\mathrm{\pi}\cdot\operatorname{rni}\frac{\eta-% \arctan x}{\mathrm{\pi}}\,.
  149. rni \operatorname{rni}
  150. π \pi
  151. π \pi
  152. π \pi

Invertible_sheaf.html

  1. S T S\otimes T
  2. Pic ( X ) \mathrm{Pic}(X)

Ion_source.html

  1. M + e - M + + 2 e - M+e^{-}\to M^{+\bullet}+2e^{-}
  2. e - e^{-}
  3. M + M^{+\bullet}
  4. A + e - 𝑀 A - A+e^{-}\overset{M}{\to}A^{-}
  5. C H 4 + e - C H 4 + + 2 e - CH_{4}+e^{-}\to CH_{4}^{+}+2e^{-}
  6. C H 4 + C H 4 + C H 5 + + C H 3 CH_{4}+CH_{4}^{+}\to CH_{5}^{+}+CH_{3}
  7. M + C H 5 + C H 4 + [ M + H ] + M+CH_{5}^{+}\to CH_{4}+[M+H]^{+}
  8. A + + B A + B + A^{+}+B\to A+B^{+}
  9. G * + M M + + e - + G G^{*}+M\to M^{+\bullet}+e^{-}+G
  10. A * + B A B + + e - A^{*}+B\to AB^{+\bullet}+e^{-}
  11. G * + M M + + e - + G G^{*}+M\to M^{+\bullet}+e^{-}+G
  12. G * + M M G + + e - G^{*}+M\to MG^{+\bullet}+e^{-}
  13. G * + S G + S + e - G^{*}+S\to G+S+e^{-}
  14. M + X + + A M X + + A M+X^{+}+A\to MX^{+}+A
  15. [ M + H ] + [M+H]^{+}
  16. [ M + H ] + [M+H]^{+}
  17. [ M + N a ] + [M+Na]^{+}
  18. [ M - H ] - [M-H]^{-}
  19. [ M + 2 H ] 2 + [M+2H]^{2+}
  20. [ M + 25 H ] 25 + [M+25H]^{25+}

Irradiance.html

  1. E e = Φ e A , E_{\mathrm{e}}=\frac{\partial\Phi_{\mathrm{e}}}{\partial A},
  2. E e , ν = E e ν , E_{\mathrm{e},\nu}=\frac{\partial E_{\mathrm{e}}}{\partial\nu},
  3. E e , λ = E e λ , E_{\mathrm{e},\lambda}=\frac{\partial E_{\mathrm{e}}}{\partial\lambda},
  4. E e = | 𝐒 | cos α , E_{\mathrm{e}}=\langle|\mathbf{S}|\rangle\cos\alpha,
  5. E e = n 2 μ 0 c E m 2 cos α = n ε 0 c 2 E m 2 cos α , E_{\mathrm{e}}=\frac{n}{2\mu_{0}\mathrm{c}}E_{\mathrm{m}}^{2}\cos\alpha=\frac{% n\varepsilon_{0}\mathrm{c}}{2}E_{\mathrm{m}}^{2}\cos\alpha,
  6. E e = E e , dir + E e , diff + E e , refl . E_{\mathrm{e}}=E_{\mathrm{e},\mathrm{dir}}+E_{\mathrm{e},\mathrm{diff}}+E_{% \mathrm{e},\mathrm{refl}}.

Irreducible_representation.html

  1. ( Π , V ) (\Pi,V)
  2. A A
  3. ( Π , W ) , W V (\Pi,W),W\subset V
  4. Π ( a ) , a A \Pi(a),a\in A
  5. V V
  6. K K
  7. ρ \rho
  8. ρ : G G L ( V ) \rho:G\to GL(V)
  9. G G
  10. V V
  11. F F
  12. B B
  13. V V
  14. ρ \rho
  15. V V
  16. W V W\subset V
  17. G G
  18. ρ ( g ) w W \rho(g)w\in W
  19. g G g\in G
  20. w W w\in W
  21. ρ \rho
  22. G G
  23. W V W\subset V
  24. ρ : G G L ( V ) \rho:G\to GL(V)
  25. G G
  26. V V
  27. ρ \rho
  28. a , b , c a,b,c...
  29. G G
  30. a b ab
  31. a a
  32. b b
  33. G G
  34. D D
  35. D ( a ) = ( D ( a ) 11 D ( a ) 12 D ( a ) 1 n D ( a ) 21 D ( a ) 22 D ( a ) 2 n D ( a ) n 1 D ( a ) n 2 D ( a ) n n ) D(a)=\begin{pmatrix}D(a)_{11}&D(a)_{12}&\cdots&D(a)_{1n}\\ D(a)_{21}&D(a)_{22}&\cdots&D(a)_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ D(a)_{n1}&D(a)_{n2}&\cdots&D(a)_{nn}\\ \end{pmatrix}
  36. D ( a b ) = D ( a ) D ( b ) D(ab)=D(a)D(b)
  37. e e
  38. a e = e a = a ae=ea=a
  39. D ( e ) D(e)
  40. D ( e a ) = D ( a e ) = D ( a ) D ( e ) = D ( e ) D ( a ) = D ( a ) D(ea)=D(ae)=D(a)D(e)=D(e)D(a)=D(a)
  41. P P
  42. D ( a ) P - 1 D ( a ) P D(a)\rightarrow P^{-1}D(a)P
  43. D ( a ) D(a)
  44. D ( a ) = ( D ( 1 ) ( a ) 0 0 0 D ( 2 ) ( a ) 0 0 0 D ( k ) ( a ) ) = D ( 1 ) ( a ) D ( 2 ) ( a ) D ( k ) ( a ) D(a)=\begin{pmatrix}D^{(1)}(a)&0&\cdots&0\\ 0&D^{(2)}(a)&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&D^{(k)}(a)\\ \end{pmatrix}=D^{(1)}(a)\oplus D^{(2)}(a)\oplus\cdots\oplus D^{(k)}(a)
  45. D ( a ) D(a)
  46. n = 1 , 2 , , k n=1,2,...,k
  47. D ( a ) D(a)
  48. dim [ D ( a ) ] = dim [ D ( 1 ) ( a ) ] + dim [ D ( 2 ) ( a ) ] + + dim [ D ( k ) ( a ) ] \mathrm{dim}[D(a)]=\mathrm{dim}[D^{(1)}(a)]+\mathrm{dim}[D^{(2)}(a)]+\ldots+% \mathrm{dim}[D^{(k)}(a)]
  49. D ( 𝐊 ) D(\mathbf{K})
  50. D ( 𝐉 ) D(\mathbf{J})
  51. 𝐉 \mathbf{J}
  52. 𝐊 \mathbf{K}

Irving_Fisher.html

  1. M V = P T MV=PT
  2. r = ( 1 + i ) ( 1 + π ) - 1 i - π r=\frac{(1+i)}{(1+\pi)}-1\simeq i-\pi
  3. r r
  4. i i
  5. π \pi

Isentropic_process.html

  1. T d S δ Q TdS\geq\delta Q
  2. δ Q \delta Q
  3. T T
  4. d S dS
  5. s = 0 \bigtriangleup s=0
  6. s 1 = s 2 s_{1}=s_{2}
  7. η T = Actual Turbine Work Isentropic Turbine Work = W a W s h 1 - h 2 a h 1 - h 2 s \eta_{T}=\frac{\,\text{Actual Turbine Work}}{\,\text{Isentropic Turbine Work}}% =\frac{W_{a}}{W_{s}}\cong\frac{h_{1}-h_{2a}}{h_{1}-h_{2s}}
  8. η C = Isentropic Compressor Work Actual Compressor Work = W s W a h 2 s - h 1 h 2 a - h 1 \eta_{C}=\frac{\,\text{Isentropic Compressor Work}}{\,\text{Actual Compressor % Work}}=\frac{W_{s}}{W_{a}}\cong\frac{h_{2s}-h_{1}}{h_{2a}-h_{1}}
  9. η N = Actual KE at Nozzle Exit Isentropic KE at Nozzle Exit = V 2 a 2 V 2 s 2 h 1 - h 2 a h 1 - h 2 s \eta_{N}=\frac{\,\text{Actual KE at Nozzle Exit}}{\,\text{Isentropic KE at % Nozzle Exit}}=\frac{V_{2a}^{2}}{V_{2s}^{2}}\cong\frac{h_{1}-h_{2a}}{h_{1}-h_{2% s}}
  10. h 1 h_{1}
  11. h 2 a h_{2a}
  12. h 2 s h_{2s}
  13. d U = δ W + δ Q dU=\delta W+\delta Q\,\!
  14. d W = - p d V dW=-pdV\,\!
  15. p p
  16. V V
  17. H = U + p V H=U+pV\,\!
  18. d H = d U + p d V + V d p dH=dU+pdV+Vdp\,\!
  19. δ Q r e v = 0 \delta Q_{rev}=0\,\!
  20. d S = δ Q r e v / T = 0 dS=\delta Q_{rev}/T=0\,\!
  21. d U = δ W + δ Q = - p d V + 0 dU=\delta W+\delta Q=-pdV+0\,\!
  22. d H = δ W + δ Q + p d V + V d p = - p d V + 0 + p d V + V d p = V d p dH=\delta W+\delta Q+pdV+Vdp=-pdV+0+pdV+Vdp=Vdp\,\!
  23. d U = n C v d T dU=nC_{v}dT\,\!
  24. d H = n C p d T dH=nC_{p}dT\,\!
  25. d U dU
  26. d H dH
  27. d U = n C v d T = - p d V dU=nC_{v}dT=-pdV\,\!
  28. d H = n C p d T = V d p dH=nC_{p}dT=Vdp\,\!
  29. γ = C p C V = - d p / p d V / V \gamma=\frac{C_{p}}{C_{V}}=-\frac{dp/p}{dV/V}\,\!
  30. γ \gamma\,\!
  31. p V γ = constant pV^{\gamma}=\mbox{constant}~{}\,
  32. p 2 p 1 = ( V 1 V 2 ) γ \frac{p_{2}}{p_{1}}=\left(\frac{V_{1}}{V_{2}}\right)^{\gamma}
  33. p V = n R T pV=nRT\,\!
  34. T V γ - 1 = constant TV^{\gamma-1}=\mbox{constant}~{}\,
  35. p γ - 1 T γ = constant \frac{p^{\gamma-1}}{T^{\gamma}}=\mbox{constant}~{}
  36. C p = C v + R C_{p}=C_{v}+R
  37. V T = n R p \frac{V}{T}=\frac{nR}{p}
  38. p = n R T V p=\frac{nRT}{V}
  39. S 2 - S 1 = n C p ln ( T 2 T 1 ) - n R ln ( p 2 p 1 ) S_{2}-S_{1}=nC_{p}\ln\left(\frac{T_{2}}{T_{1}}\right)-nR\ln\left(\frac{p_{2}}{% p_{1}}\right)
  40. S 2 - S 1 n = C p ln ( T 2 T 1 ) - R ln ( T 2 V 1 T 1 V 2 ) = C v ln ( T 2 T 1 ) + R ln ( V 2 V 1 ) \frac{S_{2}-S_{1}}{n}=C_{p}\ln\left(\frac{T_{2}}{T_{1}}\right)-R\ln\left(\frac% {T_{2}V_{1}}{T_{1}V_{2}}\right)=C_{v}\ln\left(\frac{T_{2}}{T_{1}}\right)+R\ln% \left(\frac{V_{2}}{V_{1}}\right)
  41. T 2 = T 1 ( V 1 V 2 ) ( R / C v ) T_{2}=T_{1}\left(\frac{V_{1}}{V_{2}}\right)^{(R/C_{v})}
  42. V 2 = V 1 ( T 1 T 2 ) ( C v / R ) V_{2}=V_{1}\left(\frac{T_{1}}{T_{2}}\right)^{(C_{v}/R)}
  43. T 2 T 1 \frac{T_{2}}{T_{1}}
  44. = =\,\!
  45. ( p 2 p 1 ) γ - 1 γ \left(\frac{p_{2}}{p_{1}}\right)^{\frac{\gamma-1}{\gamma}}
  46. = =\,\!
  47. ( V 1 V 2 ) ( γ - 1 ) \left(\frac{V_{1}}{V_{2}}\right)^{(\gamma-1)}
  48. = =\,\!
  49. ( ρ 2 ρ 1 ) ( γ - 1 ) \left(\frac{\rho_{2}}{\rho_{1}}\right)^{(\gamma-1)}
  50. ( T 2 T 1 ) γ γ - 1 \left(\frac{T_{2}}{T_{1}}\right)^{\frac{\gamma}{\gamma-1}}
  51. = =\,\!
  52. p 2 p 1 \frac{p_{2}}{p_{1}}
  53. = =\,\!
  54. ( V 1 V 2 ) γ \left(\frac{V_{1}}{V_{2}}\right)^{\gamma}
  55. = =\,\!
  56. ( ρ 2 ρ 1 ) γ \left(\frac{\rho_{2}}{\rho_{1}}\right)^{\gamma}
  57. ( T 1 T 2 ) 1 γ - 1 \left(\frac{T_{1}}{T_{2}}\right)^{\frac{1}{\gamma-1}}
  58. = =\,\!
  59. ( p 1 p 2 ) 1 γ \left(\frac{p_{1}}{p_{2}}\right)^{\frac{1}{\gamma}}
  60. = =\,\!
  61. V 2 V 1 \frac{V_{2}}{V_{1}}
  62. = =\,\!
  63. ρ 1 ρ 2 \frac{\rho_{1}}{\rho_{2}}
  64. ( T 2 T 1 ) 1 γ - 1 \left(\frac{T_{2}}{T_{1}}\right)^{\frac{1}{\gamma-1}}
  65. = =\,\!
  66. ( p 2 p 1 ) 1 γ \left(\frac{p_{2}}{p_{1}}\right)^{\frac{1}{\gamma}}
  67. = =\,\!
  68. V 1 V 2 \frac{V_{1}}{V_{2}}
  69. = =\,\!
  70. ρ 2 ρ 1 \frac{\rho_{2}}{\rho_{1}}
  71. p V γ = constant pV^{\gamma}=\,\text{constant}\,\!
  72. p V = m R s T pV=mR_{s}T\,\!
  73. p = ρ R s T p=\rho R_{s}T\,\!\,\!
  74. p p\,\!
  75. V V\,\!
  76. γ \gamma\,\!
  77. C p / C v C_{p}/C_{v}\,\!
  78. T T\,\!
  79. m m\,\!
  80. R s R_{s}\,\!
  81. R / M R/M\,\!
  82. R R\,\!
  83. M M\,\!
  84. ρ \rho\,\!
  85. C p C_{p}\,\!
  86. C v C_{v}\,\!

Isobaric_process.html

  1. Q = Δ U + W Q=\Delta U+W\,
  2. W = p d V W=\int\!p\,dV\,
  3. W = p Δ V W=p\Delta V\,
  4. W = n R Δ T W=n\,R\,\Delta T
  5. Δ U = n c V Δ T \Delta U=n\,c_{V}\,\Delta T
  6. c V c_{V}
  7. Q = n c V Δ T + n R Δ T Q=n\,c_{V}\,\Delta T+n\,R\,\Delta T
  8. = n ( c V + R ) Δ T =n\,(c_{V}+R)\,\Delta T
  9. = n c P Δ T =n\,c_{P}\,\Delta T
  10. c P c_{P}
  11. γ \gamma
  12. γ \gamma
  13. c V = R γ - 1 c_{V}=\frac{R}{\gamma-1}
  14. c p = γ R γ - 1 c_{p}=\frac{\gamma R}{\gamma-1}
  15. γ \gamma
  16. γ = 1.4 \gamma=1.4
  17. γ = 5 3 \gamma=\frac{5}{3}
  18. c V = 3 R 2 c_{V}=\frac{3R}{2}
  19. c P = 5 R 2 c_{P}=\frac{5R}{2}
  20. c V = 5 R 2 c_{V}=\frac{5R}{2}
  21. c P = 7 R 2 c_{P}=\frac{7R}{2}
  22. Q = Δ U + Δ ( p V ) = Δ ( U + p V ) Q=\Delta U+\Delta(p\,V)=\Delta(U+p\,V)
  23. Q = Δ H Q=\Delta H\,
  24. R ( T ρ ) = M P R(T\,\rho)=MP

Isochoric_process.html

  1. Δ V = 0 \Delta V=0
  2. Δ W = P Δ V \Delta W=P\Delta V
  3. d U = d Q - d W dU=dQ-dW
  4. d U = d Q - P d V dU=dQ-PdV
  5. d V = 0 dV=0
  6. d U = d Q dU=dQ
  7. C v = d U / d T C_{v}=dU/dT
  8. d Q = m c v d T dQ=mc_{v}dT
  9. Δ Q = m T 1 T 2 c v d T . \Delta Q\ =m\int_{T_{1}}^{T_{2}}\!c_{v}\,dT.
  10. c v c_{v}
  11. T 1 T_{1}
  12. T 2 T_{2}
  13. Δ Q = m c v Δ T \Delta Q\ =mc_{v}\Delta T

Isolated_point.html

  1. X X
  2. Y Y
  3. S = { 0 } [ 1 , 2 ] S=\{0\}\cup[1,2]
  4. S = { 0 } { 1 , 1 / 2 , 1 / 3 , } S=\{0\}\cup\{1,1/2,1/3,\dots\}
  5. = { 0 , 1 , 2 , } {\mathbb{N}}=\{0,1,2,\ldots\}

Isolated_singularity.html

  1. f f
  2. 1 z \frac{1}{z}
  3. csc ( π z ) \csc\left(\pi z\right)
  4. tan ( 1 z ) \tan\left(\frac{1}{z}\right)
  5. \ { 0 } \mathbb{C}\backslash\{0\}
  6. z n = ( π 2 + n π ) - 1 z_{n}=\left(\frac{\pi}{2}+n\pi\right)^{-1}
  7. n 0 n\in\mathbb{N}_{0}
  8. z n 0 z_{n}\rightarrow 0
  9. 0
  10. tan ( 1 z ) \tan\left(\frac{1}{z}\right)
  11. 0
  12. csc ( π z ) \csc\left(\frac{\pi}{z}\right)
  13. n = 0 z 2 n \sum_{n=0}^{\infty}z^{2^{n}}
  14. 0

Isomorphism_of_categories.html

  1. F G FG
  2. 1 D 1_{D}
  3. 1 D 1_{D}
  4. G F GF
  5. 1 C 1_{C}
  6. ( g G a g g ) v = g G a g ρ ( g ) ( v ) (\sum_{g\in G}a_{g}g)v=\sum_{g\in G}a_{g}\rho(g)(v)
  7. \land
  8. \lor

Isoperimetric_inequality.html

  1. 4 π A L 2 , 4\pi A\leq L^{2},
  2. 4 π A L 2 , 4\pi A\leq L^{2},
  3. Q = 4 π A L 2 Q=\frac{4\pi A}{L^{2}}
  4. Q n = π n tan π n . Q_{n}=\frac{\pi}{n\tan\tfrac{\pi}{n}}.
  5. L 2 A ( 4 π - A ) , L^{2}\,\geq\,A(4\pi-A),
  6. L 2 4 π A - A 2 / R 2 . L^{2}\,\geq\,4\pi A-A^{2}/R^{2}.
  7. n ω n 1 / n L n ( S ¯ ) ( n - 1 ) / n M * n - 1 ( S ) n\omega_{n}^{1/n}L^{n}(\bar{S})^{(n-1)/n}\leq M^{n-1}_{*}(\partial S)
  8. ( n | u | n n - 1 ) n - 1 n n - 1 ω n - 1 / n n | u | \left(\int_{\mathbb{R}^{n}}|u|^{\frac{n}{n-1}}\right)^{\frac{n-1}{n}}\leq n^{-% 1}\omega_{n}^{-1/n}\int_{\mathbb{R}^{n}}|\nabla u|
  9. ( X , μ , d ) \scriptstyle(X,\,\mu,\,d)
  10. μ + ( A ) = lim inf ε 0 + μ ( A ε ) - μ ( A ) ε , \mu^{+}(A)=\liminf_{\varepsilon\to 0+}\frac{\mu(A_{\varepsilon})-\mu(A)}{% \varepsilon},
  11. A ε = { x X | d ( x , A ) ε } A_{\varepsilon}=\{x\in X\,|\,d(x,A)\leq\varepsilon\}
  12. μ + ( A ) \scriptstyle\mu^{+}(A)
  13. I ( a ) = inf { μ + ( A ) | μ ( A ) = a } I(a)\,=\,\inf\{\mu^{+}(A)\,|\,\mu(A)\,=\,a\}
  14. ( X , μ , d ) \scriptstyle(X,\,\mu,\,d)
  15. G G
  16. k k
  17. Φ E ( G , k ) = min S V { | E ( S , S ¯ ) | : | S | = k } \Phi_{E}(G,k)=\min_{S\subseteq V}\left\{|E(S,\overline{S})|:|S|=k\right\}
  18. Φ V ( G , k ) = min S V { | Γ ( S ) S | : | S | = k } \Phi_{V}(G,k)=\min_{S\subseteq V}\left\{|\Gamma(S)\setminus S|:|S|=k\right\}
  19. E ( S , S ¯ ) E(S,\overline{S})
  20. S S
  21. Γ ( S ) \Gamma(S)
  22. S S
  23. Φ E \Phi_{E}
  24. Φ V \Phi_{V}
  25. d d
  26. Q d Q_{d}
  27. d d
  28. { 0 , 1 } d \{0,1\}^{d}
  29. Q d Q_{d}
  30. Φ E ( Q d , k ) k ( d - log 2 k ) \Phi_{E}(Q_{d},k)\geq k(d-\log_{2}k)
  31. S S
  32. Q d Q_{d}
  33. r r
  34. r + 1 r+1
  35. r r
  36. S V S\subseteq V
  37. | S | i = 0 r ( n i ) |S|\geq\sum_{i=0}^{r}{n\choose i}
  38. | S Γ ( S ) | i = 0 r + 1 ( n i ) . |S\cup\Gamma(S)|\geq\sum_{i=0}^{r+1}{n\choose i}.
  39. k = | S | k=|S|
  40. k = ( d 0 ) + ( d 1 ) + + ( d r ) k={d\choose 0}+{d\choose 1}+\dots+{d\choose r}
  41. r r
  42. Φ V ( Q d , k ) = ( d r + 1 ) . \Phi_{V}(Q_{d},k)={d\choose r+1}.
  43. p 2 12 3 T , p^{2}\geq 12\sqrt{3}\cdot T,

Isosbestic_point.html

  1. X Y X\rightarrow Y
  2. c X + c Y = c c_{X}+c_{Y}=c\,
  3. A = l ( ϵ X c X + ϵ Y c Y ) A=l\cdot(\epsilon_{X}c_{X}+\epsilon_{Y}c_{Y})
  4. ϵ X = ϵ Y = ϵ \epsilon_{X}=\epsilon_{Y}=\epsilon\,
  5. A = l ( ϵ X c X + ϵ Y c Y ) = l ϵ ( c X + c Y ) = l ϵ c A=l\cdot(\epsilon_{X}c_{X}+\epsilon_{Y}c_{Y})=l\cdot\epsilon\cdot(c_{X}+c_{Y})% =l\cdot\epsilon\cdot c

Isosceles_trapezoid.html

  1. A E E C = D E E B = A D B C . \frac{AE}{EC}=\frac{DE}{EB}=\frac{AD}{BC}.
  2. p = a b + c 2 p=\sqrt{ab+c^{2}}
  3. h = p 2 - ( a + b 2 ) 2 = 1 2 4 c 2 - ( a - b ) 2 . h=\sqrt{p^{2}-\left(\frac{a+b}{2}\right)^{2}}=\tfrac{1}{2}\sqrt{4c^{2}-(a-b)^{% 2}}.
  4. d = a h a + b d=\frac{ah}{a+b}
  5. K = h 2 ( a + b ) . K=\frac{h}{2}\left(a+b\right).
  6. K = ( s - a ) ( s - b ) ( s - c ) 2 , K=\sqrt{(s-a)(s-b)(s-c)^{2}},
  7. s = 1 2 ( a + b + 2 c ) s=\tfrac{1}{2}(a+b+2c)
  8. K = 1 4 ( a + b ) 2 ( a - b + 2 c ) ( b - a + 2 c ) . K=\frac{1}{4}\sqrt{(a+b)^{2}(a-b+2c)(b-a+2c)}.
  9. R = c a b + c 2 4 c 2 - ( a - b ) 2 . R=c\sqrt{\frac{ab+c^{2}}{4c^{2}-(a-b)^{2}}}.
  10. R = 1 2 a 2 + c 2 R=\tfrac{1}{2}\sqrt{a^{2}+c^{2}}

Isothermal_process.html

  1. p = n R T V = constant V p={nRT\over V}={\,\text{constant}\over V}
  2. W A B = - V A V B p d V W_{A\to B}=-\int_{V_{A}}^{V_{B}}p\,dV
  3. W A B = - V A V B p d V = - V A V B n R T V d V = - n R T V A V B 1 V d V = - n R T ln V B V A W_{A\to B}=-\int_{V_{A}}^{V_{B}}p\,dV=-\int_{V_{A}}^{V_{B}}\frac{nRT}{V}dV=-% nRT\int_{V_{A}}^{V_{B}}\frac{1}{V}dV=-nRT\ln{\frac{V_{B}}{V_{A}}}
  4. Δ U = 0 \Delta U=0
  5. Δ U = Q - W \Delta U=Q-W
  6. Q = W Q=W

Issai_Schur.html

  1. G L ( n , ) GL(n,\mathbb{C})
  2. \mathbb{C}
  3. G L ( n , 𝕂 ) GL(n,\mathbb{K})
  4. 𝕂 \mathbb{K}
  5. r 0 r\geq 0
  6. G L n ( ) GL_{n}(\mathbb{C})
  7. r r
  8. λ = ( λ 1 , , λ n ) \lambda=(\lambda_{1},\ldots,\lambda_{n})
  9. r r
  10. n n
  11. λ \lambda
  12. S ¯ λ {\underline{S}}_{\lambda}
  13. n n
  14. V V
  15. n n
  16. \mathbb{C}
  17. G L ( n , ) GL(n,\mathbb{C})
  18. r r
  19. r r
  20. V r V^{\otimes r}
  21. \mathbb{C}
  22. G L ( n , ) GL(n,\mathbb{C})
  23. S r S_{r}
  24. r r
  25. v 1 v r v_{1}\otimes\ldots\otimes v_{r}
  26. V r V^{\otimes r}
  27. S r - G L ( n , ) S_{r}-GL(n,\mathbb{C})
  28. V r V^{\otimes r}
  29. G L ( n ) GL(n)

Item_response_theory.html

  1. θ {\theta}
  2. p i ( θ ) = c i + 1 - c i 1 + e - a i ( θ - b i ) p_{i}({\theta})=c_{i}+\frac{1-c_{i}}{1+e^{-a_{i}({\theta}-b_{i})}}
  3. θ {\theta}
  4. a i a_{i}
  5. b i b_{i}
  6. c i c_{i}
  7. P ( t ) = 1 1 + e - t . P(t)=\frac{1}{1+e^{-t}}.
  8. p ( b ) = ( 1 + c ) / 2 , p(b)=(1+c)/2,
  9. c i c_{i}
  10. p ( b ) = a ( 1 - c ) / 4. p^{\prime}(b)=a\cdot(1-c)/4.
  11. p ( - ) = c . p(-\infty)=c.
  12. c = 0 , c=0,
  13. p ( b ) = 1 / 2 p(b)=1/2
  14. p ( b ) = a / 4 , p^{\prime}(b)=a/4,
  15. a ( θ - b ) a(\theta-b)
  16. c = 0 c=0
  17. c = 0 c=0
  18. b = 0 b=0
  19. P ( 0 ) = 1 / 2 P(0)=1/2
  20. P ( 0 ) = 1 / 4. P^{\prime}(0)=1/4.
  21. a a
  22. b b
  23. c c
  24. [ 0 , 1 ] [0,1]
  25. [ c , 1 ] . [c,1].
  26. b i b_{i}
  27. θ {\theta}
  28. c i c_{i}
  29. b i b_{i}
  30. a i a_{i}
  31. a i a_{i}
  32. c i c_{i}
  33. c i c_{i}
  34. c i c_{i}
  35. θ {\theta}
  36. b i b_{i}
  37. a i a_{i}
  38. b i b_{i}
  39. d i , d_{i},
  40. 1 - c i 1-c_{i}
  41. d i - c i d_{i}-c_{i}
  42. b i b_{i}
  43. b i b_{i}
  44. b i b_{i}
  45. a i a_{i}
  46. c i c_{i}
  47. d i d_{i}
  48. c i = 0 c_{i}=0
  49. p i ( θ ) = Φ ( θ - b i σ i ) p_{i}(\theta)=\Phi\left(\frac{\theta-b_{i}}{\sigma_{i}}\right)
  50. b i b_{i}
  51. σ i {\sigma}_{i}
  52. a i a_{i}
  53. θ {\theta}
  54. I ( θ ) = p i ( θ ) q i ( θ ) . I(\theta)=p_{i}(\theta)q_{i}(\theta).\,
  55. SE ( θ ) = 1 I ( θ ) . \,\text{SE}(\theta)=\frac{1}{\sqrt{I(\theta)}}.
  56. I ( θ ) = a i 2 p i ( θ ) q i ( θ ) . I(\theta)=a_{i}^{2}p_{i}(\theta)q_{i}(\theta).\,
  57. I ( θ ) = a i 2 ( p i ( θ ) - c i ) 2 ( 1 - c i ) 2 q i ( θ ) p i ( θ ) . I(\theta)=a_{i}^{2}\frac{(p_{i}(\theta)-c_{i})^{2}}{(1-c_{i})^{2}}\frac{q_{i}(% \theta)}{p_{i}(\theta)}.
  58. θ {\theta}
  59. θ {\theta}
  60. θ \theta
  61. a i ρ i t 1 - ρ i t 2 a_{i}\cong\frac{\rho_{it}}{\sqrt{1-\rho_{it}^{2}}}
  62. ρ i t \rho_{it}
  63. θ ^ = θ + ε \hat{\theta}=\theta+\varepsilon
  64. θ \theta
  65. ϵ \epsilon
  66. SE ( θ ) \mbox{SE}~{}({\theta})
  67. ϵ \epsilon
  68. R θ = var [ θ ] var [ θ ^ ] = var [ θ ^ ] - var [ ϵ ] var [ θ ^ ] R_{\theta}=\frac{\,\text{var}[\theta]}{\,\text{var}[\hat{\theta}]}=\frac{\,% \text{var}[\hat{\theta}]-\,\text{var}[\epsilon]}{\,\text{var}[\hat{\theta}]}
  69. ϵ n \epsilon_{n}

Iterated_function_system.html

  1. { f i : X X | i = 1 , 2 , , N } , N \{f_{i}:X\to X|i=1,2,\dots,N\},\ N\in\mathbb{N}
  2. f i f_{i}
  3. X X
  4. n \mathbb{R}^{n}
  5. S X S\subseteq X
  6. S = i = 1 N f i ( S ) . S=\bigcup_{i=1}^{N}f_{i}(S).
  7. H ( A ) = i = 1 N f i ( A ) . H(A)=\bigcup_{i=1}^{N}f_{i}(A).
  8. lim n H n ( A ) = S \lim_{n\to\infty}H^{\circ n}(A)=S
  9. A A
  10. X X
  11. A A
  12. f i f_{i}
  13. f i f_{i}

Iterated_integral.html

  1. f ( x , y ) f(x,y)
  2. f ( x , y , z ) f(x,y,z)
  3. f ( x , y ) f(x,y)
  4. y y
  5. x x
  6. f ( x , y ) d x \int f(x,y)dx
  7. y y
  8. ( f ( x , y ) d x ) d y . \int\left(\int f(x,y)\,dx\right)\,dy.
  9. f ( x , y ) d x d y . \iint f(x,y)\,dx\,dy.
  10. d y f ( x , y ) d x \int dy\int f(x,y)\,dx
  11. ( ( x + y ) d x ) d y \int\left(\int(x+y)\,dx\right)\,dy
  12. ( x + y ) d x = x 2 2 + y x \int(x+y)\,dx=\frac{x^{2}}{2}+yx
  13. ( x 2 2 + y x ) d y = y x 2 2 + x y 2 2 \int(\frac{x^{2}}{2}+yx)\,dy=\frac{yx^{2}}{2}+\frac{xy^{2}}{2}
  14. 0 < a 1 < a 2 < 0<a_{1}<a_{2}<\cdots
  15. a n 1 a_{n}\rightarrow 1
  16. g n g_{n}
  17. ( a n , a n + 1 ) (a_{n},a_{n+1})
  18. 0 1 g n = 1 \int_{0}^{1}g_{n}=1
  19. n n
  20. f ( x , y ) = n = 0 ( g n ( x ) - g n + 1 ( x ) ) g n ( y ) . f(x,y)=\sum_{n=0}^{\infty}(g_{n}(x)-g_{n+1}(x))g_{n}(y).
  21. ( x , y ) (x,y)
  22. 0 1 ( 0 1 f ( x , y ) d y ) d x = 1 0 = 0 1 ( 0 1 f ( x , y ) d x ) d y \int_{0}^{1}\left(\int_{0}^{1}f(x,y)\,dy\right)\,dx=1\neq 0=\int_{0}^{1}\left(% \int_{0}^{1}f(x,y)\,dx\right)\,dy

Iterative_deepening_depth-first_search.html

  1. d d
  2. O ( b d ) O(bd)
  3. b b
  4. d d
  5. O ( b d ) O(bd)
  6. O ( d ) O(d)
  7. d d
  8. d d
  9. O ( b d ) O(b^{d})
  10. d + 1 d+1
  11. ( d ) b + ( d - 1 ) b 2 + + 3 b d - 2 + 2 b d - 1 + b d (d)b+(d-1)b^{2}+\cdots+3b^{d-2}+2b^{d-1}+b^{d}
  12. i = 1 d ( d + 1 - i ) b i \sum_{i=1}^{d}(d+1-i)b^{i}
  13. b = 10 b=10
  14. d = 5 d=5
  15. d d
  16. d d
  17. b = 10 b=10
  18. O ( b d ) O(b^{d})
  19. O ( d ) O(d)
  20. f f

Ivan_Matveyevich_Vinogradov.html

  1. S = p P exp ( 2 π i f ( p ) ) . S=\sum_{p\leq P}\exp(2\pi if(p)).

Iwasawa_theory.html

  1. p \mathbb{Z}_{p}
  2. F F
  3. Γ \Gamma
  4. Γ \Gamma
  5. Γ p n \Gamma^{p^{n}}
  6. p \mathbb{Z}_{p}
  7. F / F F_{\infty}/F
  8. F = F 0 F 1 F 2 F F=F_{0}\subset F_{1}\subset F_{2}\subset\ldots\subset F_{\infty}
  9. Gal ( F n / F ) / p n \textrm{Gal}(F_{n}/F)\cong\mathbb{Z}/p^{n}\mathbb{Z}
  10. F n F_{n}
  11. F F_{\infty}
  12. K = K 0 K 1 K , K=K_{0}\subset K_{1}\subset\cdots\subset K_{\infty},
  13. K n K_{n}
  14. K K
  15. K = K n K_{\infty}=\bigcup K_{n}
  16. Gal ( K n / K ) / p n \textrm{Gal}(K_{n}/K)\simeq\mathbb{Z}/p^{n}\mathbb{Z}
  17. Gal ( K / K ) \textrm{Gal}(K_{\infty}/K)
  18. p \mathbb{Z}_{p}
  19. K n K_{n}
  20. I n I_{n}
  21. I m I n I_{m}\rightarrow I_{n}
  22. m > n m>n
  23. I = lim I n I=\underleftarrow{\lim}I_{n}
  24. I I
  25. p \mathbb{Z}_{p}
  26. I I
  27. Λ = p [ [ Γ ] ] \Lambda=\mathbb{Z}_{p}[[\Gamma]]
  28. K K
  29. K K

J-invariant.html

  1. S L ( 2 , 𝐙 ) SL(2,\mathbf{Z})
  2. j ( e 2 3 π i ) = 0 , j ( i ) = 1728. j\left(e^{\frac{2}{3}\pi i}\right)=0,\quad j(i)=1728.
  3. j j
  4. j j
  5. 𝐂 \mathbf{C}
  6. j j
  7. E E
  8. 𝐂 \mathbf{C}
  9. 𝐂 \mathbf{C}
  10. 1 1
  11. τ τ
  12. 𝐇 \mathbf{H}
  13. 𝐇 \mathbf{H}
  14. g 2 = 60 ( m , n ) ( 0 , 0 ) ( m + n τ ) - 4 , g 3 = 140 ( m , n ) ( 0 , 0 ) ( m + n τ ) - 6 , \begin{aligned}\displaystyle g_{2}&\displaystyle=60\sum_{(m,n)\neq(0,0)}(m+n% \tau)^{-4},\\ \displaystyle g_{3}&\displaystyle=140\sum_{(m,n)\neq(0,0)}(m+n\tau)^{-6},\end{aligned}
  15. 𝐂 \mathbf{C}
  16. j j
  17. j ( τ ) = 1728 g 2 3 Δ j(\tau)=1728\frac{g_{2}^{3}}{\Delta}
  18. Δ Δ
  19. Δ = g 2 3 - 27 g 3 2 \Delta=g_{2}^{3}-27g_{3}^{2}
  20. Δ Δ
  21. j j
  22. 𝐇 𝐂 \mathbf{H}→\mathbf{C}
  23. S L ( 2 , 𝐙 ) SL(2,\mathbf{Z})
  24. j j
  25. 𝐂 \mathbf{C}
  26. τ τ + 1 τ→τ+1
  27. P S L ( 2 , 𝐙 ) PSL(2,\mathbf{Z})
  28. τ a τ + b c τ + d , a d - b c = 1 , \tau\mapsto\frac{a\tau+b}{c\tau+d},\qquad ad-bc=1,
  29. τ τ
  30. j j
  31. j j
  32. τ τ
  33. | τ | 1 - 1 2 < ( τ ) 1 2 - 1 2 < ( τ ) < 0 | τ | > 1 \begin{aligned}\displaystyle|\tau|&\displaystyle\geq 1\\ \displaystyle-\tfrac{1}{2}&\displaystyle<\mathfrak{R}(\tau)\leq\tfrac{1}{2}\\ \displaystyle-\tfrac{1}{2}&\displaystyle<\mathfrak{R}(\tau)<0\Rightarrow|\tau|% >1\end{aligned}
  34. j ( τ ) j(τ)
  35. 𝐂 \mathbf{C}
  36. c c
  37. 𝐂 \mathbf{C}
  38. c = j ( τ ) c=j(τ)
  39. j j
  40. 0
  41. j j
  42. j j
  43. 𝐂 ( j ) \mathbf{C}(j)
  44. j j
  45. j j
  46. τ τ
  47. j j
  48. j ( τ ) j(τ)
  49. 𝐐 j j ( τ ) , τ / / 𝐐 ( τ ) \mathbf{Q}jj(τ),τ//\mathbf{Q}(τ)
  50. Λ Λ
  51. 𝐂 \mathbf{C}
  52. 𝐐 ( τ ) \mathbf{Q}(τ)
  53. Λ Λ
  54. j ( τ ) j(τ′)
  55. j ( τ ) j(τ)
  56. 𝐐 ( τ ) \mathbf{Q}(τ)
  57. 𝐐 ( τ ) \mathbf{Q}(τ)
  58. 𝐐 ( τ ) \mathbf{Q}(τ)
  59. τ τ
  60. 𝐐 ( τ ) \mathbf{Q}(τ)
  61. τ τ
  62. j ( τ ) j(τ)
  63. τ τ
  64. j ( τ ) j(τ)
  65. j j
  66. τ τ
  67. e x p ( 2 π i τ ) exp(2πiτ)
  68. j ( τ ) j(τ)
  69. e x p ( 2 π i τ ) exp(2πiτ)
  70. j ( τ ) , j ( τ ) π , j ′′ ( τ ) π 2 j(\tau),\frac{j^{\prime}(\tau)}{\pi},\frac{j^{\prime\prime}(\tau)}{\pi^{2}}
  71. q q
  72. j j
  73. q q
  74. q = e x p ( 2 π i τ ) q=exp(2πiτ)
  75. j ( τ ) = 1 q + 744 + 196884 q + 21493760 q 2 + 864299970 q 3 + 20245856256 q 4 + j(\tau)={1\over q}+744+196884q+21493760q^{2}+864299970q^{3}+20245856256q^{4}+\cdots
  76. j j
  77. q q
  78. e π 163 640320 3 + 744 e^{\pi\sqrt{163}}\approx 640320^{3}+744
  79. e 4 π n 2 n 3 / 4 \frac{e^{4\pi\sqrt{n}}}{\sqrt{2}n^{3/4}}
  80. q q
  81. n n
  82. 196 , 884 196,884
  83. 196884 q 196884q
  84. q - 1 + O ( q ) q^{-1}+{O}(q)
  85. j ( τ ) = 256 ( 1 - x ) 3 x 2 j(\tau)=\frac{256(1-x)^{3}}{x^{2}}
  86. x = λ ( 1 λ ) x=λ(1−λ)
  87. λ λ
  88. λ ( τ ) = θ 2 4 ( 0 , τ ) θ 3 4 ( 0 , τ ) = k 2 ( τ ) \lambda(\tau)=\frac{\theta_{2}^{4}(0,\tau)}{\theta_{3}^{4}(0,\tau)}=k^{2}(\tau)
  89. θ m \theta_{m}
  90. k ( τ ) k(\tau)
  91. j j
  92. { λ , 1 1 - λ , λ - 1 λ , 1 λ , λ λ - 1 , 1 - λ } \left\{{\lambda,\frac{1}{1-\lambda},\frac{\lambda-1}{\lambda},\frac{1}{\lambda% },\frac{\lambda}{\lambda-1},1-\lambda}\right\}
  93. j j
  94. j j
  95. q = e π i τ q=e^{\pi i\tau}
  96. ϑ ( 0 ; τ ) = ϑ 00 ( 0 ; τ ) = 1 + 2 n = 1 ( e π i τ ) n 2 = n = - q n 2 \vartheta(0;\tau)=\vartheta_{00}(0;\tau)=1+2\sum_{n=1}^{\infty}\left(e^{\pi i% \tau}\right)^{n^{2}}=\sum_{n=-\infty}^{\infty}q^{n^{2}}
  97. a = θ 2 ( 0 ; q ) = ϑ 10 ( 0 ; τ ) b = θ 3 ( 0 ; q ) = ϑ 00 ( 0 ; τ ) c = θ 4 ( 0 ; q ) = ϑ 01 ( 0 ; τ ) \begin{aligned}\displaystyle a&\displaystyle=\theta_{2}(0;q)=\vartheta_{10}(0;% \tau)\\ \displaystyle b&\displaystyle=\theta_{3}(0;q)=\vartheta_{00}(0;\tau)\\ \displaystyle c&\displaystyle=\theta_{4}(0;q)=\vartheta_{01}(0;\tau)\end{aligned}
  98. θ m \theta_{m}
  99. ϑ n \vartheta_{n}
  100. a 4 - b 4 + c 4 = 0 a^{4}-b^{4}+c^{4}=0
  101. g 2 ( τ ) = 2 3 π 4 ( a 8 + b 8 + c 8 ) g 3 ( τ ) = 4 27 π 6 ( a 8 + b 8 + c 8 ) 3 - 54 ( a b c ) 8 2 Δ = g 2 3 - 27 g 3 2 = ( 2 π ) 12 ( 1 2 a b c ) 8 = ( 2 π ) 12 η ( τ ) 24 \begin{aligned}\displaystyle g_{2}(\tau)&\displaystyle=\tfrac{2}{3}\pi^{4}% \left(a^{8}+b^{8}+c^{8}\right)\\ \displaystyle g_{3}(\tau)&\displaystyle=\tfrac{4}{27}\pi^{6}\sqrt{\frac{(a^{8}% +b^{8}+c^{8})^{3}-54(abc)^{8}}{2}}\\ \displaystyle\Delta&\displaystyle=g_{2}^{3}-27g_{3}^{2}=(2\pi)^{12}\left(% \tfrac{1}{2}abc\right)^{8}=(2\pi)^{12}\eta(\tau)^{24}\end{aligned}
  102. j ( τ ) = 1728 g 2 3 g 2 3 - 27 g 3 2 = 32 ( a 8 + b 8 + c 8 ) 3 ( a b c ) 8 j(\tau)=1728\frac{g_{2}^{3}}{g_{2}^{3}-27g_{3}^{2}}=32{(a^{8}+b^{8}+c^{8})^{3}% \over(abc)^{8}}
  103. j j
  104. y 2 + a 1 x y + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}
  105. b 2 = a 1 2 + 4 a 2 , b 4 = a 1 a 3 + 2 a 4 b_{2}=a_{1}^{2}+4a_{2},\quad b_{4}=a_{1}a_{3}+2a_{4}
  106. b 6 = a 3 2 + 4 a 6 , b 8 = a 1 2 a 6 - a 1 a 3 a 4 + a 2 a 3 2 + 4 a 2 a 6 - a 4 2 b_{6}=a_{3}^{2}+4a_{6},\quad b_{8}=a_{1}^{2}a_{6}-a_{1}a_{3}a_{4}+a_{2}a_{3}^{% 2}+4a_{2}a_{6}-a_{4}^{2}
  107. c 4 = b 2 2 - 24 b 4 , c 6 = - b 2 3 + 36 b 2 b 4 - 216 b 6 c_{4}=b_{2}^{2}-24b_{4},\quad c_{6}=-b_{2}^{3}+36b_{2}b_{4}-216b_{6}
  108. Δ = - b 2 2 b 8 + 9 b 2 b 4 b 6 - 8 b 4 3 - 27 b 6 2 \Delta=-b_{2}^{2}b_{8}+9b_{2}b_{4}b_{6}-8b_{4}^{3}-27b_{6}^{2}
  109. j j
  110. j = c 4 3 Δ j={c_{4}^{3}\over\Delta}
  111. j = 1728 c 4 3 c 4 3 - c 6 2 j=1728{c_{4}^{3}\over c_{4}^{3}-c_{6}^{2}}
  112. j j
  113. N N
  114. j ( τ ) = N j(τ)=N
  115. τ τ
  116. λ λ
  117. j ( τ ) = 256 ( 1 - λ ( 1 - λ ) ) 3 ( λ ( 1 - λ ) ) 2 j(\tau)=\frac{256(1-\lambda(1-\lambda))^{3}}{(\lambda(1-\lambda))^{2}}
  118. λ λ
  119. x = λ ( 1 λ ) x=λ(1−λ)
  120. x x
  121. τ = i F 1 2 ( 1 2 , 1 2 , 1 , 1 - λ ) F 1 2 ( 1 2 , 1 2 , 1 , λ ) \tau=i\ \frac{{}_{2}F_{1}\left(\tfrac{1}{2},\tfrac{1}{2},1,1-\lambda\right)}{{% }_{2}F_{1}\left(\tfrac{1}{2},\tfrac{1}{2},1,\lambda\right)}
  122. λ λ
  123. γ γ
  124. j ( τ ) = 27 ( 1 + 8 γ ) 3 γ ( 1 - γ ) 3 j(\tau)=\frac{27(1+8\gamma)^{3}}{\gamma(1-\gamma)^{3}}
  125. τ = i 3 F 1 2 ( 1 3 , 2 3 , 1 , 1 - γ ) F 1 2 ( 1 3 , 2 3 , 1 , γ ) \tau=\frac{i}{\sqrt{3}}\frac{{}_{2}F_{1}\left(\tfrac{1}{3},\tfrac{2}{3},1,1-% \gamma\right)}{{}_{2}F_{1}\left(\tfrac{1}{3},\tfrac{2}{3},1,\gamma\right)}
  126. β β
  127. j ( τ ) = 64 ( 1 + 3 β ) 3 β ( 1 - β ) 2 j(\tau)=\frac{64(1+3\beta)^{3}}{\beta(1-\beta)^{2}}
  128. τ = i 2 F 1 2 ( 1 4 , 3 4 , 1 , 1 - β ) F 1 2 ( 1 4 , 3 4 , 1 , β ) \tau=\frac{i}{\sqrt{2}}\frac{{}_{2}F_{1}\left(\tfrac{1}{4},\tfrac{3}{4},1,1-% \beta\right)}{{}_{2}F_{1}\left(\tfrac{1}{4},\tfrac{3}{4},1,\beta\right)}
  129. α α
  130. j ( τ ) = 1728 4 α ( 1 - α ) j(\tau)=\frac{1728}{4\alpha(1-\alpha)}
  131. τ = i F 1 2 ( 1 6 , 5 6 , 1 , 1 - α ) F 1 2 ( 1 6 , 5 6 , 1 , α ) \tau=i\ \frac{{}_{2}F_{1}\left(\tfrac{1}{6},\tfrac{5}{6},1,1-\alpha\right)}{{}% _{2}F_{1}\left(\tfrac{1}{6},\tfrac{5}{6},1,\alpha\right)}
  132. τ τ
  133. 1 / τ 1/τ
  134. j ( τ ) = j ( 1 / τ ) j(τ)=j(1/τ)
  135. α α
  136. j j
  137. 1 π = 12 640320 3 / 2 k = 0 ( 6 k ) ! ( 163 3344418 k + 13591409 ) ( 3 k ) ! ( k ! ) 3 ( - 640320 ) 3 k \frac{1}{\pi}=\frac{12}{640320^{3/2}}\sum_{k=0}^{\infty}\frac{(6k)!(163\cdot 3% 344418k+13591409)}{(3k)!(k!)^{3}(-640320)^{3k}}
  138. j ( 1 + - 163 2 ) = - 640320 3 j\big(\tfrac{1+\sqrt{-163}}{2}\big)=-640320^{3}
  139. j j
  140. 1 2 ( 1 + i 3 ) . \tfrac{1}{2}\left(1+i\sqrt{3}\right).
  141. j j
  142. J / 1728 J/1728
  143. j ( i ) \displaystyle j(i)
  144. j ( 5 i + 1 2 ) = ( 2927 - 1323 5 2 ) 3 , j ( 5 i ) = ( 2927 + 1323 5 2 ) 3 , j ( 5 i + 2 4 ) = ( ( 1 + 5 ) 37 2 39 ( 1190448488 - 858585699 2 + 540309076 5 - 374537880 10 - 5 4 ( 693172512 - 595746414 2 + 407357424 5 - 240819696 10 ) ) ) 3 , j ( 10 i + 1 2 ) = ( ( 1 + 5 ) 37 2 39 ( 1190448488 - 858585699 2 + 540309076 5 - 374537880 10 + 5 4 ( 693172512 - 595746414 2 + 407357424 5 - 240819696 10 ) ) ) 3 , j ( 5 i 4 ) = ( ( 1 + 5 ) 37 2 39 ( 1190448488 + 858585699 2 + 540309076 5 + 374537880 10 - 5 4 ( 693172512 + 595746414 2 + 407357424 5 + 240819696 10 ) ) ) 3 , j ( 20 i ) = ( ( 1 + 5 ) 37 2 39 ( 1190448488 + 858585699 2 + 540309076 5 + 374537880 10 + 5 4 ( 693172512 + 595746414 2 + 407357424 5 + 240819696 10 ) ) ) 3 . \begin{aligned}\displaystyle j\left(\frac{5\,i+1}{2}\right)&\displaystyle=% \left(\frac{2927-1323\,\sqrt{5}}{2}\right)^{3},\\ \displaystyle j\left(5\,i\right)&\displaystyle=\left(\frac{2927+1323\,\sqrt{5}% }{2}\right)^{3},\\ \displaystyle j\left(\frac{5\,i+2}{4}\right)&\displaystyle=\Bigg(\frac{\left(1% +\sqrt{5}\,\right)^{37}}{2^{39}}\Bigg(1190448488-858585699\,\sqrt{2}+540309076% \,\sqrt{5}-374537880\,\sqrt{10}\,-\,\sqrt[4]{5}\left(693172512-595746414\,% \sqrt{2}+407357424\,\sqrt{5}-240819696\,\sqrt{10}\,\right)\Bigg)\Bigg)^{3},\\ \displaystyle j\left(\frac{10\,i+1}{2}\right)&\displaystyle=\Bigg(\frac{\left(% 1+\sqrt{5}\,\right)^{37}}{2^{39}}\Bigg(1190448488-858585699\,\sqrt{2}+54030907% 6\,\sqrt{5}-374537880\,\sqrt{10}\,+\,\sqrt[4]{5}\left(693172512-595746414\,% \sqrt{2}+407357424\,\sqrt{5}-240819696\,\sqrt{10}\,\right)\Bigg)\Bigg)^{3},\\ \displaystyle j\left(\frac{5\,i}{4}\right)&\displaystyle=\Bigg(\frac{\left(1+% \sqrt{5}\,\right)^{37}}{2^{39}}\Bigg(1190448488+858585699\,\sqrt{2}+540309076% \,\sqrt{5}+374537880\,\sqrt{10}\,-\,\sqrt[4]{5}\left(693172512+595746414\,% \sqrt{2}+407357424\,\sqrt{5}+240819696\,\sqrt{10}\,\right)\Bigg)\Bigg)^{3},\\ \displaystyle j(20\,i)&\displaystyle=\Bigg(\frac{\left(1+\sqrt{5}\,\right)^{37% }}{2^{39}}\Bigg(1190448488+858585699\,\sqrt{2}+540309076\,\sqrt{5}+374537880\,% \sqrt{10}\,+\,\sqrt[4]{5}\left(693172512+595746414\,\sqrt{2}+407357424\,\sqrt{% 5}+240819696\,\sqrt{10}\,\right)\Bigg)\Bigg)^{3}.\end{aligned}
  145. j ( 10 i ) j(10i)
  146. j ( 5 i / 2 ) j(5i/2)
  147. j ( 5 i ± 1 4 ) \displaystyle j\left(\frac{5\,i\pm 1}{4}\right)
  148. j ( 4 ( 5 i ± 1 ) 13 ) = ( ( 1 - 5 ) 37 2 39 ( 1190448488 - 858585699 2 - 540309076 5 + 374537880 10 ± 𝑖 5 4 ( 693172512 - 595746414 2 - 407357424 5 + 240819696 10 ) ) ) 3 , j ( 5 ( 4 i ± 1 ) 17 ) = ( ( 1 - 5 ) 37 2 39 ( 1190448488 + 858585699 2 - 540309076 5 - 374537880 10 ± 𝑖 5 4 ( 693172512 + 595746414 2 - 407357424 5 - 240819696 10 ) ) ) 3 \begin{aligned}\displaystyle j\left(\frac{4\left(5\,i\pm 1\right)}{13}\right)=% \Bigg(\frac{\left(1-\sqrt{5}\,\right)^{37}}{2^{39}}\Bigg(1190448488-858585699% \,\sqrt{2}-540309076\,\sqrt{5}+374537880\,\sqrt{10}\,\pm\,\,\textit{i}\,\sqrt[% 4]{5}\left(693172512-595746414\,\sqrt{2}-407357424\,\sqrt{5}+240819696\,\sqrt{% 10}\,\right)\Bigg)\Bigg)^{3},\\ \displaystyle j\left(\frac{5\left(4\,i\pm 1\right)}{17}\right)=\Bigg(\frac{% \left(1-\sqrt{5}\,\right)^{37}}{2^{39}}\Bigg(1190448488+858585699\,\sqrt{2}-54% 0309076\,\sqrt{5}-374537880\,\sqrt{10}\,\pm\,\,\textit{i}\,\sqrt[4]{5}\left(69% 3172512+595746414\,\sqrt{2}-407357424\,\sqrt{5}-240819696\,\sqrt{10}\,\right)% \Bigg)\Bigg)^{3}\end{aligned}

J2.html

  1. J 2 J_{2}\,

Jacobi_elliptic_functions.html

  1. u = 0 ϕ d θ 1 - m sin 2 θ . u=\int_{0}^{\phi}\frac{\mathrm{d}\theta}{\sqrt{1-m\sin^{2}\theta}}\,.
  2. sn u = sin ϕ \operatorname{sn}\;u=\sin\phi\,
  3. cn u = cos ϕ \operatorname{cn}\;u=\cos\phi
  4. dn u = 1 - m sin 2 ϕ . \operatorname{dn}\;u=\sqrt{1-m\sin^{2}\phi}\,.
  5. ϕ \phi
  6. ϕ \phi
  7. ϕ = π / 2 \phi=\pi/2
  8. cos θ , sin θ \cos\theta,\sin\theta
  9. x 2 + y 2 b 2 = 1 , m = 1 - 1 b 2 < 1 , x = r cos θ , y = r sin θ \begin{aligned}&\displaystyle x^{2}+\frac{y^{2}}{b^{2}}=1,\\ &\displaystyle m=1-\frac{1}{b^{2}}<1,\\ &\displaystyle x=r\cos\theta,\quad y=r\sin\theta\end{aligned}
  10. r ( θ , m ) = 1 1 - m sin 2 θ . r(\theta,m)=\frac{1}{\sqrt{1-m\sin^{2}\theta}}\,.
  11. cos φ = x , \cos\varphi=x,
  12. u = u ( φ , m ) = 0 φ r ( θ , m ) d θ u=u(\varphi,m)=\int_{0}^{\varphi}r(\theta,m)\,d\theta
  13. cn ( u , m ) = x , sn ( u , m ) = y b , dn ( u , m ) = 1 r ( φ , m ) . \operatorname{cn}(u,m)=x,\quad\operatorname{sn}(u,m)=\frac{y}{b},\quad% \operatorname{dn}(u,m)=\frac{1}{r(\varphi,m)}.
  14. ϑ ( 0 ; τ ) \vartheta(0;\tau)
  15. ϑ \vartheta
  16. ϑ 01 ( 0 ; τ ) , ϑ 10 ( 0 ; τ ) , ϑ 11 ( 0 ; τ ) \vartheta_{01}(0;\tau),\vartheta_{10}(0;\tau),\vartheta_{11}(0;\tau)
  17. ϑ 01 , ϑ 10 , ϑ 11 \vartheta_{01},\vartheta_{10},\vartheta_{11}
  18. k = ( ϑ 10 ϑ ) 2 k=\left({\vartheta_{10}\over\vartheta}\right)^{2}
  19. u = π ϑ 2 z u=\pi\vartheta^{2}z
  20. sn ( u ; k ) = - ϑ ϑ 11 ( z ; τ ) ϑ 10 ϑ 01 ( z ; τ ) \mathrm{sn}(u;k)=-{\vartheta\vartheta_{11}(z;\tau)\over\vartheta_{10}\vartheta% _{01}(z;\tau)}
  21. cn ( u ; k ) = ϑ 01 ϑ 10 ( z ; τ ) ϑ 10 ϑ 01 ( z ; τ ) \mathrm{cn}(u;k)={\vartheta_{01}\vartheta_{10}(z;\tau)\over\vartheta_{10}% \vartheta_{01}(z;\tau)}
  22. dn ( u ; k ) = ϑ 01 ϑ ( z ; τ ) ϑ ϑ 01 ( z ; τ ) \mathrm{dn}(u;k)={\vartheta_{01}\vartheta(z;\tau)\over\vartheta\vartheta_{01}(% z;\tau)}
  23. k = 1 - k 2 k^{\prime}=\sqrt{1-k^{2}}
  24. k ( τ ) = ( ϑ 01 ϑ ) 2 . k^{\prime}(\tau)=\left({\vartheta_{01}\over\vartheta}\right)^{2}.
  25. = 1 2 1 - k 1 + k = 1 2 ϑ - ϑ 01 ϑ + ϑ 01 . \ell={1\over 2}{1-\sqrt{k^{\prime}}\over 1+\sqrt{k^{\prime}}}={1\over 2}{% \vartheta-\vartheta_{01}\over\vartheta+\vartheta_{01}}.
  26. q = exp ( π i τ ) q=\exp(\pi i\tau)
  27. \ell
  28. = q + q 9 + q 25 + 1 + 2 q 4 + 2 q 16 + . \ell={q+q^{9}+q^{25}+\cdots\over 1+2q^{4}+2q^{16}+\cdots}.
  29. q = + 2 5 + 15 9 + 150 13 + 1707 17 + 20910 21 + 268616 25 + . q=\ell+2\ell^{5}+15\ell^{9}+150\ell^{13}+1707\ell^{17}+20910\ell^{21}+268616% \ell^{25}+\cdots.
  30. ns ( u ) \displaystyle\operatorname{ns}(u)
  31. sc ( u ) = sn ( u ) cn ( u ) sd ( u ) = sn ( u ) dn ( u ) dc ( u ) = dn ( u ) cn ( u ) ds ( u ) = dn ( u ) sn ( u ) cs ( u ) = cn ( u ) sn ( u ) cd ( u ) = cn ( u ) dn ( u ) \begin{aligned}\displaystyle\operatorname{sc}(u)&\displaystyle=\frac{% \operatorname{sn}(u)}{\operatorname{cn}(u)}\\ \displaystyle\operatorname{sd}(u)&\displaystyle=\frac{\operatorname{sn}(u)}{% \operatorname{dn}(u)}\\ \displaystyle\operatorname{dc}(u)&\displaystyle=\frac{\operatorname{dn}(u)}{% \operatorname{cn}(u)}\\ \displaystyle\operatorname{ds}(u)&\displaystyle=\frac{\operatorname{dn}(u)}{% \operatorname{sn}(u)}\\ \displaystyle\operatorname{cs}(u)&\displaystyle=\frac{\operatorname{cn}(u)}{% \operatorname{sn}(u)}\\ \displaystyle\operatorname{cd}(u)&\displaystyle=\frac{\operatorname{cn}(u)}{% \operatorname{dn}(u)}\end{aligned}
  32. pq ( u ) = pr ( u ) qr ( u ) \operatorname{pq}(u)=\frac{\operatorname{pr}(u)}{\operatorname{qr}(u)}
  33. cn 2 ( u , k ) + sn 2 ( u , k ) = 1 , \operatorname{cn}^{2}(u,k)+\operatorname{sn}^{2}(u,k)=1,\,
  34. dn 2 ( u , k ) + k 2 sn 2 ( u , k ) = 1. \operatorname{dn}^{2}(u,k)+k^{2}\ \operatorname{sn}^{2}(u,k)=1.\,
  35. cn ( x + y ) = cn ( x ) cn ( y ) - sn ( x ) sn ( y ) dn ( x ) dn ( y ) 1 - k 2 sn 2 ( x ) sn 2 ( y ) , sn ( x + y ) = sn ( x ) cn ( y ) dn ( y ) + sn ( y ) cn ( x ) dn ( x ) 1 - k 2 sn 2 ( x ) sn 2 ( y ) , dn ( x + y ) = dn ( x ) dn ( y ) - k 2 sn ( x ) sn ( y ) cn ( x ) cn ( y ) 1 - k 2 sn 2 ( x ) sn 2 ( y ) . \begin{aligned}\displaystyle\operatorname{cn}(x+y)&\displaystyle={% \operatorname{cn}(x)\;\operatorname{cn}(y)-\operatorname{sn}(x)\;\operatorname% {sn}(y)\;\operatorname{dn}(x)\;\operatorname{dn}(y)\over{1-k^{2}\;% \operatorname{sn}^{2}(x)\;\operatorname{sn}^{2}(y)}},\\ \displaystyle\operatorname{sn}(x+y)&\displaystyle={\operatorname{sn}(x)\;% \operatorname{cn}(y)\;\operatorname{dn}(y)+\operatorname{sn}(y)\;\operatorname% {cn}(x)\;\operatorname{dn}(x)\over{1-k^{2}\;\operatorname{sn}^{2}(x)\;% \operatorname{sn}^{2}(y)}},\\ \displaystyle\operatorname{dn}(x+y)&\displaystyle={\operatorname{dn}(x)\;% \operatorname{dn}(y)-k^{2}\;\operatorname{sn}(x)\;\operatorname{sn}(y)\;% \operatorname{cn}(x)\;\operatorname{cn}(y)\over{1-k^{2}\;\operatorname{sn}^{2}% (x)\;\operatorname{sn}^{2}(y)}}.\end{aligned}
  36. - dn 2 ( u ) + m 1 = - m cn 2 ( u ) = m sn 2 ( u ) - m -\operatorname{dn}^{2}(u)+m_{1}=-m\;\operatorname{cn}^{2}(u)=m\;\operatorname{% sn}^{2}(u)-m
  37. - m 1 nd 2 ( u ) + m 1 = - m m 1 sd 2 ( u ) = m cd 2 ( u ) - m -m_{1}\;\operatorname{nd}^{2}(u)+m_{1}=-mm_{1}\;\operatorname{sd}^{2}(u)=m\;% \operatorname{cd}^{2}(u)-m
  38. m 1 sc 2 ( u ) + m 1 = m 1 nc 2 ( u ) = dc 2 ( u ) - m m_{1}\;\operatorname{sc}^{2}(u)+m_{1}=m_{1}\;\operatorname{nc}^{2}(u)=% \operatorname{dc}^{2}(u)-m
  39. cs 2 ( u ) + m 1 = ds 2 ( u ) = ns 2 ( u ) - m \operatorname{cs}^{2}(u)+m_{1}=\operatorname{ds}^{2}(u)=\operatorname{ns}^{2}(% u)-m
  40. q = exp ( - π K / K ) q=\exp(-\pi K^{\prime}/K)
  41. v = π u / ( 2 K ) v=\pi u/(2K)
  42. sn ( u ) = 2 π K m n = 0 q n + 1 / 2 1 - q 2 n + 1 sin ( ( 2 n + 1 ) v ) , \operatorname{sn}(u)=\frac{2\pi}{K\sqrt{m}}\sum_{n=0}^{\infty}\frac{q^{n+1/2}}% {1-q^{2n+1}}\sin((2n+1)v),
  43. cn ( u ) = 2 π K m n = 0 q n + 1 / 2 1 + q 2 n + 1 cos ( ( 2 n + 1 ) v ) , \operatorname{cn}(u)=\frac{2\pi}{K\sqrt{m}}\sum_{n=0}^{\infty}\frac{q^{n+1/2}}% {1+q^{2n+1}}\cos((2n+1)v),
  44. dn ( u ) = π 2 K + 2 π K n = 1 q n 1 + q 2 n cos ( 2 n v ) . \operatorname{dn}(u)=\frac{\pi}{2K}+\frac{2\pi}{K}\sum_{n=1}^{\infty}\frac{q^{% n}}{1+q^{2n}}\cos(2nv).
  45. d d z sn ( z ) = cn ( z ) dn ( z ) , \frac{\mathrm{d}}{\mathrm{d}z}\,\mathrm{sn}\,(z)=\mathrm{cn}\,(z)\,\mathrm{dn}% \,(z),
  46. d d z cn ( z ) = - sn ( z ) dn ( z ) , \frac{\mathrm{d}}{\mathrm{d}z}\,\mathrm{cn}\,(z)=-\mathrm{sn}\,(z)\,\mathrm{dn% }\,(z),
  47. d d z dn ( z ) = - k 2 sn ( z ) cn ( z ) . \frac{\mathrm{d}}{\mathrm{d}z}\,\mathrm{dn}\,(z)=-k^{2}\mathrm{sn}\,(z)\,% \mathrm{cn}\,(z).
  48. d 2 y d x 2 + ( 1 + k 2 ) y - 2 k 2 y 3 = 0 \frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}+(1+k^{2})y-2k^{2}y^{3}=0
  49. ( d y d x ) 2 = ( 1 - y 2 ) ( 1 - k 2 y 2 ) \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^{2}=(1-y^{2})(1-k^{2}y^{2})
  50. cn ( x ) \mathrm{cn}\,(x)
  51. d 2 y d x 2 + ( 1 - 2 k 2 ) y + 2 k 2 y 3 = 0 \frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}+(1-2k^{2})y+2k^{2}y^{3}=0
  52. ( d y d x ) 2 = ( 1 - y 2 ) ( 1 - k 2 + k 2 y 2 ) \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^{2}=(1-y^{2})(1-k^{2}+k^{2}y^{2})
  53. dn ( x ) \mathrm{dn}\,(x)
  54. d 2 y d x 2 - ( 2 - k 2 ) y + 2 y 3 = 0 \frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}-(2-k^{2})y+2y^{3}=0
  55. ( d y d x ) 2 = ( y 2 - 1 ) ( 1 - k 2 - y 2 ) \left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^{2}=(y^{2}-1)(1-k^{2}-y^{2})
  56. x = sn ( ξ , k ) x=\mathrm{sn}(\xi,k)
  57. ξ = arcsn ( x , k ) \xi=\mathrm{arcsn}(x,k)

Jacobian_variety.html

  1. [ γ ] : ω γ ω [\gamma]:\ \omega\mapsto\int_{\gamma}\omega
  2. J ( C ) = H 0 ( Ω C 1 ) * / H 1 ( C ) , J(C)=H^{0}(\Omega_{C}^{1})^{*}/H_{1}(C),
  3. H 1 ( C ) H_{1}(C)
  4. H 0 ( Ω C 1 ) * H^{0}(\Omega_{C}^{1})^{*}

Jacobson_density_theorem.html

  1. R R
  2. R R
  3. U U
  4. R R
  5. u u
  6. U U
  7. u R = U u•R=U
  8. u R u•R
  9. U U
  10. u u
  11. u , v u,v
  12. U U
  13. R R
  14. U U
  15. u u
  16. v v
  17. R R
  18. i i
  19. D D
  20. R R
  21. U U
  22. D D
  23. D D
  24. U U
  25. R R
  26. X U X⊂U
  27. D D
  28. A A
  29. D D
  30. U U
  31. r R r∈R
  32. A ( x ) = x r A(x)=x•r
  33. x x
  34. X X
  35. R R
  36. U U
  37. D D
  38. g u = g ( u ) g•u=g(u)
  39. U U
  40. D D
  41. U U
  42. U U
  43. D D
  44. U U
  45. R R
  46. X U X⊂U
  47. X X
  48. R R
  49. u u
  50. U U
  51. u I = 0 u•I=0
  52. u u
  53. X D XD
  54. D D
  55. X X
  56. | X | |X|
  57. X X
  58. X X
  59. x x
  60. X X
  61. A A
  62. D D
  63. U U
  64. s R s∈R
  65. A ( y ) = y s A(y)=y•s
  66. y y
  67. Y Y
  68. x I x•I
  69. U U
  70. x I = 0 x•I=0
  71. x x
  72. D D
  73. Y Y
  74. D D
  75. X X
  76. x I 0 x•I≠0
  77. U U
  78. x I = U x•I=U
  79. A ( x ) x s U = x I A(x)−x•s∈U=x•I
  80. i i
  81. I I
  82. x i = A ( x ) x s x•i=A(x)−x•s
  83. r = s + i r=s+i
  84. y y
  85. Y Y
  86. y r \displaystyle y\cdot r
  87. x x
  88. x r \displaystyle x\cdot r
  89. A ( z ) = z r A(z)=z•r
  90. z z
  91. X X
  92. X X
  93. R R
  94. R R
  95. U U
  96. R R
  97. R R
  98. E n d ( < s u b > D U ) End(<sub>DU)
  99. D - R D-R

Jean-Robert_Argand.html

  1. i i
  2. a b \overrightarrow{ab}

John_Pell.html

  1. a x 2 + 1 = y 2 , ax^{2}+1=y^{2},

Josephson_effect.html

  1. U ( t ) = 2 e ϕ t U(t)=\frac{\hbar}{2e}\frac{\partial\phi}{\partial t}
  2. I ( t ) = I c sin ( ϕ ( t ) ) \frac{}{}I(t)=I_{c}\sin(\phi(t))
  3. ϕ ( t ) \phi(t)
  4. h 2 e \frac{h}{2e}
  5. - I c \scriptstyle-I_{c}
  6. I c \scriptstyle I_{c}
  7. U D C \scriptstyle U_{DC}
  8. I c \scriptstyle I_{c}
  9. 2 e h U D C \frac{2e}{h}U_{DC}
  10. I ext I\text{ext}
  11. I ext = C J d v d t + I J sin ϕ + V R I\text{ext}\;=\;C_{J}\frac{dv}{dt}\,+\,I_{J}\sin\phi\,+\,\frac{V}{R}
  12. ϕ ( t ) = ϕ 0 + n ω t + a sin ( ω t ) \scriptstyle\phi(t)\;=\;\phi_{0}\,+\,n\omega t\,+\,a\sin(\omega t)
  13. U ( t ) = 2 e ω ( n + a cos ( ω t ) ) , I ( t ) = I c m = - J n ( a ) sin ( ϕ 0 + ( n + m ) ω t ) . U(t)=\frac{\hbar}{2e}\omega(n+a\cos(\omega t)),\ \ \ I(t)=I_{c}\sum_{m\,=\,-% \infty}^{\infty}J_{n}(a)\sin(\phi_{0}+(n+m)\omega t).
  14. U D C = n 2 e ω , I ( t ) = I c J - n ( a ) sin ϕ 0 . U_{DC}=n\frac{\hbar}{2e}\omega,\ \ \ I(t)=I_{c}J_{-n}(a)\sin\phi_{0}.
  15. 1 2 e h \scriptstyle\frac{1}{2e}h

Joule_heating.html

  1. H I 2 R t H\propto I^{2}\cdot R\cdot t
  2. P = ( V A - V B ) I P=(VA-VB)I
  3. P = I V = I 2 R = V 2 / R P=IV=I^{2}R=V^{2}/R
  4. P ( t ) = I ( t ) V ( t ) P(t)=I(t)V(t)
  5. P a v g = I r m s V r m s = I r m s 2 R = V r m s 2 / R P_{avg}=I_{rms}V_{rms}=I_{rms}^{2}R=V_{rms}^{2}/R
  6. P a v g = I r m s V r m s cos ϕ = I r m s 2 Re ( Z ) = V r m s 2 Re ( Y * ) P_{avg}=I_{rms}V_{rms}\cos\phi=I_{rms}^{2}\operatorname{Re}(Z)=V_{rms}^{2}% \operatorname{Re}(Y^{*})
  7. ϕ \phi
  8. Re \operatorname{Re}
  9. d P / d V = 𝐉 𝐄 dP/dV=\mathbf{J}\cdot\mathbf{E}
  10. 𝐉 \mathbf{J}
  11. 𝐄 \mathbf{E}
  12. σ \sigma
  13. 𝐉 = σ 𝐄 \mathbf{J}=\sigma\mathbf{E}
  14. d P / d V = 𝐉 𝐄 = 𝐉 𝐉 / σ = J 2 ρ dP/dV=\mathbf{J}\cdot\mathbf{E}=\mathbf{J}\cdot\mathbf{J}/\sigma=J^{2}\rho
  15. ρ = 1 / σ \rho=1/\sigma
  16. I 2 R I^{2}R
  17. d E d x = v x 2 K {dE\over dx}={v_{x}^{2}\over K}
  18. d E / d x dE/dx
  19. E E
  20. x x
  21. Q / t Q/t
  22. v x v_{x}
  23. x x
  24. I I
  25. K K
  26. R R

Julius_von_Mayer.html

  1. C P , m - C V , m = R C_{P,m}-C_{V,m}=R
  2. C P - C V = V T β 2 α T C_{P}-C_{V}=VT\frac{\beta^{2}}{\alpha_{T}}\,
  3. C P C_{P}
  4. C V C_{V}
  5. V V
  6. T T
  7. β \beta
  8. α T \alpha_{T}
  9. α T \alpha_{T}
  10. β {\beta}
  11. C P , m C_{P,m}
  12. C V , m C_{V,m}

Just-noticeable_difference.html

  1. Δ I I = k , \frac{\Delta I}{I}=k,
  2. I I\!
  3. Δ I \Delta I\!

Kademlia.html

  1. O ( log ( n ) ) O(\log(n))
  2. n n
  3. n n
  4. x 1 , , x n x_{1},\ldots,x_{n}
  5. d d
  6. x { x 1 , , x n } x\in\{x_{1},\ldots,x_{n}\}
  7. 𝒟 i ( x ) \mathcal{D}_{i}(x)
  8. x x
  9. d - i d-i
  10. i i
  11. x x
  12. x x
  13. k k
  14. 𝒟 i ( x ) \mathcal{D}_{i}(x)
  15. T x y T_{xy}
  16. x x
  17. y y
  18. sup x 1 , , x n sup x sup y 𝔼 T x y c k log n \sup_{x_{1},\ldots,x_{n}}\sup_{x}\sup_{y}\mathbb{E}T_{xy}\leq c_{k}\log n
  19. c k c_{k}
  20. 1 / log k 1/\log k
  21. log k n \log_{k}n

Kahan_summation_algorithm.html

  1. n \sqrt{n}
  2. S n = i = 1 n x i S_{n}=\sum_{i=1}^{n}x_{i}
  3. S n + E n S_{n}+E_{n}
  4. E n E_{n}
  5. | E n | [ 2 ε + O ( n ε 2 ) ] i = 1 n | x i | |E_{n}|\leq\left[2\varepsilon+O(n\varepsilon^{2})\right]\sum_{i=1}^{n}|x_{i}|
  6. | E n | / | S n | |E_{n}|/|S_{n}|
  7. | E n | | S n | [ 2 ε + O ( n ε 2 ) ] i = 1 n | x i | | i = 1 n x i | . \frac{|E_{n}|}{|S_{n}|}\leq\left[2\varepsilon+O(n\varepsilon^{2})\right]\frac{% \sum_{i=1}^{n}|x_{i}|}{\left|\sum_{i=1}^{n}x_{i}\right|}.
  8. n \sqrt{n}
  9. n n\to\infty
  10. O ( ε n ) O(\varepsilon n)
  11. O ( ε n ) O(\varepsilon\sqrt{n})
  12. E n E_{n}
  13. E n E_{n}
  14. O ( ε n ) O(\varepsilon\sqrt{n})
  15. S n S_{n}
  16. n \sqrt{n}
  17. O ( 1 ) O(1)
  18. O ( log n ) O(\log n)
  19. O ( log n ) O(\sqrt{\log n})

Kakeya_set.html

  1. K 0 K 1 K 2 K_{0}\supseteq K_{1}\supseteq K_{2}\cdots
  2. π 24 ( 5 - 2 2 ) \tfrac{\pi}{24}(5-2\sqrt{2})
  3. T e δ ( a ) T_{e}^{\delta}(a)
  4. f * δ ( e ) = sup a 𝐑 n 1 m ( T e δ ( a ) ) T e δ ( a ) | f ( y ) | d m ( y ) f_{*}^{\delta}(e)=\sup_{a\in\mathbf{R}^{n}}\frac{1}{m(T_{e}^{\delta}(a))}\int_% {T_{e}^{\delta}(a)}|f(y)|dm(y)
  5. f * δ f_{*}^{\delta}
  6. f * δ L n ( 𝐒 n - 1 ) C ϵ δ - ϵ f L n ( 𝐑 n ) . \left\|f_{*}^{\delta}\right\|_{L^{n}(\mathbf{S}^{n-1})}\leqslant C_{\epsilon}% \delta^{-\epsilon}\|f\|_{L^{n}(\mathbf{R}^{n})}.
  7. ( 2 - 2 ) ( n - 4 ) + 3 (2-\sqrt{2})(n-4)+3
  8. ( | 𝐅 | + n - 1 n ) | 𝐅 | n n ! . {|\mathbf{F}|+n-1\choose n}\geq\frac{|\mathbf{F}|^{n}}{n!}.

Kaon.html

  1. 1 / 2 {1}/{2}
  2. 1 / 2 {1}/{2}
  3. d s ¯ - s d ¯ 2 \mathrm{\tfrac{d\bar{s}-s\bar{d}}{\sqrt{2}}}\,
  4. 1 / 2 {1}/{2}
  5. d s ¯ + s d ¯ 2 \mathrm{\tfrac{d\bar{s}+s\bar{d}}{\sqrt{2}}}\,
  6. 1 / 2 {1}/{2}
  7. ψ ( t ) = U ( t ) ψ ( 0 ) = e i H t ( a b ) , H = ( M Δ Δ M ) \psi(t)=U(t)\psi(0)={\rm e}^{iHt}\begin{pmatrix}a\\ b\end{pmatrix},\qquad H=\begin{pmatrix}M&\Delta\\ \Delta&M\end{pmatrix}

Kaprekar_number.html

  1. a m 1 ( mod b ) am\equiv 1\;\;(\mathop{{\rm mod}}b)

KASUMI.html

  1. K = K 1 K 2 K 3 K 4 K 5 K 6 K 7 K 8 K=K_{1}\|K_{2}\|K_{3}\|K_{4}\|K_{5}\|K_{6}\|K_{7}\|K_{8}\,
  2. K L i , 1 = ROL ( K i , 1 ) K L i , 2 = K i + 2 K O i , 1 = ROL ( K i + 1 , 5 ) K O i , 2 = ROL ( K i + 5 , 8 ) K O i , 3 = ROL ( K i + 6 , 13 ) K I i , 1 = K i + 4 K I i , 2 = K i + 3 K I i , 3 = K i + 7 \begin{array}[]{lcl}KL_{i,1}&=&{\rm ROL}(K_{i},1)\\ KL_{i,2}&=&K^{\prime}_{i+2}\\ KO_{i,1}&=&{\rm ROL}(K_{i+1},5)\\ KO_{i,2}&=&{\rm ROL}(K_{i+5},8)\\ KO_{i,3}&=&{\rm ROL}(K_{i+6},13)\\ KI_{i,1}&=&K^{\prime}_{i+4}\\ KI_{i,2}&=&K^{\prime}_{i+3}\\ KI_{i,3}&=&K^{\prime}_{i+7}\end{array}
  3. L i L_{i}
  4. R i R_{i}
  5. input = R 0 L 0 {\rm input}=R_{0}\|L_{0}\,
  6. L i = F i ( K L i , K O i , K I i , L i - 1 ) R i - 1 R i = L i - 1 \begin{array}[]{rcl}L_{i}&=&F_{i}(KL_{i},KO_{i},KI_{i},L_{i-1})\oplus R_{i-1}% \\ R_{i}&=&L_{i-1}\end{array}
  7. F i ( K i , L i - 1 ) = F O ( K O i , K I i , F L ( K L i , L i - 1 ) ) F_{i}(K_{i},L_{i-1})=FO(KO_{i},KI_{i},FL(KL_{i},L_{i-1}))\,
  8. F i ( K i , L i - 1 ) = F L ( K L i , F O ( K O i , K I i , L i - 1 ) ) F_{i}(K_{i},L_{i-1})=FL(KL_{i},FO(KO_{i},KI_{i},L_{i-1}))\,
  9. output = R 8 L 8 {\rm output}=R_{8}\|L_{8}\,
  10. F L ( K L i , x ) FL(KL_{i},x)
  11. x = l r x=l\|r
  12. l l
  13. K L i , 1 KL_{i,1}
  14. r r
  15. r r^{\prime}
  16. r = ROL ( l K L i , 1 , 1 ) r r^{\prime}={\rm ROL}(l\wedge KL_{i,1},1)\oplus r
  17. r r^{\prime}
  18. K L i , 2 KL_{i,2}
  19. l l
  20. l l^{\prime}
  21. l = ROL ( r K L i , 2 , 1 ) l l^{\prime}={\rm ROL}(r^{\prime}\vee KL_{i,2},1)\oplus l
  22. x = l r x^{\prime}=l^{\prime}\|r^{\prime}
  23. F O ( K O i , K I i , x ) FO(KO_{i},KI_{i},x)
  24. x = l 0 r 0 x=l_{0}\|r_{0}
  25. r j = F I ( K I i , j , l j - 1 K O i , j ) r j - 1 l j = r j - 1 \begin{array}[]{lcl}r_{j}&=&FI(KI_{i,j},l_{j-1}\oplus KO_{i,j})\oplus r_{j-1}% \\ l_{j}&=&r_{j-1}\end{array}
  26. x = l 3 r 3 x^{\prime}=l_{3}\|r_{3}
  27. x x
  28. F I ( K i , x ) FI(Ki,x)
  29. x = l 0 r 0 x=l_{0}\|r_{0}
  30. l 0 l_{0}
  31. r 0 r_{0}
  32. l 0 l_{0}
  33. r 0 r_{0}
  34. r 1 r_{1}
  35. r 1 = S 9 ( l 0 ) ( 00 r 0 ) r_{1}=S9(l_{0})\oplus(00\|r_{0})\,
  36. r 0 r_{0}
  37. r 1 r_{1}
  38. l 1 l_{1}
  39. l 1 = S 7 ( r 0 ) L S 7 ( r 1 ) l_{1}=S7(r_{0})\oplus LS7(r_{1})\,
  40. x 1 = l 1 r 1 x_{1}=l_{1}\|r_{1}
  41. x 2 = l 2 r 2 x_{2}=l_{2}\|r_{2}
  42. l 2 l_{2}
  43. r 2 r_{2}
  44. x 2 = K I x 1 x_{2}=KI\oplus x_{1}
  45. r 2 r_{2}
  46. l 2 l_{2}
  47. r 3 r_{3}
  48. r 3 = S 9 ( r 2 ) ( 00 l 2 ) r_{3}=S9(r_{2})\oplus(00\|l_{2})\,
  49. l 2 l_{2}
  50. r 3 r_{3}
  51. l 3 l_{3}
  52. l 3 = S 7 ( l 2 ) L S 7 ( r 3 ) l_{3}=S7(l_{2})\oplus LS7(r_{3})\,
  53. x = l 3 r 3 x^{\prime}=l_{3}\|r_{3}

Kazimierz_Zarankiewicz.html

  1. K m , n K_{m,n}
  2. c r ( K m , n ) = n / 2 ( n - 1 ) / 2 m / 2 ( m - 1 ) / 2 . cr(K_{m,n})=\lfloor n/2\rfloor\lfloor(n-1)/2\rfloor\lfloor m/2\rfloor\lfloor(m% -1)/2\rfloor.
  3. c r ( K m , n ) cr(K_{m,n})

Kähler_differential.html

  1. d : S M \mathrm{d}\colon S\to M
  2. d ( f g ) = f d g + g d f \mathrm{d}(fg)=f\mathrm{\,}\mathrm{d}g+g\,\mathrm{d}f
  3. d : S Ω S / R \mathrm{d}\colon S\to\Omega_{S/R}
  4. S R S S\otimes_{R}S
  5. S R S S S\otimes_{R}S\to S
  6. Σ s i t i Σ s i . t i \Sigma s_{i}\otimes t_{i}\mapsto\Sigma s_{i}.t_{i}
  7. d s = 1 s - s 1. \mathrm{d}s=1\otimes s-s\otimes 1.\,
  8. S R S S R R S\otimes_{R}S\to S\otimes_{R}R
  9. Σ s i t i Σ s i . t i 1 \Sigma s_{i}\otimes t_{i}\mapsto\Sigma s_{i}.t_{i}\otimes 1
  10. S R S I S R R . S\otimes_{R}S\equiv\,\,{}I\,{}\oplus S\otimes_{R}R.\,
  11. S R S / S R R S\otimes_{R}S/S\otimes_{R}R
  12. Σ s i t i Σ s i t i - Σ s i . t i 1 \Sigma s_{i}\otimes t_{i}\mapsto\Sigma s_{i}\otimes t_{i}-\Sigma s_{i}.t_{i}\otimes 1
  13. Der R ( S , M ) Hom S ( Ω S / R , M ) , \operatorname{Der}_{R}(S,M)\cong\operatorname{Hom}_{S}(\Omega_{S/R},M),\,
  14. δ L / K = { x O : x d y = 0 for all y O } . \delta_{L/K}=\{x\in O:x\mathrm{d}y=0\,\text{ for all }y\in O\}.

Kähler_manifold.html

  1. ( K , ω ) (K,\omega)
  2. K K
  3. h h
  4. J J
  5. h h
  6. ω = i 2 ( h - h ¯ ) \omega=\frac{i}{2}(h-\bar{h})
  7. J J
  8. ( 1 , 1 ) (1,1)
  9. ( U , z i ) (U,z_{i})
  10. ω = i 2 j , k h j k d z j d z k ¯ \omega=\frac{i}{2}\sum_{j,k}h_{jk}dz_{j}\wedge d\bar{z_{k}}
  11. h j k C ( U , ) h_{jk}\in C^{\infty}(U,\mathbb{C})
  12. ω \omega
  13. h j k h_{jk}
  14. K K
  15. h h
  16. ω \omega
  17. J J
  18. ω \omega
  19. J J
  20. g ( u , v ) = ω ( u , J v ) g(u,v)=\omega(u,Jv)
  21. h = g + i ω h=g+i\omega
  22. K K
  23. ρ C ( K ; ) \rho\in C^{\infty}(K;\mathbb{R})
  24. ω = i 2 ¯ ρ \omega=\frac{i}{2}\partial\bar{\partial}\rho
  25. , ¯ \partial,\bar{\partial}
  26. ρ \rho
  27. ( K , ω ) (K,\omega)
  28. p K p\in K
  29. U U
  30. p p
  31. ρ C ( U , ) \rho\in C^{\infty}(U,\mathbb{R})
  32. ω | U = i ¯ ρ \omega|_{U}=i\partial\bar{\partial}\rho
  33. ρ \rho
  34. Δ d = d d * + d * d \Delta_{d}=dd^{*}+d^{*}d
  35. d d
  36. d * = - ( - 1 ) n k d d^{*}=-(-1)^{nk}\star d\star
  37. \star
  38. d * d^{*}
  39. d d
  40. L 2 L^{2}
  41. d d
  42. d * d^{*}
  43. d = + ¯ , d * = * + ¯ * d=\partial+\bar{\partial},\ \ \ \ d^{*}=\partial^{*}+\bar{\partial}^{*}
  44. Δ ¯ = ¯ ¯ * + ¯ * ¯ , Δ = * + * \Delta_{\bar{\partial}}=\bar{\partial}\bar{\partial}^{*}+\bar{\partial}^{*}% \bar{\partial},\ \ \ \ \Delta_{\partial}=\partial\partial^{*}+\partial^{*}\partial
  45. Δ d = 2 Δ ¯ = 2 Δ . \Delta_{d}=2\Delta_{\bar{\partial}}=2\Delta_{\partial}.
  46. 𝐇 𝐫 = p + q = r 𝐇 p , q \mathbf{H^{r}}=\bigoplus_{p+q=r}\mathbf{H}^{p,q}
  47. 𝐇 𝐫 \mathbf{H^{r}}
  48. 𝐇 p , q \mathbf{H}^{p,q}
  49. α \alpha
  50. α i , j \alpha^{i,j}
  51. H p ( X , Ω q ) H ¯ p , q ( X ) 𝐇 p , q H^{p}(X,\Omega^{q})\simeq H^{p,q}_{\bar{\partial}}(X)\simeq\mathbf{H}^{p,q}
  52. H ¯ p , q ( X ) H^{p,q}_{\bar{\partial}}(X)
  53. ¯ \bar{\partial}
  54. α \alpha
  55. h p , q = dim H p , q h^{p,q}=\,\text{dim}H^{p,q}
  56. b r = p + q = r h p , q , h p , q = h q , p , h p , q = h n - p , n - q . b_{r}=\sum_{p+q=r}h^{p,q},\ \ \ \ h^{p,q}=h^{q,p},\ \ \ \ h^{p,q}=h^{n-p,n-q}.
  57. Δ d \Delta_{d}
  58. H p , q = H q , p ¯ H^{p,q}=\overline{H^{q,p}}
  59. R = λ g R=\lambda g
  60. Γ β γ α \Gamma^{\alpha}_{\beta\gamma}

Keith_number.html

  1. N = i = 0 n - 1 10 i d i , N=\sum_{i=0}^{n-1}10^{i}{d_{i}},
  2. S N S_{N}
  3. d n - 1 , d n - 2 , , d 1 , d 0 d_{n-1},d_{n-2},\ldots,d_{1},d_{0}
  4. S N S_{N}
  5. 9 10 log 2 10 2.99 \textstyle\frac{9}{10}\log_{2}{10}\approx 2.99

Kepler_conjecture.html

  1. π 3 2 = 0.740480489 \frac{\pi}{3\sqrt{2}}=0.740480489\ldots

Kernel_(set_theory).html

  1. { { w X f ( x ) = f ( w ) } x X } . \left\{\,\left\{\,w\in X\mid f(x)=f(w)\,\right\}\mid x\in X\,\right\}.
  2. X / = f X/\mathord{=_{f}}
  3. f f
  4. coim f \operatorname{coim}f
  5. im f \operatorname{im}f
  6. x x
  7. X X
  8. coim f \operatorname{coim}f
  9. f ( x ) f(x)
  10. Y Y
  11. im f \operatorname{im}f
  12. ker f := { ( x , x ) f ( x ) = f ( x ) } \operatorname{ker}f:=\{(x,x^{\prime})\mid f(x)=f(x^{\prime})\}
  13. f f

Kerr_metric.html

  1. c 2 d τ 2 = ( 1 - r s r ρ 2 ) c 2 d t 2 - ρ 2 Δ d r 2 - ρ 2 d θ 2 - ( r 2 + α 2 + r s r α 2 ρ 2 sin 2 θ ) sin 2 θ d ϕ 2 + 2 r s r α sin 2 θ ρ 2 c d t d ϕ \begin{aligned}\displaystyle c^{2}d\tau^{2}=&\displaystyle\left(1-\frac{r_{s}r% }{\rho^{2}}\right)c^{2}dt^{2}-\frac{\rho^{2}}{\Delta}dr^{2}-\rho^{2}d\theta^{2% }\\ &\displaystyle-\left(r^{2}+\alpha^{2}+\frac{r_{s}r\alpha^{2}}{\rho^{2}}\sin^{2% }\theta\right)\sin^{2}\theta\ d\phi^{2}+\frac{2r_{s}r\alpha\sin^{2}\theta}{% \rho^{2}}\,c\,dt\,d\phi\end{aligned}
  2. r , θ , ϕ r,\theta,\phi
  3. r s = 2 G M c 2 r_{s}=\frac{2GM}{c^{2}}
  4. α = J M c \alpha=\frac{J}{Mc}
  5. ρ 2 = r 2 + α 2 cos 2 θ \rho^{2}=r^{2}+\alpha^{2}\cos^{2}\theta
  6. Δ = r 2 - r s r + α 2 \Delta=r^{2}-r_{s}r+\alpha^{2}
  7. c 2 d τ 2 = c 2 d t 2 - ρ 2 r 2 + α 2 d r 2 - ρ 2 d θ 2 - ( r 2 + α 2 ) sin 2 θ d ϕ 2 c^{2}d\tau^{2}=c^{2}dt^{2}-\frac{\rho^{2}}{r^{2}+\alpha^{2}}dr^{2}-\rho^{2}d% \theta^{2}-\left(r^{2}+\alpha^{2}\right)\sin^{2}\theta d\phi^{2}
  8. x = r 2 + α 2 sin θ cos ϕ {x}=\sqrt{r^{2}+\alpha^{2}}\sin\theta\cos\phi
  9. y = r 2 + α 2 sin θ sin ϕ {y}=\sqrt{r^{2}+\alpha^{2}}\sin\theta\sin\phi
  10. z = r cos θ {z}=r\cos\theta
  11. g i k g^{ik}
  12. g μ ν x μ x ν = \displaystyle g^{\mu\nu}\frac{\partial}{\partial{x^{\mu}}}\frac{\partial}{% \partial{x^{\nu}}}=
  13. c 2 d τ 2 = ( g t t - g t ϕ 2 g ϕ ϕ ) d t 2 + g r r d r 2 + g θ θ d θ 2 + g ϕ ϕ ( d ϕ + g t ϕ g ϕ ϕ d t ) 2 . c^{2}d\tau^{2}=\left(g_{tt}-\frac{g_{t\phi}^{2}}{g_{\phi\phi}}\right)dt^{2}+g_% {rr}dr^{2}+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}\left(d\phi+\frac{g_{t\phi}% }{g_{\phi\phi}}dt\right)^{2}.
  14. Ω = - g t ϕ g ϕ ϕ = r s r α c ρ 2 ( r 2 + α 2 ) + r s r α 2 sin 2 θ . \Omega=-\frac{g_{t\phi}}{g_{\phi\phi}}=\frac{r_{s}r\alpha c}{\rho^{2}\left(r^{% 2}+\alpha^{2}\right)+r_{s}r\alpha^{2}\sin^{2}\theta}.
  15. r 𝑖𝑛𝑛𝑒𝑟 = r s + r s 2 - 4 α 2 2 r_{\mathit{inner}}=\frac{r_{s}+\sqrt{r_{s}^{2}-4\alpha^{2}}}{2}
  16. r 𝑜𝑢𝑡𝑒𝑟 = r s + r s 2 - 4 α 2 cos 2 θ 2 r_{\mathit{outer}}=\frac{r_{s}+\sqrt{r_{s}^{2}-4\alpha^{2}\cos^{2}\theta}}{2}
  17. ( r - G M ) 2 = G 2 M 2 - J 2 cos 2 θ (r-GM)^{2}=G^{2}M^{2}-J^{2}\cos^{2}\theta
  18. ϕ \phi\,
  19. θ \theta\,
  20. Δ = 0 \Delta=0
  21. r s / 2 < α {r_{s}/2}<\alpha
  22. G M 2 < J c GM^{2}<Jc
  23. r r
  24. r r
  25. r r
  26. α = 0 \alpha=0\,
  27. M n = M ( i α ) n M_{n}=M\,(i\,\alpha)^{n}
  28. a 0 = M , a 1 = 0 , a 2 = M ( M 2 3 - α 2 ) a_{0}=M,\;\;a_{1}=0,\;\;a_{2}=M\,\left(\frac{M^{2}}{3}-\alpha^{2}\right)
  29. α = M \alpha=M
  30. S = - E 0 t + L ϕ + S r ( r ) + S θ ( θ ) \ S=-E_{0}t+L\phi+S_{r}(r)+S_{\theta}(\theta)
  31. E 0 E_{0}
  32. ( d S θ d θ ) 2 + ( α E 0 sin θ - L sin θ ) 2 + α 2 m 2 cos 2 θ = K \left(\frac{dS_{\theta}}{d\theta}\right)^{2}+\left(\alpha E_{0}\sin\theta-% \frac{L}{\sin\theta}\right)^{2}+\alpha^{2}m^{2}\cos^{2}\theta=K
  33. Δ ( d S r d r ) 2 - 1 Δ [ ( r 2 + α 2 ) E 0 - α L ] 2 + m 2 r 2 = - K \Delta\left(\frac{dS_{r}}{dr}\right)^{2}-\frac{1}{\Delta}\left[\left(r^{2}+% \alpha^{2}\right)E_{0}-\alpha L\right]^{2}+m^{2}r^{2}=-K
  34. S E 0 = c o n s t {\frac{\partial{S}}{\partial{E_{0}}}}=const
  35. S L = c o n s t {\frac{\partial{S}}{\partial{L}}}=const
  36. S K = c o n s t {\frac{\partial{S}}{\partial{K}}}=const

Key_derivation_function.html

  1. DK = KDF ( Key , Salt , Iterations ) \mathrm{DK}=\mathrm{KDF}(\mathrm{Key},\mathrm{Salt},\mathrm{Iterations})
  2. DK \mathrm{DK}
  3. KDF \mathrm{KDF}
  4. Key \mathrm{Key}
  5. Salt \mathrm{Salt}
  6. Iterations \mathrm{Iterations}

Khinchin's_constant.html

  1. x = a 0 + 1 a 1 + 1 a 2 + 1 a 3 + 1 x=a_{0}+\cfrac{1}{a_{1}+\cfrac{1}{a_{2}+\cfrac{1}{a_{3}+\cfrac{1}{\ddots}}}}\;
  2. lim n ( a 1 a 2 a n ) 1 / n = K 0 \lim_{n\rightarrow\infty}\left(a_{1}a_{2}...a_{n}\right)^{1/n}=K_{0}
  3. K 0 K_{0}
  4. K 0 = r = 1 ( 1 + 1 r ( r + 2 ) ) log 2 r 2.6854520010 K_{0}=\prod_{r=1}^{\infty}{\left(1+{1\over r(r+2)}\right)}^{\log_{2}r}\approx 2% .6854520010\dots
  5. \prod
  6. I = [ 0 , 1 ] \scriptstyle I=[0,1]\setminus\mathbb{Q}
  7. T ( [ a 1 , a 2 , ] ) = [ a 2 , a 3 , ] . T([a_{1},a_{2},\dots])=[a_{2},a_{3},\dots].\,
  8. μ ( E ) = 1 log 2 E d x 1 + x . \mu(E)=\frac{1}{\log 2}\int_{E}\frac{dx}{1+x}.
  9. f ( T k x ) f\left(T^{k}x\right)
  10. x x
  11. lim n 1 n k = 0 n - 1 ( f T k ) ( x ) = I f d μ for μ -almost all x I . \lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}(f\circ T^{k})(x)=\int_{I}fd\mu% \quad\,\text{for }\mu\,\text{-almost all }x\in I.
  12. lim n 1 n k = 1 n log ( a k ) = I f d μ = r = 1 log ( r ) log ( 1 + 1 r ( r + 2 ) ) log 2 \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}\log(a_{k})=\int_{I}f\,d\mu=\sum_{r=% 1}^{\infty}\log(r)\frac{\log\bigl(1+\frac{1}{r(r+2)}\bigr)}{\log 2}
  13. log K 0 = 1 log 2 n = 1 ζ ( 2 n ) - 1 n k = 1 2 n - 1 ( - 1 ) k + 1 k \log K_{0}=\frac{1}{\log 2}\sum_{n=1}^{\infty}\frac{\zeta(2n)-1}{n}\sum_{k=1}^% {2n-1}\frac{(-1)^{k+1}}{k}
  14. log K 0 = 1 log 2 [ k = 3 N log ( k - 1 k ) log ( k + 1 k ) + n = 1 ζ ( 2 n , N ) n k = 1 2 n - 1 ( - 1 ) k + 1 k ] \log K_{0}=\frac{1}{\log 2}\left[\sum_{k=3}^{N}\log\left(\frac{k-1}{k}\right)% \log\left(\frac{k+1}{k}\right)+\sum_{n=1}^{\infty}\frac{\zeta(2n,N)}{n}\sum_{k% =1}^{2n-1}\frac{(-1)^{k+1}}{k}\right]
  15. log K 0 = log 2 + 1 log 2 [ Li ( - 1 2 ) 2 + 1 2 k = 2 ( - 1 ) k Li ( 4 k 2 ) 2 ] . \log K_{0}=\log 2+\frac{1}{\log 2}\left[\mbox{Li}~{}_{2}\left(\frac{-1}{2}% \right)+\frac{1}{2}\sum_{k=2}^{\infty}(-1)^{k}\mbox{Li}~{}_{2}\left(\frac{4}{k% ^{2}}\right)\right].
  16. K p = lim n [ 1 n k = 1 n a k p ] 1 / p . K_{p}=\lim_{n\to\infty}\left[\frac{1}{n}\sum_{k=1}^{n}a_{k}^{p}\right]^{1/p}.
  17. K p = [ k = 1 - k p log 2 ( 1 - 1 ( k + 1 ) 2 ) ] 1 / p . K_{p}=\left[\sum_{k=1}^{\infty}-k^{p}\log_{2}\left(1-\frac{1}{(k+1)^{2}}\right% )\right]^{1/p}.
  18. K - 1 = 1.74540566240 K_{-1}=1.74540566240\dots
  19. π \pi

Kirchhoff's_circuit_laws.html

  1. k = 1 n I k = 0 \sum_{k=1}^{n}{I}_{k}=0
  2. k = 1 n I ~ k = 0 \sum_{k=1}^{n}\tilde{I}_{k}=0
  3. k = 1 n V k = 0 \sum_{k=1}^{n}V_{k}=0
  4. k = 1 n V ~ k = 0 \sum_{k=1}^{n}\tilde{V}_{k}=0
  5. Q \scriptstyle Q
  6. i 1 - i 2 - i 3 = 0 i_{1}-i_{2}-i_{3}=0\,
  7. - R 2 i 2 + 1 - R 1 i 1 = 0 -R_{2}i_{2}+\mathcal{E}_{1}-R_{1}i_{1}=0
  8. - R 3 i 3 - 2 - 1 + R 2 i 2 = 0 -R_{3}i_{3}-\mathcal{E}_{2}-\mathcal{E}_{1}+R_{2}i_{2}=0
  9. i 1 , i 2 , i 3 i_{1},i_{2},i_{3}
  10. { i 1 - i 2 - i 3 = 0 - R 2 i 2 + 1 - R 1 i 1 = 0 - R 3 i 3 - 2 - 1 + R 2 i 2 = 0 \begin{cases}i_{1}-i_{2}-i_{3}&=0\\ -R_{2}i_{2}+\mathcal{E}_{1}-R_{1}i_{1}&=0\\ -R_{3}i_{3}-\mathcal{E}_{2}-\mathcal{E}_{1}+R_{2}i_{2}&=0\end{cases}
  11. R 1 = 100 , R 2 = 200 , R 3 = 300 (ohms) ; 1 = 3 , 2 = 4 (volts) R_{1}=100,\ R_{2}=200,\ R_{3}=300\,\text{ (ohms)};\ \mathcal{E}_{1}=3,\ % \mathcal{E}_{2}=4\,\text{ (volts)}
  12. { i 1 = 1 1100 i 2 = 4 275 i 3 = - 3 220 \begin{cases}i_{1}=\frac{1}{1100}\\ i_{2}=\frac{4}{275}\\ i_{3}=-\frac{3}{220}\end{cases}
  13. i 3 i_{3}
  14. i 3 i_{3}

Kirchhoff's_law_of_thermal_radiation.html

  1. T T
  2. T T
  3. λ \lambda
  4. α λ \alpha_{\lambda}
  5. α λ E b λ ( λ , T ) \alpha_{\lambda}E_{b\lambda}(\lambda,T)
  6. E b λ ( λ , T ) E_{b\lambda}(\lambda,T)
  7. λ \lambda
  8. T T
  9. ϵ λ E b λ ( λ , T ) \epsilon_{\lambda}E_{b\lambda}(\lambda,T)
  10. ϵ λ \epsilon_{\lambda}
  11. λ \lambda
  12. α λ = ϵ λ \alpha_{\lambda}=\epsilon_{\lambda}
  13. α sun = 0 α λ I λ sun ( λ ) d λ 0 I λ sun ( λ ) d λ \alpha_{\mathrm{sun}}=\displaystyle\frac{\int_{0}^{\infty}\alpha_{\lambda}I_{% \lambda\mathrm{sun}}(\lambda)\,d\lambda}{\int_{0}^{\infty}I_{\lambda\mathrm{% sun}}(\lambda)\,d\lambda}
  14. ϵ paint = 0 ϵ λ ( λ , T ) E b λ ( λ , T ) d λ 0 E b λ ( λ , T ) d λ \epsilon_{\mathrm{paint}}=\frac{\int_{0}^{\infty}\epsilon_{\lambda}(\lambda,T)% E_{b\lambda}(\lambda,T)\,d\lambda}{\int_{0}^{\infty}E_{b\lambda}(\lambda,T)\,d\lambda}
  15. I λ sun I_{\lambda\mathrm{sun}}
  16. ϵ λ E b λ ( λ , T ) \epsilon_{\lambda}E_{b\lambda}(\lambda,T)
  17. ϵ λ = α λ \epsilon_{\lambda}=\alpha_{\lambda}
  18. α sun \alpha_{\mathrm{sun}}
  19. ϵ paint \epsilon_{\mathrm{paint}}
  20. I I
  21. I I
  22. I I
  23. E / A E/A
  24. E E
  25. A A
  26. E / A E/A
  27. I I

Kirszbraun_theorem.html

  1. m \mathbb{R}^{m}
  2. p \ell_{p}
  3. p 2 p\neq 2
  4. f ~ ( x ) := inf u U f ( u ) + Lip ( f ) d ( x , u ) , \tilde{f}(x):=\inf_{u\in U}f(u)+\,\text{Lip}(f)\cdot d(x,u),
  5. Lip ( f ) \,\text{Lip}(f)

Kissing_number_problem.html

  1. x n x_{n}
  2. \exist x { n { x n T x n = 1 } and n , m { ( x n - x m ) T ( x n - x m ) 1 } } \exist x\{\forall_{n}\{x_{n}^{T}x_{n}=1\}\and\forall_{n,m}\{(x_{n}-x_{m})^{T}(% x_{n}-x_{m})\geq 1\}\}
  3. y n m y_{nm}
  4. \exist x y { n ( x n T x n - 1 ) 2 + n m ( ( x n - x m ) T ( x n - x m ) - 1 - ( y n m ) 2 ) 2 = 0 } \exist xy\{\sum_{n}(x_{n}^{T}x_{n}-1)^{2}+\sum_{n\neq m}((x_{n}-x_{m})^{T}(x_{% n}-x_{m})-1-(y_{nm})^{2})^{2}=0\}
  5. R n m R_{nm}
  6. \exist R { n { R 0 n = 1 } and n , m { R n m 1 } } \exist R\{\forall_{n}\{R_{0n}=1\}\and\forall_{n,m}\{R_{nm}\geq 1\}\}
  7. R n m = 1 + y n m 2 R_{nm}=1+y_{nm}^{2}

Klein_quartic.html

  1. x 3 y + y 3 z + z 3 x = 0. x^{3}y+y^{3}z+z^{3}x=0.
  2. I = η - 2 I=\langle\eta-2\rangle
  3. ( η ) \mathbb{Z}(\eta)
  4. ( η ) \mathbb{Q}(\eta)
  5. η = 2 cos ( 2 π / 7 ) \eta=2\cos(2\pi/7)
  6. ( 2 - η ) 3 = 7 ( η - 1 ) 2 , (2-\eta)^{3}=7(\eta-1)^{2},
  7. 2 - η 2-\eta
  8. i 2 = j 2 = η , i j = - j i i^{2}=j^{2}=\eta,\quad ij=-ji
  9. 𝒬 Hur \mathcal{Q}_{\mathrm{Hur}}
  10. 1 + I 𝒬 Hur 1+I\mathcal{Q}_{\mathrm{Hur}}
  11. η 2 + 3 η + 2 \eta^{2}+3\eta+2

Knot_(mathematics).html

  1. S j S^{j}
  2. S n S^{n}
  3. j = n - 2 j=n-2
  4. H 2 × R H^{2}\times R
  5. H 3 H^{3}
  6. M M
  7. N N
  8. N N
  9. M M
  10. N N
  11. M M
  12. N N
  13. N = S 1 N=S^{1}
  14. M = 3 M=\mathbb{R}^{3}
  15. M = S 3 M=S^{3}
  16. n n
  17. n + 1 n+1
  18. n n
  19. n n
  20. n + 1 n+1
  21. n 3 n\neq 3
  22. n = 3 n=3
  23. S n S^{n}
  24. 2 n - 3 j - 3 > 0 2n-3j-3>0
  25. n > j 1 n>j\geq 1
  26. 2 n - 3 j - 3 = 0 2n-3j-3=0
  27. n - j n-j
  28. S j S^{j}
  29. S n S^{n}

Knudsen_number.html

  1. Kn = λ L \mathrm{Kn}=\frac{\lambda}{L}
  2. λ \lambda
  3. L L
  4. Kn = k B T 2 π d 2 p L \mathrm{Kn}=\frac{k_{B}T}{\sqrt{2}\pi d^{2}pL}
  5. k B k_{B}
  6. T T
  7. d d
  8. p p
  9. λ \lambda
  10. μ = 1 2 ρ c ¯ λ . \mu=\frac{1}{2}\rho\bar{c}\lambda.
  11. c ¯ = 8 k B T π m \bar{c}=\sqrt{\frac{8k_{B}T}{\pi m}}
  12. λ = μ ρ π m 2 k B T \lambda=\frac{\mu}{\rho}\sqrt{\frac{\pi m}{2k_{B}T}}
  13. λ L = μ ρ L π m 2 k B T \frac{\lambda}{L}=\frac{\mu}{\rho L}\sqrt{\frac{\pi m}{2k_{B}T}}
  14. c ¯ \bar{c}
  15. Ma = U c s \mathrm{Ma}=\frac{U_{\infty}}{c_{s}}
  16. c s = γ R T M = γ k B T m c_{s}=\sqrt{\frac{\gamma RT}{M}}=\sqrt{\frac{\gamma k_{B}T}{m}}
  17. γ \gamma
  18. Re = ρ U L μ . \mathrm{Re}=\frac{\rho U_{\infty}L}{\mu}.
  19. Ma Re = U / c s ρ U L / μ = μ ρ L c s = μ ρ L γ k B T m = μ ρ L m γ k B T \frac{\mathrm{Ma}}{\mathrm{Re}}=\frac{U_{\infty}/c_{s}}{\rho U_{\infty}L/\mu}=% \frac{\mu}{\rho Lc_{s}}=\frac{\mu}{\rho L\sqrt{\frac{\gamma k_{B}T}{m}}}=\frac% {\mu}{\rho L}\sqrt{\frac{m}{\gamma k_{B}T}}
  20. γ π 2 \sqrt{\frac{\gamma\pi}{2}}
  21. μ ρ L m γ k B T γ π 2 = μ ρ L π m 2 k B T = Kn \frac{\mu}{\rho L}\sqrt{\frac{m}{\gamma k_{B}T}}\sqrt{\frac{\gamma\pi}{2}}=% \frac{\mu}{\rho L}\sqrt{\frac{\pi m}{2k_{B}T}}=\mathrm{Kn}
  22. Kn = Ma Re γ π 2 . \mathrm{Kn}=\frac{\mathrm{Ma}}{\mathrm{Re}}\;\sqrt{\frac{\gamma\pi}{2}}.

Knuth–Bendix_completion_algorithm.html

  1. M = X R M=\langle X\mid R\rangle
  2. X R \langle X\mid R\rangle
  3. X X
  4. R R
  5. < <
  6. X X
  7. P i = Q i P_{i}=Q_{i}
  8. R R
  9. Q i < P i Q_{i}<P_{i}
  10. P i Q i P_{i}\rightarrow Q_{i}
  11. P i = Q i P_{i}=Q_{i}
  12. P i P_{i}
  13. Q i Q_{i}
  14. P i P_{i}
  15. P j P_{j}
  16. i j i\neq j
  17. P i P_{i}
  18. P j P_{j}
  19. P i = B C P_{i}=BC
  20. P j = A B P_{j}=AB
  21. P i = A B P_{i}=AB
  22. P j = B C P_{j}=BC
  23. P i P_{i}
  24. P j P_{j}
  25. P i = B P_{i}=B
  26. P j = A B C P_{j}=ABC
  27. P i = A B C P_{i}=ABC
  28. P j = B P_{j}=B
  29. A B C ABC
  30. P i P_{i}
  31. P j P_{j}
  32. r 1 , r 2 r_{1},r_{2}
  33. r 1 r 2 r_{1}\neq r_{2}
  34. max r 1 , r 2 min r 1 , r 2 \max r_{1},r_{2}\rightarrow\min r_{1},r_{2}
  35. R R
  36. R R
  37. R R
  38. x , y x 3 = y 3 = ( x y ) 3 = 1 \langle x,y\mid x^{3}=y^{3}=(xy)^{3}=1\rangle
  39. x 3 1 x^{3}\rightarrow 1
  40. y 3 1 y^{3}\rightarrow 1
  41. ( x y ) 3 1 (xy)^{3}\rightarrow 1
  42. x 3 y x y x y x^{3}yxyxy
  43. y x y x y yxyxy
  44. x 2 x^{2}
  45. y x y x y = x 2 yxyxy=x^{2}
  46. y x y x y x 2 yxyxy\rightarrow x^{2}
  47. x y x y x y 3 xyxyxy^{3}
  48. x y x y x = y 2 xyxyx=y^{2}
  49. x y x y x y 2 xyxyx\rightarrow y^{2}
  50. x 3 y x y x x^{3}yxyx
  51. y x y x = x 2 y 2 yxyx=x^{2}y^{2}
  52. y x y x x 2 y 2 yxyx\rightarrow x^{2}y^{2}
  53. x y x y x 3 xyxyx^{3}
  54. x y x y = y 2 x 2 xyxy=y^{2}x^{2}
  55. y 2 x 2 x y x y y^{2}x^{2}\rightarrow xyxy
  56. x 3 1 x^{3}\rightarrow 1
  57. y 3 1 y^{3}\rightarrow 1
  58. y x y x x 2 y 2 yxyx\rightarrow x^{2}y^{2}
  59. y 2 x 2 x y x y y^{2}x^{2}\rightarrow xyxy
  60. x , y , x - 1 , y - 1 | x y = y x , x x - 1 = x - 1 x = y y - 1 = y - 1 y = 1 . \langle x,y,x^{-1},y^{-1}\,|\,xy=yx,xx^{-1}=x^{-1}x=yy^{-1}=y^{-1}y=1\rangle.
  61. x < x - 1 < y < y - 1 x<x^{-1}<y<y^{-1}
  62. x < y < x - 1 < y - 1 x<y<x^{-1}<y^{-1}

Kondo_effect.html

  1. ρ ( T ) = ρ 0 + a T 2 + c m ln μ T + b T 5 , \rho(T)=\rho_{0}+aT^{2}+c_{m}\ln\frac{\mu}{T}+bT^{5},

Korteweg–de_Vries_equation.html

  1. ϕ \phi
  2. t ϕ + x 3 ϕ + 6 ϕ x ϕ = 0 \partial_{t}\phi+\partial^{3}_{x}\phi+6\,\phi\,\partial_{x}\phi=0\,
  3. ϕ \phi
  4. ϕ ( x , t ) = ϕ ( x ± t ) \phi(x,t)=\phi(x\pm t)
  5. ± x ϕ + x 3 ϕ + 6 ϕ x ϕ = 0 \pm\partial_{x}\phi+\partial^{3}_{x}\phi+6\,\phi\,\partial_{x}\phi=0\,
  6. ± x ϕ + x ( x 2 ϕ + 3 ϕ 2 ) = 0 \pm\partial_{x}\phi+\partial_{x}(\partial^{2}_{x}\phi+3\phi^{2})=0\,
  7. - x 2 ϕ - 3 ϕ 2 = ± ϕ -\partial^{2}_{x}\phi-3\phi^{2}=\pm\phi\,
  8. ( λ = 1 ) (\lambda=1)
  9. - x 2 ϕ - 3 ϕ λ ϕ = ± ϕ -\partial^{2}_{x}\phi-3\phi^{\lambda}\phi=\pm\phi\,
  10. λ = 4 \lambda=4
  11. λ = 2 \lambda=2
  12. 3 3
  13. ϕ \phi
  14. ϕ \phi
  15. - c d f d X + d 3 f d X 3 + 6 f d f d X = 0 , -c\frac{df}{dX}+\frac{d^{3}f}{dX^{3}}+6f\frac{df}{dX}=0,
  16. - c f + d 2 f d X 2 + 3 f 2 = A -cf+\frac{d^{2}f}{dX^{2}}+3f^{2}=A
  17. ϕ ( x , t ) = 1 2 c sech 2 [ c 2 ( x - c t - a ) ] \phi(x,t)=\frac{1}{2}\,c\,\mathrm{sech}^{2}\left[{\sqrt{c}\over 2}(x-c\,t-a)\right]
  18. - + P 2 n - 1 ( ϕ , x ϕ , x 2 ϕ , ) d x \int_{-\infty}^{+\infty}P_{2n-1}(\phi,\,\partial_{x}\phi,\,\partial_{x}^{2}% \phi,\,\ldots)\,\,\text{d}x\,
  19. P 1 = ϕ , P n = - d P n - 1 d x + i = 1 n - 2 P i P n - 1 - i for n 2. \begin{aligned}\displaystyle P_{1}&\displaystyle=\phi,\\ \displaystyle P_{n}&\displaystyle=-\frac{dP_{n-1}}{dx}+\sum_{i=1}^{n-2}\,P_{i}% \,P_{n-1-i}\quad\,\text{ for }n\geq 2.\end{aligned}
  20. ϕ d x , \int\phi\,\,\text{d}x,
  21. ϕ 2 d x , \int\phi^{2}\,\,\text{d}x,
  22. 1 3 ϕ 3 - ( x ϕ ) 2 d x . \int\frac{1}{3}\phi^{3}-\left(\partial_{x}\phi\right)^{2}\,\,\text{d}x.
  23. t ϕ = 6 ϕ x ϕ - x 3 ϕ \partial_{t}\phi=6\,\phi\,\partial_{x}\phi-\partial_{x}^{3}\phi
  24. L t = [ L , A ] L A - A L L_{t}=[L,A]\equiv LA-AL\,
  25. L \displaystyle L
  26. t ϕ - 6 ϕ x ϕ + x 3 ϕ = 0 , \partial_{t}\phi-6\phi\,\partial_{x}\phi+\partial_{x}^{3}\phi=0,\,
  27. \mathcal{L}\,
  28. = 1 2 x ψ t ψ + ( x ψ ) 3 - 1 2 ( x 2 ψ ) 2 ( 1 ) \mathcal{L}=\frac{1}{2}\partial_{x}\psi\,\partial_{t}\psi+\left(\partial_{x}% \psi\right)^{3}-\frac{1}{2}\left(\partial_{x}^{2}\psi\right)^{2}\quad\quad% \quad\quad(1)\,
  29. ϕ \phi
  30. ϕ = ψ x = x ψ . \phi=\frac{\partial\psi}{\partial x}=\partial_{x}\psi.\,
  31. μ μ ( ( μ μ ψ ) ) - μ ( ( μ ψ ) ) + ψ = 0. ( 2 ) \partial_{\mu\mu}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{\mu\mu}% \psi)}\right)-\partial_{\mu}\left(\frac{\partial\mathcal{L}}{\partial(\partial% _{\mu}\psi)}\right)+\frac{\partial\mathcal{L}}{\partial\psi}=0.\quad\quad\quad% \quad\quad\quad\quad(2)\,
  32. \partial
  33. μ \mu
  34. μ \mu
  35. t t ( ( t t ψ ) ) + x x ( ( x x ψ ) ) - t ( ( t ψ ) ) - x ( ( x ψ ) ) + ψ = 0. ( 3 ) \partial_{tt}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{tt}\psi)}% \right)+\partial_{xx}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{xx}% \psi)}\right)-\partial_{t}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{% t}\psi)}\right)-\partial_{x}\left(\frac{\partial\mathcal{L}}{\partial(\partial% _{x}\psi)}\right)+\frac{\partial\mathcal{L}}{\partial\psi}=0.\quad\quad(3)\,
  36. t t ( ( t t ψ ) ) = 0 \partial_{tt}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{tt}\psi)}% \right)=0\,
  37. x x ( ( x x ψ ) ) = x x ( - x x ψ ) \partial_{xx}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{xx}\psi)}% \right)=\partial_{xx}\left(-\partial_{xx}\psi\right)\,
  38. t ( ( t ψ ) ) = t ( 1 2 x ψ ) \partial_{t}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{t}\psi)}\right% )=\partial_{t}\left(\frac{1}{2}\partial_{x}\psi\right)\,
  39. x ( ( x ψ ) ) = x ( 1 2 t ψ + 3 ( x ψ ) 2 ) \partial_{x}\left(\frac{\partial\mathcal{L}}{\partial(\partial_{x}\psi)}\right% )=\partial_{x}\left(\frac{1}{2}\partial_{t}\psi+3(\partial_{x}\psi)^{2}\right)\,
  40. ψ = 0 \frac{\partial\mathcal{L}}{\partial\psi}=0\,
  41. ϕ = x ψ \phi=\partial_{x}\psi\,
  42. x x ( - x x ψ ) = - x x x ϕ \partial_{xx}\left(-\partial_{xx}\psi\right)=-\partial_{xxx}\phi\,
  43. t ( 1 2 x ψ ) = 1 2 t ϕ \partial_{t}\left(\frac{1}{2}\partial_{x}\psi\right)=\frac{1}{2}\partial_{t}\phi\,
  44. x ( 1 2 t ψ + 3 ( x ψ ) 2 ) = 1 2 t ϕ + 3 x ( ϕ ) 2 = 1 2 t ϕ + 6 ϕ x ϕ \partial_{x}\left(\frac{1}{2}\partial_{t}\psi+3(\partial_{x}\psi)^{2}\right)=% \frac{1}{2}\partial_{t}\phi+3\partial_{x}(\phi)^{2}=\frac{1}{2}\partial_{t}% \phi+6\phi\partial_{x}\phi\,
  45. ( - x x x ϕ ) - ( 1 2 t ϕ ) - ( 1 2 t ϕ + 6 ϕ x ϕ ) = 0 , \left(-\partial_{xxx}\phi\right)-\left(\frac{1}{2}\partial_{t}\phi\right)-% \left(\frac{1}{2}\partial_{t}\phi+6\phi\partial_{x}\phi\right)=0,\,
  46. t ϕ + 6 ϕ x ϕ + x 3 ϕ = 0. \partial_{t}\phi+6\phi\,\partial_{x}\phi+\partial_{x}^{3}\phi=0.\,
  47. t ϕ + x 3 ϕ + 6 ϕ x ϕ = 0 \displaystyle\partial_{t}\phi+\partial^{3}_{x}\phi+6\,\phi\,\partial_{x}\phi=0
  48. t u + x 3 u - 6 u x u + u / 2 t = 0 \displaystyle\partial_{t}u+\partial_{x}^{3}u-6\,u\,\partial_{x}u+u/2t=0
  49. t u + x ( x 2 u - 2 η u 3 - 3 u ( x u ) 2 / 2 ( η + u 2 ) ) = 0 \displaystyle\partial_{t}u+\partial_{x}(\partial_{x}^{2}u-2\,\eta\,u^{3}-3\,u% \,(\partial_{x}u)^{2}/2(\eta+u^{2}))=0
  50. t u + x 3 u = x 5 u \displaystyle\partial_{t}u+\partial_{x}^{3}u=\partial_{x}^{5}u
  51. t u + x 3 u + x f ( u ) = 0 \displaystyle\partial_{t}u+\partial_{x}^{3}u+\partial_{x}f(u)=0
  52. t u + x { 35 u 4 + 70 ( u 2 x 2 u + u ( x u ) 2 ) + 7 [ 2 u x 4 u + 3 ( x 2 u ) 2 + 4 x x 3 u ] + x 6 u } = 0 \begin{aligned}\displaystyle\partial_{t}u+\partial_{x}&\displaystyle\left\{35u% ^{4}+70\left(u^{2}\partial_{x}^{2}u+u\left(\partial_{x}u\right)^{2}\right)% \right.\\ &\displaystyle\left.\quad+7\left[2u\partial_{x}^{4}u+3\left(\partial_{x}^{2}u% \right)^{2}+4\partial_{x}\partial_{x}^{3}u\right]+\partial_{x}^{6}u\right\}=0% \end{aligned}
  53. t u + x 3 u ± 6 u 2 x u = 0 \displaystyle\partial_{t}u+\partial_{x}^{3}u\pm 6\,u^{2}\,\partial_{x}u=0
  54. t u + x 3 u - ( x u ) 3 / 8 + ( x u ) ( A e a u + B + C e - a u ) = 0 \displaystyle\partial_{t}u+\partial_{x}^{3}u-(\partial_{x}u)^{3}/8+(\partial_{% x}u)(Ae^{au}+B+Ce^{-au})=0
  55. t u + x 3 u - 6 u x u + u / t = 0 \displaystyle\partial_{t}u+\partial_{x}^{3}u-6\,u\,\partial_{x}u+u/t=0
  56. t u = 6 u x u - x 3 u + 3 w x 2 w \displaystyle\partial_{t}u=6\,u\,\partial_{x}u-\partial_{x}^{3}u+3\,w\,% \partial_{x}^{2}w
  57. t w = 3 ( x u ) w + 6 u x w - 4 x 3 w \displaystyle\partial_{t}w=3\,(\partial_{x}u)\,w+6\,u\,\partial_{x}w-4\,% \partial_{x}^{3}w
  58. t u + x 3 u - 6 f ( t ) u x u = 0 \displaystyle\partial_{t}u+\partial_{x}^{3}u-6\,f(t)\,u\,\partial_{x}u=0
  59. t u + β t n x 3 u + α t n u x u = 0 \displaystyle\partial_{t}u+\beta\,t^{n}\,\partial_{x}^{3}u+\alpha\,t^{n}u\,% \partial_{x}u=0
  60. t u + μ x 3 u + 2 u x u - ν x 2 u = 0 \displaystyle\partial_{t}u+\mu\,\partial_{x}^{3}u+2\,u\,\partial_{x}u-\nu\,% \partial_{x}^{2}u=0

Koszul_complex.html

  1. 0 R 𝑥 R 0. 0\to R\xrightarrow{\ x\ }R\to 0.
  2. e i 1 i p e_{i_{1}...i_{p}}
  3. \mapsto
  4. d ( e i 1 i p ) := j = 1 p ( - 1 ) j - 1 x i j e i 1 i j ^ i p . d(e_{i_{1}...i_{p}}):=\sum_{j=1}^{p}(-1)^{j-1}x_{i_{j}}e_{i_{1}...\widehat{i_{% j}}...i_{p}}.
  5. 0 R d 2 R 2 d 1 R 0 , 0\to R\xrightarrow{\ d_{2}\ }R^{2}\xrightarrow{\ d_{1}\ }R\to 0,
  6. d 1 d_{1}
  7. d 2 d_{2}
  8. d 1 = [ x y ] d_{1}=\begin{bmatrix}x&y\\ \end{bmatrix}
  9. d 2 = [ - y x ] . d_{2}=\begin{bmatrix}-y\\ x\\ \end{bmatrix}.

Kriging.html

  1. N N
  2. x 1 x_{1}
  3. z ( x 1 ) z(x_{1})
  4. Z ( x 1 ) Z(x_{1})
  5. A A
  6. N N
  7. Z ( x 1 ) , Z ( x 2 ) , , Z ( x N ) Z(x_{1}),Z(x_{2}),\cdots,Z(x_{N})
  8. z ( x i ) z(x_{i})
  9. A A
  10. C ( Z ( x 1 ) , Z ( x 2 ) ) = C ( Z ( x i ) , Z ( x i + 𝐡 ) ) = C ( 𝐡 ) C(Z(x_{1}),Z(x_{2}))=C(Z(x_{i}),Z(x_{i}+\mathbf{h}))=C(\mathbf{h})
  11. γ ( Z ( x 1 ) , Z ( x 2 ) ) = γ ( Z ( x i ) , Z ( x i + 𝐡 ) ) = γ ( 𝐡 ) \gamma(Z(x_{1}),Z(x_{2}))=\gamma(Z(x_{i}),Z(x_{i}+\mathbf{h}))=\gamma(\mathbf{% h})
  12. 𝐡 = ( x 1 , x 2 ) = ( x i , x i + 𝐡 ) \mathbf{h}=(x_{1},x_{2})=(x_{i},x_{i}+\mathbf{h})
  13. N N
  14. γ ( 𝐡 ) = 1 2 N ( 𝐡 ) i = 1 N ( 𝐡 ) ( Z ( x i ) - Z ( x i + 𝐡 ) ) 2 \gamma(\mathbf{h})=\frac{1}{2N(\mathbf{h})}\sum^{N(\mathbf{h})}_{i=1}\left(Z(x% _{i})-Z(x_{i}+\mathbf{h})\right)^{2}
  15. C ( 𝐡 ) = 1 N ( 𝐡 ) i = 1 N ( 𝐡 ) ( Z ( x i ) Z ( x i + 𝐡 ) ) - m ( x i ) m ( x i + 𝐡 ) C(\mathbf{h})=\frac{1}{N(\mathbf{h})}\sum^{N(\mathbf{h})}_{i=1}\left(Z(x_{i})Z% (x_{i}+\mathbf{h})\right)-m(x_{i})m(x_{i}+\mathbf{h})
  16. m ( x i ) = 1 N ( 𝐡 ) i = 1 N ( 𝐡 ) Z ( x i ) m(x_{i})=\frac{1}{N(\mathbf{h})}\sum^{N(\mathbf{h})}_{i=1}Z(x_{i})
  17. Z : n Z:\mathbb{R}^{n}\rightarrow\mathbb{R}
  18. x 0 x_{0}
  19. z i = Z ( x i ) z_{i}=Z(x_{i})
  20. w i ( x 0 ) , i = 1 , , N w_{i}(x_{0}),\;i=1,\ldots,N
  21. Z ^ ( x 0 ) = [ w 1 w 2 w N ] [ z 1 z 2 z N ] = i = 1 n w i ( x 0 ) × Z ( x i ) \hat{Z}(x_{0})=\begin{bmatrix}w_{1}&w_{2}&\cdots&w_{N}\end{bmatrix}\cdot\begin% {bmatrix}z_{1}\\ z_{2}\\ \vdots\\ z_{N}\end{bmatrix}=\sum_{i=1}^{n}w_{i}(x_{0})\times Z(x_{i})
  22. w i w_{i}
  23. x 0 x_{0}
  24. w i w_{i}
  25. Z ( x 0 ) Z(x_{0})
  26. Z ^ ( x 0 ) \hat{Z}(x_{0})
  27. ( Z ^ ( x ) - Z ( x ) ) (\hat{Z}(x)-Z(x))
  28. E { Z ( x ) } = E { Z ( x 0 ) } = m E\{Z(x)\}=E\{Z(x_{0})\}=m
  29. m m
  30. x 0 x_{0}
  31. E { Z ( x ) } = k = 0 p β k f k ( x ) E\{Z(x)\}=\sum_{k=0}^{p}\beta_{k}f_{k}(x)
  32. E { Z ( x ) } E\{Z(x)\}
  33. x x
  34. Z ( x 0 ) Z(x_{0})
  35. x 0 x_{0}
  36. Z ( x i ) , i = 1 , , N Z(x_{i}),i=1,\cdots,N
  37. Z ^ ( x 0 ) \hat{Z}(x_{0})
  38. x 0 x_{0}
  39. Z ( x ) Z(x)
  40. x 0 x_{0}
  41. ϵ ( x 0 ) = Z ^ ( x 0 ) - Z ( x 0 ) = [ W T - 1 ] [ Z ( x i ) Z ( x N ) Z ( x 0 ) ] T = i = 1 N w i ( x 0 ) × Z ( x i ) - Z ( x 0 ) \epsilon(x_{0})=\hat{Z}(x_{0})-Z(x_{0})=\begin{bmatrix}W^{T}&-1\end{bmatrix}% \cdot\begin{bmatrix}Z(x_{i})&\cdots&Z(x_{N})&Z(x_{0})\end{bmatrix}^{T}=\sum^{N% }_{i=1}w_{i}(x_{0})\times Z(x_{i})-Z(x_{0})
  42. ϵ ( x 0 ) \epsilon(x_{0})
  43. E ( Z ( x i ) ) = E ( Z ( x 0 ) ) = m E(Z(x_{i}))=E(Z(x_{0}))=m
  44. E ( ϵ ( x 0 ) ) = 0 i = 1 N w i ( x 0 ) × E ( Z ( x i ) ) - E ( Z ( x 0 ) ) = 0 E\left(\epsilon(x_{0})\right)=0\Leftrightarrow\sum^{N}_{i=1}w_{i}(x_{0})\times E% (Z(x_{i}))-E(Z(x_{0}))=0\Leftrightarrow
  45. m i = 1 N w i ( x 0 ) - m = 0 i = 1 N w i ( x 0 ) = 1 𝟏 T W = 1 \Leftrightarrow m\sum^{N}_{i=1}w_{i}(x_{0})-m=0\Leftrightarrow\sum^{N}_{i=1}w_% {i}(x_{0})=1\Leftrightarrow\mathbf{1}^{T}\cdot W=1
  46. E [ ϵ ( x 0 ) ] = 0 E\left[\epsilon(x_{0})\right]=0
  47. E ( ϵ ( x 0 ) 2 ) E\left(\epsilon(x_{0})^{2}\right)
  48. V a r ( ϵ ( x 0 ) ) = V a r ( [ W T - 1 ] [ Z ( x i ) Z ( x N ) Z ( x 0 ) ] T ) = = * [ W T - 1 ] V a r ( [ Z ( x i ) Z ( x N ) Z ( x 0 ) ] T ) [ W - 1 ] \begin{array}[]{rl}Var(\epsilon(x_{0}))&=Var\left(\begin{bmatrix}W^{T}&-1\end{% bmatrix}\cdot\begin{bmatrix}Z(x_{i})&\cdots&Z(x_{N})&Z(x_{0})\end{bmatrix}^{T}% \right)=\\ &\overset{*}{=}\begin{bmatrix}W^{T}&-1\end{bmatrix}\cdot Var\left(\begin{% bmatrix}Z(x_{i})&\cdots&Z(x_{N})&Z(x_{0})\end{bmatrix}^{T}\right)\cdot\begin{% bmatrix}W\\ -1\end{bmatrix}\end{array}
  49. V a r ( ϵ ( x 0 ) ) = * [ W T - 1 ] [ V a r x i C o v x i x 0 C o v x i x 0 T V a r x 0 ] [ W - 1 ] Var(\epsilon(x_{0}))\overset{*}{=}\begin{bmatrix}W^{T}&-1\end{bmatrix}\cdot% \begin{bmatrix}Var_{x_{i}}&Cov_{x_{i}x_{0}}\\ Cov_{x_{i}x_{0}}^{T}&Var_{x_{0}}\end{bmatrix}\cdot\begin{bmatrix}W\\ -1\end{bmatrix}
  50. { V a r x i , V a r x 0 , C o v x i x 0 } \left\{Var_{x_{i}},Var_{x_{0}},Cov_{x_{i}x_{0}}\right\}
  51. { V a r ( [ Z ( x 1 ) Z ( x N ) ] T ) , V a r ( Z ( x 0 ) ) , C o v ( [ Z ( x 1 ) Z ( x N ) ] T , Z ( x 0 ) ) } \left\{Var\left(\begin{bmatrix}Z(x_{1})&\cdots&Z(x_{N})\end{bmatrix}^{T}\right% ),Var\left(Z(x_{0})\right),Cov\left(\begin{bmatrix}Z(x_{1})&\cdots&Z(x_{N})% \end{bmatrix}^{T},Z(x_{0})\right)\right\}
  52. C ( 𝐡 ) C(\mathbf{h})
  53. γ ( 𝐡 ) \gamma(\mathbf{h})
  54. Z ( x ) Z(x)
  55. { V a r ( ϵ ( x 0 ) ) = W T V a r x i W - C o v x i x 0 T W - W T C o v x i x 0 + V a r x 0 V a r ( ϵ ( x 0 ) ) = C o v ( 0 ) + i j w i w j C o v ( x i , x j ) - 2 i w i C ( x i , x 0 ) \left\{\begin{array}[]{l}Var(\epsilon(x_{0}))=W^{T}\cdot Var_{x_{i}}\cdot W-% Cov_{x_{i}x_{0}}^{T}\cdot W-W^{T}\cdot Cov_{x_{i}x_{0}}+Var_{x_{0}}\\ Var(\epsilon(x_{0}))=Cov(0)+\sum_{i}\sum_{j}w_{i}w_{j}Cov(x_{i},x_{j})-2\sum_{% i}w_{i}C(x_{i},x_{0})\end{array}\right.
  56. C ( x i , x j ) C(x_{i},x_{j})
  57. x 0 x_{0}
  58. C ( 0 ) C(0)
  59. Z ( x ) Z(x)
  60. A A
  61. A A
  62. minimize 𝑊 W T V a r x i W - C o v x i x 0 T W - W T C o v x i x 0 + V a r x 0 subject to 𝟏 T W = 1 \begin{aligned}&\displaystyle\underset{W}{\operatorname{minimize}}&&% \displaystyle W^{T}\cdot Var_{x_{i}}\cdot W-Cov_{x_{i}x_{0}}^{T}\cdot W-W^{T}% \cdot Cov_{x_{i}x_{0}}+Var_{x_{0}}\\ &\displaystyle\operatorname{subject\;to}&&\displaystyle\mathbf{1}^{T}\cdot W=1% \end{aligned}
  63. [ W ^ μ ] = [ V a r x i 𝟏 𝟏 T 0 ] - 1 [ C o v x i x 0 1 ] = [ γ ( x 1 , x 1 ) γ ( x 1 , x n ) 1 γ ( x n , x 1 ) γ ( x n , x n ) 1 1 1 0 ] - 1 [ γ ( x 1 , x * ) γ ( x n , x * ) 1 ] \begin{bmatrix}\hat{W}\\ \mu\end{bmatrix}=\begin{bmatrix}Var_{x_{i}}&\mathbf{1}\\ \mathbf{1}^{T}&0\end{bmatrix}^{-1}\cdot\begin{bmatrix}Cov_{x_{i}x_{0}}\\ 1\end{bmatrix}=\begin{bmatrix}\gamma(x_{1},x_{1})&\cdots&\gamma(x_{1},x_{n})&1% \\ \vdots&\ddots&\vdots&\vdots\\ \gamma(x_{n},x_{1})&\cdots&\gamma(x_{n},x_{n})&1\\ 1&\cdots&1&0\end{bmatrix}^{-1}\begin{bmatrix}\gamma(x_{1},x^{*})\\ \vdots\\ \gamma(x_{n},x^{*})\\ 1\end{bmatrix}
  64. μ \mu
  65. σ k 2 ( x ) \sigma_{k}^{2}(x)
  66. μ ( x ) = 0 \mu(x)=0
  67. c ( x , y ) = Cov ( Z ( x ) , Z ( y ) ) c(x,y)=\mathrm{Cov}(Z(x),Z(y))
  68. ( w 1 w n ) = ( c ( x 1 , x 1 ) c ( x 1 , x n ) c ( x n , x 1 ) c ( x n , x n ) ) - 1 ( c ( x 1 , x 0 ) c ( x n , x 0 ) ) \begin{pmatrix}w_{1}\\ \vdots\\ w_{n}\end{pmatrix}=\begin{pmatrix}c(x_{1},x_{1})&\cdots&c(x_{1},x_{n})\\ \vdots&\ddots&\vdots\\ c(x_{n},x_{1})&\cdots&c(x_{n},x_{n})\end{pmatrix}^{-1}\begin{pmatrix}c(x_{1},x% _{0})\\ \vdots\\ c(x_{n},x_{0})\end{pmatrix}
  69. Z ( x 0 ) Z(x_{0})
  70. z 1 , , z n z_{1},\ldots,z_{n}
  71. Z ^ ( x 0 ) = ( z 1 z n ) ( c ( x 1 , x 1 ) c ( x 1 , x n ) c ( x n , x 1 ) c ( x n , x n ) ) - 1 ( c ( x 1 , x 0 ) c ( x n , x 0 ) ) \hat{Z}(x_{0})=\begin{pmatrix}z_{1}\\ \vdots\\ z_{n}\end{pmatrix}^{\prime}\begin{pmatrix}c(x_{1},x_{1})&\cdots&c(x_{1},x_{n})% \\ \vdots&\ddots&\vdots\\ c(x_{n},x_{1})&\cdots&c(x_{n},x_{n})\end{pmatrix}^{-1}\begin{pmatrix}c(x_{1},x% _{0})\\ \vdots\\ c(x_{n},x_{0})\end{pmatrix}
  72. Var ( Z ^ ( x 0 ) - Z ( x 0 ) ) = c ( x 0 , x 0 ) Var ( Z ( x 0 ) ) - ( c ( x 1 , x 0 ) c ( x n , x 0 ) ) ( c ( x 1 , x 1 ) c ( x 1 , x n ) c ( x n , x 1 ) c ( x n , x n ) ) - 1 ( c ( x 1 , x 0 ) c ( x n , x 0 ) ) Var ( Z ^ ( x 0 ) ) \mathrm{Var}\left(\hat{Z}(x_{0})-Z(x_{0})\right)=\underbrace{c(x_{0},x_{0})}_{% \mathrm{Var}(Z(x_{0}))}-\underbrace{\begin{pmatrix}c(x_{1},x_{0})\\ \vdots\\ c(x_{n},x_{0})\end{pmatrix}^{\prime}\begin{pmatrix}c(x_{1},x_{1})&\cdots&c(x_{% 1},x_{n})\\ \vdots&\ddots&\vdots\\ c(x_{n},x_{1})&\cdots&c(x_{n},x_{n})\end{pmatrix}^{-1}\begin{pmatrix}c(x_{1},x% _{0})\\ \vdots\\ c(x_{n},x_{0})\end{pmatrix}}_{\mathrm{Var}(\hat{Z}(x_{0}))}
  73. Var ( Z ( x 0 ) ) = Var ( Z ^ ( x 0 ) ) + Var ( Z ^ ( x 0 ) - Z ( x 0 ) ) . \mathrm{Var}(Z(x_{0}))=\mathrm{Var}(\hat{Z}(x_{0}))+\mathrm{Var}\left(\hat{Z}(% x_{0})-Z(x_{0})\right).
  74. E [ Z ^ ( x i ) ] = E [ Z ( x i ) ] E[\hat{Z}(x_{i})]=E[Z(x_{i})]
  75. Z ^ ( x i ) = Z ( x i ) \hat{Z}(x_{i})=Z(x_{i})
  76. Z ^ ( x ) \hat{Z}(x)
  77. Z ( x ) Z(x)
  78. σ k 2 \sigma_{k}^{2}

Kronecker's_theorem.html

  1. α i = ( α i 1 , , α i n ) n , i = 1 , , m \alpha_{i}=(\alpha_{i_{1}},\cdots,\alpha_{i_{n}})\in\mathbb{R}^{n},i=1,\cdots,m
  2. β j = ( β 1 , , β n ) n \beta_{j}=(\beta_{1},\cdots,\beta_{n})\in\mathbb{R}^{n}
  3. ϵ > 0 \epsilon>0
  4. p i p_{i}
  5. q j q_{j}
  6. | i = 1 m q i α i j - p j - β j | < ϵ , 1 j n \biggl|\sum^{m}_{i=1}q_{i}\alpha_{ij}-p_{j}-\beta_{j}\biggr|<\epsilon,\ \ \ \ % 1\leq j\leq n
  7. r 1 , , r n , i = 1 , , m r_{1},\dots,r_{n}\in\mathbb{Z},\ i=1,\dots,m
  8. j = 1 n α i j r j , i = 1 , , m , \sum^{n}_{j=1}\alpha_{ij}r_{j}\in\mathbb{Z},\ \ i=1,\dots,m\ ,
  9. j = 1 n β j r j \sum^{n}_{j=1}\beta_{j}r_{j}

Kullback–Leibler_divergence.html

  1. D KL ( P Q ) = i P ( i ) ln P ( i ) Q ( i ) . D_{\mathrm{KL}}(P\|Q)=\sum_{i}P(i)\,\ln\frac{P(i)}{Q(i)}.
  2. lim x 0 x ln ( x ) = 0 \lim_{x\to 0}x\ln(x)=0
  3. D KL ( P Q ) = - p ( x ) ln p ( x ) q ( x ) d x , D_{\mathrm{KL}}(P\|Q)=\int_{-\infty}^{\infty}p(x)\,\ln\frac{p(x)}{q(x)}\,{\rm d% }x,\!
  4. D KL ( P Q ) = X ln d P d Q d P , D_{\mathrm{KL}}(P\|Q)=\int_{X}\ln\frac{{\rm d}P}{{\rm d}Q}\,{\rm d}P,\!
  5. d P d Q \frac{{\rm d}P}{{\rm d}Q}
  6. D KL ( P Q ) = X ln ( d P d Q ) d P d Q d Q , D_{\mathrm{KL}}(P\|Q)=\int_{X}\ln\!\left(\frac{{\rm d}P}{{\rm d}Q}\right)\frac% {{\rm d}P}{{\rm d}Q}\,{\rm d}Q,
  7. μ \mu
  8. p = d P d μ p=\frac{{\rm d}P}{{\rm d}\mu}
  9. q = d Q d μ q=\frac{{\rm d}Q}{{\rm d}\mu}
  10. μ \mu
  11. D KL ( P Q ) = X p ln p q d μ . D_{\mathrm{KL}}(P\|Q)=\int_{X}p\,\ln\frac{p}{q}\,{\rm d}\mu.\!
  12. x i x_{i}
  13. X X
  14. q ( x i ) = 2 - l i q(x_{i})=2^{-l_{i}}
  15. X X
  16. l i l_{i}
  17. x i x_{i}
  18. Q Q
  19. P P
  20. D KL ( P Q ) = - x p ( x ) log q ( x ) + x p ( x ) log p ( x ) = H ( P , Q ) - H ( P ) \begin{matrix}D_{\mathrm{KL}}(P\|Q)&=&-\sum_{x}p(x)\log q(x)&+&\sum_{x}p(x)% \log p(x)\\ &=&H(P,Q)&-&H(P)\end{matrix}
  21. D KL ( P Q ) 0 , D_{\mathrm{KL}}(P\|Q)\geq 0,\,
  22. D KL ( P Q ) = x a x b P ( x ) log ( P ( x ) Q ( x ) ) d x = y a y b P ( y ) log ( P ( y ) d y / d x Q ( y ) d y / d x ) d y = y a y b P ( y ) log ( P ( y ) Q ( y ) ) d y D_{\mathrm{KL}}(P\|Q)=\int_{x_{a}}^{x_{b}}P(x)\log\left(\frac{P(x)}{Q(x)}% \right)\,dx=\int_{y_{a}}^{y_{b}}P(y)\log\left(\frac{P(y)dy/dx}{Q(y)dy/dx}% \right)\,dy=\int_{y_{a}}^{y_{b}}P(y)\log\left(\frac{P(y)}{Q(y)}\right)\,dy
  23. y a = y ( x a ) y_{a}=y(x_{a})
  24. y b = y ( x b ) y_{b}=y(x_{b})
  25. P 1 , P 2 P_{1},P_{2}
  26. P ( x , y ) = P 1 ( x ) P 2 ( y ) P(x,y)=P_{1}(x)P_{2}(y)
  27. Q , Q 1 , Q 2 Q,Q_{1},Q_{2}
  28. D KL ( P Q ) = D KL ( P 1 Q 1 ) + D KL ( P 2 Q 2 ) . D_{\mathrm{KL}}(P\|Q)=D_{\mathrm{KL}}(P_{1}\|Q_{1})+D_{\mathrm{KL}}(P_{2}\|Q_{% 2}).
  29. μ 0 , μ 1 \mu_{0},\mu_{1}
  30. Σ 0 , Σ 1 \Sigma_{0},\Sigma_{1}
  31. D KL ( 𝒩 0 𝒩 1 ) = 1 2 ( tr ( Σ 1 - 1 Σ 0 ) + ( μ 1 - μ 0 ) Σ 1 - 1 ( μ 1 - μ 0 ) - k + ln ( det Σ 1 det Σ 0 ) ) . D\text{KL}(\mathcal{N}_{0}\|\mathcal{N}_{1})={1\over 2}\left(\mathrm{tr}\left(% \Sigma_{1}^{-1}\Sigma_{0}\right)+\left(\mu_{1}-\mu_{0}\right)^{\top}\Sigma_{1}% ^{-1}(\mu_{1}-\mu_{0})-k+\ln\left({\det\Sigma_{1}\over\det\Sigma_{0}}\right)% \right).
  32. D KL ( P Q ) D KL ( Q P ) D_{\mathrm{KL}}(P\|Q)\neq D_{\mathrm{KL}}(Q\|P)
  33. { P 1 , P 2 , } \{P_{1},P_{2},\cdots\}
  34. lim n D KL ( P n Q ) = 0 \lim_{n\rightarrow\infty}D_{\mathrm{KL}}(P_{n}\|Q)=0
  35. P n 𝐷 Q P_{n}\xrightarrow{D}Q
  36. P n D P P n TV P P_{n}\xrightarrow{\mathrm{D}}P\Rightarrow P_{n}\xrightarrow{\mathrm{TV}}P
  37. θ \theta
  38. P = P ( θ ) P=P(\theta)
  39. Q = P ( θ 0 ) Q=P(\theta_{0})
  40. θ \theta
  41. θ 0 \theta_{0}
  42. P ( θ ) = P ( θ 0 ) + Δ θ j P j ( θ 0 ) + P(\theta)=P(\theta_{0})+\Delta\theta^{j}P_{j}(\theta_{0})+\cdots
  43. Δ θ j = ( θ - θ 0 ) j \Delta\theta^{j}=(\theta-\theta_{0})^{j}
  44. θ \theta
  45. P j ( θ 0 ) = P θ j ( θ 0 ) P_{j}(\theta_{0})=\frac{\partial P}{\partial\theta^{j}}(\theta_{0})
  46. θ = θ 0 \theta=\theta_{0}
  47. Δ θ j \Delta\theta^{j}
  48. θ j | θ = θ 0 D K L ( P ( θ ) P ( θ 0 ) ) = 0 , \left.\frac{\partial}{\partial\theta^{j}}\right|_{\theta=\theta_{0}}D_{KL}(P(% \theta)\|P(\theta_{0}))=0,
  49. D KL ( P ( θ ) P ( θ 0 ) ) = 1 2 Δ θ j Δ θ k g j k ( θ 0 ) + D_{\mathrm{KL}}(P(\theta)\|P(\theta_{0}))=\frac{1}{2}\Delta\theta^{j}\Delta% \theta^{k}g_{jk}(\theta_{0})+\cdots
  50. g j k ( θ 0 ) = 2 θ j θ k | θ = θ 0 D K L ( P ( θ ) P ( θ 0 ) ) g_{jk}(\theta_{0})=\left.\frac{\partial^{2}}{\partial\theta^{j}\partial\theta^% {k}}\right|_{\theta=\theta_{0}}D_{KL}(P(\theta)\|P(\theta_{0}))
  51. θ 0 \theta_{0}
  52. g j k ( θ ) g_{jk}(\theta)
  53. θ \theta
  54. I ( m ) = D KL ( δ i m { p i } ) , I(m)=D_{\mathrm{KL}}(\delta_{im}\|\{p_{i}\}),
  55. I ( X ; Y ) = D KL ( P ( X , Y ) P ( X ) P ( Y ) ) = E X { D KL ( P ( Y | X ) P ( Y ) ) } = E Y { D KL ( P ( X | Y ) P ( X ) ) } \begin{aligned}\displaystyle I(X;Y)&\displaystyle=D_{\mathrm{KL}}(P(X,Y)\|P(X)% P(Y))\\ &\displaystyle=\operatorname{E}_{X}\{D_{\mathrm{KL}}(P(Y|X)\|P(Y))\}\\ &\displaystyle=\operatorname{E}_{Y}\{D_{\mathrm{KL}}(P(X|Y)\|P(X))\}\end{aligned}
  56. H ( X ) = ( i ) E x { I ( x ) } = ( ii ) log N - D KL ( P ( X ) P U ( X ) ) \begin{aligned}\displaystyle H(X)&\displaystyle=\mathrm{(i)}\,\operatorname{E}% _{x}\{I(x)\}\\ &\displaystyle=\mathrm{(ii)}\log N-D_{\mathrm{KL}}(P(X)\|P_{U}(X))\end{aligned}
  57. H ( X Y ) = log N - D KL ( P ( X , Y ) P U ( X ) P ( Y ) ) = ( i ) log N - D KL ( P ( X , Y ) P ( X ) P ( Y ) ) - D KL ( P ( X ) P U ( X ) ) = H ( X ) - I ( X ; Y ) = ( ii ) log N - E Y { D KL ( P ( X | Y ) P U ( X ) ) } \begin{aligned}\displaystyle H(X\mid Y)&\displaystyle=\log N-D_{\mathrm{KL}}(P% (X,Y)\|P_{U}(X)P(Y))\\ &\displaystyle=\mathrm{(i)}\,\,\log N-D_{\mathrm{KL}}(P(X,Y)\|P(X)P(Y))-D_{% \mathrm{KL}}(P(X)\|P_{U}(X))\\ &\displaystyle=H(X)-I(X;Y)\\ &\displaystyle=\mathrm{(ii)}\,\log N-\operatorname{E}_{Y}\{D_{\mathrm{KL}}(P(X% |Y)\|P_{U}(X))\}\end{aligned}
  58. q q
  59. p p
  60. p p
  61. q q
  62. H ( p , q ) = E p [ - log q ] = H ( p ) + D KL ( p q ) . H(p,q)=\operatorname{E}_{p}[-\log q]=H(p)+D_{\mathrm{KL}}(p\|q).\!
  63. p ( x y , I ) = p ( y x , I ) p ( x I ) p ( y I ) p(x\mid y,I)=\frac{p(y\mid x,I)p(x\mid I)}{p(y\mid I)}
  64. H ( p ( y , I ) ) = x p ( x y , I ) log p ( x y , I ) , H\big(p(\cdot\mid y,I)\big)=\sum_{x}p(x\mid y,I)\log p(x\mid y,I),
  65. D KL ( p ( y , I ) p ( I ) ) = x p ( x y , I ) log p ( x y , I ) p ( x I ) D_{\mathrm{KL}}\big(p(\cdot\mid y,I)\mid p(\cdot\mid I)\big)=\sum_{x}p(x\mid y% ,I)\log\frac{p(x\mid y,I)}{p(x\mid I)}
  66. x p ( x y 1 , y 2 , I ) log p ( x y 1 , y 2 , I ) p ( x I ) \sum_{x}p(x\mid y_{1},y_{2},I)\log\frac{p(x\mid y_{1},y_{2},I)}{p(x\mid I)}
  67. x p ( x y 1 , I ) log p ( x y 1 , I ) p ( x I ) \displaystyle\sum_{x}p(x\mid y_{1},I)\log\frac{p(x\mid y_{1},I)}{p(x\mid I)}
  68. D KL ( p ( y 1 , y 2 , I ) p ( I ) ) D_{\mathrm{KL}}\big(p(\cdot\mid y_{1},y_{2},I)\big\|p(\cdot\mid I)\big)
  69. D KL ( p ( y 1 , y 2 , I ) p ( | y 1 , I ) ) + D KL ( p ( y 1 , I ) p ( x I ) ) D_{\mathrm{KL}}\big(p(\cdot\mid y_{1},y_{2},I)\big\|p(\cdot|y_{1},I)\big)+D_{% \mathrm{KL}}\big(p(\cdot\mid y_{1},I)\big\|p(x\mid I)\big)
  70. \neq
  71. D KL ( q ( x | a ) u ( a ) p ( x , a ) ) = E u ( a ) { D KL ( q ( x | a ) p ( x | a ) ) } + D KL ( u ( a ) p ( a ) ) , D_{\mathrm{KL}}(q(x|a)u(a)\|p(x,a))=\operatorname{E}_{u(a)}\{D_{\mathrm{KL}}(q% (x|a)\|p(x|a))\}+D_{\mathrm{KL}}(u(a)\|p(a)),
  72. H ( p , m ) = H ( p ) + D KL ( p m ) , H(p,m)=H(p)+D_{\mathrm{KL}}(p\|m),
  73. p p
  74. s = k ln ( 1 / p ) s=k\ln(1/p)
  75. k k
  76. { 1 , 1 / ln 2 , 1.38 × 10 - 23 } \{1,1/\ln 2,1.38\times 10^{-23}\}
  77. { \{
  78. J / K } J/K\}
  79. N N
  80. N N
  81. S S
  82. P P
  83. V V
  84. A - k ln Z A\equiv-k\ln Z
  85. Z Z
  86. T T
  87. T × A T\times A
  88. T , V T,V
  89. N N
  90. F U - T S F\equiv U-TS
  91. U U
  92. T T
  93. P P
  94. G = U + P V - T S G=U+PV-TS
  95. T o T_{o}
  96. P o P_{o}
  97. W = Δ G = N k T o Θ ( V / V o ) W=\Delta G=NkT_{o}\Theta(V/V_{o})
  98. V o = N k T o / P o V_{o}=NkT_{o}/P_{o}
  99. Θ ( x ) = x - 1 - ln x 0 \Theta(x)=x-1-\ln x\geq 0
  100. T o T_{o}
  101. Δ I 0 \Delta I\geq 0
  102. k ln ( p / p o ) k\ln(p/p_{o})
  103. p o p_{o}
  104. V o V_{o}
  105. T o T_{o}
  106. W = T o Δ I W=T_{o}\Delta I
  107. Δ I = N k [ Θ ( V / V o ) + 3 2 Θ ( T / T o ) ] \Delta I=Nk[\Theta(V/V_{o})+\frac{3}{2}\Theta(T/T_{o})]
  108. D KL ( P Q ) = Tr ( P ( log ( P ) - log ( Q ) ) ) . D_{\mathrm{KL}}(P\|Q)=\operatorname{Tr}(P(\log(P)-\log(Q))).\!
  109. D KL ( P Q ) D_{\mathrm{KL}}(P\|Q)
  110. D KL ( P Q ) + D KL ( Q P ) D_{\mathrm{KL}}(P\|Q)+D_{\mathrm{KL}}(Q\|P)\,\!
  111. D λ ( P Q ) = λ D KL ( P λ P + ( 1 - λ ) Q ) + ( 1 - λ ) D KL ( Q λ P + ( 1 - λ ) Q ) , D_{\lambda}(P\|Q)=\lambda D_{\mathrm{KL}}(P\|\lambda P+(1-\lambda)Q)+(1-% \lambda)D_{\mathrm{KL}}(Q\|\lambda P+(1-\lambda)Q),\,\!
  112. D JS = 1 2 D KL ( P M ) + 1 2 D KL ( Q M ) D_{\mathrm{JS}}=\tfrac{1}{2}D_{\mathrm{KL}}\left(P\|M\right)+\tfrac{1}{2}D_{% \mathrm{KL}}\left(Q\|M\right)\,\!
  113. M = 1 2 ( P + Q ) . M=\tfrac{1}{2}(P+Q).\,
  114. δ ( p , q ) \delta(p,q)
  115. δ ( P , Q ) 1 2 D KL ( P Q ) \delta(P,Q)\leq\sqrt{\frac{1}{2}D_{\mathrm{KL}}(P\|Q)}
  116. α \alpha

Kummer_theory.html

  1. a n . \sqrt[n]{a}.
  2. K × / ( K × ) n , K^{\times}/(K^{\times})^{n},\,\!
  3. Δ K × / ( K × ) n , \Delta\subseteq K^{\times}/(K^{\times})^{n},\,\!
  4. K ( Δ 1 / n ) , K(\Delta^{1/n}),\,\!
  5. Δ 1 / n = { a n : a K × , a ( K × ) n Δ } \Delta^{1/n}=\{\sqrt[n]{a}:a\in K^{\times},a\cdot(K^{\times})^{n}\in\Delta\}
  6. Δ = ( K × ( L × ) n ) / ( K × ) n . \Delta=(K^{\times}\cap(L^{\times})^{n})/(K^{\times})^{n}.\,\!
  7. Δ Hom c ( Gal ( L / K ) , μ n ) \Delta\cong\operatorname{Hom}_{\,\text{c}}(\operatorname{Gal}(L/K),\mu_{n})
  8. a ( σ σ ( α ) α ) , a\mapsto\biggl(\sigma\mapsto\frac{\sigma(\alpha)}{\alpha}\biggr),
  9. μ n \mu_{n}
  10. Hom c ( Gal ( L / K ) , μ n ) \operatorname{Hom}_{\,\text{c}}(\operatorname{Gal}(L/K),\mu_{n})
  11. Gal ( L / K ) \operatorname{Gal}(L/K)
  12. μ n \mu_{n}
  13. Gal ( L / K ) \operatorname{Gal}(L/K)
  14. μ n \mu_{n}
  15. Gal ( L / K ) \operatorname{Gal}(L/K)
  16. Δ Hom ( Gal ( L / K ) , μ n ) Gal ( L / K ) , \Delta\cong\operatorname{Hom}(\operatorname{Gal}(L/K),\mu_{n})\cong% \operatorname{Gal}(L/K),

Kuratowski_closure_axioms.html

  1. X X
  2. 𝒫 ( X ) \mathcal{P}(X)
  3. cl : 𝒫 ( X ) 𝒫 ( X ) \operatorname{cl}:\mathcal{P}(X)\to\mathcal{P}(X)
  4. cl ( ) = \operatorname{cl}(\varnothing)=\varnothing
  5. A cl ( A ) for every subset A X A\subseteq\operatorname{cl}(A)\,\text{ for every subset }A\subseteq X
  6. cl ( A B ) = cl ( A ) cl ( B ) for any subsets A , B X \operatorname{cl}(A\cup B)=\operatorname{cl}(A)\cup\operatorname{cl}(B)\,\text% { for any subsets }A,B\subseteq X
  7. cl ( cl ( A ) ) = cl ( A ) for every subset A X \operatorname{cl}(\operatorname{cl}(A))=\operatorname{cl}(A)\,\text{ for every% subset }A\subseteq X
  8. A B cl ( A ) cl ( B ) A\subseteq B\Rightarrow\operatorname{cl}(A)\subseteq\operatorname{cl}(B)
  9. A cl ( A ) cl ( cl ( B ) ) = cl ( A B ) cl ( ) for all subsets A , B X . A\cup\operatorname{cl}(A)\cup\operatorname{cl}(\operatorname{cl}(B))=% \operatorname{cl}(A\cup B)\setminus\operatorname{cl}(\varnothing)\,\text{ for % all subsets }A,B\subseteq X.
  10. C X C\subseteq X
  11. cl ( C ) = C \operatorname{cl}(C)=C
  12. X cl ( X ) X\subseteq\operatorname{cl}(X)
  13. X X
  14. X X
  15. cl ( X ) X \operatorname{cl}(X)\subseteq X
  16. X = cl ( X ) X=\operatorname{cl}(X)
  17. X X
  18. cl ( ) = \operatorname{cl}(\varnothing)=\varnothing
  19. \varnothing
  20. \mathcal{I}
  21. C i C_{i}
  22. i i\in\mathcal{I}
  23. i C i cl ( i C i ) . \bigcap_{i\in\mathcal{I}}C_{i}\subseteq\operatorname{cl}(\bigcap_{i\in\mathcal% {I}}C_{i}).
  24. i C i C i i cl ( i C i ) cl ( C i ) = C i i cl ( i C i ) i C i . \bigcap_{i\in\mathcal{I}}C_{i}\subseteq C_{i}\forall i\in\mathcal{I}% \Rightarrow\operatorname{cl}(\bigcap_{i\in\mathcal{I}}C_{i})\subseteq% \operatorname{cl}(C_{i})=C_{i}\forall i\in\mathcal{I}\Rightarrow\operatorname{% cl}(\bigcap_{i\in\mathcal{I}}C_{i})\subseteq\bigcap_{i\in\mathcal{I}}C_{i}.
  25. i C i = cl ( i C i ) \bigcap_{i\in\mathcal{I}}C_{i}=\operatorname{cl}(\bigcap_{i\in\mathcal{I}}C_{i})
  26. i C i \bigcap_{i\in\mathcal{I}}C_{i}
  27. \mathcal{I}
  28. C i C_{i}
  29. i i\in\mathcal{I}
  30. i C i = cl ( i C i ) \bigcup_{i\in\mathcal{I}}C_{i}=\operatorname{cl}(\bigcup_{i\in\mathcal{I}}C_{i})
  31. i C i \bigcup_{i\in\mathcal{I}}C_{i}
  32. cl A ( B ) = A cl X ( B ) for all B A . \operatorname{cl_{A}}(B)=A\cap\operatorname{cl_{X}}(B)\,\text{ for all }B% \subseteq A.
  33. p p
  34. A A
  35. p cl ( A ) p\in\operatorname{cl}(A)
  36. f : X Y f:X\to Y
  37. p p
  38. p cl ( A ) f ( p ) cl ( f ( A ) ) p\in\operatorname{cl}(A)\Rightarrow f(p)\in\operatorname{cl}(f(A))

Kwela.html

  1. 4 6 {}^{6}_{4}

Lab_color_space.html

  1. L = 116 f ( Y / Y n ) - 16 a = 500 [ f ( X / X n ) - f ( Y / Y n ) ] b = 200 [ f ( Y / Y n ) - f ( Z / Z n ) ] \begin{aligned}\displaystyle L^{\star}&\displaystyle=116f(Y/Y_{n})-16\\ \displaystyle a^{\star}&\displaystyle=500\left[f(X/X_{n})-f(Y/Y_{n})\right]\\ \displaystyle b^{\star}&\displaystyle=200\left[f(Y/Y_{n})-f(Z/Z_{n})\right]% \end{aligned}
  2. f ( t ) = { t 1 / 3 if t > ( 6 29 ) 3 1 3 ( 29 6 ) 2 t + 4 29 otherwise f(t)=\begin{cases}t^{1/3}&\,\text{if }t>(\frac{6}{29})^{3}\\ \frac{1}{3}\left(\frac{29}{6}\right)^{2}t+\frac{4}{29}&\,\text{otherwise}\end{cases}
  3. X n = 0.95047 , Y n = 1.00000 , Z n = 1.08883 X_{n}=0.95047,Y_{n}=1.00000,Z_{n}=1.08883
  4. t 0 1 / 3 \displaystyle t_{0}^{1/3}
  5. a \displaystyle a
  6. X \displaystyle X
  7. f - 1 ( t ) = { t 3 if t > 6 29 3 ( 6 29 ) 2 ( t - 4 29 ) otherwise f^{-1}(t)=\begin{cases}t^{3}&\,\text{if }t>\tfrac{6}{29}\\ 3\left(\tfrac{6}{29}\right)^{2}\left(t-\tfrac{4}{29}\right)&\,\text{otherwise}% \end{cases}
  8. L = 100 Y / Y n L=100\sqrt{Y/Y_{n}}
  9. 100 25 / 100 = 100 1 / 2 100\sqrt{25/100}=100\cdot 1/2
  10. a = K a ( X / X n - Y / Y n Y / Y n ) a=K_{a}\left(\frac{X/X_{n}-Y/Y_{n}}{\sqrt{Y/Y_{n}}}\right)
  11. K a K_{a}
  12. X n X_{n}
  13. b = K b ( Y / Y n - Z / Z n Y / Y n ) b=K_{b}\left(\frac{Y/Y_{n}-Z/Z_{n}}{\sqrt{Y/Y_{n}}}\right)
  14. K a 175 198.04 ( X n + Y n ) K_{a}\approx\frac{175}{198.04}(X_{n}+Y_{n})
  15. K b 70 218.11 ( Y n + Z n ) K_{b}\approx\frac{70}{218.11}(Y_{n}+Z_{n})
  16. L = 100 Y / Y n L=100\sqrt{Y/Y_{n}}
  17. c a = X / X n Y / Y n - 1 = X / X n - Y / Y n Y / Y n c_{a}=\frac{X/X_{n}}{Y/Y_{n}}-1=\frac{X/X_{n}-Y/Y_{n}}{Y/Y_{n}}
  18. c b = k e ( 1 - Z / Z n Y / Y n ) = k e Y / Y n - Z / Z n Y / Y n c_{b}=k_{e}\left(1-\frac{Z/Z_{n}}{Y/Y_{n}}\right)=k_{e}\frac{Y/Y_{n}-Z/Z_{n}}{% Y/Y_{n}}
  19. a = K L c a = K 100 X / X n - Y / Y n Y / Y n a=K\cdot L\cdot c_{a}=K\cdot 100\frac{X/X_{n}-Y/Y_{n}}{\sqrt{Y/Y_{n}}}
  20. b = K L c b = K 100 k e Y / Y n - Z / Z n Y / Y n b=K\cdot L\cdot c_{b}=K\cdot 100k_{e}\frac{Y/Y_{n}-Z/Z_{n}}{\sqrt{Y/Y_{n}}}
  21. C a b * = a * 2 + b * 2 , h a b = arctan ( b * a * ) C_{ab}^{*}=\sqrt{{a^{*}}^{2}+{b^{*}}^{2}},\qquad h_{ab}^{\circ}=\arctan\left(% \frac{b^{*}}{a^{*}}\right)
  22. a * = C a b * cos ( h a b ) , b * = C a b * sin ( h a b ) a^{*}=C_{ab}^{*}\cdot\cos(h_{ab}^{\circ}),\qquad b^{*}=C_{ab}^{*}\cdot\sin(h_{% ab}^{\circ})

Lagrange's_four-square_theorem.html

  1. p = a 0 2 + a 1 2 + a 2 2 + a 3 2 p=a_{0}^{2}+a_{1}^{2}+a_{2}^{2}+a_{3}^{2}
  2. a 0 , a 1 , a 2 , a 3 a_{0},a_{1},a_{2},a_{3}
  3. 3 \displaystyle 3
  4. 4 k ( 8 m + 7 ) 4^{k}(8m+7)
  5. k k
  6. m m
  7. Z / p Z Z/pZ
  8. α = 1 2 E 0 ( 1 + 𝐢 + 𝐣 + 𝐤 ) + E 1 𝐢 + E 2 𝐣 + E 3 𝐤 = a 0 + a 1 𝐢 + a 2 𝐣 + a 3 𝐤 \alpha=\frac{1}{2}E_{0}(1+\mathbf{i}+\mathbf{j}+\mathbf{k})+E_{1}\mathbf{i}+E_% {2}\mathbf{j}+E_{3}\mathbf{k}=a_{0}+a_{1}\mathbf{i}+a_{2}\mathbf{j}+a_{3}% \mathbf{k}
  9. E 0 , E 1 , E 2 , E 3 E_{0},E_{1},E_{2},E_{3}
  10. a 0 , a 1 , a 2 , a 3 a_{0},a_{1},a_{2},a_{3}
  11. E 0 E_{0}
  12. N ( α ) \mathrm{N}(\alpha)
  13. α \alpha
  14. N ( α ) = α α ¯ = a 0 2 + a 1 2 + a 2 2 + a 3 2 \mathrm{N}(\alpha)=\alpha\bar{\alpha}=a_{0}^{2}+a_{1}^{2}+a_{2}^{2}+a_{3}^{2}
  15. α ¯ = a 0 - a 1 𝐢 - a 2 𝐣 - a 3 𝐤 \bar{\alpha}=a_{0}-a_{1}\mathbf{i}-a_{2}\mathbf{j}-a_{3}\mathbf{k}
  16. α \alpha
  17. 1 4 + n : n \tfrac{1}{4}+n:n\in\mathbb{Z}
  18. N ( α β ) = α β ( α β ¯ ) = α β β ¯ α ¯ = α N ( β ) α ¯ = α α ¯ N ( β ) = N ( α ) N ( β ) . \mathrm{N}(\alpha\beta)=\alpha\beta(\overline{\alpha\beta})=\alpha\beta\bar{% \beta}\bar{\alpha}=\alpha\mathrm{N}(\beta)\bar{\alpha}=\alpha\bar{\alpha}% \mathrm{N}(\beta)=\mathrm{N}(\alpha)\mathrm{N}(\beta).
  19. α 0 \alpha\neq 0
  20. α - 1 = α ¯ N ( α ) - 1 \alpha^{-1}=\bar{\alpha}\mathrm{N}(\alpha)^{-1}
  21. α \alpha
  22. N ( α ) = 1 \mathrm{N}(\alpha)=1
  23. 2 = 1 2 + 1 2 + 0 2 + 0 2 2=1^{2}+1^{2}+0^{2}+0^{2}
  24. p p
  25. ( p , 0 , 0 , 0 ) (p,0,0,0)
  26. p = α β . p=\alpha\beta.
  27. p , α , β p,\alpha,\beta
  28. N ( p ) = p 2 = N ( α β ) = N ( α ) N ( β ) \mathrm{N}(p)=p^{2}=\mathrm{N}(\alpha\beta)=\mathrm{N}(\alpha)\mathrm{N}(\beta)
  29. N ( α ) , N ( β ) > 1 \mathrm{N}(\alpha),\mathrm{N}(\beta)>1
  30. N ( α ) \mathrm{N}(\alpha)
  31. N ( β ) \mathrm{N}(\beta)
  32. p p
  33. p p
  34. p = N ( α ) = a 0 2 + a 1 2 + a 2 2 + a 3 2 . p=\mathrm{N}(\alpha)=a_{0}^{2}+a_{1}^{2}+a_{2}^{2}+a_{3}^{2}.
  35. α \alpha
  36. ω = ( ± 1 ± 𝐢 ± 𝐣 ± 𝐤 ) / 2 \omega=(\pm 1\pm\mathbf{i}\pm\mathbf{j}\pm\mathbf{k})/2
  37. γ ω + α \gamma\equiv\omega+\alpha
  38. p = ( γ ¯ - ω ¯ ) ω ω ¯ ( γ - ω ) = ( γ ¯ ω - 1 ) ( ω ¯ γ - 1 ) . p=(\bar{\gamma}-\bar{\omega})\omega\bar{\omega}(\gamma-\omega)=(\bar{\gamma}% \omega-1)(\bar{\omega}\gamma-1).
  39. γ \gamma
  40. ( ω ¯ γ - 1 ) (\bar{\omega}\gamma-1)
  41. α \alpha
  42. p p
  43. p p
  44. p p
  45. u = 1 + l 2 + m 2 u=1+l^{2}+m^{2}
  46. l l
  47. m m
  48. p p
  49. a 2 b 2 ( mod p ) a^{2}\equiv b^{2}\;\;(\mathop{{\rm mod}}p)
  50. a , b a,b
  51. a ± b ( mod p ) a\equiv\pm b\;\;(\mathop{{\rm mod}}p)
  52. X = { 0 2 , 1 2 , , ( ( p - 1 ) / 2 ) 2 } X=\{0^{2},1^{2},\dots,((p-1)/2)^{2}\}
  53. ( p + 1 ) / 2 (p+1)/2
  54. p p
  55. Y = { - ( 1 + x ) : x X } Y=\{-(1+x):x\in X\}
  56. ( p + 1 ) / 2 (p+1)/2
  57. p p
  58. | X | + | Y | = p + 1 > p |X|+|Y|=p+1>p
  59. X X
  60. Y Y
  61. u u
  62. 1 + l 2 + m 2 = ( 1 + l 𝐢 + m 𝐣 ) ( 1 - l 𝐢 - m 𝐣 ) . 1+l^{2}+m^{2}=(1+l\;\mathbf{i}+m\;\mathbf{j})(1-l\;\mathbf{i}-m\;\mathbf{j}).
  63. α = a 0 + a 1 𝐢 + a 2 𝐣 + a 3 𝐤 \alpha=a_{0}+a_{1}\mathbf{i}+a_{2}\mathbf{j}+a_{3}\mathbf{k}
  64. β = b 0 + b 1 𝐢 + b 2 𝐣 + b 3 𝐤 \beta=b_{0}+b_{1}\mathbf{i}+b_{2}\mathbf{j}+b_{3}\mathbf{k}
  65. N ( α - β ) < 1 \mathrm{N}(\alpha-\beta)<1
  66. b 0 b_{0}
  67. | a 0 - b 0 | 1 / 4 |a_{0}-b_{0}|\leq 1/4
  68. b 1 , b 2 , b 3 b_{1},b_{2},b_{3}
  69. | a i - b i | 1 / 2 |a_{i}-b_{i}|\leq 1/2
  70. i = 1 , 2 , 3 i=1,2,3
  71. N ( α - β ) = ( a 0 - b 0 ) 2 + ( a 1 - b 1 ) 2 + ( a 2 - b 2 ) 2 + ( a 3 - b 3 ) 2 ( 1 4 ) 2 + ( 1 2 ) 2 + ( 1 2 ) 2 + ( 1 2 ) 2 = 13 16 < 1. \begin{aligned}\displaystyle\mathrm{N}(\alpha-\beta)&\displaystyle=(a_{0}-b_{0% })^{2}+(a_{1}-b_{1})^{2}+(a_{2}-b_{2})^{2}+(a_{3}-b_{3})^{2}\\ &\displaystyle\leq\left(\frac{1}{4}\right)^{2}+\left(\frac{1}{2}\right)^{2}+% \left(\frac{1}{2}\right)^{2}+\left(\frac{1}{2}\right)^{2}=\frac{13}{16}<1.\end% {aligned}
  72. α , β \alpha,\beta
  73. α 0 \alpha\neq 0
  74. γ \gamma
  75. N ( β - α γ ) < N ( α ) . \mathrm{N}(\beta-\alpha\gamma)<\mathrm{N}(\alpha).
  76. H H
  77. α \alpha
  78. α H = p H + ( 1 - l 𝐢 - m 𝐣 ) H . \alpha H=pH+(1-l\;\mathbf{i}-m\;\mathbf{j})H.
  79. p = α β p=\alpha\beta
  80. β \beta
  81. β \beta
  82. 1 - l 𝐢 - m 𝐣 1-l\;\mathbf{i}-m\;\mathbf{j}
  83. p p
  84. 1 / p - l / p 𝐢 - m / p 𝐣 1/p-l/p\;\mathbf{i}-m/p\;\mathbf{j}
  85. p > 2 p>2
  86. α \alpha
  87. ( 1 + l 𝐢 + m 𝐣 ) H = ( 1 + l 𝐢 + m 𝐣 ) p H + ( 1 + l 𝐢 + m 𝐣 ) ( 1 - l 𝐢 - m 𝐣 ) H p H (1+l\;\mathbf{i}+m\;\mathbf{j})H=(1+l\;\mathbf{i}+m\;\mathbf{j})pH+(1+l\;% \mathbf{i}+m\;\mathbf{j})(1-l\;\mathbf{i}-m\;\mathbf{j})H\subseteq pH
  88. p p
  89. 1 + l 𝐢 + m 𝐣 1+l\;\mathbf{i}+m\;\mathbf{j}
  90. 1 / p - l / p 𝐢 - m / p 𝐣 1/p-l/p\;\mathbf{i}-m/p\;\mathbf{j}
  91. p p
  92. a , b , c , d a,b,c,d
  93. n = a x 1 2 + b x 2 2 + c x 3 2 + d x 4 2 n=ax_{1}^{2}+bx_{2}^{2}+cx_{3}^{2}+dx_{4}^{2}
  94. n n
  95. x 1 , x 2 , x 3 , x 4 x_{1},x_{2},x_{3},x_{4}
  96. a = b = c = d = 1 a=b=c=d=1
  97. a b c d a\leq b\leq c\leq d
  98. a , b , c , d a,b,c,d
  99. x 1 , x 2 , x 3 , x 4 x_{1},x_{2},x_{3},x_{4}
  100. n n
  101. a = 1 , b = 2 , c = 5 , d = 5 a=1,b=2,c=5,d=5
  102. n = 15 n=15
  103. n = x 1 2 + x 2 2 + x 3 2 + x 4 2 n=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}
  104. n n
  105. O ( ( log n ) 2 ) \mathrm{O}((\log n)^{2})
  106. r 4 ( n ) = { 8 m n m if n is odd 24 m | n m odd m if n is even . r_{4}(n)=\begin{cases}8\sum\limits_{m\mid n}m&\,\text{if }n\,\text{ is odd}\\ 24\sum\limits_{\begin{smallmatrix}m|n\\ m\,\text{ odd}\end{smallmatrix}}m&\,\text{if }n\,\text{ is even}.\end{cases}
  107. r 4 ( n ) = 8 m : 4 m n m . r_{4}(n)=8\sum_{m\,:\,4\nmid m\mid n}m.
  108. r 4 ( n ) = 8 σ ( n ) - 32 σ ( n / 4 ) , r_{4}(n)=8\sigma(n)-32\sigma(n/4)\ ,
  109. 2 ( 4 k ) , 6 ( 4 k ) 2(4^{k}),6(4^{k})
  110. 14 ( 4 k ) 14(4^{k})
  111. 2 ( 4 k ) , 6 ( 4 k ) 2(4^{k}),6(4^{k})
  112. 14 ( 4 k ) 14(4^{k})

Landau–Ramanujan_constant.html

  1. x / ln ( x ) . x/{\sqrt{\ln(x)}}.
  2. lim x N ( x ) ln ( x ) x 0.76422365358922066299069873125. \lim_{x\rightarrow\infty}\frac{N(x)\sqrt{\ln(x)}}{x}\approx 0.7642236535892206% 6299069873125.

Landing_gear.html

  1. F m F\text{m}
  2. F m = l n l m + l n W . F\text{m}=\frac{l\text{n}}{l\text{m}+l\text{n}}W.
  3. W W
  4. l m l\text{m}
  5. l n l\text{n}
  6. F n F\text{n}
  7. F n = l n l m + l n ( W - L ) + h cg l m + l n ( a x g W - D + T ) . F\text{n}=\frac{l\text{n}}{l\text{m}+l\text{n}}(W-L)+\frac{h\text{cg}}{l\text{% m}+l\text{n}}\left(\frac{a\text{x}}{g}W-D+T\right).
  8. L L
  9. D D
  10. T T
  11. h cg h\text{cg}
  12. a x g \frac{a\text{x}}{g}
  13. L L
  14. D D
  15. T = 0 T=0
  16. F n = l m + h cg ( a x g ) l m + l n W . F\text{n}=\frac{l\text{m}+h\text{cg}(\frac{a\text{x}}{g})}{l\text{m}+l\text{n}% }W.

Lane–Emden_equation.html

  1. 1 ξ 2 d d ξ ( ξ 2 d θ d ξ ) + θ n = 0 \frac{1}{\xi^{2}}\frac{d}{d\xi}\left({\xi^{2}\frac{d\theta}{d\xi}}\right)+% \theta^{n}=0
  2. ξ \xi
  3. θ \theta
  4. ρ = ρ c θ n \rho=\rho_{c}\theta^{n}
  5. ρ c \rho_{c}
  6. n n
  7. P = K ρ 1 + 1 n P=K\rho^{1+\frac{1}{n}}\,
  8. P P
  9. ρ \rho
  10. K K
  11. θ ( 0 ) = 1 \theta(0)=1
  12. θ ( 0 ) = 0 \theta^{\prime}(0)=0
  13. n n
  14. d m d r = 4 π r 2 ρ \frac{dm}{dr}=4\pi r^{2}\rho
  15. ρ \rho
  16. r r
  17. 1 ρ d P d r = - G m r 2 \frac{1}{\rho}\frac{dP}{dr}=-\frac{Gm}{r^{2}}
  18. m m
  19. r r
  20. d d r ( 1 ρ d P d r ) \displaystyle\frac{d}{dr}\left(\frac{1}{\rho}\frac{dP}{dr}\right)
  21. r 2 r^{2}
  22. P P
  23. r 2 d d r ( 1 ρ d P d r ) + 2 r ρ d P d r = d d r ( r 2 ρ d P d r ) = - 4 π G r 2 ρ r^{2}\frac{d}{dr}\left(\frac{1}{\rho}\frac{dP}{dr}\right)+\frac{2r}{\rho}\frac% {dP}{dr}=\frac{d}{dr}\left(\frac{r^{2}}{\rho}\frac{dP}{dr}\right)=-4\pi Gr^{2}\rho
  24. r 2 r^{2}
  25. P = K ρ c 1 + 1 n θ n + 1 P=K\rho_{c}^{1+\frac{1}{n}}\theta^{n+1}
  26. ρ = ρ c θ n \rho=\rho_{c}\theta^{n}
  27. 1 r 2 d d r ( r 2 K ρ c 1 n ( n + 1 ) d θ d r ) = - 4 π G ρ c θ n \frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}K\rho_{c}^{\frac{1}{n}}(n+1)\frac{d% \theta}{dr}\right)=-4\pi G\rho_{c}\theta^{n}
  28. r = α ξ r=\alpha\xi
  29. α 2 = ( n + 1 ) K ρ c 1 n - 1 / 4 π G \alpha^{2}=(n+1)K\rho_{c}^{\frac{1}{n}-1}/4\pi G
  30. 1 ξ 2 d d ξ ( ξ 2 d θ d ξ ) + θ n = 0 \frac{1}{\xi^{2}}\frac{d}{d\xi}\left({\xi^{2}\frac{d\theta}{d\xi}}\right)+% \theta^{n}=0
  31. 2 Φ = 1 r 2 d d r ( r 2 d Φ d r ) = 4 π G ρ \nabla^{2}\Phi=\frac{1}{r^{2}}\frac{d}{dr}\left({r^{2}\frac{d\Phi}{dr}}\right)% =4\pi G\rho
  32. d Φ d r = - 1 ρ d P d r \frac{d\Phi}{dr}=-\frac{1}{\rho}\frac{dP}{dr}
  33. n n
  34. θ n ( ξ ) \theta_{n}(\xi)
  35. θ n \theta_{n}
  36. n n
  37. n = 0 , 1 , 5 n=0,1,5
  38. n n
  39. R = α ξ 1 R=\alpha\xi_{1}
  40. θ n ( ξ 1 ) = 0 \theta_{n}(\xi_{1})=0
  41. θ n \theta_{n}
  42. ρ = ρ c θ n n \rho=\rho_{c}\theta_{n}^{n}
  43. M M
  44. ξ 1 \xi_{1}
  45. P = K ρ 1 + 1 n P=K\rho^{1+\frac{1}{n}}
  46. P = K ρ c 1 + 1 n θ n n + 1 P=K\rho_{c}^{1+\frac{1}{n}}\theta_{n}^{n+1}
  47. P = k B ρ T / μ P=k_{B}\rho T/\mu
  48. k B k_{B}
  49. m m
  50. T = K μ k B ρ c 1 / n θ n T=\frac{K\mu}{k_{B}}\rho_{c}^{1/n}\theta_{n}
  51. n n
  52. n = 0 n=0
  53. 1 ξ 2 d d ξ ( ξ 2 d θ d ξ ) + 1 = 0 \frac{1}{\xi^{2}}\frac{d}{d\xi}\left({\xi^{2}\frac{d\theta}{d\xi}}\right)+1=0
  54. ξ 2 d θ d ξ = C 1 - 1 3 ξ 3 \xi^{2}\frac{d\theta}{d\xi}=C_{1}-\frac{1}{3}\xi^{3}
  55. ξ 2 \xi^{2}
  56. θ ( ξ ) = C 0 - C 1 ξ - 1 6 ξ 2 \theta(\xi)=C_{0}-\frac{C_{1}}{\xi}-\frac{1}{6}\xi^{2}
  57. θ ( 0 ) = 1 \theta(0)=1
  58. θ ( 0 ) = 0 \theta^{\prime}(0)=0
  59. C 0 = 1 C_{0}=1
  60. C 1 = 0 C_{1}=0
  61. n = 1 n=1
  62. d 2 θ d ξ 2 + 2 ξ d θ d ξ + θ = 0 \frac{d^{2}\theta}{d\xi^{2}}+\frac{2}{\xi}\frac{d\theta}{d\xi}+\theta=0
  63. θ ( ξ ) = n = 0 a n ξ n \theta(\xi)=\sum\limits_{n=0}^{\infty}a_{n}\xi^{n}
  64. a n + 2 = - a n ( n + 3 ) ( n + 2 ) a_{n+2}=-\frac{a_{n}}{(n+3)(n+2)}
  65. θ ( ξ ) = a 0 sin ξ ξ + a 1 cos ξ ξ \theta(\xi)=a_{0}\frac{\sin\xi}{\xi}+a_{1}\frac{\cos\xi}{\xi}
  66. θ ( ξ ) 1 \theta(\xi)\rightarrow 1
  67. ξ 0 \xi\rightarrow 0
  68. a 0 = 1 , a 1 = 0 a_{0}=1,a_{1}=0
  69. θ ( ξ ) = sin ξ ξ \theta(\xi)=\frac{\sin\xi}{\xi}
  70. θ ( ξ ) = 1 1 + ξ 2 / 3 \theta(\xi)=\frac{1}{\sqrt{1+\xi^{2}/3}}
  71. n = 5 n=5
  72. d θ d ξ = - ϕ ξ 2 \frac{d\theta}{d\xi}=-\frac{\phi}{\xi^{2}}
  73. d ϕ d ξ = θ n ξ 2 \frac{d\phi}{d\xi}=\theta^{n}\xi^{2}
  74. ϕ ( ξ ) \phi(\xi)
  75. m ( r ) = 4 π α 3 ρ c ϕ ( ξ ) m(r)=4\pi\alpha^{3}\rho_{c}\phi(\xi)
  76. ϕ ( 0 ) = 0 \phi(0)=0
  77. θ ( 0 ) = 1 \theta(0)=1
  78. θ ( ξ ) \theta(\xi)
  79. C 2 / n + 1 θ ( C ξ ) C^{2/n+1}\theta(C\xi)
  80. U = d log m d log r = ξ 3 θ n ϕ U=\frac{d\log m}{d\log r}=\frac{\xi^{3}\theta^{n}}{\phi}
  81. V = d log P d log r = ( n + 1 ) ϕ ξ θ V=\frac{d\log P}{d\log r}=(n+1)\frac{\phi}{\xi\theta}
  82. ξ \xi
  83. 1 U d U d ξ = 1 ξ ( 3 - n ( n + 1 ) - 1 V - U ) \frac{1}{U}\frac{dU}{d\xi}=\frac{1}{\xi}(3-n(n+1)^{-1}V-U)
  84. 1 V d V d ξ = 1 ξ ( - 1 + U + ( n + 1 ) - 1 V ) \frac{1}{V}\frac{dV}{d\xi}=\frac{1}{\xi}(-1+U+(n+1)^{-1}V)
  85. ξ \xi
  86. d V d U = - V U ( U + ( n + 1 ) - 1 V - 1 U + n ( n + 1 ) - 1 V - 3 ) \frac{dV}{dU}=-\frac{V}{U}\left(\frac{U+(n+1)^{-1}V-1}{U+n(n+1)^{-1}V-3}\right)
  87. d U d log ξ = - U ( U + n ( n + 1 ) - 1 V - 3 ) \frac{dU}{d\log\xi}=-U(U+n(n+1)^{-1}V-3)
  88. d V d log ξ = V ( U + ( n + 1 ) - 1 V - 1 ) \frac{dV}{d\log\xi}=V(U+(n+1)^{-1}V-1)
  89. d V / d log ξ = d U / d log ξ = 0 dV/d\log\xi=dU/d\log\xi=0
  90. ( 0 , 0 ) (0,0)
  91. 3 3
  92. - 1 -1
  93. ( 1 , 0 ) (1,0)
  94. ( 0 , 1 ) (0,1)
  95. ( 3 , 0 ) (3,0)
  96. - 3 -3
  97. 2 2
  98. ( 1 , 0 ) (1,0)
  99. ( - 3 n , 5 + 5 n ) (-3n,5+5n)
  100. ( 0 , n + 1 ) (0,n+1)
  101. 1 1
  102. 3 - n 3-n
  103. ( 0 , 1 ) (0,1)
  104. ( 2 - n , 1 + n ) (2-n,1+n)
  105. ( n - 3 n - 1 , 2 n + 1 n - 1 ) \left(\frac{n-3}{n-1},2\frac{n+1}{n-1}\right)
  106. n - 5 ± Δ n 2 - 2 n \frac{n-5\pm\Delta_{n}}{2-2n}
  107. ( 1 - n Δ n , 4 + 4 n ) \left(1-n\mp\Delta_{n},4+4n\right)