wpmath0000002_7

Free_fall.html

  1. g g
  2. v ( t ) = - g t + v 0 v(t)=-gt+v_{0}\,
  3. y ( t ) = - 1 2 g t 2 + v 0 t + y 0 y(t)=-\frac{1}{2}gt^{2}+v_{0}t+y_{0}
  4. v 0 v_{0}\,
  5. v ( t ) v(t)\,
  6. y 0 y_{0}\,
  7. y ( t ) y(t)\,
  8. t t\,
  9. g g\,
  10. m m
  11. A A
  12. v v
  13. m d v d t = 1 2 ρ C D A v 2 - m g , m\frac{dv}{dt}=\frac{1}{2}\rho C_{\mathrm{D}}Av^{2}-mg\,,
  14. ρ \rho
  15. C D C_{\mathrm{D}}
  16. v ( t ) = - v tanh ( g t v ) , v(t)=-v_{\infty}\tanh\left(\frac{gt}{v_{\infty}}\right),
  17. v = 2 m g ρ C D A . v_{\infty}=\sqrt{\frac{2mg}{\rho C_{D}A}}\,.
  18. y = y 0 - v 2 g ln cosh ( g t v ) . y=y_{0}-\frac{v_{\infty}^{2}}{g}\ln\cosh\left(\frac{gt}{v_{\infty}}\right).
  19. t ( y ) = y 0 3 2 μ ( y y 0 ( 1 - y y 0 ) + arccos y y 0 ) t(y)=\sqrt{\frac{{y_{0}}^{3}}{2\mu}}\left(\sqrt{\frac{y}{y_{0}}\left(1-\frac{y% }{y_{0}}\right)}+\arccos{\sqrt{\frac{y}{y_{0}}}}\right)
  20. y ( t ) = n = 1 [ lim r 0 ( x n n ! d n - 1 d r n - 1 [ r n ( 7 2 ( arcsin ( r ) - r - r 2 ) ) - 2 3 n ] ) ] y(t)=\sum_{n=1}^{\infty}\left[\lim_{r\to 0}\left({\frac{x^{n}}{n!}}\frac{% \mathrm{d}^{\,n-1}}{\mathrm{d}r^{\,n-1}}\left[r^{n}\left(\frac{7}{2}(\arcsin(% \sqrt{r})-\sqrt{r-r^{2}})\right)^{-\frac{2}{3}n}\right]\right)\right]
  21. y ( t ) = y 0 ( x - 1 5 x 2 - 3 175 x 3 - 23 7875 x 4 - 1894 3931875 x 5 - 3293 21896875 x 6 - 2418092 62077640625 x 7 - ) y(t)=y_{0}\left(x-\frac{1}{5}x^{2}-\frac{3}{175}x^{3}-\frac{23}{7875}x^{4}-% \frac{1894}{3931875}x^{5}-\frac{3293}{21896875}x^{6}-\frac{2418092}{6207764062% 5}x^{7}-\cdots\right)
  22. x = [ 3 2 ( π 2 - t 2 μ y 0 3 ) ] 2 / 3 x=\left[\frac{3}{2}\left(\frac{\pi}{2}-t\sqrt{\frac{2\mu}{{y_{0}}^{3}}}\right)% \right]^{2/3}

Free_variables_and_bound_variables.html

  1. k = 1 10 f ( k , n ) , \sum_{k=1}^{10}f(k,n),
  2. 0 x y - 1 e - x d x , \int_{0}^{\infty}x^{y-1}e^{-x}\,dx,
  3. lim h 0 f ( x + h ) - f ( x ) h , \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h},
  4. x y [ φ ( x , y , z ) ] , \forall x\ \exists y\ \Big[\varphi(x,y,z)\Big],
  5. x S x S 0 d x lim x 0 x x \sum_{x\in S}\quad\quad\prod_{x\in S}\quad\quad\int_{0}^{\infty}\cdots\,dx% \quad\quad\lim_{x\to 0}\quad\quad\forall x\quad\quad\exists x
  6. 1 10 ( k f ( k , n ) ) \sum_{1\,\ldots\,10}\left(k\mapsto f(k,n)\right)
  7. D ( x x 2 + 2 x + 1 ) D\left(x\mapsto x^{2}+2x+1\right)\,
  8. x ( y A ( x ) B ( z ) ) \forall x\,(\exists y\,A(x)\vee B(z))
  9. f = [ ( x 1 , , x n ) t ] f=\left[(x_{1},\ldots,x_{n})\mapsto\operatorname{t}\right]
  10. x 𝔸 x 2 \sum_{x\in\mathbb{A}}{x^{2}}
  11. 𝔸 ( x x 2 ) \sum_{\mathbb{A}}{(x\mapsto x^{2})}
  12. 𝕊 f \sum_{\mathbb{S}}{f}
  13. x 𝕊 P ( x ) \forall x\in\mathbb{S}\ P(x)

Freiling's_axiom_of_symmetry.html

  1. κ \kappa\,
  2. 0 \aleph_{0}\,
  3. 𝙰𝚇 κ . \texttt{AX}_{\kappa}.\,
  4. f : 𝒫 ( κ ) 𝒫 𝒫 ( κ ) f:\mathcal{P}(\kappa)\to\mathcal{P}\mathcal{P}(\kappa)\,
  5. κ \leq\kappa
  6. ( x , y 𝒫 ( κ ) ) (\forall{x,y\in\mathcal{P}(\kappa)})\,
  7. x f ( y ) x\in f(y)\,
  8. y f ( x ) y\in f(x)\,
  9. 𝚉𝙵𝙲 2 κ = κ + ¬ 𝙰𝚇 κ . \texttt{ZFC}\vdash 2^{\kappa}=\kappa^{+}\leftrightarrow\neg\texttt{AX}_{\kappa% }.\,
  10. \Rightarrow\,
  11. 2 κ = κ + 2^{\kappa}=\kappa^{+}\,
  12. σ : κ + 𝒫 ( κ ) \sigma:\kappa^{+}\to\mathcal{P}(\kappa)\,
  13. f : 𝒫 ( κ ) 𝒫 𝒫 ( κ ) f:\mathcal{P}(\kappa)\to\mathcal{P}\mathcal{P}(\kappa)\,
  14. : σ ( α ) { σ ( β ) : β α } :\sigma(\alpha)\mapsto\{\sigma(\beta):\beta\preceq\alpha\}\,
  15. \Leftarrow\,
  16. f f\,
  17. 𝒫 ( κ ) \mathcal{P}(\kappa)\,
  18. A f B A\leq_{f}B
  19. A f ( B ) A\in f(B)
  20. κ \leq\kappa
  21. ( A α 𝒫 ( κ ) ) α < κ + (A_{\alpha}\in\mathcal{P}(\kappa))_{\alpha<\kappa^{+}}
  22. A α 𝒫 ( κ ) ξ < α f ( A ξ ) A_{\alpha}\in\mathcal{P}(\kappa)\setminus\bigcup_{\xi<\alpha}f(A_{\xi})
  23. α < κ + \alpha<\kappa^{+}\,
  24. ξ < α f ( A ξ ) \bigcup_{\xi<\alpha}f(A_{\xi})\,
  25. κ \leq\kappa\,
  26. κ \leq\kappa\,
  27. κ < 2 κ \leq\kappa<2^{\kappa}\,
  28. 𝒫 ( κ ) \mathcal{P}(\kappa)\,
  29. 𝒫 ( κ ) \mathcal{P}(\kappa)\,
  30. f \leq_{f}\,
  31. A α A_{\alpha}\,
  32. B 𝒫 ( κ ) B\in\mathcal{P}(\kappa)\,
  33. f \leq_{f}\,
  34. A α A_{\alpha}
  35. ( α < κ + ) A α f B (\forall{\alpha<\kappa^{+}})A_{\alpha}\leq_{f}B\,
  36. B B\,
  37. κ + > κ \geq\kappa^{+}>\kappa\,
  38. g : 𝒫 ( κ ) κ + g:\mathcal{P}(\kappa)\to\kappa^{+}\,
  39. B min { α < κ + : B f ( A α ) } B\mapsto\operatorname{min}\{\alpha<\kappa^{+}:B\in f(A_{\alpha})\}
  40. 𝒫 ( κ ) = α < κ + g - 1 { α } = α < κ + f ( A α ) \mathcal{P}(\kappa)=\bigcup_{\alpha<\kappa^{+}}g^{-1}\{\alpha\}=\bigcup_{% \alpha<\kappa^{+}}f(A_{\alpha})\,
  41. κ + \kappa^{+}\,
  42. κ \leq\kappa\,
  43. 2 κ κ + κ = κ + 2^{\kappa}\leq\kappa^{+}\cdot\kappa=\kappa^{+}\,
  44. | [ 0 , 1 ] | = | 𝒫 ( 0 ) | |[0,1]|=|\mathcal{P}(\aleph_{0})|\,
  45. ¬ 𝙲𝙷 \neg\texttt{CH}\Leftrightarrow\,
  46. 𝚉𝙵 ( 𝙰𝙲 𝒫 ( κ ) + ¬ 𝙰𝚇 κ ) 𝙲𝙷 κ \texttt{ZF}\vdash(\texttt{AC}_{\mathcal{P}(\kappa)}+\neg\texttt{AX}_{\kappa})% \leftrightarrow\texttt{CH}_{\kappa}\,
  47. 2 κ = κ + ¬ 𝙰𝚇 κ 2^{\kappa}=\kappa^{+}\Leftrightarrow\neg\texttt{AX}_{\kappa}\,
  48. 2 κ = κ + 2^{\kappa}=\kappa^{+}\,
  49. 𝒫 ( κ ) \mathcal{P}(\kappa)\,
  50. κ \kappa\,
  51. κ = 0 \kappa=\aleph_{0}\,

Frequency_distribution.html

  1. N u m b e r o f C l a s s e s = C = 1 + 3.3 l o g ( n ) NumberofClasses=C=1+3.3log(n)
  2. C = n ( a p p r o x i m a t e l y ) C=\sqrt{n}(approximately)
  3. h = R a n g e N u m b e r o f C l a s s e s h=\frac{Range}{NumberofClasses}

Fresnel_integral.html

  1. π 2 t 2 \frac{\pi}{2}t^{2}
  2. t 2 t^{2}
  3. S ( x ) = 0 x sin ( t 2 ) d t , C ( x ) = 0 x cos ( t 2 ) d t . S(x)=\int_{0}^{x}\sin(t^{2})\,\mathrm{d}t,\quad C(x)=\int_{0}^{x}\cos(t^{2})\,% \mathrm{d}t.
  4. S ( x ) = 0 x sin ( t 2 ) d t = n = 0 ( - 1 ) n x 4 n + 3 ( 2 n + 1 ) ! ( 4 n + 3 ) S(x)=\int_{0}^{x}\sin(t^{2})\,\mathrm{d}t=\sum_{n=0}^{\infty}(-1)^{n}\frac{x^{% 4n+3}}{(2n+1)!(4n+3)}
  5. C ( x ) = 0 x cos ( t 2 ) d t = n = 0 ( - 1 ) n x 4 n + 1 ( 2 n ) ! ( 4 n + 1 ) C(x)=\int_{0}^{x}\cos(t^{2})\,\mathrm{d}t=\sum_{n=0}^{\infty}(-1)^{n}\frac{x^{% 4n+1}}{(2n)!(4n+1)}
  6. π 2 t 2 \frac{\pi}{2}t^{2}
  7. 2 π \sqrt{\frac{2}{\pi}}
  8. π 2 \sqrt{\frac{\pi}{2}}
  9. d x = C ( t ) d t = cos ( t 2 ) d t \mathrm{d}x=C^{\prime}(t)\,\mathrm{d}t=\cos(t^{2})\,\mathrm{d}t\,
  10. d y = S ( t ) d t = sin ( t 2 ) d t \mathrm{d}y=S^{\prime}(t)\,\mathrm{d}t=\sin(t^{2})\,\mathrm{d}t\,
  11. L = 0 t 0 d x 2 + d y 2 = 0 t 0 d t = t 0 L=\int_{0}^{t_{0}}{\sqrt{\mathrm{d}x^{2}+\mathrm{d}y^{2}}}=\int_{0}^{t_{0}}{% \mathrm{d}t}=t_{0}
  12. t t
  13. c o s ( t ² ) , s i n ( t ² ) ) cos(t²),sin(t²))
  14. t ²
  15. κ \kappa
  16. κ = 1 R = d θ d t = 2 t \kappa=\tfrac{1}{R}=\tfrac{\mathrm{d}\theta}{\mathrm{d}t}=2t
  17. d κ d t = d 2 θ d t 2 = 2 \tfrac{\mathrm{d}\kappa}{\mathrm{d}t}=\tfrac{\mathrm{d}^{2}\theta}{\mathrm{d}t% ^{2}}=2
  18. t t
  19. x x\to\infty
  20. S ( x ) = π 2 ( sign ( x ) 2 - [ 1 + O ( x - 4 ) ] ( cos ( x 2 ) x 2 π + sin ( x 2 ) x 3 8 π ) ) , S(x)=\sqrt{\frac{\pi}{2}}\left(\frac{\mbox{sign}~{}{(x)}}{2}-\left[1+O(x^{-4})% \right]\left(\frac{\cos{(x^{2})}}{x\sqrt{2\pi}}+\frac{\sin{(x^{2})}}{x^{3}% \sqrt{8\pi}}\right)\right),
  21. C ( x ) = π 2 ( sign ( x ) 2 + [ 1 + O ( x - 4 ) ] ( sin ( x 2 ) x 2 π - cos ( x 2 ) x 3 8 π ) ) . C(x)=\sqrt{\frac{\pi}{2}}\left(\frac{\mbox{sign}~{}{(x)}}{2}+\left[1+O(x^{-4})% \right]\left(\frac{\sin{(x^{2})}}{x\sqrt{2\pi}}-\frac{\cos{(x^{2})}}{x^{3}% \sqrt{8\pi}}\right)\right).
  22. S ( z ) = π 2 1 + i 4 [ erf ( 1 + i 2 z ) - i erf ( 1 - i 2 z ) ] , S(z)=\sqrt{\frac{\pi}{2}}\frac{1+i}{4}\left[\operatorname{erf}\left(\frac{1+i}% {\sqrt{2}}z\right)-i\operatorname{erf}\left(\frac{1-i}{\sqrt{2}}z\right)\right],
  23. C ( z ) = π 2 1 - i 4 [ erf ( 1 + i 2 z ) + i erf ( 1 - i 2 z ) ] . C(z)=\sqrt{\frac{\pi}{2}}\frac{1-i}{4}\left[\operatorname{erf}\left(\frac{1+i}% {\sqrt{2}}z\right)+i\operatorname{erf}\left(\frac{1-i}{\sqrt{2}}z\right)\right].
  24. S ( z ) + i C ( z ) = π 2 1 + i 2 erf ( 1 + i 2 z ) S(z)+iC(z)=\sqrt{\frac{\pi}{2}}\frac{1+i}{2}\operatorname{erf}\left(\frac{1+i}% {\sqrt{2}}z\right)
  25. 0 cos t 2 d t = 0 sin t 2 d t = 2 π 4 = π 8 . \int_{0}^{\infty}\cos t^{2}\,\mathrm{d}t=\int_{0}^{\infty}\sin t^{2}\,\mathrm{% d}t=\frac{\sqrt{2\pi}}{4}=\sqrt{\frac{\pi}{8}}.
  26. C C
  27. S S
  28. e - t 2 e^{-t^{2}}
  29. x x
  30. y = x y=x
  31. x 0 x≥0
  32. R R
  33. R R
  34. 0
  35. 0 e - t 2 d t = π 2 , \int_{0}^{\infty}e^{-t^{2}}\mathrm{d}t=\frac{\sqrt{\pi}}{2},
  36. x m exp ( i x n ) d x = l = 0 i l x m + n l l ! d x = l = 0 i l ( m + n l + 1 ) x m + n l + 1 l ! \int x^{m}\exp(ix^{n})\,\mathrm{d}x=\int\sum_{l=0}^{\infty}\frac{i^{l}x^{m+nl}% }{l!}\,\mathrm{d}x=\sum_{l=0}^{\infty}\frac{i^{l}}{(m+nl+1)}\frac{x^{m+nl+1}}{% l!}
  37. x m exp ( i x n ) d x = x m + 1 m + 1 1 F 1 ( m + 1 n 1 + m + 1 n i x n ) = 1 n i ( m + 1 ) / n γ ( m + 1 n , - i x n ) , \int x^{m}\exp(ix^{n})\,\mathrm{d}x=\frac{x^{m+1}}{m+1}\,_{1}F_{1}\left(\begin% {array}[]{c}\frac{m+1}{n}\\ 1+\frac{m+1}{n}\end{array}\mid ix^{n}\right)=\frac{1}{n}i^{(m+1)/n}\gamma\left% (\frac{m+1}{n},-ix^{n}\right),
  38. x m sin ( x n ) d x = x m + n + 1 m + n + 1 1 F 2 ( 1 2 + m + 1 2 n 3 2 + m + 1 2 n , 3 2 - x 2 n 4 ) \int x^{m}\sin(x^{n})\,\mathrm{d}x=\frac{x^{m+n+1}}{m+n+1}\,_{1}F_{2}\left(% \begin{array}[]{c}\frac{1}{2}+\frac{m+1}{2n}\\ \frac{3}{2}+\frac{m+1}{2n},\frac{3}{2}\end{array}\mid-\frac{x^{2n}}{4}\right)
  39. F 1 1 ( m + 1 n 1 + m + 1 n i x n ) m + 1 n Γ ( m + 1 n ) e i π ( m + 1 ) / ( 2 n ) x - m - 1 {}_{1}F_{1}\left(\begin{array}[]{c}\frac{m+1}{n}\\ 1+\frac{m+1}{n}\end{array}\mid ix^{n}\right)\sim\frac{m+1}{n}\,\Gamma\left(% \frac{m+1}{n}\right)e^{i\pi(m+1)/(2n)}x^{-m-1}
  40. 0 x m exp ( i x n ) d x = 1 n Γ ( m + 1 n ) e i π ( m + 1 ) / ( 2 n ) \int_{0}^{\infty}x^{m}\exp(ix^{n})\,\mathrm{d}x=\frac{1}{n}\,\Gamma\left(\frac% {m+1}{n}\right)e^{i\pi(m+1)/(2n)}
  41. 0 sin ( x a ) d x = Γ ( 1 + 1 a ) sin ( π 2 a ) \int_{0}^{\infty}\sin(x^{a})\,\mathrm{d}x=\Gamma\left(1+\frac{1}{a}\right)\sin% \left(\frac{\pi}{2a}\right)
  42. Γ ( a - 1 ) \Gamma(a^{-1})
  43. x m exp ( i x n ) d x = V n , m ( x ) e i x n \int x^{m}\exp(ix^{n})\,\mathrm{d}x=V_{n,m}(x)e^{ix^{n}}
  44. V n , m := x m + 1 m + 1 1 F 1 ( 1 1 + m + 1 n - i x n ) V_{n,m}:=\frac{x^{m+1}}{m+1}\,_{1}F_{1}\left(\begin{array}[]{c}1\\ 1+\frac{m+1}{n}\end{array}\mid-ix^{n}\right)
  45. π / 2 \sqrt{\pi/2}

Fret.html

  1. 2 12 \sqrt[12]{2}

Fréchet_space.html

  1. x 0 \|x\|\geq 0
  2. x + y x + y \|x+y\|\leq\|x\|+\|y\|
  3. c x = | c | x \|c\cdot x\|=|c|\|x\|
  4. d ( x , y ) = k = 0 2 - k x - y k 1 + x - y k x , y X . d(x,y)=\sum_{k=0}^{\infty}2^{-k}\frac{\|x-y\|_{k}}{1+\|x-y\|_{k}}\qquad x,y\in X.
  5. f k = sup { | f ( k ) ( x ) | : x [ 0 , 1 ] } \|f\|_{k}=\sup\{|f^{(k)}(x)|:x\in[0,1]\}
  6. f n ( k ) f_{n}^{(k)}
  7. f k , n = sup { | f ( k ) ( x ) | : x [ - n , n ] } \|f\|_{k,n}=\sup\{|f^{(k)}(x)|:x\in[-n,n]\}
  8. f k , n = sup { | f ( k ) ( x ) | : x [ - n , n ] } \|f\|_{k,n}=\sup\{|f^{(k)}(x)|:x\in[-n,n]\}
  9. f n = sup { | f ( z ) | : | z | n } \|f\|_{n}=\sup\{|f(z)|:|z|\leq n\}
  10. f n = sup z exp [ - ( τ + 1 n ) | z | ] | f ( z ) | \|f\|_{n}=\sup_{z\in\mathbb{C}}\exp\left[-\left(\tau+\frac{1}{n}\right)|z|% \right]|f(z)|
  11. K n = { y M | d ( x , y ) n } K_{n}=\{y\in M|d(x,y)\leq n\}
  12. s n = j = 0 n sup x M | D j s | \|s\|_{n}=\sum_{j=0}^{n}\sup_{x\in M}|D^{j}s|
  13. D ( P ) ( x ) ( h ) = lim t 0 1 t ( P ( x + t h ) - P ( x ) ) D(P)(x)(h)=\lim_{t\to 0}\,\frac{1}{t}\Big(P(x+th)-P(x)\Big)
  14. D ( P ) : U × X Y D(P):U\times X\to Y
  15. D ( P ) ( f ) ( h ) = h D(P)(f)(h)=h^{\prime}
  16. x ( t ) = P ( x ( t ) ) , x ( 0 ) = x 0 U x^{\prime}(t)=P(x(t)),\quad x(0)=x_{0}\in U

Frustum.html

  1. V = 1 3 h ( a 2 + a b + b 2 ) . V=\frac{1}{3}h(a^{2}+ab+b^{2}).
  2. V = h 1 B 1 - h 2 B 2 3 V=\frac{h_{1}B_{1}-h_{2}B_{2}}{3}
  3. B 1 h 1 2 = B 2 h 2 2 = B 1 B 2 h 1 h 2 = α \frac{B_{1}}{h_{1}^{2}}=\frac{B_{2}}{h_{2}^{2}}=\frac{\sqrt{B_{1}B_{2}}}{h_{1}% h_{2}}=\alpha
  4. V = h 1 a h 1 2 - h 2 a h 2 2 3 = a 3 ( h 1 3 - h 2 3 ) V=\frac{h_{1}ah_{1}^{2}-h_{2}ah_{2}^{2}}{3}=\frac{a}{3}(h_{1}^{3}-h_{2}^{3})
  5. V = h 3 ( B 1 + B 1 B 2 + B 2 ) V=\frac{h}{3}(B_{1}+\sqrt{B_{1}B_{2}}+B_{2})
  6. V = π h 3 ( R 1 2 + R 1 R 2 + R 2 2 ) V=\frac{\pi h}{3}(R_{1}^{2}+R_{1}R_{2}+R_{2}^{2})
  7. V = n h 12 ( a 1 2 + a 1 a 2 + a 2 2 ) cot π n V=\frac{nh}{12}(a_{1}^{2}+a_{1}a_{2}+a_{2}^{2})\cot\frac{\pi}{n}
  8. Lateral Surface Area = π ( R 1 + R 2 ) s = π ( R 1 + R 2 ) ( R 1 - R 2 ) 2 + h 2 \begin{aligned}\displaystyle\,\text{Lateral Surface Area}&\displaystyle=\pi(R_% {1}+R_{2})s\\ &\displaystyle=\pi(R_{1}+R_{2})\sqrt{(R_{1}-R_{2})^{2}+h^{2}}\end{aligned}
  9. Total Surface Area = π ( ( R 1 + R 2 ) s + R 1 2 + R 2 2 ) = π ( ( R 1 + R 2 ) ( R 1 - R 2 ) 2 + h 2 + R 1 2 + R 2 2 ) \begin{aligned}\displaystyle\,\text{Total Surface Area}&\displaystyle=\pi((R_{% 1}+R_{2})s+R_{1}^{2}+R_{2}^{2})\\ &\displaystyle=\pi((R_{1}+R_{2})\sqrt{(R_{1}-R_{2})^{2}+h^{2}}+R_{1}^{2}+R_{2}% ^{2})\end{aligned}
  10. A = n 4 [ ( a 1 2 + a 2 2 ) cot π n + ( a 1 2 - a 2 2 ) 2 sec 2 π n + 4 h 2 ( a 1 + a 2 ) 2 ] A=\frac{n}{4}\left[(a_{1}^{2}+a_{2}^{2})\cot\frac{\pi}{n}+\sqrt{(a_{1}^{2}-a_{% 2}^{2})^{2}\sec^{2}\frac{\pi}{n}+4h^{2}(a_{1}+a_{2})^{2}}\right]

Fubini's_theorem.html

  1. X ( Y f ( x , y ) d y ) d x = Y ( X f ( x , y ) d x ) d y = X × Y f ( x , y ) d ( x , y ) \int_{X}\left(\int_{Y}f(x,y)\,\,\text{d}y\right)\,\,\text{d}x=\int_{Y}\left(% \int_{X}f(x,y)\,\,\text{d}x\right)\,\,\text{d}y=\int_{X\times Y}f(x,y)\,\,% \text{d}(x,y)
  2. X × Y | f ( x , y ) | d ( x , y ) < , \int_{X\times Y}|f(x,y)|\,\,\text{d}(x,y)<\infty,
  3. X ( Y f ( x , y ) d y ) d x = Y ( X f ( x , y ) d x ) d y = X × Y f ( x , y ) d ( x , y ) . \int_{X}\left(\int_{Y}f(x,y)\,\,\text{d}y\right)\,\,\text{d}x=\int_{Y}\left(% \int_{X}f(x,y)\,\,\text{d}x\right)\,\,\text{d}y=\int_{X\times Y}f(x,y)\,\,% \text{d}(x,y).
  4. Y f ( x , y ) d y , X f ( x , y ) d x \int_{Y}f(x,y)\,\,\text{d}y,\int_{X}f(x,y)\,\,\text{d}x
  5. | f | |f|
  6. f f
  7. X ( Y f ( x , y ) d y ) d x = Y ( X f ( x , y ) d x ) d y = X × Y f ( x , y ) d ( x , y ) . \int_{X}\left(\int_{Y}f(x,y)\,\,\text{d}y\right)\,\,\text{d}x=\int_{Y}\left(% \int_{X}f(x,y)\,\,\text{d}x\right)\,\,\text{d}y=\int_{X\times Y}f(x,y)\,\,% \text{d}(x,y).
  8. x y a x y = y x a x y \sum_{x}\sum_{y}a_{xy}=\sum_{y}\sum_{x}a_{xy}
  9. a x y a_{xy}
  10. + +\infty
  11. - -\infty
  12. X ( Y | f ( x , y ) | d y ) d x \int_{X}\left(\int_{Y}|f(x,y)|\,\,\text{d}y\right)\,\,\text{d}x
  13. Y ( X | f ( x , y ) | d x ) d y \int_{Y}\left(\int_{X}|f(x,y)|\,\,\text{d}x\right)\,\,\text{d}y
  14. X × Y | f ( x , y ) | d ( x , y ) \int_{X\times Y}|f(x,y)|\,\,\text{d}(x,y)
  15. X ( Y f ( x , y ) d y ) d x = Y ( X f ( x , y ) d x ) d y = X × Y f ( x , y ) d ( x , y ) . \int_{X}\left(\int_{Y}f(x,y)\,\,\text{d}y\right)\,\,\text{d}x=\int_{Y}\left(% \int_{X}f(x,y)\,\,\text{d}x\right)\,\,\text{d}y=\int_{X\times Y}f(x,y)\,\,% \text{d}(x,y).
  16. x 2 - y 2 ( x 2 + y 2 ) 2 = - 2 x y arctan ( y / x ) . \frac{x^{2}-y^{2}}{(x^{2}+y^{2})^{2}}=-\frac{\partial^{2}}{\partial x\partial y% }\arctan(y/x).
  17. x = 0 1 ( y = 0 1 x 2 - y 2 ( x 2 + y 2 ) 2 d y ) d x = π 4 \int_{x=0}^{1}\left(\int_{y=0}^{1}\frac{x^{2}-y^{2}}{(x^{2}+y^{2})^{2}}\,\,% \text{d}y\right)\,\,\text{d}x=\frac{\pi}{4}
  18. y = 0 1 ( x = 0 1 x 2 - y 2 ( x 2 + y 2 ) 2 d x ) d y = - π 4 \int_{y=0}^{1}\left(\int_{x=0}^{1}\frac{x^{2}-y^{2}}{(x^{2}+y^{2})^{2}}\,\,% \text{d}x\right)\,\,\text{d}y=-\frac{\pi}{4}
  19. 0 1 0 1 | x 2 - y 2 ( x 2 + y 2 ) 2 | d y d x = . \int_{0}^{1}\int_{0}^{1}\left|\frac{x^{2}-y^{2}}{(x^{2}+y^{2})^{2}}\right|\,\,% \text{d}y\,\,\text{d}x=\infty.

Full_moon_cycle.html

  1. F C = S M × A M S M - A M = 411.78443 d FC=\frac{SM\times AM}{SM-AM}=411.78443d

Function_(mathematics).html

  1. f : X Y f\colon X\rightarrow Y
  2. X f Y . X\stackrel{f}{\rightarrow}Y.
  3. y = f ( x ) . y=f(x).
  4. \mapsto
  5. f : x 4 - x . \begin{aligned}\displaystyle f\colon\mathbb{N}&\displaystyle\to\mathbb{Z}\\ \displaystyle x&\displaystyle\mapsto 4-x.\end{aligned}
  6. \mathbb{N}
  7. \mathbb{Z}
  8. \mathbb{Z}
  9. \mathbb{N}
  10. g : \displaystyle g\colon\mathbb{Z}
  11. f ( x ) = x 2 - 5 x + 6 f(x)=\sqrt{x^{2}-5x+6}
  12. a ( ) 2 \scriptstyle a(\cdot)^{2}
  13. x a x 2 \textstyle x\mapsto ax^{2}
  14. a f ( u ) d u \scriptstyle\int_{a}^{\,\cdot}f(u)du
  15. x a x f ( u ) d u \scriptstyle x\mapsto\int_{a}^{x}f(u)du
  16. n ( n - 1 ) ! n(n-1)!
  17. f - 1 ( B ) = { x X : f ( x ) B } . f^{-1}(B)=\{x\in X:f(x)\in B\}.
  18. f - 1 ( b ) = { x X : f ( x ) = b } . f^{-1}(b)=\{x\in X:f(x)=b\}.
  19. g f : X Z g\circ f\colon X\rightarrow Z
  20. ( g f ) ( x ) = g ( f ( x ) ) . (g\circ f)(x)=g(f(x)).
  21. g f g\circ f
  22. g f g\circ f
  23. f g f\circ g
  24. g f = f g . g\circ f=f\circ g.
  25. f id X = f , id Y f = f . \begin{aligned}\displaystyle f\circ\operatorname{id}_{X}&\displaystyle=f,\\ \displaystyle\operatorname{id}_{Y}\circ f&\displaystyle=f.\end{aligned}
  26. f f - 1 = id Y , f - 1 f = id X . f\circ f^{-1}=\operatorname{id}_{Y},f^{-1}\circ f=\operatorname{id}_{X}.
  27. f ( C ) = 9 5 C + 32 f - 1 ( F ) = 5 9 ( F - 32 ) \begin{aligned}\displaystyle f(C)&\displaystyle=\frac{9}{5}C+32\\ \displaystyle f^{-1}(F)&\displaystyle=\frac{5}{9}(F-32)\end{aligned}
  28. g f g\circ f
  29. g f gf
  30. ( f + g ) ( x ) \displaystyle(f+g)(x)
  31. f ( x ) = x f(x)=\sqrt{x}

Function_composition.html

  1. f : X Y f:X→Y
  2. g : Y Z g:Y→Z
  3. x x
  4. X X
  5. g ( f ( x ) ) g(f(x))
  6. Z Z
  7. z z
  8. y y
  9. y y
  10. x x
  11. z z
  12. x x
  13. g f : X Z g∘f:X→Z
  14. ( g f ) ( x ) = g ( f ( x ) ) (g∘f)(x)=g(f(x))
  15. x x
  16. X X
  17. g f g∘f
  18. g g
  19. f f
  20. g g
  21. f f
  22. g g
  23. f f
  24. g g
  25. f f
  26. g g
  27. f f
  28. g g
  29. f f
  30. g g
  31. f f
  32. g f g∘f
  33. f f
  34. g g
  35. ( g f ) ( c ) = # (g∘f)(c)=\#
  36. f : f:ℝ→ℝ
  37. f ( x ) = 2 x + 4 f(x)=2x+4
  38. g : g:ℝ→ℝ
  39. t t
  40. h ( t ) h(t)
  41. x x
  42. c ( x ) c(x)
  43. ( c h ) ( t ) (c∘h)(t)
  44. t t
  45. f f
  46. g g
  47. h h
  48. f ( g h ) = ( f g ) h f∘(g∘h)=(f∘g)∘h
  49. g f g∘f
  50. f f
  51. g g
  52. f f
  53. f f
  54. g g
  55. g f g∘f
  56. f : ( , + 9 f:ℝ→(−∞,+9
  57. g : 0 , + ) ) g:0,+∞))→ℝ
  58. g ( x ) = x ¯ g(x)=√\overline{x}
  59. 3 , + 33 −3,+33
  60. g g
  61. f f
  62. g f = f g g∘f=f∘g
  63. | x | + 3 = | x + 3 | |x|+3=|x+3|
  64. x 0 x≥0
  65. f : X X , f:X→X,
  66. g : X X g:X→X
  67. f f g f f∘f∘g∘f
  68. f : X X f:X→X
  69. X X
  70. H H
  71. R R
  72. R R
  73. H H
  74. f : X X f:X→X
  75. Y X Y⊆X
  76. f : X Y f:X→Y
  77. n 2 n≥2
  78. n n
  79. f f
  80. Y = X Y=X
  81. f : X X f:X→X
  82. n > 0 n>0
  83. f f
  84. f f
  85. n n
  86. f f
  87. f f
  88. g g = f g∘g=f
  89. g g
  90. f f
  91. n > 0 n>0
  92. g f gf
  93. g f g∘f
  94. g f g∘f
  95. f f
  96. g g
  97. x f xf
  98. f ( x ) f(x)
  99. ( x f ) g (xf)g
  100. g ( f ( x ) ) g(f(x))
  101. x x
  102. f f
  103. g g
  104. f g fg
  105. f f
  106. g g
  107. f g fg
  108. f ; g f;g
  109. g g
  110. C g f = f g . C_{g}f=f\circ g.
  111. f f
  112. g g
  113. f f
  114. g g
  115. f | x i = g = f ( x 1 , , x i - 1 , g ( x 1 , x 2 , , x n ) , x i + 1 , , x n ) . f|_{x_{i}=g}=f(x_{1},\ldots,x_{i-1},g(x_{1},x_{2},\ldots,x_{n}),x_{i+1},\ldots% ,x_{n}).
  116. g g
  117. b b
  118. f | x i = b = f ( x 1 , , x i - 1 , b , x i + 1 , , x n ) . f|_{x_{i}=b}=f(x_{1},\ldots,x_{i-1},b,x_{i+1},\ldots,x_{n}).
  119. f f
  120. n n
  121. n n
  122. m m
  123. f f
  124. m m
  125. h ( x 1 , , x m ) = f ( g 1 ( x 1 , , x m ) , , g n ( x 1 , , x m ) ) h(x_{1},\ldots,x_{m})=f(g_{1}(x_{1},\ldots,x_{m}),\ldots,g_{n}(x_{1},\ldots,x_% {m}))
  126. f ( g ( a 11 , , a 1 m ) , , g ( a n 1 , , a n m ) ) = g ( f ( a 11 , , a n 1 ) , , g ( a 1 m , , a m n ) ) f(g(a_{11},\ldots,a_{1m}),\ldots,g(a_{n1},\ldots,a_{nm}))=g(f(a_{11},\ldots,a_% {n1}),\ldots,g(a_{1m},\ldots,a_{mn}))
  127. R X × Y R⊆X×Y
  128. S Y × Z S⊆Y×Z
  129. S R S∘R
  130. ( f g ) < s u p > 1 = ( g 1 f 1 ) (f∘g)<sup>−1=(g^{−1}∘f^{−1})
  131. f g : X Z f∘g:X→Z
  132. ( f g ) ( x ) = g ( f ( x ) ) (f∘g)(x)=g(f(x))

Functional_dependency.html

  1. π X , Y R \pi_{X,Y}R
  2. π X Y ( R ) π X Z ( R ) = R \pi_{XY}(R)\bowtie\pi_{XZ}(R)=R
  3. Σ \Sigma
  4. Γ \Gamma
  5. Σ \Sigma
  6. Γ \Gamma
  7. Σ Γ \Sigma\models\Gamma
  8. X X\rightarrow\varnothing
  9. X \vdash X\rightarrow\varnothing
  10. X Y X Z Y Z X\rightarrow Y\vdash XZ\rightarrow YZ
  11. X Y , Y Z X Z X\rightarrow Y,Y\rightarrow Z\vdash X\rightarrow Z
  12. R R
  13. F F
  14. R R
  15. F F
  16. R R
  17. F F
  18. F F
  19. F F
  20. F F
  21. F F
  22. G G
  23. G G
  24. F F
  25. F F
  26. G G
  27. G G
  28. F F
  29. F F
  30. G G
  31. R R
  32. F F
  33. G G
  34. F F
  35. G G
  36. F F
  37. G G
  38. F F
  39. G G
  40. F F
  41. F F^{\prime}
  42. F F
  43. F F^{\prime}
  44. F F
  45. F F^{\prime}
  46. F F
  47. F F
  48. G G
  49. F F
  50. G G
  51. F F
  52. F F
  53. F F
  54. F F
  55. \models
  56. F F
  57. F F
  58. F F
  59. \models
  60. R = π X Y ( R ) π X Z ( R ) R=\pi_{XY}(R)\bowtie\pi_{XZ}(R)
  61. π X Y ( R ) π X Z ( R ) \pi_{XY}(R)\bowtie\pi_{XZ}(R)
  62. π X Y ( R ) \pi_{XY}(R)
  63. π X Z ( R ) \pi_{XZ}(R)
  64. π X Z ( R ) \pi_{XZ}(R)
  65. π X Y ( R ) \pi_{XY}(R)

Functional_derivative.html

  1. L L
  2. f f
  3. δ f δf
  4. δ f δf
  5. δ f δf
  6. J [ f ] = a b L [ x , f ( x ) , f ( x ) ] d x , J[f]=\int\limits_{a}^{b}L[\,x,f(x),f\,^{\prime}(x)\,]\,dx\ ,
  7. f ( x ) d f / d x f′(x)≡df/dx
  8. f f
  9. δ f δf
  10. L ( x , f + δ f , f + δ f f ) L(x,f+δf,f+δff′)
  11. δ f δf
  12. J J
  13. δ f δf
  14. δ J = a b δ J δ f ( x ) δ f ( x ) d x . \delta J=\int_{a}^{b}\frac{\delta J}{\delta f(x)}{\delta f(x)}\,dx\,.
  15. δ f ( x ) δf(x)
  16. δ J / δ f ( x ) δJ/δf(x)
  17. J J
  18. f f
  19. x x
  20. δ J δ f ( x ) = L f - d d x L f . \frac{\delta J}{\delta f(x)}=\frac{\partial L}{\partial f}-\frac{d}{dx}\frac{% \partial L}{\partial f^{\prime}}\,.
  21. M M
  22. ρ ρ
  23. F F
  24. F : M or F : M , F\colon M\rightarrow\mathbb{R}\quad\mbox{or}~{}\quad F\colon M\rightarrow% \mathbb{C}\,,
  25. F F
  26. δ F / δ ρ δF/δρ
  27. δ F δ ρ ( x ) ϕ ( x ) d x = lim ε 0 F [ ρ + ε ϕ ] - F [ ρ ] ε = [ d d ϵ F [ ρ + ϵ ϕ ] ] ϵ = 0 , \begin{aligned}\displaystyle\int\frac{\delta F}{\delta\rho}(x)\phi(x)\;dx&% \displaystyle=\lim_{\varepsilon\to 0}\frac{F[\rho+\varepsilon\phi]-F[\rho]}{% \varepsilon}\\ &\displaystyle=\left[\frac{d}{d\epsilon}F[\rho+\epsilon\phi]\right]_{\epsilon=% 0},\end{aligned}
  28. ϕ ϕ
  29. ε ϕ εϕ
  30. ρ ρ
  31. ϕ [ d d ϵ F [ ρ + ϵ ϕ ] ] ϵ = 0 \phi\mapsto\left[\frac{d}{d\epsilon}F[\rho+\epsilon\phi]\right]_{\epsilon=0}
  32. δ F / δ ρ δF/δρ
  33. δ F / δ ρ δF/δρ
  34. F F
  35. ρ ρ
  36. δ F δ ρ ( x ) ϕ ( x ) d x \int\frac{\delta F}{\delta\rho}(x)\phi(x)\;dx
  37. ρ ρ
  38. ϕ ϕ
  39. F F
  40. δ F ( ρ , ϕ ) = δ F δ ρ ( x ) ϕ ( x ) d x , \delta F(\rho,\phi)=\int\frac{\delta F}{\delta\rho}(x)\ \phi(x)\ dx\ ,
  41. ϕ ϕ
  42. ρ ρ
  43. ϕ = δ ρ ϕ=δρ
  44. d F = i = 1 n F ρ i d ρ i , dF=\sum_{i=1}^{n}\frac{\partial F}{\partial\rho_{i}}\ d\rho_{i}\ ,
  45. δ F / δ ρ ( x ) δF/δρ(x)
  46. x x
  47. i i
  48. F F
  49. G G
  50. δ ( λ F + μ G ) [ ρ ] δ ρ ( x ) = λ δ F [ ρ ] δ ρ ( x ) + μ δ G [ ρ ] δ ρ ( x ) , \frac{\delta(\lambda F+\mu G)[\rho]}{\delta\rho(x)}=\lambda\frac{\delta F[\rho% ]}{\delta\rho(x)}+\mu\frac{\delta G[\rho]}{\delta\rho(x)},
  51. λ , μ λ,μ
  52. δ ( F G ) [ ρ ] δ ρ ( x ) = δ F [ ρ ] δ ρ ( x ) G [ ρ ] + F [ ρ ] δ G [ ρ ] δ ρ ( x ) , \frac{\delta(FG)[\rho]}{\delta\rho(x)}=\frac{\delta F[\rho]}{\delta\rho(x)}G[% \rho]+F[\rho]\frac{\delta G[\rho]}{\delta\rho(x)}\,,
  53. F F
  54. G G
  55. δ F [ G [ ρ ] ] δ ρ ( y ) = d x δ F [ G ( ρ ) ] δ G [ ρ ( x ) ] δ G [ ρ ] δ ρ ( y ) . \displaystyle\frac{\delta F[G[\rho]]}{\delta\rho(y)}=\int dx\frac{\delta F[G(% \rho)]}{\delta G[\rho(x)]}\ \frac{\delta G[\rho]}{\delta\rho(y)}\ .
  56. G G
  57. g g
  58. δ F [ g ( ρ ) ] δ ρ ( y ) = δ F [ g ( ρ ) ] δ g [ ρ ( x ) ] d g ( ρ ) d ρ ( y ) . \displaystyle\frac{\delta F[g(\rho)]}{\delta\rho(y)}=\frac{\delta F[g(\rho)]}{% \delta g[\rho(x)]}\ \frac{dg(\rho)}{d\rho(y)}\ .
  59. F [ ρ ] = f ( s y m b o l r , ρ ( s y m b o l r ) , ρ ( s y m b o l r ) ) d s y m b o l r , F[\rho]=\int f(symbol{r},\rho(symbol{r}),\nabla\rho(symbol{r}))\,dsymbol{r},
  60. ϕ ϕ
  61. r r
  62. δ F δ ρ ( s y m b o l r ) ϕ ( s y m b o l r ) d s y m b o l r = [ d d ε f ( s y m b o l r , ρ + ε ϕ , ρ + ε ϕ ) d s y m b o l r ] ε = 0 = ( f ρ ϕ + f ρ ϕ ) d s y m b o l r = [ f ρ ϕ + ( f ρ ϕ ) - ( f ρ ) ϕ ] d s y m b o l r = [ f ρ ϕ - ( f ρ ) ϕ ] d s y m b o l r = ( f ρ - f ρ ) ϕ ( s y m b o l r ) d s y m b o l r . \begin{aligned}\displaystyle\int\frac{\delta F}{\delta\rho(symbol{r})}\,\phi(% symbol{r})\,dsymbol{r}&\displaystyle=\left[\frac{d}{d\varepsilon}\int f(symbol% {r},\rho+\varepsilon\phi,\nabla\rho+\varepsilon\nabla\phi)\,dsymbol{r}\right]_% {\varepsilon=0}\\ &\displaystyle=\int\left(\frac{\partial f}{\partial\rho}\,\phi+\frac{\partial f% }{\partial\nabla\rho}\cdot\nabla\phi\right)dsymbol{r}\\ &\displaystyle=\int\left[\frac{\partial f}{\partial\rho}\,\phi+\nabla\cdot% \left(\frac{\partial f}{\partial\nabla\rho}\,\phi\right)-\left(\nabla\cdot% \frac{\partial f}{\partial\nabla\rho}\right)\phi\right]dsymbol{r}\\ &\displaystyle=\int\left[\frac{\partial f}{\partial\rho}\,\phi-\left(\nabla% \cdot\frac{\partial f}{\partial\nabla\rho}\right)\phi\right]dsymbol{r}\\ &\displaystyle=\int\left(\frac{\partial f}{\partial\rho}-\nabla\cdot\frac{% \partial f}{\partial\nabla\rho}\right)\phi(symbol{r})\ dsymbol{r}\,.\end{aligned}
  63. f / ∂f/∂∇
  64. f ρ = f ρ x 𝐢 ^ + f ρ y 𝐣 ^ + f ρ z 𝐤 ^ , where ρ x = ρ x , ρ y = ρ y , ρ z = ρ z and 𝐢 ^ , 𝐣 ^ , 𝐤 ^ are unit vectors along the x, y, z axes. \begin{aligned}\displaystyle\frac{\partial f}{\partial\nabla\rho}=\frac{% \partial f}{\partial\rho_{x}}\mathbf{\hat{i}}+\frac{\partial f}{\partial\rho_{% y}}\mathbf{\hat{j}}+\frac{\partial f}{\partial\rho_{z}}\mathbf{\hat{k}}\,,&% \displaystyle\,\text{where}\ \rho_{x}=\frac{\partial\rho}{\partial x}\,,\ \rho% _{y}=\frac{\partial\rho}{\partial y}\,,\ \rho_{z}=\frac{\partial\rho}{\partial z% }\\ &\displaystyle\,\text{and}\ \ \mathbf{\hat{i}},\ \mathbf{\hat{j}},\ \mathbf{% \hat{k}}\ \ \text{are unit vectors along the x, y, z axes.}\end{aligned}
  65. ϕ = 0 ϕ=0
  66. ϕ ϕ

Functional_integration.html

  1. G [ f ] [ D f ] - - G [ f ] x d f ( x ) \int{G[f][Df]}\equiv\int\limits_{-\infty}^{\infty}{...\int\limits_{-\infty}^{% \infty}{G[f]}}\prod_{x}df(x)
  2. f ( x ) = f n H n ( x ) f(x)=f_{n}H_{n}(x)
  3. G [ f ] [ D f ] - - G ( f 1 , f 2 , . . ) n d f n \int{G[f][Df]}\equiv\int\limits_{-\infty}^{\infty}{...\int\limits_{-\infty}^{% \infty}{G(f_{1},f_{2},..)}}\prod_{n}df_{n}
  4. e i - 1 2 f ( x ) K ( x , y ) f ( y ) d x d y + J ( x ) f ( x ) d x [ D f ] e i - 1 2 f ( x ) K ( x , y ) f ( y ) d x d y [ D f ] = e i 1 2 J ( x ) K - 1 ( x , y ) J ( y ) d x d y \frac{\int{e^{i\int{-\frac{1}{2}f(x)\cdot K(x,y)\cdot f(y)dxdy}+\int{J(x)\cdot f% (x)dx}}[Df]}}{\int{e^{i\int{-\frac{1}{2}f(x)\cdot K(x,y)\cdot f(y)dxdy}}[Df]}}% =e^{i\frac{1}{2}\int{J(x)\cdot K^{-1}(x,y)\cdot J(y)dxdy}}
  5. K ( x , y ) = δ ( x - y ) K(x,y)=\Box\delta(x-y)
  6. f ( a ) f ( b ) e i f ( x ) f ( x ) d x 4 [ D f ] e i f ( x ) f ( x ) d x 4 [ D f ] = K - 1 ( a , b ) = 1 | a - b | 2 \frac{\int{f(a)f(b)e^{i\int{f(x)\Box f(x)dx^{4}}}}[Df]}{\int{e^{i\int{f(x)\Box f% (x)dx^{4}}}}[Df]}=K^{-1}(a,b)=\frac{1}{|a-b|^{2}}
  7. e i f ( x ) g ( x ) d x [ D f ] = δ [ g ] = x δ ( g ( x ) ) \int{e^{i\int{f(x)g(x)dx}}}[Df]=\delta[g]=\prod_{x}\delta(g(x))
  8. ψ ( x ) \psi(x)
  9. ψ ( x ) ψ ( y ) = - ψ ( y ) ψ ( x ) \psi(x)\psi(y)=-\psi(y)\psi(x)
  10. Pr ( w ( s + t ) , t | w ( s ) , s ) = 1 2 π D t exp ( - w ( s + t ) - w ( s ) 2 2 D t ) \mathrm{Pr}(w(s+t),t|w(s),s)=\frac{1}{\sqrt{2\pi Dt}}\exp\left({-\frac{\|w(s+t% )-w(s)\|^{2}}{2Dt}}\right)

Functional_predicate.html

  1. [ F ] := { ( [ X ] , [ F ( X ) ] ) : [ X ] [ 𝐓 ] } , [F]:=\big\{([X],[F(X)]):[X]\in[\mathbf{T}]\big\},
  2. A , B , C , C A F ( C ) B . \forall A,\exists B,\forall C,C\in A\rightarrow F(C)\in B.
  3. A , B , C , C A D B . \forall A,\exists B,\forall C,C\in A\rightarrow D\in B.
  4. A , B , C , D , C A D B . \forall A,\exists B,\forall C,\forall D,C\in A\rightarrow D\in B.
  5. A , B , C , D , P ( C , D ) ( C A D B ) . \forall A,\exists B,\forall C,\forall D,P(C,D)\rightarrow(C\in A\rightarrow D% \in B).
  6. ( X , ! Y , P ( X , Y ) ) ( A , B , C , D , P ( C , D ) ( C A D B ) ) . (\forall X,\exists!Y,P(X,Y))\rightarrow(\forall A,\exists B,\forall C,\forall D% ,P(C,D)\rightarrow(C\in A\rightarrow D\in B)).

Futurama.html

  1. 0 \aleph_{0}

Futures_contract.html

  1. F ( t , T ) = S ( t ) × ( 1 + r ) ( T - t ) F(t,T)=S(t)\times(1+r)^{(T-t)}
  2. F ( t , T ) = S ( t ) e r ( T - t ) F(t,T)=S(t)e^{r(T-t)}\,
  3. F ( t ) = E t { S ( T ) } F(t)=E_{t}\left\{S(T)\right\}
  4. [ t , s ] [t,s]
  5. F ( s , T ) - F ( t , T ) F(s,T)-F(t,T)

G.711.html

  1. s ¯ \overline{s}
  2. m m
  3. e e
  4. s s
  5. y y
  6. y = ( - 1 ) s ( 16 min { e , 1 } + m + 0.5 ) 2 max { e , 1 } , y=(-1)^{s}\cdot(16\cdot\min\{e,1\}+m+0.5)\cdot 2^{\max\{e,1\}},
  7. 12 {}^{12}
  8. 6 {}^{6}
  9. m m
  10. e e
  11. s s
  12. y y
  13. y = ( - 1 ) s [ ( 16.5 + m ) 2 e + 1 - 33 ] , y=(-1)^{s}\cdot[(16.5+m)\cdot 2^{e+1}-33],

G2_(mathematics).html

  1. 𝔤 2 \mathfrak{g}_{2}
  2. GL ( 7 ) \operatorname{GL}(\mathbb{R}^{7})
  3. d x 124 + d x 235 + d x 346 + d x 450 + d x 561 + d x 602 + d x 013 , dx^{124}+dx^{235}+dx^{346}+dx^{450}+dx^{561}+dx^{602}+dx^{013},
  4. d x i j k dx^{ijk}
  5. d x i d x j d x k . dx^{i}\wedge dx^{j}\wedge dx^{k}.
  6. [ 2 - 1 - 3 2 ] \left[\begin{smallmatrix}\;\,\,2&-1\\ -3&\;\,\,2\end{smallmatrix}\right]
  7. C 1 = t 2 + u 2 + v 2 + w 2 + x 2 + y 2 + z 2 C_{1}=t^{2}+u^{2}+v^{2}+w^{2}+x^{2}+y^{2}+z^{2}
  8. C 2 = t u v + w t x + y w u + z y t + v z w + x v y + u x z C_{2}=tuv+wtx+ywu+zyt+vzw+xvy+uxz
  9. A λ 1 + + N λ 14 = [ 0 C - B E - D - G - F + M - C 0 A F - G + N D - K E + L B - A 0 - N M L K - E - F N 0 - A + H - B + I - C + J D G - N - M A - H 0 J - I G K - D - L B - I - J 0 H F - M - E - L - K C - J I - H 0 ] A\lambda_{1}+...+N\lambda_{14}=\begin{bmatrix}0&C&-B&E&-D&-G&-F+M\\ -C&0&A&F&-G+N&D-K&E+L\\ B&-A&0&-N&M&L&K\\ -E&-F&N&0&-A+H&-B+I&-C+J\\ D&G-N&-M&A-H&0&J&-I\\ G&K-D&-L&B-I&-J&0&H\\ F-M&-E-L&-K&C-J&I&-H&0\\ \end{bmatrix}

Galaxy_rotation_curve.html

  1. ρ ( r ) = 3 v ( r ) 2 4 π G r 2 ( 1 + 2 d log v ( r ) d log r ) \rho(r)=\frac{3v(r)^{2}}{4\pi Gr^{2}}\left(1+2~{}\frac{d\log~{}v(r)}{d\log~{}r% }\right)
  2. ρ r - 2 \rho\propto r^{-2}
  3. ρ ( r ) = ρ 0 r R s ( 1 + r R s ) 2 \rho(r)=\frac{\rho_{0}}{\frac{r}{R_{s}}\left(1~{}+~{}\frac{r}{R_{s}}\right)^{2}}
  4. v ( r ) {v(r)}
  5. v ( r ) = ( r d Φ / d r ) 1 / 2 v(r)=(r\,d\Phi/dr)^{1/2}
  6. Φ \Phi

Galilean_invariance.html

  1. r ( t ) = r ( t ) - v t . r^{\prime}(t)=r(t)-vt.\,
  2. u ( t ) = d d t r ( t ) = d d t r ( t ) - v = u ( t ) - v . u^{\prime}(t)=\frac{d}{dt}r^{\prime}(t)=\frac{d}{dt}r(t)-v=u(t)-v.
  3. a ( t ) = d d t u ( t ) = d d t u ( t ) - 0 = a ( t ) . a^{\prime}(t)=\frac{d}{dt}u^{\prime}(t)=\frac{d}{dt}u(t)-0=a(t).

Galilean_transformation.html

  1. x = x - v t x^{\prime}=x-vt
  2. y = y y^{\prime}=y
  3. z = z z^{\prime}=z
  4. t = t . t^{\prime}=t.
  5. ( x t ) = ( 1 - v 0 1 ) ( x t ) \begin{pmatrix}x^{\prime}\\ t^{\prime}\end{pmatrix}=\begin{pmatrix}1&-v\\ 0&1\end{pmatrix}\begin{pmatrix}x\\ t\end{pmatrix}
  6. ( x , t ) ( x + t v , t ) ({x},t)\mapsto({x}+t{v},t)
  7. ( x , t ) ( x + a , t + b ) ({x},t)\mapsto({x}+{a},t+b)
  8. ( x , t ) ( G x , t ) ({x},t)\mapsto(G{x},t)
  9. ( t , x , 1 ) ( 1 v 0 0 R 0 s y 1 ) = ( t + s , t v + x R + y , 1 ) , (t,\ x,\ 1)\begin{pmatrix}1&v&0\\ 0&R&0\\ s&y&1\end{pmatrix}=(t+s,tv+xR+y,1),
  10. G 1 = { m : s = 0 , y = 0 } , G_{1}=\{m:s=0,y=0\},
  11. G 2 = { m : v = 0 , R = I 3 } ( 𝐑 4 , + ) , G_{2}=\{m:v=0,R=I_{3}\}\cong(\mathbf{R}^{4},+),
  12. G 3 = { m : s = 0 , y = 0 , v = 0 } SO ( 3 ) , G_{3}=\{m:s=0,y=0,v=0\}\cong\mathrm{SO}(3),
  13. G 4 = { m : s = 0 , y = 0 , R = I 3 } ( 𝐑 3 , + ) , G_{4}=\{m:s=0,y=0,R=I_{3}\}\cong(\mathbf{R}^{3},+),
  14. A B A\rtimes B
  15. G 2 SGal ( 3 ) G_{2}\triangleleft\mathrm{SGal}(3)
  16. SGal ( 3 ) G 2 G 1 \mathrm{SGal}(3)\cong G_{2}\rtimes G_{1}
  17. G 4 G 1 G_{4}\trianglelefteq G_{1}
  18. G 1 G 4 G 3 G_{1}\cong G_{4}\rtimes G_{3}
  19. SGal ( 3 ) 𝐑 4 ( 𝐑 3 SO ( 3 ) ) . \mathrm{SGal}(3)\cong\mathbf{R}^{4}\rtimes(\mathbf{R}^{3}\rtimes\mathrm{SO}(3)).
  20. [ H , P i ] = 0 [H,P_{i}]=0
  21. [ P i , P j ] = 0 [P_{i},P_{j}]=0
  22. [ L i j , H ] = 0 [L_{ij},H]=0
  23. [ C i , C j ] = 0 [C_{i},C_{j}]=0
  24. [ L i j , L k l ] = i [ δ i k L j l - δ i l L j k - δ j k L i l + δ j l L i k ] [L_{ij},L_{kl}]=i[\delta_{ik}L_{jl}-\delta_{il}L_{jk}-\delta_{jk}L_{il}+\delta% _{jl}L_{ik}]
  25. [ L i j , P k ] = i [ δ i k P j - δ j k P i ] [L_{ij},P_{k}]=i[\delta_{ik}P_{j}-\delta_{jk}P_{i}]
  26. [ L i j , C k ] = i [ δ i k C j - δ j k C i ] [L_{ij},C_{k}]=i[\delta_{ik}C_{j}-\delta_{jk}C_{i}]
  27. [ C i , H ] = i P i [C_{i},H]=iP_{i}\,\!
  28. [ C i , P j ] = 0 . [C_{i},P_{j}]=0~{}.
  29. H H
  30. c c→∞
  31. c c
  32. c c→∞
  33. M M
  34. [ H , P i ] = 0 [H^{\prime},P^{\prime}_{i}]=0\,\!
  35. [ P i , P j ] = 0 [P^{\prime}_{i},P^{\prime}_{j}]=0\,\!
  36. [ L i j , H ] = 0 [L^{\prime}_{ij},H^{\prime}]=0\,\!
  37. [ C i , C j ] = 0 [C^{\prime}_{i},C^{\prime}_{j}]=0\,\!
  38. [ L i j , L k l ] = i [ δ i k L j l - δ i l L j k - δ j k L i l + δ j l L i k ] [L^{\prime}_{ij},L^{\prime}_{kl}]=i[\delta_{ik}L^{\prime}_{jl}-\delta_{il}L^{% \prime}_{jk}-\delta_{jk}L^{\prime}_{il}+\delta_{jl}L^{\prime}_{ik}]\,\!
  39. [ L i j , P k ] = i [ δ i k P j - δ j k P i ] [L^{\prime}_{ij},P^{\prime}_{k}]=i[\delta_{ik}P^{\prime}_{j}-\delta_{jk}P^{% \prime}_{i}]\,\!
  40. [ L i j , C k ] = i [ δ i k C j - δ j k C i ] [L^{\prime}_{ij},C^{\prime}_{k}]=i[\delta_{ik}C^{\prime}_{j}-\delta_{jk}C^{% \prime}_{i}]\,\!
  41. [ C i , H ] = i P i [C^{\prime}_{i},H^{\prime}]=iP^{\prime}_{i}\,\!
  42. [ C i , P j ] = i M δ i j . [C^{\prime}_{i},P^{\prime}_{j}]=iM\delta_{ij}~{}.

Galois_connection.html

  1. ( A , ) (A,≤)
  2. ( B , ) (B,≤)
  3. F : A B F:A→B
  4. G : B A G:B→A
  5. a a
  6. A A
  7. b b
  8. B B
  9. F ( a ) b F(a)≤b
  10. a G ( b ) a≤G(b)
  11. F F
  12. G G
  13. G G
  14. F ( a ) F(a)
  15. [ u o v e r s e t , u , u b ] [u^{\prime}overset^{\prime},u^{\prime}~{}^{\prime},u^{\prime}b^{\prime}]
  16. a G ( [ u o v e r s e t , u , u b ] ) a≤G([u^{\prime}overset^{\prime},u^{\prime}~{}^{\prime},u^{\prime}b^{\prime}])
  17. G ( b ) G(b)
  18. [ u o v e r s e t , u , u a ] [u^{\prime}overset^{\prime},u^{\prime}~{}^{\prime},u^{\prime}a^{\prime}]
  19. F ( [ u o v e r s e t , u , u a ] ) b F([u^{\prime}overset^{\prime},u^{\prime}~{}^{\prime},u^{\prime}a^{\prime}])≤b
  20. F F
  21. G G
  22. F F
  23. G G
  24. G F : A A GF:A→A
  25. F G : B B FG:B→B
  26. a G F ( a ) a≤GF(a)
  27. a a
  28. A A
  29. F G ( b ) b FG(b)≤b
  30. b b
  31. B B
  32. A A
  33. B B
  34. G F GF
  35. A A
  36. F : A B F:A→B
  37. G : B A G:B→A
  38. A A
  39. B B
  40. b F ( a ) b≤F(a)
  41. a G ( b ) a≤G(b)
  42. F F
  43. G G
  44. F ( a ) F(a)
  45. b b
  46. a G ( b ) a≤G(b)
  47. G ( b ) G(b)
  48. a a
  49. b F ( a ) b≤F(a)
  50. G F : A A GF:A→A
  51. F G : B B FG:B→B
  52. a G F ( a ) a≤GF(a)
  53. a a
  54. A A
  55. b F G ( b ) b≤FG(b)
  56. b b
  57. B B
  58. A A
  59. B B
  60. A A
  61. B B
  62. U U
  63. A A
  64. B B
  65. U U
  66. L L
  67. U U
  68. F F
  69. G G
  70. F ( M ) = L M F(M)=L∩M
  71. G ( N ) = N ( U L ) G(N)=N∪(U\ L)
  72. F F
  73. F ( x ) = ( a x ) F(x)=(a∧x)
  74. G ( y ) = ( y ¬ a ) = ( a y ) G(y)=(y∨¬a)=(a⇒y)
  75. a a
  76. a a
  77. X X × X X→X×X
  78. G G
  79. X X
  80. x x
  81. X X
  82. = { B X : x B ; g G , g B = B or g B B = } , \mathcal{B}=\{B\subseteq X:x\in B;\forall g\in G,gB=B\ \mathrm{or}\ gB\cap B=% \emptyset\},
  83. x x
  84. 𝒢 \mathcal{G}
  85. G G
  86. x x
  87. 𝒢 \mathcal{B}\to\mathcal{G}
  88. B H B = { g G : g x B } B\mapsto H_{B}=\{g\in G:gx\in B\}
  89. X X
  90. G G
  91. f : X Y f:X→Y
  92. M M
  93. X X
  94. N N
  95. Y Y
  96. F F
  97. G G
  98. X X
  99. Y Y
  100. M M
  101. X X
  102. G G
  103. H H
  104. Y Y
  105. X X
  106. G G
  107. G G
  108. G / N G/N
  109. G G
  110. H ¯ = H N \overline{H}=HN
  111. X X
  112. S S
  113. X X
  114. F ( S ) F(S)
  115. X X
  116. S S
  117. S S
  118. U U
  119. X X
  120. G ( U ) G(U)
  121. U U
  122. X X
  123. F ( S ) F(S)
  124. S S
  125. X X
  126. X X
  127. F F
  128. G G
  129. X X
  130. X X
  131. F F
  132. A A
  133. B B
  134. T A T∈A
  135. F ( T ) F(T)
  136. T T
  137. S S
  138. G ( S ) G(S)
  139. S S
  140. F ( T ) F(T)
  141. S S
  142. T T
  143. G ( S ) G(S)
  144. F F
  145. G G
  146. L / K L/K
  147. A A
  148. L L
  149. K K
  150. E E
  151. G a l ( L / E ) Gal(L/E)
  152. L L
  153. E E
  154. B B
  155. G a l ( L / K ) Gal(L/K)
  156. G G
  157. F i x ( G ) Fix(G)
  158. L L
  159. G G
  160. E G a l ( L / E ) E\mapsto Gal(L/E)
  161. G F i x ( G ) G\mapsto Fix(G)
  162. X X
  163. X X
  164. X X
  165. G G
  166. G G
  167. V V
  168. F ( X ) F(X)
  169. X X
  170. V V
  171. V V
  172. F F
  173. V V
  174. X X
  175. V V
  176. F ( X ) F(X)
  177. V V
  178. X X
  179. Y Y
  180. V V
  181. n n
  182. K K
  183. A A
  184. B B
  185. S S
  186. V ( S ) = { x K n : f ( x ) = 0 for all f S } , V(S)=\{x\in K^{n}:f(x)=0\mbox{ for all }~{}f\in S\},
  187. S S
  188. U U
  189. I ( U ) I(U)
  190. U U
  191. I ( U ) = { f K [ X 1 , , X n ] : f ( x ) = 0 for all x U } . I(U)=\{f\in K[X_{1},\cdots,X_{n}]:f(x)=0\mbox{ for all }~{}x\in U\}.
  192. V V
  193. K < s u p > n K<sup>n
  194. x x
  195. A A
  196. x g ( f ( x ) ) x≤g(f(x))

Galvanic_cell.html

  1. \rightarrow
  2. ln K = n F E 0 R T \ln K=\frac{nFE^{0}}{RT}
  3. E half-cell = E 0 - R T n F ln e Q E_{\,\text{half-cell}}=E^{0}-\frac{RT}{nF}\ln_{e}Q
  4. E half-cell = E 0 - 2.303 R T n F log 10 { M n + } E_{\,\text{half-cell}}=E^{0}-2.303\frac{RT}{nF}\log_{10}\{M^{n+}\}
  5. 0.05918 V / n {0.05918V}/{n}
  6. E half-cell = E 0 - 0.05918 V n log 10 [ M n + ] E_{\,\text{half-cell}}=E^{0}-\frac{0.05918V}{n}\log_{10}[M^{n+}]

Gamma_distribution.html

  1. 1 Γ ( k ) γ ( k , x θ ) \frac{1}{\Gamma(k)}\gamma\left(k,\,\frac{x}{\theta}\right)
  2. 𝐄 [ X ] = k θ \scriptstyle\mathbf{E}[X]=k\theta
  3. 𝐄 [ ln X ] = ψ ( k ) + ln ( θ ) \scriptstyle\mathbf{E}[\ln X]=\psi(k)+\ln(\theta)
  4. ( k - 1 ) θ for k 1 \scriptstyle(k\,-\,1)\theta\,\text{ for }k\;{\geq}\;1
  5. Var [ X ] = k θ 2 \scriptstyle\operatorname{Var}[X]=k\theta^{2}
  6. Var [ ln X ] = ψ 1 ( k ) \scriptstyle\operatorname{Var}[\ln X]=\psi_{1}(k)
  7. 2 k \scriptstyle\frac{2}{\sqrt{k}}
  8. 6 k \scriptstyle\frac{6}{k}
  9. k + ln θ + ln [ Γ ( k ) ] + ( 1 - k ) ψ ( k ) \scriptstyle\begin{aligned}\displaystyle\scriptstyle k&\displaystyle% \scriptstyle\,+\,\ln\theta\,+\,\ln[\Gamma(k)]\\ &\displaystyle\scriptstyle\,+\,(1\,-\,k)\psi(k)\end{aligned}
  10. ( 1 - θ t ) - k for t < 1 θ \scriptstyle(1\,-\,\theta t)^{-k}\,\text{ for }t\;<\;\frac{1}{\theta}
  11. ( 1 - θ i t ) - k \scriptstyle(1\,-\,\theta i\,t)^{-k}
  12. x ( 0 , ) \scriptstyle x\;\in\;(0,\,\infty)
  13. β α Γ ( α ) x α - 1 e - β x \frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{\alpha\,-\,1}e^{-\beta x}
  14. 1 Γ ( α ) γ ( α , β x ) \frac{1}{\Gamma(\alpha)}\gamma(\alpha,\,\beta x)
  15. 𝐄 [ X ] = α β \scriptstyle\mathbf{E}[X]=\frac{\alpha}{\beta}
  16. 𝐄 [ ln X ] = ψ ( α ) - ln ( β ) \scriptstyle\mathbf{E}[\ln X]=\psi(\alpha)-\ln(\beta)
  17. α - 1 β for α 1 \scriptstyle\frac{\alpha\,-\,1}{\beta}\,\text{ for }\alpha\;{\geq}\;1
  18. Var [ X ] = α β 2 \scriptstyle\operatorname{Var}[X]=\frac{\alpha}{\beta^{2}}
  19. Var [ ln X ] = ψ 1 ( α ) \scriptstyle\operatorname{Var}[\ln X]=\psi_{1}(\alpha)
  20. 2 α \scriptstyle\frac{2}{\sqrt{\alpha}}
  21. 6 α \scriptstyle\frac{6}{\alpha}
  22. α - ln β + ln [ Γ ( α ) ] + ( 1 - α ) ψ ( α ) \scriptstyle\begin{aligned}\displaystyle\scriptstyle\alpha&\displaystyle% \scriptstyle\,-\,\ln\beta\,+\,\ln[\Gamma(\alpha)]\\ &\displaystyle\scriptstyle\,+\,(1\,-\,\alpha)\psi(\alpha)\end{aligned}
  23. ( 1 - t β ) - α for t < β \scriptstyle\left(1\,-\,\frac{t}{\beta}\right)^{-\alpha}\,\text{ for }t\;<\;\beta
  24. ( 1 - i t β ) - α \scriptstyle\left(1\,-\,\frac{i\,t}{\beta}\right)^{-\alpha}
  25. X Γ ( k , θ ) Gamma ( k , θ ) X\sim\Gamma(k,\theta)\equiv\textrm{Gamma}(k,\theta)
  26. f ( x ; k , θ ) = x k - 1 e - x θ θ k Γ ( k ) for x > 0 and k , θ > 0. f(x;k,\theta)=\frac{x^{k-1}e^{-\frac{x}{\theta}}}{\theta^{k}\Gamma(k)}\quad\,% \text{ for }x>0\,\text{ and }k,\theta>0.
  27. F ( x ; k , θ ) = 0 x f ( u ; k , θ ) d u = γ ( k , x θ ) Γ ( k ) F(x;k,\theta)=\int_{0}^{x}f(u;k,\theta)\,du=\frac{\gamma\left(k,\frac{x}{% \theta}\right)}{\Gamma(k)}
  28. F ( x ; k , θ ) = 1 - i = 0 k - 1 1 i ! ( x θ ) i e - x θ = e - x θ i = k 1 i ! ( x θ ) i F(x;k,\theta)=1-\sum_{i=0}^{k-1}\frac{1}{i!}\left(\frac{x}{\theta}\right)^{i}e% ^{-\frac{x}{\theta}}=e^{-\frac{x}{\theta}}\sum_{i=k}^{\infty}\frac{1}{i!}\left% (\frac{x}{\theta}\right)^{i}
  29. X Γ ( α , β ) Gamma ( α , β ) X\sim\Gamma(\alpha,\beta)\equiv\textrm{Gamma}(\alpha,\beta)
  30. g ( x ; α , β ) = β α x α - 1 e - x β Γ ( α ) for x 0 and α , β > 0 g(x;\alpha,\beta)=\frac{\beta^{\alpha}x^{\alpha-1}e^{-x\beta}}{\Gamma(\alpha)}% \quad\,\text{ for }x\geq 0\,\text{ and }\alpha,\beta>0
  31. F ( x ; α , β ) = 0 x f ( u ; α , β ) d u = γ ( α , β x ) Γ ( α ) F(x;\alpha,\beta)=\int_{0}^{x}f(u;\alpha,\beta)\,du=\frac{\gamma(\alpha,\beta x% )}{\Gamma(\alpha)}
  32. F ( x ; α , β ) = 1 - i = 0 α - 1 ( β x ) i i ! e - β x = e - β x i = α ( β x ) i i ! F(x;\alpha,\beta)=1-\sum_{i=0}^{\alpha-1}\frac{(\beta x)^{i}}{i!}e^{-\beta x}=% e^{-\beta x}\sum_{i=\alpha}^{\infty}\frac{(\beta x)^{i}}{i!}
  33. 2 / k 2/\sqrt{k}
  34. 1 Γ ( k ) θ k 0 ν x k - 1 e - x θ d x = 1 2 . \frac{1}{\Gamma(k)\theta^{k}}\int_{0}^{\nu}x^{k-1}e^{-\frac{x}{\theta}}dx=% \tfrac{1}{2}.
  35. ν μ 3 k - 0.8 3 k + 0.2 , \nu\approx\mu\frac{3k-0.8}{3k+0.2},
  36. μ ( = k θ ) \mu(=k\theta)
  37. m - 1 3 < λ ( m ) < m , m-\frac{1}{3}<\lambda(m)<m,
  38. λ ( m ) \lambda(m)
  39. Gamma ( m , 1 ) \,\text{Gamma}(m,1)
  40. λ ( m ) = m - 1 3 + 8 405 m + 184 25515 m 2 + 2248 3444525 m 3 - \lambda(m)=m-\frac{1}{3}+\frac{8}{405m}+\frac{184}{25515m^{2}}+\frac{2248}{344% 4525m^{3}}-\cdots
  41. θ \theta
  42. λ ( m ) \lambda(m)
  43. m m
  44. i = 1 N X i Gamma ( i = 1 N k i , θ ) \sum_{i=1}^{N}X_{i}\sim\mathrm{Gamma}\left(\sum_{i=1}^{N}k_{i},\theta\right)
  45. X Gamma ( k , θ ) , X\sim\mathrm{Gamma}(k,\theta),
  46. c X Gamma ( k , c θ ) . cX\sim\mathrm{Gamma}(k,c\theta).
  47. 𝐄 [ ln ( X ) ] = ψ ( α ) - ln ( β ) \mathbf{E}[\ln(X)]=\psi(\alpha)-\ln(\beta)
  48. 𝐄 [ ln ( X ) ] = ψ ( k ) + ln ( θ ) \mathbf{E}[\ln(X)]=\psi(k)+\ln(\theta)
  49. H ( X ) = 𝐄 [ - ln ( p ( X ) ) ] = 𝐄 [ - α ln ( β ) + ln ( Γ ( α ) ) - ( α - 1 ) ln ( X ) + β X ] = α - ln ( β ) + ln ( Γ ( α ) ) + ( 1 - α ) ψ ( α ) . \operatorname{H}(X)=\mathbf{E}[-\ln(p(X))]=\mathbf{E}[-\alpha\ln(\beta)+\ln(% \Gamma(\alpha))-(\alpha-1)\ln(X)+\beta X]=\alpha-\ln(\beta)+\ln(\Gamma(\alpha)% )+(1-\alpha)\psi(\alpha).
  50. H ( X ) = k + ln ( θ ) + ln ( Γ ( k ) ) + ( 1 - k ) ψ ( k ) . \operatorname{H}(X)=k+\ln(\theta)+\ln(\Gamma(k))+(1-k)\psi(k).
  51. D KL ( α p , β p ; α q , β q ) = ( α p - α q ) ψ ( α p ) - log Γ ( α p ) + log Γ ( α q ) + α q ( log β p - log β q ) + α p β q - β p β p D_{\mathrm{KL}}(\alpha_{p},\beta_{p};\alpha_{q},\beta_{q})=(\alpha_{p}-\alpha_% {q})\psi(\alpha_{p})-\log\Gamma(\alpha_{p})+\log\Gamma(\alpha_{q})+\alpha_{q}(% \log\beta_{p}-\log\beta_{q})+\alpha_{p}\frac{\beta_{q}-\beta_{p}}{\beta_{p}}
  52. D KL ( k p , θ p ; k q , θ q ) = ( k p - k q ) ψ ( k p ) - log Γ ( k p ) + log Γ ( k q ) + k q ( log θ q - log θ p ) + k p θ p - θ q θ q D_{\mathrm{KL}}(k_{p},\theta_{p};k_{q},\theta_{q})=(k_{p}-k_{q})\psi(k_{p})-% \log\Gamma(k_{p})+\log\Gamma(k_{q})+k_{q}(\log\theta_{q}-\log\theta_{p})+k_{p}% \frac{\theta_{p}-\theta_{q}}{\theta_{q}}
  53. F ( s ) = ( 1 + θ s ) - k = β α ( s + β ) α . F(s)=(1+\theta s)^{-k}=\frac{\beta^{\alpha}}{(s+\beta)^{\alpha}}.
  54. { β x f ( x ) + f ( x ) ( - α β + β + x ) = 0 ; f ( 1 ) = e - 1 / β β - α Γ ( α ) } \left\{\beta xf^{\prime}(x)+f(x)(-\alpha\beta+\beta+x)=0;f(1)=\frac{e^{-1/% \beta}\beta^{-\alpha}}{\Gamma(\alpha)}\right\}
  55. { x f ( x ) + f ( x ) ( - k + θ x + 1 ) = 0 ; f ( 1 ) = e - θ ( 1 θ ) - k Γ ( k ) } \left\{xf^{\prime}(x)+f(x)(-k+\theta x+1)=0;f(1)=\frac{e^{-\theta}\left(\frac{% 1}{\theta}\right)^{-k}}{\Gamma(k)}\right\}
  56. L ( k , θ ) = i = 1 N f ( x i ; k , θ ) L(k,\theta)=\prod_{i=1}^{N}f(x_{i};k,\theta)
  57. ( k , θ ) = ( k - 1 ) i = 1 N ln ( x i ) - i = 1 N x i θ - N k ln ( θ ) - N ln ( Γ ( k ) ) \ell(k,\theta)=(k-1)\sum_{i=1}^{N}\ln{(x_{i})}-\sum_{i=1}^{N}\frac{x_{i}}{% \theta}-Nk\ln(\theta)-N\ln(\Gamma(k))
  58. θ ^ = 1 k N i = 1 N x i \hat{\theta}=\frac{1}{kN}\sum_{i=1}^{N}x_{i}
  59. = ( k - 1 ) i = 1 N ln ( x i ) - N k - N k ln ( x i k N ) - N ln ( Γ ( k ) ) \ell=(k-1)\sum_{i=1}^{N}\ln{(x_{i})}-Nk-Nk\ln{\left(\frac{\sum x_{i}}{kN}% \right)}-N\ln(\Gamma(k))
  60. ln ( k ) - ψ ( k ) = ln ( 1 N i = 1 N x i ) - 1 N i = 1 N ln ( x i ) \ln(k)-\psi(k)=\ln\left(\frac{1}{N}\sum_{i=1}^{N}x_{i}\right)-\frac{1}{N}\sum_% {i=1}^{N}\ln(x_{i})
  61. ln ( k ) - ψ ( k ) 1 2 k ( 1 + 1 6 k + 1 ) \ln(k)-\psi(k)\approx\frac{1}{2k}\left(1+\frac{1}{6k+1}\right)
  62. s = ln ( 1 N i = 1 N x i ) - 1 N i = 1 N ln ( x i ) s=\ln{\left(\frac{1}{N}\sum_{i=1}^{N}x_{i}\right)}-\frac{1}{N}\sum_{i=1}^{N}% \ln{(x_{i})}
  63. k 3 - s + ( s - 3 ) 2 + 24 s 12 s k\approx\frac{3-s+\sqrt{(s-3)^{2}+24s}}{12s}
  64. k k - ln ( k ) - ψ ( k ) - s 1 k - ψ ( k ) . k\leftarrow k-\frac{\ln(k)-\psi(k)-s}{\frac{1}{k}-\psi^{\prime}(k)}.
  65. P ( θ | k , x 1 , , x N ) 1 θ i = 1 N f ( x i ; k , θ ) P(\theta|k,x_{1},\dots,x_{N})\propto\frac{1}{\theta}\prod_{i=1}^{N}f(x_{i};k,\theta)
  66. y i = 1 N x i , P ( θ | k , x 1 , , x N ) = C ( x i ) θ - N k - 1 e - y θ y\equiv\sum_{i=1}^{N}x_{i},\qquad P(\theta|k,x_{1},\dots,x_{N})=C(x_{i})\theta% ^{-Nk-1}e^{-\frac{y}{\theta}}
  67. 0 θ - N k - 1 + m e - y θ d θ = 0 x N k - 1 - m e - x y d x = y - ( N k - m ) Γ ( N k - m ) \int_{0}^{\infty}\theta^{-Nk-1+m}e^{-\frac{y}{\theta}}\,d\theta=\int_{0}^{% \infty}x^{Nk-1-m}e^{-xy}\,dx=y^{-(Nk-m)}\Gamma(Nk-m)\!
  68. 𝐄 [ x m ] = Γ ( N k - m ) Γ ( N k ) y m \mathbf{E}[x^{m}]=\frac{\Gamma(Nk-m)}{\Gamma(Nk)}y^{m}
  69. y N k - 1 ± y 2 ( N k - 1 ) 2 ( N k - 2 ) \frac{y}{Nk-1}\pm\frac{y^{2}}{(Nk-1)^{2}(Nk-2)}
  70. V 3 m - 2 v 0 V_{3m-2}\leq v_{0}
  71. v 0 = e e + δ v_{0}=\frac{e}{e+\delta}
  72. ξ m = V 3 m - 1 1 / δ , η m = V 3 m ξ m δ - 1 \xi_{m}=V_{3m-1}^{1/\delta},\ \eta_{m}=V_{3m}\xi_{m}^{\delta-1}
  73. ξ m = 1 - ln V 3 m - 1 , η m = V 3 m e - ξ m \xi_{m}=1-\ln{V_{3m-1}},\ \eta_{m}=V_{3m}e^{-\xi_{m}}
  74. η m > ξ m δ - 1 e - ξ m \eta_{m}>\xi_{m}^{\delta-1}e^{-\xi_{m}}
  75. θ ( ξ - i = 1 k ln ( U i ) ) Γ ( k , θ ) \theta\left(\xi-\prod_{i=1}^{\lfloor{k}\rfloor}{\ln(U_{i})}\right)\sim\Gamma(k% ,\theta)
  76. k \scriptstyle\lfloor{k}\rfloor
  77. 1 a = α = k 1\leq a=\alpha=k
  78. p ( k , θ | p , q , r , s ) = 1 Z p k - 1 e - θ - 1 q Γ ( k ) r θ k s , p(k,\theta|p,q,r,s)=\frac{1}{Z}\frac{p^{k-1}e^{-\theta^{-1}q}}{\Gamma(k)^{r}% \theta^{ks}},
  79. p \displaystyle p^{\prime}
  80. X Γ ( k 𝐙 , θ ) , Y Pois ( x θ ) , X\sim\Gamma(k\in\mathbf{Z},\theta),\qquad Y\sim\mathrm{Pois}\left(\frac{x}{% \theta}\right),
  81. P ( X > x ) = P ( Y < k ) . P(X>x)=P(Y<k).
  82. X 2 Γ ( 3 2 , 2 a 2 ) X^{2}\sim\Gamma\left(\tfrac{3}{2},2a^{2}\right)
  83. X \sqrt{X}
  84. a = θ a=\sqrt{\theta}
  85. X q X^{q}
  86. q > 0 q>0
  87. a = θ q a=\theta^{q}

Gasification.html

  1. C + O 2 CO 2 {\rm C}+{\rm O}_{2}\rightarrow{\rm CO}_{2}
  2. C + H 2 O H 2 + CO {\rm C}+{\rm H}_{2}{\rm O}\rightarrow{\rm H}_{2}+{\rm CO}
  3. CO + H 2 O \lrarr CO 2 + H 2 {\rm CO}+{\rm H}_{2}{\rm O}\lrarr{\rm CO}_{2}+{\rm H}_{2}

Gauss's_law.html

  1. 1 / ε {1}/{ε}
  2. Φ E = Q ε 0 \Phi_{E}=\frac{Q}{\varepsilon_{0}}
  3. 𝐄 = ρ ε 0 \nabla\cdot\mathbf{E}=\frac{\rho}{\varepsilon_{0}}
  4. = Q ε 0 =\frac{Q}{\varepsilon_{0}}
  5. V 𝐄 d V = Q ε 0 \iiint\limits_{V}\nabla\cdot\mathbf{E}\ \mathrm{d}V=\frac{Q}{\varepsilon_{0}}
  6. V 𝐄 d V = V ρ ε 0 d V \iiint\limits_{V}\nabla\cdot\mathbf{E}\ \mathrm{d}V=\iiint\limits_{V}\frac{% \rho}{\varepsilon_{0}}\ \mathrm{d}V
  7. 𝐄 = ρ ε 0 . \nabla\cdot\mathbf{E}=\frac{\rho}{\varepsilon_{0}}.
  8. Φ D = Q free \Phi_{D}=Q\text{free}\!
  9. 𝐃 = ρ free \mathbf{\nabla}\cdot\mathbf{D}=\rho\text{free}
  10. 𝐄 = ρ / ε 0 \nabla\cdot\mathbf{E}=\rho/\varepsilon_{0}
  11. 𝐃 = ρ free \nabla\cdot\mathbf{D}=\rho_{\mathrm{free}}
  12. 𝐃 = ε 0 𝐄 + 𝐏 \mathbf{D}=\varepsilon_{0}\mathbf{E}+\mathbf{P}
  13. ρ bound = - 𝐏 \rho_{\mathrm{bound}}=-\nabla\cdot\mathbf{P}
  14. ρ bound = ( - 𝐏 ) \rho_{\mathrm{bound}}=\nabla\cdot(-\mathbf{P})
  15. ρ free = 𝐃 \rho_{\mathrm{free}}=\nabla\cdot\mathbf{D}
  16. ρ = ( ε 0 𝐄 ) \rho=\nabla\cdot(\varepsilon_{0}\mathbf{E})
  17. 𝐃 = ε 𝐄 \mathbf{D}=\varepsilon\mathbf{E}
  18. Φ E = Q free ε \Phi_{E}=\frac{Q\text{free}}{\varepsilon}
  19. 𝐄 = ρ free ε \mathbf{\nabla}\cdot\mathbf{E}=\frac{\rho\text{free}}{\varepsilon}
  20. 𝐄 ( 𝐫 ) = q 4 π ε 0 𝐞 𝐫 r 2 \mathbf{E}(\mathbf{r})=\frac{q}{4\pi\varepsilon_{0}}\frac{\mathbf{e_{r}}}{r^{2}}
  21. ε 0 \varepsilon_{0}
  22. 𝐄 ( 𝐫 ) = 1 4 π ε 0 ρ ( 𝐬 ) ( 𝐫 - 𝐬 ) | 𝐫 - 𝐬 | 3 d 3 𝐬 \mathbf{E}(\mathbf{r})=\frac{1}{4\pi\varepsilon_{0}}\int\frac{\rho(\mathbf{s})% (\mathbf{r}-\mathbf{s})}{|\mathbf{r}-\mathbf{s}|^{3}}\,d^{3}\mathbf{s}
  23. ρ \rho
  24. ( 𝐫 | 𝐫 | 3 ) = 4 π δ ( 𝐫 ) \nabla\cdot\left(\frac{\mathbf{r}}{|\mathbf{r}|^{3}}\right)=4\pi\delta(\mathbf% {r})
  25. 𝐄 ( 𝐫 ) = 1 ε 0 ρ ( 𝐬 ) δ ( 𝐫 - 𝐬 ) d 3 𝐬 \nabla\cdot\mathbf{E}(\mathbf{r})=\frac{1}{\varepsilon_{0}}\int\rho(\mathbf{s}% )\ \delta(\mathbf{r}-\mathbf{s})\,d^{3}\mathbf{s}
  26. 𝐄 ( 𝐫 ) = ρ ( 𝐫 ) ε 0 , \nabla\cdot\mathbf{E}(\mathbf{r})=\frac{\rho(\mathbf{r})}{\varepsilon_{0}},
  27. S 𝐄 d 𝐀 = Q ε 0 \oint_{S}\mathbf{E}\cdot d\mathbf{A}=\frac{Q}{\varepsilon_{0}}
  28. 4 π r 2 𝐫 ^ 𝐄 ( 𝐫 ) = Q ε 0 4\pi r^{2}\hat{\mathbf{r}}\cdot\mathbf{E}(\mathbf{r})=\frac{Q}{\varepsilon_{0}}
  29. 𝐫 ^ \hat{\mathbf{r}}
  30. 𝐄 ( 𝐫 ) = Q 4 π ε 0 𝐫 ^ r 2 \mathbf{E}(\mathbf{r})=\frac{Q}{4\pi\varepsilon_{0}}\frac{\hat{\mathbf{r}}}{r^% {2}}

Gaussian_curvature.html

  1. K = κ 1 κ 2 , K=\kappa_{1}\kappa_{2},
  2. K = ( 2 1 - 1 2 ) 𝐞 1 , 𝐞 2 det g , K=\frac{\langle(\nabla_{2}\nabla_{1}-\nabla_{1}\nabla_{2})\mathbf{e}_{1},% \mathbf{e}_{2}\rangle}{\det g},
  3. i = 𝐞 i \nabla_{i}=\nabla_{{\mathbf{e}}_{i}}
  4. K ( 𝐩 ) = det ( S ( 𝐩 ) ) , K(\mathbf{p})=\det(S(\mathbf{p})),
  5. i = 1 3 θ i = π + T K d A . \sum_{i=1}^{3}\theta_{i}=\pi+\iint_{T}K\,dA.
  6. K = det I I det I = L N - M 2 E G - F 2 . K=\frac{\det II}{\det I}=\frac{LN-M^{2}}{EG-F^{2}}.
  7. K = det | - 1 2 E v v + F u v - 1 2 G u u 1 2 E u F u - 1 2 E v F v - 1 2 G u E F 1 2 G v F G | - det | 0 1 2 E v 1 2 G u 1 2 E v E F 1 2 G u F G | ( E G - F 2 ) 2 K=\frac{\det\begin{vmatrix}-\frac{1}{2}E_{vv}+F_{uv}-\frac{1}{2}G_{uu}&\frac{1% }{2}E_{u}&F_{u}-\frac{1}{2}E_{v}\\ F_{v}-\frac{1}{2}G_{u}&E&F\\ \frac{1}{2}G_{v}&F&G\end{vmatrix}-\det\begin{vmatrix}0&\frac{1}{2}E_{v}&\frac{% 1}{2}G_{u}\\ \frac{1}{2}E_{v}&E&F\\ \frac{1}{2}G_{u}&F&G\end{vmatrix}}{(EG-F^{2})^{2}}
  8. K = - 1 2 E G ( u G u E G + v E v E G ) . K=-\frac{1}{2\sqrt{EG}}\left(\frac{\partial}{\partial u}\frac{G_{u}}{\sqrt{EG}% }+\frac{\partial}{\partial v}\frac{E_{v}}{\sqrt{EG}}\right).
  9. K = F x x F y y - F x y 2 ( 1 + F x 2 + F y 2 ) 2 K=\frac{F_{xx}\cdot F_{yy}-F_{xy}^{2}}{(1+F_{x}^{2}+F_{y}^{2})^{2}}
  10. F \nabla F
  11. H ( F ) H(F)
  12. K = - det | H ( F ) F 𝖳 F 0 | | F | 4 = - det | F x x F x y F x z F x F x y F y y F y z F y F x z F y z F z z F z F x F y F z 0 | | F | 4 K=-\frac{\det\begin{vmatrix}H(F)&\nabla F^{\mathsf{T}}\\ \nabla F&0\end{vmatrix}}{|\nabla F|^{4}}=-\frac{\det\begin{vmatrix}F_{xx}&F_{% xy}&F_{xz}&F_{x}\\ F_{xy}&F_{yy}&F_{yz}&F_{y}\\ F_{xz}&F_{yz}&F_{zz}&F_{z}\\ F_{x}&F_{y}&F_{z}&0\\ \end{vmatrix}}{|\nabla F|^{4}}
  13. K = - 1 2 e σ Δ σ , K=-\frac{1}{2e^{\sigma}}\Delta\sigma,
  14. K = lim r 0 + 3 2 π r - C ( r ) π r 3 K=\lim_{r\to 0^{+}}3\frac{2\pi r-C(r)}{\pi r^{3}}
  15. K = lim r 0 + 12 π r 2 - A ( r ) π r 4 K=\lim_{r\to 0^{+}}12\frac{\pi r^{2}-A(r)}{\pi r^{4}}
  16. K = - 1 E ( u Γ 12 2 - v Γ 11 2 + Γ 12 1 Γ 11 2 - Γ 11 1 Γ 12 2 + Γ 12 2 Γ 12 2 - Γ 11 2 Γ 22 2 ) K=-\frac{1}{E}\left(\frac{\partial}{\partial u}\Gamma_{12}^{2}-\frac{\partial}% {\partial v}\Gamma_{11}^{2}+\Gamma_{12}^{1}\Gamma_{11}^{2}-\Gamma_{11}^{1}% \Gamma_{12}^{2}+\Gamma_{12}^{2}\Gamma_{12}^{2}-\Gamma_{11}^{2}\Gamma_{22}^{2}\right)

Gaussian_function.html

  1. f ( x ) = a exp ( - ( x - b ) 2 2 c 2 ) f\left(x\right)=a\exp{\left(-{\frac{(x-b)^{2}}{2c^{2}}}\right)}
  2. FWHM = 2 2 ln 2 c 2.35482 c . \mathrm{FWHM}=2\sqrt{2\ln 2}\ c\approx 2.35482c.
  3. FWTM = 2 2 ln 10 c 4.29193 c . \mathrm{FWTM}=2\sqrt{2\ln 10}\ c\approx 4.29193c.
  4. - e - x 2 d x = π \int_{-\infty}^{\infty}e^{-x^{2}}\,dx=\sqrt{\pi}
  5. - a e - ( x - b ) 2 2 c 2 d x = a c 2 π . \int_{-\infty}^{\infty}ae^{-{(x-b)^{2}\over 2c^{2}}}\,dx=ac\cdot\sqrt{2\pi}.
  6. a = 1 c 2 π a=\tfrac{1}{c\sqrt{2\pi}}
  7. g ( x ) = 1 σ 2 π e - 1 2 ( x - μ σ ) 2 . g(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}% \right)^{2}}.
  8. c 2 = c 1 2 + c 2 2 c^{2}=c_{1}^{2}+c_{2}^{2}
  9. 2 π a c \sqrt{2\pi}ac
  10. 1 2 π c \frac{1}{2\pi c}
  11. c = 1 2 π c=\frac{1}{\sqrt{2\pi}}
  12. k exp ( - π ( k c ) 2 ) = c k exp ( - π ( k c ) 2 ) . \sum_{k\in\mathbb{Z}}\exp\left(-\pi\cdot\left(\frac{k}{c}\right)^{2}\right)=c% \cdot\sum_{k\in\mathbb{Z}}\exp(-\pi\cdot(kc)^{2}).
  13. - a e - ( x - b ) 2 / c 2 d x = a | c | π . \int_{-\infty}^{\infty}a\,e^{-\left(x-b\right)^{2}/c^{2}}\,dx=a\,\left|c\right% |\,\sqrt{\pi}.
  14. - k e - f x 2 + g x + h d x = - k e - f ( x - g / ( 2 f ) ) 2 + g 2 / ( 4 f ) + h d x = k π f exp ( g 2 4 f + h ) , \int_{-\infty}^{\infty}k\,e^{-fx^{2}+gx+h}\,dx=\int_{-\infty}^{\infty}k\,e^{-f% \left(x-g/\left(2f\right)\right)^{2}+g^{2}/\left(4f\right)+h}\,dx=k\,\sqrt{% \frac{\pi}{f}}\,\exp\left(\frac{g^{2}}{4f}+h\right),
  15. - a e - ( x - b ) 2 / c 2 d x \int_{-\infty}^{\infty}ae^{-(x-b)^{2}/c^{2}}\,dx
  16. a - e - y 2 / c 2 d y , a\int_{-\infty}^{\infty}e^{-y^{2}/c^{2}}\,dy,
  17. z = y / | c | z=y/|c|
  18. a | c | - e - z 2 d z . a|c|\int_{-\infty}^{\infty}e^{-z^{2}}\,dz.
  19. - e - z 2 d z = π , \int_{-\infty}^{\infty}e^{-z^{2}}\,dz=\sqrt{\pi},
  20. - a e - ( x - b ) 2 / c 2 d x = a | c | π . \int_{-\infty}^{\infty}ae^{-(x-b)^{2}/c^{2}}\,dx=a|c|\sqrt{\pi}.
  21. f ( x , y ) = A exp ( - ( ( x - x o ) 2 2 σ x 2 + ( y - y o ) 2 2 σ y 2 ) ) . f(x,y)=A\exp\left(-\left(\frac{(x-x_{o})^{2}}{2\sigma_{x}^{2}}+\frac{(y-y_{o})% ^{2}}{2\sigma_{y}^{2}}\right)\right).
  22. V = - - f ( x , y ) d x d y = 2 π A σ x σ y . V=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f(x,y)\,dxdy=2\pi A\sigma_{x}% \sigma_{y}.
  23. f ( x , y ) = A exp ( - ( a ( x - x o ) 2 + 2 b ( x - x o ) ( y - y o ) + c ( y - y o ) 2 ) ) f(x,y)=A\exp\left(-\left(a(x-x_{o})^{2}+2b(x-x_{o})(y-y_{o})+c(y-y_{o})^{2}% \right)\right)
  24. [ a b b c ] \left[\begin{matrix}a&b\\ b&c\end{matrix}\right]
  25. a = cos 2 θ 2 σ x 2 + sin 2 θ 2 σ y 2 a=\frac{\cos^{2}\theta}{2\sigma_{x}^{2}}+\frac{\sin^{2}\theta}{2\sigma_{y}^{2}}
  26. b = - sin 2 θ 4 σ x 2 + sin 2 θ 4 σ y 2 b=-\frac{\sin 2\theta}{4\sigma_{x}^{2}}+\frac{\sin 2\theta}{4\sigma_{y}^{2}}
  27. c = sin 2 θ 2 σ x 2 + cos 2 θ 2 σ y 2 c=\frac{\sin^{2}\theta}{2\sigma_{x}^{2}}+\frac{\cos^{2}\theta}{2\sigma_{y}^{2}}
  28. θ \theta
  29. θ = 0 \theta=0
  30. θ = π / 6 \theta=\pi/6
  31. θ = π / 3 \theta=\pi/3
  32. n n
  33. f ( x ) = exp ( - x T A x ) , f(x)=\exp(-x^{T}Ax)\;,
  34. x = { x 1 , , x n } x=\{x_{1},\dots,x_{n}\}
  35. n n
  36. A A
  37. n × n n\times n
  38. T {}^{T}
  39. n n
  40. n exp ( - x T A x ) d x = π n det A . \int_{\mathbb{R}^{n}}\exp(-x^{T}Ax)\,dx=\sqrt{\frac{\pi^{n}}{\det{A}}}\;.
  41. A A
  42. A A
  43. f ( x ) = exp ( - x T A x + s T x ) , f(x)=\exp(-x^{T}Ax+s^{T}x)\;,
  44. s = { s 1 , , s n } s=\{s_{1},\dots,s_{n}\}
  45. A A
  46. A T = A A^{T}=A
  47. n e - x T A x + v T x d x = π n det A exp ( 1 4 v T A - 1 v ) . \int_{\mathbb{R}^{n}}e^{-x^{T}Ax+v^{T}x}\,dx=\sqrt{\frac{\pi^{n}}{\det{A}}}% \exp(\frac{1}{4}v^{T}A^{-1}v)\equiv\mathcal{M}\;.
  48. n e - x T A x + v T x ( a T x ) d x = ( a T u ) , where u = 1 2 A - 1 v . \int_{\mathbb{R}^{n}}e^{-x^{T}Ax+v^{T}x}\left(a^{T}x\right)\,dx=(a^{T}u)\cdot% \mathcal{M}\;,\;{\rm where}\;u=\frac{1}{2}A^{-1}v\;.
  49. n e - x T A x + v T x ( x T D x ) d x = ( u T D u + 1 2 tr ( D A - 1 ) ) . \int_{\mathbb{R}^{n}}e^{-x^{T}Ax+v^{T}x}\left(x^{T}Dx\right)\,dx=\left(u^{T}Du% +\frac{1}{2}{\rm tr}(DA^{-1})\right)\cdot\mathcal{M}\;.
  50. n e - x T A x + s T x ( - x Λ x ) e - x T A x + s T x d x = = ( 2 t r ( A Λ A B - 1 ) + 4 u T A Λ A u - 2 u T ( A Λ s + A Λ s ) + s T Λ s ) , where u = 1 2 B - 1 v , v = s + s , B = A + A . \begin{aligned}&\displaystyle\int_{\mathbb{R}^{n}}e^{-x^{T}A^{\prime}x+s^{% \prime T}x}\left(-\frac{\partial}{\partial x}\Lambda\frac{\partial}{\partial x% }\right)e^{-x^{T}Ax+s^{T}x}\,dx=\\ &\displaystyle=\left(2{\rm tr}(A^{\prime}\Lambda AB^{-1})+4u^{T}A^{\prime}% \Lambda Au-2u^{T}(A^{\prime}\Lambda s+A\Lambda s^{\prime})+s^{\prime T}\Lambda s% \right)\cdot\mathcal{M}\;,\\ &\displaystyle{\rm where}\;u=\frac{1}{2}B^{-1}v,v=s+s^{\prime},B=A+A^{\prime}% \;.\end{aligned}
  51. a a
  52. b b
  53. c c
  54. A A
  55. ( x 0 , y 0 ) (x_{0},y_{0})
  56. ( σ x , σ y ) (\sigma_{x},\sigma_{y})
  57. a a
  58. b b
  59. c c
  60. 𝐊 Gauss = σ 2 π δ x Q 2 ( 3 2 c 0 - 1 a 0 2 c a 2 0 - 1 a 0 2 c a 2 ) , 𝐊 Poiss = 1 2 π ( 3 a 2 c 0 - 1 2 0 c a 0 - 1 2 0 c 2 a ) , \mathbf{K}_{\,\text{Gauss}}=\frac{\sigma^{2}}{\sqrt{\pi}\delta_{x}Q^{2}}\begin% {pmatrix}\frac{3}{2c}&0&\frac{-1}{a}\\ 0&\frac{2c}{a^{2}}&0\\ \frac{-1}{a}&0&\frac{2c}{a^{2}}\end{pmatrix}\ ,\qquad\mathbf{K}_{\,\text{Poiss% }}=\frac{1}{\sqrt{2\pi}}\begin{pmatrix}\frac{3a}{2c}&0&-\frac{1}{2}\\ 0&\frac{c}{a}&0\\ -\frac{1}{2}&0&\frac{c}{2a}\end{pmatrix}\ ,
  61. δ x \delta_{x}
  62. Q Q
  63. σ \sigma
  64. var ( a ) = 3 σ 2 2 π δ x Q 2 c var ( b ) = 2 σ 2 c δ x π Q 2 a 2 var ( c ) = 2 σ 2 c δ x π Q 2 a 2 \begin{aligned}\displaystyle\,\text{var}(a)&\displaystyle=\frac{3\sigma^{2}}{2% \sqrt{\pi}\,\delta_{x}Q^{2}c}\\ \displaystyle\,\text{var}(b)&\displaystyle=\frac{2\sigma^{2}c}{\delta_{x}\sqrt% {\pi}\,Q^{2}a^{2}}\\ \displaystyle\,\text{var}(c)&\displaystyle=\frac{2\sigma^{2}c}{\delta_{x}\sqrt% {\pi}\,Q^{2}a^{2}}\end{aligned}
  65. var ( a ) = 3 a 2 2 π c var ( b ) = c 2 π a var ( c ) = c 2 2 π a . \begin{aligned}\displaystyle\,\text{var}(a)&\displaystyle=\frac{3a}{2\sqrt{2% \pi}\,c}\\ \displaystyle\,\text{var}(b)&\displaystyle=\frac{c}{\sqrt{2\pi}\,a}\\ \displaystyle\,\text{var}(c)&\displaystyle=\frac{c}{2\sqrt{2\pi}\,a}.\end{aligned}
  66. A A
  67. ( x 0 , y 0 ) (x_{0},y_{0})
  68. ( σ x , σ y ) (\sigma_{x},\sigma_{y})
  69. 𝐊 Gauss = σ 2 π δ x δ y Q 2 ( 2 σ x σ y 0 0 - 1 A σ y - 1 A σ x 0 2 σ x A 2 σ y 0 0 0 0 0 2 σ y A 2 σ x 0 0 - 1 A σ y 0 0 2 σ x A 2 σ y 0 - 1 A σ x 0 0 0 2 σ y A 2 σ x ) , \mathbf{K}_{\,\text{Gauss}}=\frac{\sigma^{2}}{\pi\delta_{x}\delta_{y}Q^{2}}% \begin{pmatrix}\frac{2}{\sigma_{x}\sigma_{y}}&0&0&\frac{-1}{A\sigma_{y}}&\frac% {-1}{A\sigma_{x}}\\ 0&\frac{2\sigma_{x}}{A^{2}\sigma_{y}}&0&0&0\\ 0&0&\frac{2\sigma_{y}}{A^{2}\sigma_{x}}&0&0\\ \frac{-1}{A\sigma_{y}}&0&0&\frac{2\sigma_{x}}{A^{2}\sigma_{y}}&0\\ \frac{-1}{A\sigma_{x}}&0&0&0&\frac{2\sigma_{y}}{A^{2}\sigma_{x}}\end{pmatrix}\ ,
  70. 𝐊 Poiss = 1 2 π ( 3 A σ x σ y 0 0 - 1 σ y - 1 σ x 0 σ x A σ y 0 0 0 0 0 σ y A σ x 0 0 - 1 σ y 0 0 2 σ x 3 A σ y 1 3 A - 1 σ x 0 0 1 3 A 2 σ y 3 A σ x ) . \qquad\mathbf{K}_{\,\text{Poiss}}=\frac{1}{2\pi}\begin{pmatrix}\frac{3A}{% \sigma_{x}\sigma_{y}}&0&0&\frac{-1}{\sigma_{y}}&\frac{-1}{\sigma_{x}}\\ 0&\frac{\sigma_{x}}{A\sigma_{y}}&0&0&0\\ 0&0&\frac{\sigma_{y}}{A\sigma_{x}}&0&0\\ \frac{-1}{\sigma_{y}}&0&0&\frac{2\sigma_{x}}{3A\sigma_{y}}&\frac{1}{3A}\\ \frac{-1}{\sigma_{x}}&0&0&\frac{1}{3A}&\frac{2\sigma_{y}}{3A\sigma_{x}}\end{% pmatrix}\ .
  71. t = .5 , 1 , 2 , 4. t=.5,1,2,4.
  72. T ( n , t ) = e - t I n ( t ) T(n,t)=e^{-t}I_{n}(t)\,
  73. I n ( t ) I_{n}(t)
  74. log ( x ) = - log ( 1 x ) \log\left(x\right)=-\log\left(\frac{1}{x}\right)
  75. FWHM = 2 - 2 ln 0.5 c \mathrm{FWHM}=2\sqrt{-2\ln 0.5}\ c
  76. log ( x ) = - log ( 1 x ) \log\left(x\right)=-\log\left(\frac{1}{x}\right)
  77. FWTM = 2 - 2 ln 0.1 c \mathrm{FWTM}=2\sqrt{-2\ln 0.1}\ c

Gaussian_gravitational_constant.html

  1. h = r 2 θ ˙ = 2 Δ A Δ t = p G M ( 1 + m ) h=r^{2}\dot{\theta}=2\frac{\Delta A}{\Delta t}=\sqrt{pGM\left(1+m\right)}
  2. h p ( 1 + m ) = G M = k \frac{h}{\sqrt{p\left(1+m\right)}}=\sqrt{GM}=k
  3. n = k 1 + m a 3 n=k\sqrt{\frac{1+m}{a^{3}}}

Gaussian_process.html

  1. t 1 , , t k t_{1},\ldots,t_{k}
  2. T T
  3. 𝐗 t 1 , , t k = ( 𝐗 t 1 , , 𝐗 t k ) {\mathbf{X}}_{t_{1},\ldots,t_{k}}=(\mathbf{X}_{t_{1}},\ldots,\mathbf{X}_{t_{k}})
  4. { X t ; t T } \left\{X_{t};t\in T\right\}
  5. t 1 , , t k t_{1},\ldots,t_{k}
  6. σ j \sigma_{\ell j}
  7. μ \mu_{\ell}
  8. σ i i > 0 \sigma_{ii}>0
  9. E ( exp ( i = 1 k s 𝐗 t ) ) = exp ( - 1 2 , j σ j s s j + i μ s ) . \operatorname{E}\left(\exp\left(i\ \sum_{\ell=1}^{k}s_{\ell}\ \mathbf{X}_{t_{% \ell}}\right)\right)=\exp\left(-\frac{1}{2}\,\sum_{\ell,j}\sigma_{\ell j}s_{% \ell}s_{j}+i\sum_{\ell}\mu_{\ell}s_{\ell}\right).
  10. σ j \sigma_{\ell j}
  11. μ \mu_{\ell}
  12. K C ( x , x ) = C K\text{C}(x,x^{\prime})=C
  13. K L ( x , x ) = x T x K\text{L}(x,x^{\prime})=x^{T}x^{\prime}
  14. K GN ( x , x ) = σ 2 δ x , x K\text{GN}(x,x^{\prime})=\sigma^{2}\delta_{x,x^{\prime}}
  15. K SE ( x , x ) = exp ( - | d | 2 2 l 2 ) K\text{SE}(x,x^{\prime})=\exp\Big(-\frac{|d|^{2}}{2l^{2}}\Big)
  16. K OU ( x , x ) = exp ( - | d | l ) K\text{OU}(x,x^{\prime})=\exp\Big(-\frac{|d|}{l}\Big)
  17. K Matern ( x , x ) = 2 1 - ν Γ ( ν ) ( 2 ν | d | l ) ν K ν ( 2 ν | d | l ) K\text{Matern}(x,x^{\prime})=\frac{2^{1-\nu}}{\Gamma(\nu)}\Big(\frac{\sqrt{2% \nu}|d|}{l}\Big)^{\nu}K_{\nu}\Big(\frac{\sqrt{2\nu}|d|}{l}\Big)
  18. K P ( x , x ) = exp ( - 2 sin 2 ( d 2 ) l 2 ) K\text{P}(x,x^{\prime})=\exp\Big(-\frac{2\sin^{2}(\frac{d}{2})}{l^{2}}\Big)
  19. K RQ ( x , x ) = ( 1 + | d | 2 ) - α , α 0 K\text{RQ}(x,x^{\prime})=(1+|d|^{2})^{-\alpha},\quad\alpha\geq 0
  20. d = x - x d=x-x^{\prime}
  21. l l
  22. x x
  23. x x^{\prime}
  24. K ν K_{\nu}
  25. ν \nu
  26. Γ \Gamma
  27. ν \nu
  28. l l
  29. y y
  30. log p ( f ( x ) | θ , x ) = - 1 2 f ( x ) T K ( θ , x , x ) - 1 f ( x ) - 1 2 log det ( K ( θ , x , x ) ) - | x | 2 log 2 π \log p(f(x)|\theta,x)=-\frac{1}{2}f(x)^{T}K(\theta,x,x^{\prime})^{-1}f(x)-% \frac{1}{2}\log\det(K(\theta,x,x^{\prime}))-\frac{|x|}{2}\log 2\pi
  31. A = K ( θ , x * , x ) K ( θ , x , x ) - 1 f ( x ) A=K(\theta,x^{*},x)K(\theta,x,x^{\prime})^{-1}f(x)
  32. B = K ( θ , x * , x * ) - K ( θ , x * , x ) K ( θ , x , x ) - 1 K ( θ , x * , x ) T B=K(\theta,x^{*},x^{*})-K(\theta,x^{*},x)K(\theta,x,x^{\prime})^{-1}K(\theta,x% ^{*},x)^{T}

Gaussian_year.html

  1. 1 Gaussian year = 2 π k \mbox{1 Gaussian year}~{}=\frac{2\pi}{k}\,

Gauss–Bonnet_theorem.html

  1. M M
  2. M \partial M
  3. K K
  4. M M
  5. k g k_{g}
  6. M \partial M
  7. M K d A + M k g d s = 2 π χ ( M ) , \int_{M}K\;dA+\int_{\partial M}k_{g}\;ds=2\pi\chi(M),\,
  8. χ ( M ) \chi(M)
  9. M M
  10. M \partial M
  11. M k g d s \int_{\partial M}k_{g}\;ds
  12. M k g d s \int_{\partial M}k_{g}\;ds
  13. 2 - 2 g 2-2g
  14. g g
  15. g g
  16. M M
  17. χ = 2 - 2 g \chi=2-2g
  18. 2 π χ . 2\pi\chi.
  19. M M
  20. χ ( v ) \chi(v)
  21. v v
  22. v int M ( 6 - χ ( v ) ) + v M ( 3 - χ ( v ) ) = 6 χ ( M ) , \sum_{v\in{\mathrm{int}}{M}}(6-\chi(v))+\sum_{v\in\partial M}(3-\chi(v))=6\chi% (M),
  23. M M
  24. χ ( M ) \chi(M)
  25. M M
  26. M M
  27. g = 1 + ( M 5 + 2 M 6 - M 3 ) / 8 , g=1+(M_{5}+2M_{6}-M_{3})/8,
  28. M i M_{i}
  29. i i

Gauss–Markov_process.html

  1. 𝐄 ( X 2 ( t ) ) = σ 2 \,\textbf{E}(X^{2}(t))=\sigma^{2}
  2. β - 1 \beta^{-1}
  3. 𝐑 x ( τ ) = σ 2 e - β | τ | . \,\textbf{R}_{x}(\tau)=\sigma^{2}e^{-\beta|\tau|}.\,
  4. 𝐒 x ( j ω ) = 2 σ 2 β ω 2 + β 2 . \,\textbf{S}_{x}(j\omega)=\frac{2\sigma^{2}\beta}{\omega^{2}+\beta^{2}}.\,
  5. 𝐒 x ( s ) = 2 σ 2 β - s 2 + β 2 = 2 β σ ( s + β ) 2 β σ ( - s + β ) . \,\textbf{S}_{x}(s)=\frac{2\sigma^{2}\beta}{-s^{2}+\beta^{2}}=\frac{\sqrt{2% \beta}\,\sigma}{(s+\beta)}\cdot\frac{\sqrt{2\beta}\,\sigma}{(-s+\beta)}.

Gauss–Markov_theorem.html

  1. y ¯ = X β ¯ + ε ¯ , ( y ¯ , ε ¯ n , β K and X n × K ) \underline{y}=X\underline{\beta}+\underline{\varepsilon},\quad(\underline{y},% \underline{\varepsilon}\in\mathbb{R}^{n},\beta\in\mathbb{R}^{K}\,\text{ and }X% \in\mathbb{R}^{n\times K})
  2. y i = j = 1 K β j X i j + ε i i = 1 , 2 , , n y_{i}=\sum_{j=1}^{K}\beta_{j}X_{ij}+\varepsilon_{i}\quad\forall i=1,2,\ldots,n
  3. β j \beta_{j}
  4. X i j X_{ij}
  5. ε i \varepsilon_{i}
  6. y i y_{i}
  7. ε i \varepsilon_{i}
  8. β K + 1 \beta_{K+1}
  9. X i ( K + 1 ) = 1 X_{i(K+1)}=1
  10. i i
  11. E ( ε i ) = 0 , E(\varepsilon_{i})=0,
  12. V ( ε i ) = σ 2 < , V(\varepsilon_{i})=\sigma^{2}<\infty,
  13. cov ( ε i , ε j ) = 0 , i j {\rm cov}(\varepsilon_{i},\varepsilon_{j})=0,\forall i\neq j
  14. i j i\neq j
  15. β j \beta_{j}
  16. β ^ j = c 1 j y 1 + + c n j y n \widehat{\beta}_{j}=c_{1j}y_{1}+\cdots+c_{nj}y_{n}
  17. c i j c_{ij}
  18. β j \beta_{j}
  19. X i j X_{ij}
  20. X i j X_{ij}
  21. y i y_{i}
  22. ε \varepsilon
  23. E ( β ^ j ) = β j E(\widehat{\beta}_{j})=\beta_{j}\,
  24. X i j X_{ij}
  25. j = 1 K λ j β j \sum_{j=1}^{K}\lambda_{j}\beta_{j}
  26. E ( ( j = 1 K λ j ( β ^ j - β j ) ) 2 ) ; E\left(\left(\sum_{j=1}^{K}\lambda_{j}(\widehat{\beta}_{j}-\beta_{j})\right)^{% 2}\right);
  27. β \beta
  28. β j \beta_{j}
  29. λ \lambda
  30. V ( β ~ ) - V ( β ^ ) V(\tilde{\beta})-V(\widehat{\beta})
  31. β ~ \tilde{\beta}
  32. β ^ = ( X X ) - 1 X y \widehat{\beta}=(X^{\prime}X)^{-1}X^{\prime}y
  33. y y
  34. X X
  35. X X^{\prime}
  36. X X
  37. i = 1 n ( y i - y ^ i ) 2 = i = 1 n ( y i - j = 1 K β ^ j X i j ) 2 . \sum_{i=1}^{n}\left(y_{i}-\widehat{y}_{i}\right)^{2}=\sum_{i=1}^{n}\left(y_{i}% -\sum_{j=1}^{K}\widehat{\beta}_{j}X_{ij}\right)^{2}.
  38. a 1 y 1 + + a n y n a_{1}y_{1}+\cdots+a_{n}y_{n}
  39. β \beta
  40. β ~ = C y \tilde{\beta}=Cy
  41. β \beta
  42. ( X X ) - 1 X + D (X^{\prime}X)^{-1}X^{\prime}+D
  43. k × n k\times n
  44. β ^ \hat{\beta}
  45. β ~ \tilde{\beta}
  46. E ( C y ) \displaystyle E(Cy)
  47. β ~ \tilde{\beta}
  48. D X = 0 DX=0
  49. β ~ \tilde{\beta}
  50. V ( β ~ ) = V ( C y ) = C V ( y ) C = σ 2 C C = σ 2 ( ( X X ) - 1 X + D ) ( X ( X X ) - 1 + D ) = σ 2 ( ( X X ) - 1 X X ( X X ) - 1 + ( X X ) - 1 X D + D X ( X X ) - 1 + D D ) = σ 2 ( X X ) - 1 + σ 2 ( X X ) - 1 ( D X 0 ) + σ 2 D X 0 ( X X ) - 1 + σ 2 D D = σ 2 ( X X ) - 1 V ( β ^ ) + σ 2 D D . \begin{aligned}\displaystyle V(\tilde{\beta})&\displaystyle=V(Cy)=CV(y)C^{% \prime}=\sigma^{2}CC^{\prime}\\ &\displaystyle=\sigma^{2}((X^{\prime}X)^{-1}X^{\prime}+D)(X(X^{\prime}X)^{-1}+% D^{\prime})\\ &\displaystyle=\sigma^{2}((X^{\prime}X)^{-1}X^{\prime}X(X^{\prime}X)^{-1}+(X^{% \prime}X)^{-1}X^{\prime}D^{\prime}+DX(X^{\prime}X)^{-1}+DD^{\prime})\\ &\displaystyle=\sigma^{2}(X^{\prime}X)^{-1}+\sigma^{2}(X^{\prime}X)^{-1}(% \underbrace{DX}_{0})^{\prime}+\sigma^{2}\underbrace{DX}_{0}(X^{\prime}X)^{-1}+% \sigma^{2}DD^{\prime}\\ &\displaystyle=\underbrace{\sigma^{2}(X^{\prime}X)^{-1}}_{V(\hat{\beta})}+% \sigma^{2}DD^{\prime}.\end{aligned}
  51. V ( β ~ ) V(\tilde{\beta})
  52. V ( β ^ ) V(\hat{\beta})
  53. V ( β ~ ) - V ( β ^ ) V(\tilde{\beta})-V(\widehat{\beta})
  54. l t β l^{t}\beta
  55. l t β ^ l^{t}\widehat{\beta}
  56. l t β ~ l^{t}\tilde{\beta}
  57. l t β l^{t}\beta
  58. V ( l t β ~ ) = l t V ( β ~ ) l = σ 2 l t ( X X ) - 1 l V ( l t β ^ ) + l t D D t = V ( l t β ^ ) + ( D t l ) ( D t l ) = V ( l t β ^ ) + || D t l || V ( l t β ^ ) \begin{aligned}\displaystyle V(l^{t}\tilde{\beta})&\displaystyle=l^{t}V(\tilde% {\beta})l=\underbrace{\sigma^{2}l^{t}(X^{\prime}X)^{-1}l}_{V(l^{t}\hat{\beta})% }+l^{t}DD^{t}\\ &\displaystyle={V(l^{t}\hat{\beta})}+(D^{t}l)(D^{t}l)={V(l^{t}\hat{\beta})}+||% D^{t}l||\geq{V(l^{t}\hat{\beta})}\\ \end{aligned}
  59. V ( l t β ~ ) V ( l t β ^ ) V(l^{t}\tilde{\beta})\geq V(l^{t}\hat{\beta})
  60. V ( l t β ~ ) = V ( l t β ^ ) V(l^{t}\tilde{\beta})=V(l^{t}\hat{\beta})
  61. D t l = 0 D^{t}l=0
  62. β ~ = ( ( X X ) - 1 X + D ) Y \tilde{\beta}=((X^{\prime}X)^{-1}X^{\prime}+D)Y
  63. l t β ~ = \displaystyle l^{t}\tilde{\beta}=
  64. l t β ~ = l t β ^ l^{t}\tilde{\beta}=l^{t}\widehat{\beta}
  65. y = α + β x 2 , y=\alpha+\beta x^{2},\,
  66. y = α + β 2 x y=\alpha+\beta^{2}x
  67. Y = A L α K β ε Y=AL^{\alpha}K^{\beta}\varepsilon\,
  68. l n Y = l n A + α l n L + β l n K + l n ε lnY=lnA+\alpha lnL+\beta lnK+ln\varepsilon
  69. Var [ ε | X ] = σ 2 I n , \operatorname{Var}[\,\varepsilon|X\,]=\sigma^{2}I_{n},
  70. Var [ ε | X ] = σ 2 I n \operatorname{Var}[\,\varepsilon|X\,]=\sigma^{2}I_{n}
  71. E [ ε | X ] = 0. \operatorname{E}[\,\varepsilon|X\,]=0.

GCD.html

  1. p ( x ) p(x)
  2. q ( x ) q(x)
  3. p ( x ) p(x)
  4. q ( x ) q(x)

GDP_deflator.html

  1. GDP deflator = Nominal GDP Real GDP × 100 \operatorname{GDP\ deflator}=\frac{\operatorname{Nominal\ GDP}}{\operatorname{% Real\ GDP}}\times 100

Gear.html

  1. E = β 1 + β 2 E=\beta_{1}+\beta_{2}
  2. E = β 1 - β 2 E=\beta_{1}-\beta_{2}
  3. β \beta
  4. 1 R P M = π / 30 1\mathrm{RPM}=\pi/30
  5. d = N m n cos ψ d=\frac{Nm_{n}}{\cos\psi}
  6. d = N P d cos ψ d=\frac{N}{P_{d}\cos\psi}
  7. m = p / π m=p/\pi
  8. m = 25.4 / D P m=25.4/DP
  9. a = m ( z 1 + z 2 ) / 2 a=m(z_{1}+z_{2})/2
  10. d w = 2 a u + 1 = 2 a z 2 z 1 + 1 . d_{w}=\frac{2a}{u+1}=\frac{2a}{\frac{z_{2}}{z_{1}}+1}.
  11. θ \theta
  12. D o D_{o}
  13. a = ( D o - D ) / 2 a=(D_{o}-D)/2
  14. b = ( D - root diameter ) / 2 b=(D-\,\text{root diameter})/2
  15. h t h_{t}
  16. D P = N / d = π / p DP=N/d=\pi/p
  17. D P = 25.4 / m DP=25.4/m
  18. p b p_{b}
  19. ψ \psi
  20. p n p_{n}
  21. p n = p cos ( ψ ) p_{n}=p\cos(\psi)
  22. λ \lambda
  23. d w d_{w}
  24. ϵ γ = ϵ α + ϵ β \epsilon_{\gamma}=\epsilon_{\alpha}+\epsilon_{\beta}
  25. m t = m p + m F m_{\rm t}=m_{\rm p}+m_{\rm F}
  26. m o = m p 2 + m F 2 m_{\rm o}=\sqrt{m_{\rm p}^{2}+m_{\rm F}^{2}}
  27. P d = N d = 25.4 m = π p P_{\rm d}=\frac{N}{d}=\frac{25.4}{m}=\frac{\pi}{p}
  28. P nd = P d cos ψ P_{\rm nd}=\frac{P_{\rm d}}{\cos\psi}
  29. τ = 360 z \tau=\frac{360}{z}
  30. 2 π z \frac{2\pi}{z}

Gelfand–Naimark_theorem.html

  1. π ( x ) [ f ξ f ] = f π f ( x ) ξ f . \pi(x)[\bigoplus_{f}\xi_{f}]=\bigoplus_{f}\pi_{f}(x)\xi_{f}.
  2. π f ( x ) ξ 2 = π f ( x ) ξ π f ( x ) ξ = ξ π f ( x * ) π f ( x ) ξ = ξ π f ( x * x ) ξ = f ( x * x ) > 0 , \begin{aligned}\displaystyle\|\pi_{f}(x)\xi\|^{2}&\displaystyle=\langle\pi_{f}% (x)\xi\mid\pi_{f}(x)\xi\rangle=\langle\xi\mid\pi_{f}(x^{*})\pi_{f}(x)\xi% \rangle\\ &\displaystyle=\langle\xi\mid\pi_{f}(x^{*}x)\xi\rangle=f(x^{*}x)>0,\end{aligned}
  3. x C * = sup f f ( x * x ) \|x\|_{\operatorname{C}^{*}}=\sup_{f}\sqrt{f(x^{*}x)}
  4. C * \|\cdot\|_{\operatorname{C}^{*}}
  5. sup f State ( A ) f ( x * x ) = sup f PureState ( A ) f ( x * x ) . \sup_{f\in\operatorname{State}(A)}f(x^{*}x)=\sup_{f\in\operatorname{PureState}% (A)}f(x^{*}x).
  6. A A
  7. A A

Gelfand–Naimark–Segal_construction.html

  1. { π ( x ) ξ : x A } \{\pi(x)\xi:x\in A\}
  2. a π ( a ) ξ , ξ a\mapsto\langle\pi(a)\xi,\xi\rangle
  3. ρ ( a ) = π ( a ) ξ , ξ \rho(a)=\langle\pi(a)\xi,\xi\rangle
  4. a , b = ρ ( b * a ) , a , b A . \langle a,b\rangle=\rho(b^{*}a),\;a,b\in A.
  5. ρ ( a ) = π ( a ) ξ , ξ \rho(a)=\langle\pi(a)\xi,\xi\rangle
  6. ρ ( a ) = π ( a ) ξ , ξ = π ( a ) ξ , ξ \rho(a)=\langle\pi(a)\xi,\xi\rangle=\langle\pi^{\prime}(a)\xi^{\prime},\xi^{% \prime}\rangle
  7. a A a\in A
  8. g ( x * x ) = π ( x ) ξ , π ( x ) T g ξ g(x^{*}x)=\langle\pi(x)\xi,\pi(x)T_{g}\,\xi\rangle

General_linear_group.html

  1. T e k = j = 1 n a j k e j Te_{k}=\sum_{j=1}^{n}a_{jk}e_{j}
  2. 𝔤 𝔩 n , \mathfrak{gl}_{n},
  3. ( q n - 1 ) ( q n - q ) ( q n - q 2 ) ( q n - q n - 1 ) (q^{n}-1)(q^{n}-q)(q^{n}-q^{2})\ \cdots\ (q^{n}-q^{n-1})
  4. [ n ] q ! ( q - 1 ) n q ( n 2 ) . [n]_{q}!(q-1)^{n}q^{n\choose 2}.

General_number_field_sieve.html

  1. n n
  2. log 2 n + 1 \left\lfloor\log_{2}n\right\rfloor+1
  3. exp ( ( 64 9 3 + o ( 1 ) ) ( ln n ) 1 3 ( ln ln n ) 2 3 ) = L n [ 1 3 , 64 9 3 ] \exp\left(\left(\sqrt[3]{\frac{64}{9}}+o(1)\right)(\ln n)^{\frac{1}{3}}(\ln\ln n% )^{\frac{2}{3}}\right)=L_{n}\left[\frac{1}{3},\sqrt[3]{\frac{64}{9}}\right]
  4. l n ln
  5. n n
  6. n n
  7. n n
  8. n n
  9. c c
  10. l o g n logn
  11. f f
  12. k k
  13. 𝐐 \mathbf{Q}
  14. r r
  15. f f
  16. f ( r ) = 0 f(r)=0
  17. r r
  18. k k
  19. r k r≥k
  20. r r
  21. i i
  22. ( a + b i ) ( c + d i ) = a c + ( a d + b c ) i + ( b d ) i 2 = ( a c - b d ) + ( a d + b c ) i . (a+bi)(c+di)=ac+(ad+bc)i+(bd)i^{2}=(ac-bd)+(ad+bc)i.
  23. 𝐐 r r \mathbf{Q}rr
  24. a k - 1 r k - 1 + + a 1 r 1 + a 0 r 0 , where a 0 , , a k - 1 in Q . a_{k-1}r^{k-1}+...+a_{1}r^{1}+a_{0}r^{0},\,\text{ where }a_{0},...,a_{k-1}\,% \text{ in }{Q}.
  25. r k r≥k
  26. k k
  27. f f
  28. [ - F o r m u l a E r r o r - ] [-FormulaError-]

General_topology.html

  1. f f
  2. x X x∈X
  3. V V
  4. f ( x ) f(x)
  5. U U
  6. x x
  7. f ( U ) V f(U) ⊆V
  8. V V
  9. U U
  10. x x
  11. V V
  12. f f
  13. f ( x ) f(x)
  14. Y Y
  15. X X
  16. X X
  17. f : X T f\colon X\rightarrow T
  18. T T
  19. X X
  20. T T
  21. f : ( X , cl ) ( X , cl ) f\colon(X,\mathrm{cl})\to(X^{\prime},\mathrm{cl}^{\prime})\,
  22. f ( cl ( A ) ) cl ( f ( A ) ) . f(\mathrm{cl}(A))\subseteq\mathrm{cl}^{\prime}(f(A)).
  23. f - 1 ( cl ( A ) ) cl ( f - 1 ( A ) ) . f^{-1}(\mathrm{cl}^{\prime}(A^{\prime}))\supseteq\mathrm{cl}(f^{-1}(A^{\prime}% )).
  24. f : ( X , int ) ( X , int ) f\colon(X,\mathrm{int})\to(X^{\prime},\mathrm{int}^{\prime})\,
  25. f - 1 ( int ( A ) ) int ( f - 1 ( A ) ) f^{-1}(\mathrm{int}^{\prime}(A))\subseteq\mathrm{int}(f^{-1}(A))
  26. ( X , τ X ) ( Y , τ Y ) (X,\tau_{X})\rightarrow(Y,\tau_{Y})
  27. f : X S , f\colon X\rightarrow S,\,
  28. S X S\rightarrow X
  29. X S . X\rightarrow S.
  30. { U α } α A \{U_{\alpha}\}_{\alpha\in A}
  31. X X
  32. X = α A U α , X=\bigcup_{\alpha\in A}U_{\alpha},
  33. J J
  34. A A
  35. X = i J U i . X=\bigcup_{i\in J}U_{i}.
  36. Γ x \Gamma_{x}
  37. Γ x \Gamma_{x}^{\prime}
  38. Γ x Γ x \Gamma_{x}\subset\Gamma^{\prime}_{x}
  39. X := i I X i , X:=\prod_{i\in I}X_{i},
  40. i I i\in I
  41. i I U i \prod_{i\in I}U_{i}
  42. i I X i \prod_{i\in I}X_{i}
  43. ( M , d ) (M,d)
  44. M M
  45. d d
  46. M M
  47. d : M × M d\colon M\times M\rightarrow\mathbb{R}
  48. x , y , z M x,y,z\in M
  49. d ( x , y ) 0 d(x,y)\geq 0
  50. d ( x , y ) = 0 d(x,y)=0\,
  51. x = y x=y\,
  52. d ( x , y ) = d ( y , x ) d(x,y)=d(y,x)\,
  53. d ( x , z ) d ( x , y ) + d ( y , z ) d(x,z)\leq d(x,y)+d(y,z)
  54. d d
  55. d d
  56. M M
  57. : A × A A \cdot:A\times A\longrightarrow A
  58. ( a , b ) a b (a,b)\longmapsto a\cdot b
  59. ( X , τ ) (X,\tau)
  60. d : X × X [ 0 , ) d\colon X\times X\to[0,\infty)
  61. τ \tau

Generalised_logistic_function.html

  1. Y ( t ) = A + K - A ( C + Q e - B t ) 1 / ν Y(t)=A+{K-A\over(C+Qe^{-Bt})^{1/\nu}}
  2. Y Y
  3. t t
  4. A A
  5. K K
  6. A = 0 A=0
  7. K K
  8. B B
  9. ν > 0 \nu>0
  10. Q Q
  11. Y ( 0 ) Y(0)
  12. C C
  13. Y ( t ) = A + K - A ( C + e - B ( t - M ) ) 1 / ν Y(t)=A+{K-A\over(C+e^{-B(t-M)})^{1/\nu}}
  14. M M
  15. t 0 t_{0}
  16. Y ( t 0 ) = A + K - A ( C + 1 ) 1 / ν Y(t_{0})=A+{K-A\over(C+1)^{1/\nu}}
  17. Q Q
  18. M M
  19. Y ( t ) = A + K - A ( C + Q e - B ( t - M ) ) 1 / ν Y(t)=A+{K-A\over(C+Qe^{-B(t-M)})^{1/\nu}}
  20. M M
  21. Q = ν Q=\nu
  22. Y ( t ) = K ( 1 + Q e - α ν ( t - t 0 ) ) 1 / ν Y(t)={K\over(1+Qe^{-\alpha\nu(t-t_{0})})^{1/\nu}}
  23. Y ( t ) = α ( 1 - ( Y K ) ν ) Y Y^{\prime}(t)=\alpha\left(1-\left(\frac{Y}{K}\right)^{\nu}\right)Y
  24. Y ( t 0 ) = Y 0 Y(t_{0})=Y_{0}
  25. Q = - 1 + ( K Y 0 ) ν Q=-1+\left(\frac{K}{Y_{0}}\right)^{\nu}
  26. ν 0 + \nu\rightarrow 0^{+}
  27. α = O ( 1 ν ) \alpha=O\left(\frac{1}{\nu}\right)
  28. Y ( t ) = Y r 1 - exp ( ν ln ( Y K ) ) ν r Y ln ( Y K ) Y^{\prime}(t)=Yr\frac{1-\exp\left(\nu\ln\left(\frac{Y}{K}\right)\right)}{\nu}% \approx rY\ln\left(\frac{Y}{K}\right)
  29. t t
  30. C = 1 C=1
  31. Y A = 1 - ( 1 + Q e - B ( t - M ) ) - 1 / ν Y K = ( 1 + Q e - B ( t - M ) ) - 1 / ν Y B = ( K - A ) ( t - M ) Q e - B ( t - M ) ν ( 1 + Q e - B ( t - M ) ) 1 ν + 1 Y ν = ( K - A ) ln ( 1 + Q e - B ( t - M ) ) ν 2 ( 1 + Q e - B ( t - M ) ) 1 ν Y Q = - ( K - A ) e - B ( t - M ) ν ( 1 + Q e - B ( t - M ) ) 1 ν + 1 Y M = - ( K - A ) B e - B ( t - M ) ν ( 1 + Q e - B ( t - M ) ) 1 ν + 1 \begin{aligned}\displaystyle\frac{\partial Y}{\partial A}&\displaystyle=1-(1+% Qe^{-B(t-M)})^{-1/\nu}\\ \displaystyle\frac{\partial Y}{\partial K}&\displaystyle=(1+Qe^{-B(t-M)})^{-1/% \nu}\\ \displaystyle\frac{\partial Y}{\partial B}&\displaystyle=\frac{(K-A)(t-M)Qe^{-% B(t-M)}}{\nu(1+Qe^{-B(t-M)})^{\frac{1}{\nu}+1}}\\ \displaystyle\frac{\partial Y}{\partial\nu}&\displaystyle=\frac{(K-A)\ln(1+Qe^% {-B(t-M)})}{\nu^{2}(1+Qe^{-B(t-M)})^{\frac{1}{\nu}}}\\ \displaystyle\frac{\partial Y}{\partial Q}&\displaystyle=-\frac{(K-A)e^{-B(t-M% )}}{\nu(1+Qe^{-B(t-M)})^{\frac{1}{\nu}+1}}\\ \displaystyle\frac{\partial Y}{\partial M}&\displaystyle=-\frac{(K-A)Be^{-B(t-% M)}}{\nu(1+Qe^{-B(t-M)})^{\frac{1}{\nu}+1}}\end{aligned}

Generalized_Fourier_series.html

  1. 𝔽 = or \mathbb{F}=\mathbb{C}\mbox{ or }~{}\mathbb{R}
  2. Φ = { φ n : [ a , b ] 𝔽 } n = 0 , \Phi=\{\varphi_{n}:[a,b]\rightarrow\mathbb{F}\}_{n=0}^{\infty},
  3. f , g w = a b f ( x ) g ¯ ( x ) w ( x ) d x \langle f,g\rangle_{w}=\int_{a}^{b}f(x)\,\overline{g}(x)\,w(x)\,dx
  4. ¯ \overline{\cdot}
  5. g ¯ ( x ) = g ( x ) \overline{g}(x)=g(x)
  6. 𝔽 = \mathbb{F}=\mathbb{R}
  7. 𝔽 \mathbb{F}
  8. f ( x ) n = 0 c n φ n ( x ) , f(x)\sim\sum_{n=0}^{\infty}c_{n}\varphi_{n}(x),
  9. c n = f , φ n w φ n w 2 . c_{n}={\langle f,\varphi_{n}\rangle_{w}\over\|\varphi_{n}\|_{w}^{2}}.
  10. \sim\,
  11. ( ( 1 - x 2 ) P n ( x ) ) + n ( n + 1 ) P n ( x ) = 0 \left((1-x^{2})P_{n}^{\prime}(x)\right)^{\prime}+n(n+1)P_{n}(x)=0
  12. f ( x ) n = 0 c n P n ( x ) , f(x)\sim\sum_{n=0}^{\infty}c_{n}P_{n}(x),
  13. c n = f , P n w P n w 2 c_{n}={\langle f,P_{n}\rangle_{w}\over\|P_{n}\|_{w}^{2}}
  14. c 0 = sin 1 = - 1 1 cos x d x - 1 1 ( 1 ) 2 d x c 1 = 0 = - 1 1 x cos x d x - 1 1 x 2 d x = 0 2 / 3 c 2 = 5 2 ( 6 cos 1 - 4 sin 1 ) = - 1 1 3 x 2 - 1 2 cos x d x - 1 1 9 x 4 - 6 x 2 + 1 4 d x = 6 cos 1 - 4 sin 1 2 / 5 \begin{aligned}\displaystyle c_{0}&\displaystyle=\sin{1}={\int_{-1}^{1}\cos{x}% \,dx\over\int_{-1}^{1}(1)^{2}\,dx}\\ \displaystyle c_{1}&\displaystyle=0={\int_{-1}^{1}x\cos{x}\,dx\over\int_{-1}^{% 1}x^{2}\,dx}={0\over 2/3}\\ \displaystyle c_{2}&\displaystyle={5\over 2}(6\cos{1}-4\sin{1})={\int_{-1}^{1}% {3x^{2}-1\over 2}\cos{x}\,dx\over\int_{-1}^{1}{9x^{4}-6x^{2}+1\over 4}\,dx}={6% \cos{1}-4\sin{1}\over 2/5}\end{aligned}
  15. c 2 P 2 ( x ) + c 1 P 1 ( x ) + c 0 P 0 ( x ) = 5 2 ( 6 cos 1 - 4 sin 1 ) ( 3 x 2 - 1 2 ) + sin 1 ( 1 ) c_{2}P_{2}(x)+c_{1}P_{1}(x)+c_{0}P_{0}(x)={5\over 2}(6\cos{1}-4\sin{1})\left({% 3x^{2}-1\over 2}\right)+\sin{1}(1)
  16. = ( 45 2 cos 1 - 15 sin 1 ) x 2 + 6 sin 1 - 15 2 cos 1 =\left({45\over 2}\cos{1}-15\sin{1}\right)x^{2}+6\sin{1}-{15\over 2}\cos{1}
  17. n = 0 | c n | 2 a b | f ( x ) | 2 d x . \sum_{n=0}^{\infty}|c_{n}|^{2}\leq\int_{a}^{b}|f(x)|^{2}\,dx.
  18. n = 0 | c n | 2 = a b | f ( x ) | 2 d x . \sum_{n=0}^{\infty}|c_{n}|^{2}=\int_{a}^{b}|f(x)|^{2}\,dx.

Generalized_permutation_matrix.html

  1. [ 0 0 3 0 0 - 2 0 0 1 0 0 0 0 0 0 1 ] . \begin{bmatrix}0&0&3&0\\ 0&-2&0&0\\ 1&0&0&0\\ 0&0&0&1\end{bmatrix}.
  2. A = D P . A=DP.
  3. μ m \mu_{m}
  4. N ( T ) / Z ( T ) = N ( T ) / T S n N(T)/Z(T)=N(T)/T\cong S_{n}
  5. G S n G\wr S_{n}
  6. B n B_{n}
  7. 2 n n ! 2^{n}n!
  8. D n D_{n}

Generalized_Riemann_hypothesis.html

  1. L ( χ , s ) = n = 1 χ ( n ) n s L(\chi,s)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}
  2. π ( x , a , d ) = 1 φ ( d ) 2 x 1 ln t d t + O ( x 1 / 2 + ϵ ) as x \pi(x,a,d)=\frac{1}{\varphi(d)}\int_{2}^{x}\frac{1}{\ln t}\,dt+O(x^{1/2+% \epsilon})\quad\mbox{ as }~{}\ x\to\infty
  3. O O
  4. ( / n ) × (\mathbb{Z}/n\mathbb{Z})^{\times}
  5. ( / n ) × (\mathbb{Z}/n\mathbb{Z})^{\times}
  6. O ( ( ln p ) 6 ) . O((\ln p)^{6}).\,
  7. O ( q log log q ) O\left(\sqrt{q}\log\log q\right)
  8. ζ K ( s ) = a 1 ( N a ) s \zeta_{K}(s)=\sum_{a}\frac{1}{(Na)^{s}}
  9. | C | | G | ( li ( x ) + O ( x ( n log x + log | Δ | ) ) ) , \frac{|C|}{|G|}\Bigl(\mathrm{li}(x)+O\bigl(\sqrt{x}(n\log x+\log|\Delta|)\bigr% )\Bigr),

Generating_function.html

  1. G ( a n ; x ) = n = 0 a n x n . G(a_{n};x)=\sum_{n=0}^{\infty}a_{n}x^{n}.
  2. G ( a m , n ; x , y ) = m , n = 0 a m , n x m y n . G(a_{m,n};x,y)=\sum_{m,n=0}^{\infty}a_{m,n}x^{m}y^{n}.
  3. EG ( a n ; x ) = n = 0 a n x n n ! . \operatorname{EG}(a_{n};x)=\sum_{n=0}^{\infty}a_{n}\frac{x^{n}}{n!}.
  4. PG ( a n ; x ) = n = 0 a n e - x x n n ! = e - x EG ( a n ; x ) . \operatorname{PG}(a_{n};x)=\sum_{n=0}^{\infty}a_{n}e^{-x}\frac{x^{n}}{n!}=e^{-% x}\,\operatorname{EG}(a_{n};x).
  5. LG ( a n ; x ) = n = 1 a n x n 1 - x n . \operatorname{LG}(a_{n};x)=\sum_{n=1}^{\infty}a_{n}\frac{x^{n}}{1-x^{n}}.
  6. BG p ( a n ; x ) = n = 0 a p n x n . \operatorname{BG}_{p}(a_{n};x)=\sum_{n=0}^{\infty}a_{p^{n}}x^{n}.
  7. DG ( a n ; s ) = n = 1 a n n s . \operatorname{DG}(a_{n};s)=\sum_{n=1}^{\infty}\frac{a_{n}}{n^{s}}.
  8. DG ( a n ; s ) = p BG p ( a n ; p - s ) . \operatorname{DG}(a_{n};s)=\prod_{p}\operatorname{BG}_{p}(a_{n};p^{-s})\,.
  9. e x f ( t ) = n = 0 p n ( x ) n ! t n e^{xf(t)}=\sum_{n=0}^{\infty}\frac{p_{n}(x)}{n!}t^{n}
  10. n = 0 x n = 1 1 - x . \sum_{n=0}^{\infty}x^{n}=\frac{1}{1-x}.
  11. n = 0 ( a x ) n = 1 1 - a x . \sum_{n=0}^{\infty}(ax)^{n}=\frac{1}{1-ax}.
  12. n = 0 ( - 1 ) n x n = 1 1 + x . \sum_{n=0}^{\infty}(-1)^{n}x^{n}=\frac{1}{1+x}.
  13. n = 0 x 2 n = 1 1 - x 2 . \sum_{n=0}^{\infty}x^{2n}=\frac{1}{1-x^{2}}.
  14. n = 0 ( n + 1 ) x n = 1 ( 1 - x ) 2 , \sum_{n=0}^{\infty}(n+1)x^{n}=\frac{1}{(1-x)^{2}},
  15. ( n + 2 2 ) {\textstyle\left({{n+2}\atop{2}}\right)}
  16. n = 0 ( n + 2 2 ) x n = 1 ( 1 - x ) 3 . \sum_{n=0}^{\infty}{\left({{n+2}\atop{2}}\right)}x^{n}=\frac{1}{(1-x)^{3}}.
  17. n = 0 a n ( n + k k ) x n = 1 ( 1 - a x ) k + 1 . \sum_{n=0}^{\infty}a^{n}{\left({{n+k}\atop{k}}\right)}x^{n}=\frac{1}{(1-ax)^{k% +1}}\,.
  18. 2 ( n + 2 2 ) - 3 ( n + 1 1 ) + ( n 0 ) = 2 ( n + 1 ) ( n + 2 ) 2 - 3 ( n + 1 ) + 1 = n 2 , 2{\left({{n+2}\atop{2}}\right)}-3{\left({{n+1}\atop{1}}\right)}+{\left({{n}% \atop{0}}\right)}=2\frac{(n+1)(n+2)}{2}-3(n+1)+1=n^{2},
  19. G ( n 2 ; x ) = n = 0 n 2 x n = 2 ( 1 - x ) 3 - 3 ( 1 - x ) 2 + 1 1 - x = x ( x + 1 ) ( 1 - x ) 3 . G(n^{2};x)=\sum_{n=0}^{\infty}n^{2}x^{n}=\frac{2}{(1-x)^{3}}-\frac{3}{(1-x)^{2% }}+\frac{1}{1-x}=\frac{x(x+1)}{(1-x)^{3}}.
  20. ( a 0 , a 0 + a 1 , a 0 + a 1 + a 2 , ) (a_{0},a_{0}+a_{1},a_{0}+a_{1}+a_{2},\cdots)
  21. G ( a n ; x ) 1 1 - x G(a_{n};x)\cdot\frac{1}{1-x}
  22. G ( a n ; e - i ω ) = n = 0 a n e - i ω n G\left(a_{n};e^{-i\omega}\right)=\sum_{n=0}^{\infty}a_{n}e^{-i\omega n}
  23. G ( a n ; x ) = A ( x ) + B ( x ) ( 1 - x r ) - β x α G(a_{n};x)=\frac{A(x)+B(x)\left(1-\frac{x}{r}\right)^{-\beta}}{x^{\alpha}}
  24. a n B ( r ) r α ( n + β - 1 n ) ( 1 / r ) n B ( r ) r α Γ ( β ) n β - 1 ( 1 / r ) n , a_{n}\sim\frac{B(r)}{r^{\alpha}}{\left({{n+\beta-1}\atop{n}}\right)}(1/r)^{n}% \sim\frac{B(r)}{r^{\alpha}\Gamma(\beta)}\,n^{\beta-1}(1/r)^{n}\,,
  25. G ( a n - B ( r ) r α ( n + β - 1 n ) ( 1 / r ) n , x ) = G ( a n ; x ) - B ( r ) r α ( 1 - x r ) - β . G\left(a_{n}-\frac{B(r)}{r^{\alpha}}{\left({{n+\beta-1}\atop{n}}\right)}(1/r)^% {n},x\right)=G(a_{n};x)-\frac{B(r)}{r^{\alpha}}\left(1-\frac{x}{r}\right)^{-% \beta}\,.
  26. x ( x + 1 ) ( 1 - x ) 3 . \frac{x(x+1)}{(1-x)^{3}}.
  27. a n B ( r ) r α Γ ( β ) n β - 1 ( 1 r ) n = 1 ( 1 + 1 ) 1 0 Γ ( 3 ) n 3 - 1 ( 1 / 1 ) n = n 2 . a_{n}\sim\frac{B(r)}{r^{\alpha}\Gamma(\beta)}\,n^{\beta-1}\left(\frac{1}{r}% \right)^{n}=\frac{1(1+1)}{1^{0}\,\Gamma(3)}\,n^{3-1}(1/1)^{n}=n^{2}.
  28. 1 - 1 - 4 x 2 x . \frac{1-\sqrt{1-4x}}{2x}.
  29. a n B ( r ) r α Γ ( β ) n β - 1 ( 1 r ) n = - 1 2 ( 1 4 ) 1 Γ ( - 1 2 ) n - 1 2 - 1 ( 1 1 4 ) n = 1 π n - 3 2 4 n . a_{n}\sim\frac{B(r)}{r^{\alpha}\Gamma(\beta)}\,n^{\beta-1}\left(\frac{1}{r}% \right)^{n}=\frac{-\frac{1}{2}}{(\frac{1}{4})^{1}\Gamma(-\frac{1}{2})}\,n^{-% \frac{1}{2}-1}\left(\frac{1}{\frac{1}{4}}\right)^{n}=\frac{1}{\sqrt{\pi}}n^{-% \frac{3}{2}}\,4^{n}.
  30. ( 1 + x ) n (1+x)^{n}
  31. ( n k ) {\left({{n}\atop{k}}\right)}
  32. ( 1 + x ) n (1+x)^{n}
  33. a n a^{n}
  34. 1 1 - a y , \frac{1}{1-ay},
  35. n , k ( n k ) x k y n = 1 1 - ( 1 + x ) y = 1 1 - y - x y . \sum_{n,k}{\left({{n}\atop{k}}\right)}x^{k}y^{n}=\frac{1}{1-(1+x)y}=\frac{1}{1% -y-xy}.
  36. G ( n 2 ; x ) = n = 0 n 2 x n = x ( x + 1 ) ( 1 - x ) 3 G(n^{2};x)=\sum_{n=0}^{\infty}n^{2}x^{n}=\frac{x(x+1)}{(1-x)^{3}}
  37. EG ( n 2 ; x ) = n = 0 n 2 x n n ! = x ( x + 1 ) e x \operatorname{EG}(n^{2};x)=\sum_{n=0}^{\infty}\frac{n^{2}x^{n}}{n!}=x(x+1)e^{x}
  38. BG p ( n 2 ; x ) = n = 0 ( p n ) 2 x n = 1 1 - p 2 x \operatorname{BG}_{p}(n^{2};x)=\sum_{n=0}^{\infty}(p^{n})^{2}x^{n}=\frac{1}{1-% p^{2}x}
  39. DG ( n 2 ; s ) = n = 1 n 2 n s = ζ ( s - 2 ) , \operatorname{DG}(n^{2};s)=\sum_{n=1}^{\infty}\frac{n^{2}}{n^{s}}=\zeta(s-2),
  40. DG ( a n ; s ) = ζ ( s ) m \operatorname{DG}(a_{n};s)=\zeta(s)^{m}
  41. ζ ( s ) \zeta(s)
  42. n = 1 a n x n = x + ( m 1 ) a = 2 x a + ( m 2 ) a = 2 b = 2 x a b + ( m 3 ) a = 2 b = 2 c = 2 x a b c + ( m 4 ) a = 2 b = 2 c = 2 d = 2 x a b c d + \sum_{n=1}^{\infty}a_{n}x^{n}=x+{m\choose 1}\sum_{a=2}^{\infty}x^{a}+{m\choose 2% }\sum_{a=2}^{\infty}\sum_{b=2}^{\infty}x^{ab}+{m\choose 3}\sum_{a=2}^{\infty}% \sum_{b=2}^{\infty}\sum_{c=2}^{\infty}x^{abc}+{m\choose 4}\sum_{a=2}^{\infty}% \sum_{b=2}^{\infty}\sum_{c=2}^{\infty}\sum_{d=2}^{\infty}x^{abcd}+\cdots
  43. t 1 , t r t_{1},\ldots t_{r}
  44. s 1 , s c s_{1},\ldots s_{c}
  45. x 1 t 1 x r t r y 1 s 1 y c s c x_{1}^{t_{1}}\ldots x_{r}^{t_{r}}y_{1}^{s_{1}}\ldots y_{c}^{s_{c}}
  46. i = 1 r j = 1 c 1 1 - x i y j . \prod_{i=1}^{r}\prod_{j=1}^{c}\frac{1}{1-x_{i}y_{j}}.
  47. s n = k = 0 n a k s_{n}=\sum_{k=0}^{n}{a_{k}}
  48. S ( z ) = A ( z ) 1 - z S(z)=\frac{A(z)}{1-z}
  49. s n = k = 1 n H k s_{n}=\sum_{k=1}^{n}H_{k}
  50. H k = 1 + 1 2 + + 1 k H_{k}=1+\frac{1}{2}+\cdots+\frac{1}{k}
  51. H ( z ) = n 1 H n z n H(z)=\sum_{n\geq 1}{H_{n}z^{n}}
  52. H ( z ) = n 1 1 n z n 1 - z , H(z)=\dfrac{\sum_{n\geq 1}{\frac{1}{n}z^{n}}}{1-z}\,,
  53. S ( z ) = n 1 s n z n = n 1 1 n z n ( 1 - z ) 2 . S(z)=\sum_{n\geq 1}{s_{n}z^{n}}=\dfrac{\sum_{n\geq 1}{\frac{1}{n}z^{n}}}{(1-z)% ^{2}}\,.
  54. 1 ( 1 - z ) 2 = n 0 ( n + 1 ) z n \frac{1}{(1-z)^{2}}=\sum_{n\geq 0}{(n+1)z^{n}}
  55. s n = k = 1 n 1 k ( n + 1 - k ) = ( n + 1 ) H n - n , s_{n}=\sum_{k=1}^{n}{\frac{1}{k}(n+1-k)}=(n+1)H_{n}-n\,,
  56. k = 1 n H k = ( n + 1 ) ( H n + 1 - 1 ) . \sum_{k=1}^{n}{H_{k}}=(n+1)(H_{n+1}-1)\,.
  57. C ( z ) = A ( z ) B ( z ) [ z n ] C ( z ) = a k b n - k C(z)=A(z)B(z)\Leftrightarrow[z^{n}]C(z)=\sum{a_{k}b_{n-k}}
  58. C ( z ) = A ( z ) B ( z ) [ z n ] C ( z ) = ( n k ) a k b n - k C(z)=A(z)B(z)\Leftrightarrow[z^{n}]C(z)=\sum{{\left({{n}\atop{k}}\right)}a_{k}% b_{n-k}}
  59. s n s_{n}
  60. s n s_{n}
  61. F ( z ) = s n z n F(z)=\sum{s_{n}z^{n}}
  62. s n = k 0 ( n + k m + 2 k ) ( 2 k k ) ( - 1 ) k k + 1 ( m , n 0 ) s_{n}=\sum_{k\geq 0}{{\left({{n+k}\atop{m+2k}}\right)}{\left({{2k}\atop{k}}% \right)}\frac{(-1)^{k}}{k+1}}\quad(m,n\in\mathbb{N}_{0})
  63. F ( z ) = n 0 [ k 0 ( n + k m + 2 k ) ( 2 k k ) ( - 1 ) k k + 1 ] z n F(z)=\sum_{n\geq 0}{\left[\sum_{k\geq 0}{{\left({{n+k}\atop{m+2k}}\right)}{% \left({{2k}\atop{k}}\right)}\frac{(-1)^{k}}{k+1}}\right]}z^{n}
  64. F ( z ) = k 0 ( 2 k k ) ( - 1 ) k k + 1 z - k n 0 ( n + k m + 2 k ) z n + k F(z)=\sum_{k\geq 0}{{\left({{2k}\atop{k}}\right)}\frac{(-1)^{k}}{k+1}z^{-k}}% \sum_{n\geq 0}{{\left({{n+k}\atop{m+2k}}\right)}z^{n+k}}
  65. z m + 2 k ( 1 - z ) m + 2 k + 1 \frac{z^{m+2k}}{(1-z)^{m+2k+1}}
  66. F ( z ) = z m ( 1 - z ) m + 1 k 0 1 k + 1 ( 2 k k ) ( - z ( 1 - z ) 2 ) k = z m ( 1 - z ) m + 1 k 0 C k ( - z ( 1 - z ) 2 ) k ( where , C k = k th Catalan number ) = z m ( 1 - z ) m + 1 1 - 1 + 4 z ( 1 - z ) 2 - 2 z ( 1 - z ) 2 = - z m - 1 2 ( 1 - z ) m - 1 ( 1 - 1 + z 1 - z ) = z m ( 1 - z ) m = z z m - 1 ( 1 - z ) m . \begin{aligned}\displaystyle F(z)&\displaystyle=\frac{z^{m}}{(1-z)^{m+1}}\sum_% {k\geq 0}{\frac{1}{k+1}{\left({{2k}\atop{k}}\right)}(\frac{-z}{(1-z)^{2}})^{k}% }\\ &\displaystyle=\frac{z^{m}}{(1-z)^{m+1}}\sum_{k\geq 0}{C_{k}(\frac{-z}{(1-z)^{% 2}})^{k}}\quad{\rm(where,C_{k}\,=\,k^{th}\,Catalan\,number)}\\ &\displaystyle=\frac{z^{m}}{(1-z)^{m+1}}\frac{1-\sqrt{1+\frac{4z}{(1-z)^{2}}}}% {\frac{-2z}{(1-z)^{2}}}\\ &\displaystyle=\frac{-z^{m-1}}{2(1-z)^{m-1}}(1-\frac{1+z}{1-z})\\ &\displaystyle=\frac{z^{m}}{(1-z)^{m}}=z\frac{z^{m-1}}{(1-z)^{m}}.\end{aligned}
  67. s n = ( n - 1 m - 1 ) f o r m 1 , s n = [ n = 0 ] f o r m = 0. s_{n}={\left({{n-1}\atop{m-1}}\right)}\quad for\quad m\geq 1\quad,\quad s_{n}=% [n=0]\quad for\quad m=0.

Generating_set_of_a_group.html

  1. { 7 i mod 9 | i } = { 7 , 4 , 1 } . \{7^{i}\mod{9}\ |\ i\in\mathbb{N}\}=\{7,4,1\}.
  2. { 2 i mod 9 | i } = { 2 , 4 , 8 , 7 , 5 , 1 } . \{2^{i}\mod{9}\ |\ i\in\mathbb{N}\}=\{2,4,8,7,5,1\}.

Genetic_drift.html

  1. 1 2 1 2 1 2 1 2 = 1 16 . \frac{1}{2}\cdot\frac{1}{2}\cdot\frac{1}{2}\cdot\frac{1}{2}=\frac{1}{16}.
  2. ( n k ) ( 1 2 ) k ( 1 - 1 2 ) n - k = ( n k ) ( 1 2 ) n {n\choose k}\left(\frac{1}{2}\right)^{k}\left(1-\frac{1}{2}\right)^{n-k}={n% \choose k}\left(\frac{1}{2}\right)^{n}\!
  3. ( 2 N ) ! k ! ( 2 N - k ) ! p k q 2 N - k \frac{(2N)!}{k!(2N-k)!}p^{k}q^{2N-k}
  4. ( 2 N k ) p k q 2 N - k {2N\choose k}p^{k}q^{2N-k}
  5. V t p q ( 1 - exp ( - t 2 N e ) ) V_{t}\approx pq\left(1-\exp\left(-\frac{t}{2N_{e}}\right)\right)
  6. T ¯ fixed = - 4 N e ( 1 - p ) ln ( 1 - p ) p \bar{T}\text{fixed}=\frac{-4N_{e}(1-p)\ln(1-p)}{p}
  7. T ¯ lost = - 4 N e p 1 - p ln p . \bar{T}\text{lost}=\frac{-4N_{e}p}{1-p}\ln p.
  8. T ¯ fixed = 4 N e \bar{T}\text{fixed}=4N_{e}
  9. T ¯ lost = 2 ( N e N ) ln ( 2 N ) \bar{T}\text{lost}=2\left(\frac{N_{e}}{N}\right)\ln(2N)
  10. T ¯ lost { 1 m , if m N e 1 ln ( m N e ) + γ m if m N e 1 \bar{T}\text{lost}\approx\begin{cases}\frac{1}{m},\,\text{ if }mN_{e}\ll 1\\ \frac{\ln{(mN_{e})}+\gamma}{m}\,\text{ if }mN_{e}\gg 1\end{cases}
  11. γ \gamma

Genetic_linkage.html

  1. L O D = Z \displaystyle LOD=Z

Geodesic.html

  1. d ( γ ( t 1 ) , γ ( t 2 ) ) = v | t 1 - t 2 | . d(\gamma(t_{1}),\gamma(t_{2}))=v\left|t_{1}-t_{2}\right|.
  2. d ( γ ( t 1 ) , γ ( t 2 ) ) = | t 1 - t 2 | . d(\gamma(t_{1}),\gamma(t_{2}))=\left|t_{1}-t_{2}\right|.
  3. L ( γ ) = a b g γ ( t ) ( γ ˙ ( t ) , γ ˙ ( t ) ) d t . L(\gamma)=\int_{a}^{b}\sqrt{g_{\gamma(t)}(\dot{\gamma}(t),\dot{\gamma}(t))}\,dt.
  4. E ( γ ) = 1 2 a b g γ ( t ) ( γ ˙ ( t ) , γ ˙ ( t ) ) d t . E(\gamma)=\frac{1}{2}\int_{a}^{b}g_{\gamma(t)}(\dot{\gamma}(t),\dot{\gamma}(t)% )\,dt.
  5. L ( γ ) 2 2 ( b - a ) E ( γ ) L(\gamma)^{2}\leq 2(b-a)E(\gamma)
  6. d 2 x λ d t 2 + Γ μ ν λ d x μ d t d x ν d t = 0 , \frac{d^{2}x^{\lambda}}{dt^{2}}+\Gamma^{\lambda}_{\mu\nu}\frac{dx^{\mu}}{dt}% \frac{dx^{\nu}}{dt}=0,
  7. Γ μ ν λ \Gamma^{\lambda}_{\mu\nu}
  8. δ E ( γ ) ( φ ) = t | t = 0 E ( γ + t φ ) . \delta E(\gamma)(\varphi)=\left.\frac{\partial}{\partial t}\right|_{t=0}E(% \gamma+t\varphi).
  9. δ 2 E ( γ ) ( φ , ψ ) = 2 s t | s = t = 0 E ( γ + t φ + s ψ ) . \delta^{2}E(\gamma)(\varphi,\psi)=\left.\frac{\partial^{2}}{\partial s\partial t% }\right|_{s=t=0}E(\gamma+t\varphi+s\psi).
  10. γ ˙ γ ˙ = 0 \nabla_{\dot{\gamma}}\dot{\gamma}=0
  11. γ ˙ \dot{\gamma}
  12. t t
  13. γ ˙ \dot{\gamma}
  14. γ ˙ \dot{\gamma}
  15. d 2 γ λ d t 2 + Γ μ ν λ d γ μ d t d γ ν d t = 0 , \frac{d^{2}\gamma^{\lambda}}{dt^{2}}+\Gamma^{\lambda}_{\mu\nu}\frac{d\gamma^{% \mu}}{dt}\frac{d\gamma^{\nu}}{dt}=0\ ,
  16. γ μ = x μ γ ( t ) \gamma^{\mu}=x^{\mu}\circ\gamma(t)
  17. Γ μ ν λ \Gamma^{\lambda}_{\mu\nu}
  18. γ ˙ γ ˙ = 0 \nabla_{\dot{\gamma}}\dot{\gamma}=0
  19. γ \gamma\,
  20. γ ( 0 ) = p \gamma(0)=p\,
  21. γ ˙ ( 0 ) = V \dot{\gamma}(0)=V
  22. G t ( V ) = γ ˙ V ( t ) G^{t}(V)=\dot{\gamma}_{V}(t)
  23. γ V \gamma_{V}
  24. γ ˙ V ( 0 ) = V \dot{\gamma}_{V}(0)=V
  25. g g
  26. g ( G t ( V ) , G t ( V ) ) = g ( V , V ) g(G^{t}(V),G^{t}(V))=g(V,V)
  27. γ V \gamma_{V}
  28. T T M = H V . TTM=H\oplus V.
  29. π * W v = v \pi_{*}W_{v}=v\,
  30. H λ X = d ( S λ ) X H X H_{\lambda X}=d(S_{\lambda})_{X}H_{X}\,
  31. S λ : X λ X . S_{\lambda}:X\mapsto\lambda X.
  32. t a t + b t\mapsto at+b
  33. , ¯ \nabla,\bar{\nabla}
  34. D ( X , Y ) = X Y - ¯ X Y D(X,Y)=\nabla_{X}Y-\bar{\nabla}_{X}Y
  35. \nabla
  36. ¯ \bar{\nabla}
  37. \nabla

Geodesic_dome.html

  1. ν \nu
  2. η \eta
  3. η = 2 sin ( θ 2 ) \eta=2\sin\left(\frac{\theta}{2}\right)
  4. θ \theta

Geoid.html

  1. V = G M r ( 1 + n = 2 n max ( a r ) n m = 0 n P ¯ n m ( cos ϕ ) [ C ¯ n m cos m λ + S ¯ n m sin m λ ] ) , V=\frac{GM}{r}\left(1+{\sum_{n=2}^{n\text{max}}}\left(\frac{a}{r}\right)^{n}{% \sum_{m=0}^{n}}\overline{P}_{nm}(\cos\phi)\left[\overline{C}_{nm}\cos m\lambda% +\overline{S}_{nm}\sin m\lambda\right]\right),
  2. ϕ \phi
  3. λ \lambda
  4. P ¯ n m \overline{P}_{nm}
  5. n n
  6. m m
  7. C ¯ n m \overline{C}_{nm}
  8. S ¯ n m \overline{S}_{nm}
  9. V V
  10. ϕ , λ , r , \phi,\;\lambda,\;r,
  11. r r
  12. n max = 360 n\text{max}=360
  13. C ¯ n m \overline{C}_{nm}
  14. S ¯ n m \overline{S}_{nm}
  15. sin ( 0 λ ) = 0 \sin(0\lambda)=0
  16. I = 1 L I = L ( L + 1 ) / 2 \sum_{I=1}^{L}I=L(L+1)/2
  17. n = 2 n max ( 2 n + 1 ) = n max ( n max + 1 ) + n max - 3 = 130317 \sum_{n=2}^{n\text{max}}(2n+1)=n\text{max}(n\text{max}+1)+n\text{max}-3=130317
  18. n max = 360 n\text{max}=360

Geometric_Brownian_motion.html

  1. d S t = μ S t d t + σ S t d W t dS_{t}=\mu S_{t}\,dt+\sigma S_{t}\,dW_{t}
  2. W t W_{t}
  3. μ \mu
  4. σ \sigma
  5. S t = S 0 exp ( ( μ - σ 2 2 ) t + σ W t ) . S_{t}=S_{0}\exp\left(\left(\mu-\frac{\sigma^{2}}{2}\right)t+\sigma W_{t}\right).
  6. S t S_{t}
  7. 0 t d S s S s = μ t + σ W t , assuming W 0 = 0 . \int_{0}^{t}\frac{dS_{s}}{S_{s}}=\mu\,t+\sigma\,W_{t}\,,\qquad\,\text{assuming% }W_{0}=0\,.
  8. d S t S t \frac{dS_{t}}{S_{t}}
  9. ln S t \ln S_{t}
  10. S t S_{t}
  11. d ( ln S t ) = d S t S t - 1 2 1 S t 2 d < S . > t . d(\ln S_{t})=\frac{dS_{t}}{S_{t}}-\frac{1}{2}\,\frac{1}{S_{t}^{2}}\,d<S_{.}>_{% t}\,.
  12. d < S . > t = σ 2 S t 2 d t . d<S_{.}>_{t}\,=\,\sigma^{2}\,S_{t}^{2}\,dt.
  13. ln S t S 0 = ( μ - σ 2 2 ) t + σ W t . \ln\frac{S_{t}}{S_{0}}=\left(\mu-\frac{\sigma^{2}}{2}\,\right)t+\sigma W_{t}\,.
  14. S t S_{t}
  15. 𝔼 ( S t ) = S 0 e μ t , \mathbb{E}(S_{t})=S_{0}e^{\mu t},
  16. Var ( S t ) = S 0 2 e 2 μ t ( e σ 2 t - 1 ) , \operatorname{Var}(S_{t})=S_{0}^{2}e^{2\mu t}\left(e^{\sigma^{2}t}-1\right),
  17. f S t ( s ; μ , σ , t ) = 1 2 π 1 s σ t exp ( - ( ln s - ln S 0 - ( μ - 1 2 σ 2 ) t ) 2 2 σ 2 t ) . f_{S_{t}}(s;\mu,\sigma,t)=\frac{1}{\sqrt{2\pi}}\,\frac{1}{s\sigma\sqrt{t}}\,% \exp\left(-\frac{\left(\ln s-\ln S_{0}-\left(\mu-\frac{1}{2}\sigma^{2}\right)t% \right)^{2}}{2\sigma^{2}t}\right).
  18. d log ( S ) = f ( S ) d S + 1 2 f ′′ ( S ) S 2 σ 2 d t = 1 S ( σ S d W t + μ S d t ) - 1 2 σ 2 d t = σ d W t + ( μ - σ 2 / 2 ) d t . \begin{aligned}\displaystyle d\log(S)&\displaystyle=f^{\prime}(S)\,dS+\frac{1}% {2}f^{\prime\prime}(S)S^{2}\sigma^{2}\,dt\\ &\displaystyle=\frac{1}{S}\left(\sigma S\,dW_{t}+\mu S\,dt\right)-\frac{1}{2}% \sigma^{2}\,dt\\ &\displaystyle=\sigma\,dW_{t}+(\mu-\sigma^{2}/2)\,dt.\end{aligned}
  19. 𝔼 log ( S t ) = log ( S 0 ) + ( μ - σ 2 / 2 ) t \mathbb{E}\log(S_{t})=\log(S_{0})+(\mu-\sigma^{2}/2)t
  20. log ( S t ) \displaystyle\log(S_{t})
  21. 𝔼 log ( S t ) = log ( S 0 ) + ( μ - σ 2 / 2 ) t \mathbb{E}\log(S_{t})=\log(S_{0})+(\mu-\sigma^{2}/2)t
  22. d S t i = μ i S t i d t + σ i S t i d W t i dS_{t}^{i}=\mu_{i}S_{t}^{i}\,dt+\sigma_{i}S_{t}^{i}\,dW_{t}^{i}
  23. 𝔼 ( d W t i d W t j ) = ρ i , j d t \mathbb{E}(dW_{t}^{i}dW_{t}^{j})=\rho_{i,j}dt
  24. ρ i , i = 1 \rho_{i,i}=1
  25. Cov ( S t i , S t j ) = S 0 i S 0 j e ( μ i + μ j ) t ( e ρ i , j σ i σ j t - 1 ) \mathrm{Cov}(S_{t}^{i},S_{t}^{j})=S_{0}^{i}S_{0}^{j}e^{(\mu_{i}+\mu_{j})t}% \left(e^{\rho_{i,j}\sigma_{i}\sigma_{j}t}-1\right)
  26. σ \sigma

Geometrization_conjecture.html

  1. SL ~ ( 2 , 𝐑 ) {\tilde{\rm{SL}}}(2,\mathbf{R})
  2. ( 𝐑 × SL ~ 2 ( 𝐑 ) ) / 𝐙 (\mathbf{R}\times\tilde{\rm{SL}}_{2}(\mathbf{R}))/\mathbf{Z}
  3. ( 2 1 1 1 ) \left({\begin{array}[]{*{20}c}2&1\\ 1&1\\ \end{array}}\right)

Geometry_of_numbers.html

  1. v o l ( K ) > 2 n v o l ( R n / Γ ) vol(K)>2^{n}vol(R^{n}/\Gamma)
  2. λ 1 λ 2 λ n v o l ( K ) 2 n v o l ( R n / Γ ) . \lambda_{1}\lambda_{2}\cdots\lambda_{n}vol(K)\leq 2^{n}vol(R^{n}/\Gamma).
  3. | L 1 ( x ) L n ( x ) | < | x | - ε |L_{1}(x)\cdots L_{n}(x)|<|x|^{-\varepsilon}

Geopotential.html

  1. W W
  2. U U
  3. U 0 U_{0}
  4. W 0 W_{0}
  5. T = W - U T=W-U
  6. C C
  7. C = - ( W - W 0 ) C=-(W-W_{0})
  8. W 0 W_{0}
  9. Φ ( h ) = 0 h g d z \Phi(h)=\int_{0}^{h}g\,dz
  10. Φ = 0 z [ G m ( a + z ) 2 ] d z \Phi=\int_{0}^{z}\left[\frac{Gm}{(a+z)^{2}}\right]dz
  11. Φ = G m [ 1 a - 1 a + z ] \Phi=Gm\left[\frac{1}{a}-\frac{1}{a+z}\right]

Geopotential_height.html

  1. Φ ( h ) = 0 h g ( ϕ , z ) d z , \Phi(h)=\int_{0}^{h}g(\phi,z)\,dz\,,
  2. g ( ϕ , z ) g(\phi,z)
  3. ϕ \phi
  4. Z g ( h ) = Φ ( h ) g 0 , {Z_{g}}(h)=\frac{\Phi(h)}{g_{0}}\,,

George_Peacock.html

  1. + +
  2. - -
  3. a + b a+b
  4. a a
  5. b b
  6. a - b a-b
  7. a a
  8. b b
  9. a b ab
  10. a b \frac{a}{b}
  11. a a
  12. b b
  13. a + b a+b
  14. a - b a-b
  15. b b
  16. a a
  17. a b ab
  18. a b \frac{a}{b}
  19. b b
  20. a a
  21. a b \frac{a}{b}
  22. a b ab
  23. a a
  24. b b
  25. a b = b a ab=ba
  26. m m
  27. m m
  28. 1 m \frac{1}{m}
  29. / m /m
  30. m m
  31. / n /n
  32. m / n m/n
  33. a m a^{m}
  34. a n a^{n}
  35. a m + n a^{m+n}
  36. m m
  37. n n
  38. m m
  39. n n
  40. ( a + b ) n (a+b)^{n}
  41. n n
  42. ( a + b ) n (a+b)^{n}
  43. n n
  44. a a
  45. b b
  46. c c
  47. d d
  48. b b
  49. a a
  50. d d
  51. c c
  52. ( a - b ) ( c - d ) = a c + b d - a d - b c (a-b)(c-d)=ac+bd-ad-bc
  53. a a
  54. b b
  55. c c
  56. d d
  57. a a
  58. b b
  59. c c
  60. d d
  61. d d x \frac{d}{dx}
  62. a a
  63. b b
  64. c c
  65. d d
  66. b b
  67. a a
  68. d d
  69. c c
  70. ( a - b ) ( c - d ) = a c + b d - a d - b c (a-b)(c-d)=ac+bd-ad-bc
  71. a a
  72. m m
  73. n n
  74. a m a n = a m + n a^{m}a^{n}=a^{m+n}
  75. a a
  76. m m
  77. n n
  78. a a
  79. e e
  80. p + q - 1 p+q^{\sqrt{-1}}
  81. m m
  82. n n
  83. e m e n = e m + n e^{m}e^{n}=e^{m+n}
  84. m m
  85. n n
  86. m m
  87. n n

Georgi–Glashow_model.html

  1. Y 2 = diag ( - 1 / 3 , - 1 / 3 , - 1 / 3 , 1 / 2 , 1 / 2 ) \frac{Y}{2}=\operatorname{diag}\left(-1/3,-1/3,-1/3,1/2,1/2\right)
  2. [ S U ( 3 ) × S U ( 2 ) × U ( 1 ) Y ] / 6 [SU(3)\times SU(2)\times U(1)_{Y}]/\mathbb{Z}_{6}
  3. 24 ( 8 , 1 ) 0 ( 1 , 3 ) 0 ( 1 , 1 ) 0 ( 3 , 2 ) - 5 6 ( 3 ¯ , 2 ) 5 6 24\rightarrow(8,1)_{0}\oplus(1,3)_{0}\oplus(1,1)_{0}\oplus(3,2)_{-\frac{5}{6}}% \oplus(\bar{3},2)_{\frac{5}{6}}
  4. 𝟓 ¯ 𝟏𝟎 𝟏 \mathbf{\bar{5}}\oplus\mathbf{10}\oplus\mathbf{1}
  5. 5 ¯ ( 3 ¯ , 1 ) 1 3 ( 1 , 2 ) - 1 2 \bar{5}\rightarrow(\bar{3},1)_{\frac{1}{3}}\oplus(1,2)_{-\frac{1}{2}}
  6. 10 ( 3 , 2 ) 1 6 ( 3 ¯ , 1 ) - 2 3 ( 1 , 1 ) 1 10\rightarrow(3,2)_{\frac{1}{6}}\oplus(\bar{3},1)_{-\frac{2}{3}}\oplus(1,1)_{1}
  7. 1 ( 1 , 1 ) 0 1\rightarrow(1,1)_{0}
  8. π 2 ( S U ( 5 ) [ S U ( 3 ) × S U ( 2 ) × U ( 1 ) Y ] / 6 ) = \pi_{2}\left(\frac{SU(5)}{[SU(3)\times SU(2)\times U(1)_{Y}]/\mathbb{Z}_{6}}% \right)=\mathbb{Z}
  9. 2 \mathbb{Z}_{2}
  10. 5 ¯ \bar{5}
  11. 5 ¯ \bar{5}
  12. 5 ¯ \bar{5}
  13. S U ( 5 ) × 2 SU(5)\times\mathbb{Z}_{2}
  14. Φ 2 Φ B A Φ A B Φ 3 Φ B A Φ C B Φ A C H d H u H d A H u A H d Φ H u H d A Φ B A H u B H u 𝟏𝟎 i 10 j ϵ A B C D E H u A 𝟏𝟎 i B C 𝟏𝟎 j D E H d 𝟓 ¯ i 10 j H d A 𝟓 ¯ B i 𝟏𝟎 j A B H u 𝟓 ¯ i N j c H u A 𝟓 ¯ A i N j c N i c N j c N i c N j c \begin{matrix}\Phi^{2}&\Phi^{A}_{B}\Phi^{B}_{A}\\ \Phi^{3}&\Phi^{A}_{B}\Phi^{B}_{C}\Phi^{C}_{A}\\ H_{d}H_{u}&{H_{d}}_{A}H_{u}^{A}\\ H_{d}\Phi H_{u}&{H_{d}}_{A}\Phi^{A}_{B}H_{u}^{B}\\ H_{u}\mathbf{10}_{i}\;\mathbf{10}_{j}&\epsilon_{ABCDE}H_{u}^{A}\mathbf{10}^{BC% }_{i}\mathbf{10}^{DE}_{j}\\ H_{d}\mathbf{\bar{5}}_{i}\;\mathbf{10}_{j}&{H_{d}}_{A}\mathbf{\bar{5}}_{Bi}% \mathbf{10}^{AB}_{j}\\ H_{u}\mathbf{\bar{5}}_{i}N^{c}_{j}&H_{u}^{A}\mathbf{\bar{5}}_{Ai}N^{c}_{j}\\ N^{c}_{i}N^{c}_{j}&N^{c}_{i}N^{c}_{j}\\ \end{matrix}
  15. W = T r [ a Φ 2 + b Φ 3 ] W=Tr[a\Phi^{2}+b\Phi^{3}]
  16. T r [ Φ ] = 0 Tr[\Phi]=0
  17. 2 a Φ + 3 b Φ 2 = λ 𝟏 2a\Phi+3b\Phi^{2}=\lambda\mathbf{1}
  18. Φ = { diag ( 0 , 0 , 0 , 0 , 0 ) diag ( 2 a 9 b , 2 a 9 b , 2 a 9 b , 2 a 9 b , - 8 a 9 b ) diag ( 4 a 3 b , 4 a 3 b , 4 a 3 b , - 2 a b , - 2 a b ) \Phi=\left\{\begin{matrix}\operatorname{diag}(0,0,0,0,0)\\ \operatorname{diag}(\frac{2a}{9b},\frac{2a}{9b},\frac{2a}{9b},\frac{2a}{9b},-% \frac{8a}{9b})\\ \operatorname{diag}(\frac{4a}{3b},\frac{4a}{3b},\frac{4a}{3b},-\frac{2a}{b},-% \frac{2a}{b})\end{matrix}\right.
  19. [ S U ( 4 ) × U ( 1 ) ] / 4 [SU(4)\times U(1)]/\mathbb{Z}_{4}
  20. [ S U ( 3 ) × S U ( 2 ) × U ( 1 ) ] / 6 [SU(3)\times SU(2)\times U(1)]/\mathbb{Z}_{6}
  21. ( ( 8 , 1 ) 0 ( 1 , 3 ) 0 ( 1 , 1 ) 0 ( 3 , 2 ) - 5 6 ( 3 ¯ , 2 ) 5 6 ) \begin{pmatrix}(8,1)_{0}\\ (1,3)_{0}\\ (1,1)_{0}\\ (3,2)_{-\frac{5}{6}}\\ (\bar{3},2)_{\frac{5}{6}}\end{pmatrix}
  22. ( 3 , 2 ) - 5 6 (3,2)_{-\frac{5}{6}}
  23. ( 3 ¯ , 2 ) 5 6 (\bar{3},2)_{\frac{5}{6}}
  24. ( 3 , 2 ) - 5 6 (3,2)_{-\frac{5}{6}}
  25. ( 3 ¯ , 2 ) 5 6 (\bar{3},2)_{\frac{5}{6}}
  26. ( 8 , 1 ) 0 (8,1)_{0}
  27. ( 1 , 3 ) 0 (1,3)_{0}
  28. ( 1 , 1 ) 0 (1,1)_{0}
  29. 5 ¯ \bar{5}
  30. 5 ¯ H \bar{5}_{H}
  31. 5 H 5_{H}
  32. 5 ¯ H \bar{5}_{H}
  33. ( ( 3 , 1 ) - 1 3 ( 1 , 2 ) 1 2 ) \begin{pmatrix}(3,1)_{-\frac{1}{3}}\\ (1,2)_{\frac{1}{2}}\end{pmatrix}
  34. _ _ _ ? ? ? \begin{matrix}\_\_\_\\ ???\end{matrix}
  35. ( ( 3 ¯ , 1 ) 1 3 ( 1 , 2 ) - 1 2 ) \begin{pmatrix}(\bar{3},1)_{\frac{1}{3}}\\ (1,2)_{-\frac{1}{2}}\end{pmatrix}
  36. 5 H 5 ¯ H 5_{H}\bar{5}_{H}
  37. < 24 > 5 H 5 ¯ H <24>5_{H}\bar{5}_{H}

Geostationary_transfer_orbit.html

  1. Δ \Delta
  2. Δ \Delta
  3. Δ \Delta
  4. Δ V = 2 V sin Δ i 2 \Delta V=2V\sin\frac{\Delta i}{2}
  5. Δ \Delta
  6. Δ \Delta
  7. Δ \Delta
  8. Δ \Delta
  9. Δ \Delta
  10. Δ \Delta
  11. Δ V = V t , a 2 + V G E O 2 - 2 V t , a V G E O cos Δ i \Delta V=\sqrt{V_{t,a}^{2}+V_{GEO}^{2}-2V_{t,a}V_{GEO}\cos\Delta i}
  12. V t , a V_{t,a}
  13. V G E O V_{GEO}

Gibbs_free_energy.html

  1. G ( p , T ) = U + p V - T S G(p,T)=U+pV-TS
  2. G ( p , T ) = H - T S G(p,T)=H-TS
  3. T d S = d U + p d V - i = 1 k μ i d N i + i = 1 n X i d a i + T\mathrm{d}S=\mathrm{d}U+p\mathrm{d}V-\sum_{i=1}^{k}\mu_{i}\,\mathrm{d}N_{i}+% \sum_{i=1}^{n}X_{i}\,\mathrm{d}a_{i}+\cdots
  4. d ( T S ) - S d T = d U + d ( p V ) - V d p - i = 1 k μ i d N i + i = 1 n X i d a i + \mathrm{d}(TS)-S\mathrm{d}T=\mathrm{d}U+\mathrm{d}(pV)-V\mathrm{d}p-\sum_{i=1}% ^{k}\mu_{i}\,\mathrm{d}N_{i}+\sum_{i=1}^{n}X_{i}\,\mathrm{d}a_{i}+\cdots
  5. d ( U - T S + p V ) = V d p - S d T + i = 1 k μ i d N i - i = 1 n X i d a i + \mathrm{d}(U-TS+pV)=V\mathrm{d}p-S\mathrm{d}T+\sum_{i=1}^{k}\mu_{i}\,\mathrm{d% }N_{i}-\sum_{i=1}^{n}X_{i}\,\mathrm{d}a_{i}+\cdots
  6. d G = V d p - S d T + i = 1 k μ i d N i - i = 1 n X i d a i + \mathrm{d}G=V\mathrm{d}p-S\mathrm{d}T+\sum_{i=1}^{k}\mu_{i}\,\mathrm{d}N_{i}-% \sum_{i=1}^{n}X_{i}\,\mathrm{d}a_{i}+\cdots
  7. G N = G N + k T ln p < m t p l > p \frac{G}{N}=\frac{G}{N}^{\circ}+kT\ln\frac{p}{<}mtpl>{{p^{\circ}}}
  8. G N = G N + k T ln V < m t p l > V \frac{G}{N}=\frac{G}{N}^{\circ}+kT\ln\frac{V^{\circ}}{<}mtpl>{{V}}
  9. G N = μ = μ + k T ln p < m t p l > p . \frac{G}{N}=\mu=\mu^{\circ}+kT\ln\frac{p}{<}mtpl>{{p^{\circ}}}.
  10. d U = T d S - p d V + i μ i d N i \mathrm{d}U=T\mathrm{d}S-p\,\mathrm{d}V+\sum_{i}\mu_{i}\,\mathrm{d}N_{i}\,
  11. U = T S - p V + i μ i N i U=TS-pV+\sum_{i}\mu_{i}N_{i}\,
  12. G = U + p V - T S G=U+pV-TS\,
  13. d G = d U + p d V + V d p - T d S - S d T \mathrm{d}G=\mathrm{d}U+p\,\mathrm{d}V+V\mathrm{d}p-T\mathrm{d}S-S\mathrm{d}T\,
  14. d G = T d S - p d V + i μ i d N i + p d V + V d p - T d S - S d T = V d p - S d T + i μ i d N i \begin{aligned}\displaystyle\mathrm{d}G&\displaystyle=T\mathrm{d}S-p\,\mathrm{% d}V+\sum_{i}\mu_{i}\,\mathrm{d}N_{i}+p\,\mathrm{d}V+V\mathrm{d}p-T\mathrm{d}S-% S\mathrm{d}T\\ &\displaystyle=V\mathrm{d}p-S\mathrm{d}T+\sum_{i}\mu_{i}\,\mathrm{d}N_{i}\end{aligned}
  15. G = T S - p V + i μ i N i + p V - T S = i μ i N i \begin{aligned}\displaystyle G&\displaystyle=TS-pV+\sum_{i}\mu_{i}N_{i}+pV-TS% \\ &\displaystyle=\sum_{i}\mu_{i}N_{i}\end{aligned}
  16. Δ S t o t 0 \Delta S_{tot}\geq 0\,
  17. ( Q T = Δ S = 0 ) \left({Q\over T}=\Delta S=0\right)\,
  18. Δ S i n t + Δ S e x t 0 \Delta S_{int}+\Delta S_{ext}\geq 0\,
  19. Δ S e x t = - Q T , \Delta S_{ext}=-{Q\over T},
  20. Δ S i n t - Q T 0 \Delta S_{int}-{Q\over T}\geq 0\,
  21. T Δ S i n t - Q 0 T\Delta S_{int}-Q\geq 0\,
  22. T Δ S i n t - Δ H 0 T\Delta S_{int}-\Delta H\geq 0\,
  23. Δ H - T Δ S i n t 0 \Delta H-T\Delta S_{int}\leq 0\,
  24. Δ G = Δ H - T Δ S i n t \Delta G=\Delta H-T\Delta S_{int}\,
  25. Δ G < 0 \Delta G<0\,
  26. Δ G = 0 \Delta G=0\,
  27. Δ G > 0 \Delta G>0\,
  28. G = H - T S i n t G=H-TS_{int}\,
  29. Δ H = 0 \Delta H=0\,
  30. Δ G = - T Δ S \Delta G=-T\Delta S\,
  31. Δ G = Δ H - T Δ S \Delta G=\Delta H-T\Delta S\,
  32. Δ r G = - R T ln K \Delta_{r}G^{\circ}=-RT\ln K\,
  33. Δ r G = Δ r G + R T ln Q r \Delta_{r}G=\Delta_{r}G^{\circ}+RT\ln Q_{r}\,
  34. Δ G = - n F E \Delta G=-nFE\,
  35. Δ G = - n F E \Delta G^{\circ}=-nFE^{\circ}\,
  36. n F E = R T ln K nFE^{\circ}=RT\ln K\,
  37. n F E = n F E - R T ln Q r nFE=nFE^{\circ}-RT\ln Q_{r}\,\,
  38. E = E - R T n F ln Q r E=E^{\circ}-\frac{RT}{nF}\ln Q_{r}\,\,
  39. K e q = e - Δ r G R T K_{eq}=e^{-\frac{\Delta_{r}G^{\circ}}{RT}}
  40. Δ r G = - R T ( ln K e q ) = - 2.303 R T ( log 10 K e q ) \Delta_{r}G^{\circ}=-RT(\ln K_{eq})=-2.303\,RT(\log_{10}K_{eq})

Gibbs_paradox.html

  1. U = 1 2 m i = 1 N p i x 2 + p i y 2 + p i z 2 U=\frac{1}{2m}\sum_{i=1}^{N}p_{ix}^{2}+p_{iy}^{2}+p_{iz}^{2}
  2. 0 x i j X 0\leq x_{ij}\leq X
  3. i = 1... N i=1...N
  4. j = 1 , 2 , 3 j=1,2,3
  5. ϕ ( U , V , N ) = V N ( 2 π 3 N 2 ( 2 m U ) 3 N - 1 2 Γ ( 3 N / 2 ) ) ( 1 ) \phi(U,V,N)=V^{N}\left(\frac{2\pi^{\frac{3N}{2}}(2mU)^{\frac{3N-1}{2}}}{\Gamma% (3N/2)}\right)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(1)
  6. h 3 N h^{3N}
  7. h 3 N h^{3N}
  8. δ U \delta U
  9. δ U \delta U
  10. δ U \delta U
  11. ϕ \phi
  12. δ p = δ ( 2 m U ) = m 2 U δ U \delta p=\delta\left(\sqrt{2mU}\right)=\sqrt{\frac{m}{2U}}\delta U
  13. S = k ln ( ϕ δ p / h 3 N ) \left.\right.S=k\,\ln(\phi\delta p/h^{3N})
  14. S = k N ln [ V ( U N ) 3 2 ] + 3 2 k N ( 1 + ln 4 π m 3 h 2 ) S=kN\ln\left[V\left(\frac{U}{N}\right)^{\frac{3}{2}}\right]+{\frac{3}{2}}kN% \left(1+\ln\frac{4\pi m}{3h^{2}}\right)
  15. δ S = k [ 2 N ln ( 2 V ) - N ln V - N ln V ] = 2 k N ln 2 > 0 \delta S=k\left[2N\ln(2V)-N\ln V-N\ln V\right]=2kN\ln 2>0
  16. S = k N ln [ ( V N ) ( U N ) 3 2 ] + k N ( 5 2 + 3 2 ln 4 π m 3 h 2 ) S=kN\ln\left[\left(\frac{V}{N}\right)\left(\frac{U}{N}\right)^{\frac{3}{2}}% \right]+kN\left({\frac{5}{2}}+{\frac{3}{2}}\ln\frac{4\pi m}{3h^{2}}\right)
  17. n log ( n ) n\log(n)
  18. S = k B ln Ω = k B H ( p , q ) = E ( d p ) n ( d q ) n S=k_{B}\ln\Omega=k_{B}\oint\limits_{H(\vec{p},\vec{q})=E}(d\vec{p}\,)^{n}(d% \vec{q}\,)^{n}
  19. n n
  20. v \vec{v}
  21. E = j = 1 n 1 2 m v j 2 , E=\sum_{j=1}^{n}\frac{1}{2}mv_{j}^{2}\,,
  22. n n
  23. 0 < x < 0<x<\ell
  24. m = k B = 1 m=k_{B}=1
  25. ξ = [ x 1 , , x n , v 1 , , v n ] = [ x , v ] \vec{\xi}=[x_{1},...,x_{n},v_{1},...,v_{n}]=[\vec{x},\vec{v}\,]
  26. x = [ x 1 , , x n ] \vec{x}=[x_{1},...,x_{n}]
  27. v = [ v 1 , , v n ] . \vec{v}=[v_{1},...,v_{n}]\,.
  28. Σ v j 2 = R 2 , \Sigma v_{j}^{2}=R^{2}\,,
  29. A ~ n ( R ) = n π n / 2 ( n / 2 ) ! R n - 1 . \tilde{A}_{n}(R)=\frac{n\pi^{n/2}}{(n/2)!}R^{n-1}\,.
  30. A ~ 2 ( R ) = 2 π R \tilde{A}_{2}(R)=2\pi R
  31. n n
  32. n n
  33. n \ell^{n}
  34. Ω E , = ( d x 1 d x 2 d x n ) ( d v 1 d v 2 d v n Σ v i 2 = 2 E ) \Omega_{E,\ell}=\left(\int dx_{1}\int dx_{2}...\int dx_{n}\right)\left(% \underbrace{\int dv_{1}\int dv_{2}...\int dv_{n}}_{\Sigma v_{i}^{2}=2E}\right)
  35. Ω \Omega
  36. n n
  37. n \ell^{n}
  38. 2 E \sqrt{2E}
  39. Ω E , = n n π n / 2 ( n / 2 ) ! ( 2 E ) n - 1 2 \Omega_{E,\ell}=\ell^{n}\frac{n\pi^{n/2}}{(n/2)!}(2E)^{\frac{n-1}{2}}
  40. Ω E , = n n π n / 2 ( n / 2 ) ! ( 2 E ) n - 1 2 \Omega_{E,\ell}=\ell^{n}\frac{n\pi^{n/2}}{(n/2)!}(2E)^{\frac{n-1}{2}}
  41. ln Ω E , = ln ( n n π n / 2 ( 2 E ) n - 1 2 ) - ln [ ( n / 2 ) ! ] \ln\Omega_{E,\ell}=\ln\left(\ell^{n}n\pi^{n/2}(2E)^{\frac{n-1}{2}}\right)-\ln% \left[(n/2)!\right]
  42. ln M ! M ln M \ln M!\approx M\ln M
  43. - M + ln 2 π M -M+\ln\sqrt{2\pi M}
  44. ln ( n n π n / 2 ( 2 E ) n - 1 2 ) = ln ( n E n 2 ) i m p o r t a n t + ln ( n ( 2 π ) n 2 2 E ) d r o p \ln\left(\ell^{n}n\pi^{n/2}(2E)^{\frac{n-1}{2}}\right)=\underbrace{\ln\left(% \ell^{n}E^{\frac{n}{2}}\right)}_{important}+\underbrace{\ln\left(\frac{n(2\pi)% ^{\frac{n}{2}}}{\sqrt{2E}}\right)}_{drop}
  45. - ln [ ( n / 2 ) ! ] - n 2 ln n 2 + n 2 - ln n π = - n 2 ln n k e e p + n 2 ln 2 + n 2 - ln n π d r o p \begin{aligned}\displaystyle-\ln[(n/2)!]&\displaystyle\approx-\frac{n}{2}\ln% \frac{n}{2}+\frac{n}{2}-\ln\sqrt{n\pi}\\ &\displaystyle=\underbrace{-\frac{n}{2}\ln n}_{keep}+\underbrace{\frac{n}{2}% \ln 2+\frac{n}{2}-\ln\sqrt{n\pi}}_{drop}\\ \end{aligned}
  46. S / E \partial S/\partial E
  47. E 1 2 E^{\frac{1}{2}}
  48. E n 2 E^{\frac{n}{2}}
  49. π n \pi^{n}
  50. n n n^{n}
  51. ln Ω E , \displaystyle\ln\Omega_{E,\ell}
  52. ln Ω E , n ln + n ln E n + c o n s t . = n ln n + n ln E n e x t e n s i v e + n ln n + c o n s t . \begin{aligned}\displaystyle\ln\Omega_{E,\ell}&\displaystyle\approx n\ln\ell+n% \ln\sqrt{\frac{E}{n}}+const.\\ &\displaystyle=\underbrace{n\ln\frac{\ell}{n}+n\ln\sqrt{\frac{E}{n}}}_{% extensive}+\,n\ln n+const.\\ \end{aligned}
  53. n ln n n\ln n
  54. ln - ln n = ln ( / n ) \ln\ell-\ln n=\ln(\ell/n)
  55. Ω E , \Omega_{E,\ell}
  56. n n
  57. n n
  58. f ( n ) = - n ln n f(n)=-n\ln n
  59. S = ln Ω E , \displaystyle S=\ln\Omega_{E,\ell}
  60. S ( α E , α , α n ) = α S ( E , , n ) S(\alpha E,\alpha\ell,\alpha n)=\alpha\,S(E,\ell,n)
  61. N N
  62. n A n_{A}
  63. 0 < x < A 0<x<\ell_{A}
  64. E A E_{A}
  65. n B n_{B}
  66. 0 < x < B 0<x<\ell_{B}
  67. E B E_{B}
  68. E A + E B = E E_{A}+E_{B}=E
  69. n A + n B = N n_{A}+n_{B}=N
  70. Ω E , , N = \Omega_{E,\ell,N}=
  71. ( d x ? d x ? n A t e r m s d x ? d x N n B t e r m s ) ( d v 1 d v 2 d v N Σ v 2 = 2 E A o r Σ v 2 = 2 E B ) \left(\underbrace{\int dx_{?}...\int dx_{?}}_{n_{A}\;terms}\underbrace{\int dx% _{?}...\int dx_{N}}_{n_{B}\;terms}\right)\left(\underbrace{\int dv_{1}\int dv_% {2}...\int dv_{N}}_{\Sigma v^{2}=2E_{A}\;or\;\Sigma v^{2}=2E_{B}}\right)
  72. = ( A ) n A ( B ) n B ( N ! n A ! n B ! c o m b i n a t i o n ) ( n A π n A / 2 ( n A / 2 ) ! ( 2 E A ) n A - 1 2 n A - s p h e r e ) ( n B π n B / 2 ( n B / 2 ) ! ( 2 E B ) n B - 1 2 n B - s p h e r e ) =\left(\ell_{A}\right)^{n_{A}}\left(\ell_{B}\right)^{n_{B}}\left(\underbrace{% \frac{N!}{n_{A}!n_{B}!}}_{combination}\right)\left(\underbrace{\frac{n_{A}\pi^% {n_{A}/2}}{(n_{A}/2)!}(2E_{A})^{\frac{n_{A}-1}{2}}}_{n_{A}-sphere}\right)\left% (\underbrace{\frac{n_{B}\pi^{n_{B}/2}}{(n_{B}/2)!}(2E_{B})^{\frac{n_{B}-1}{2}}% }_{n_{B}-sphere}\right)
  73. S = ln Ω E , , N S=\ln\Omega_{E,\ell,N}
  74. n A ln ( n A A E A A ) + n B ln ( n B B E B B ) + N ln N + c o n s t . \approx n_{A}\ln\left(\frac{n_{A}}{\ell_{A}}\sqrt{\frac{E_{A}}{\ell_{A}}}% \right)+n_{B}\ln\left(\frac{n_{B}}{\ell_{B}}\sqrt{\frac{E_{B}}{\ell_{B}}}% \right)+N\ln N+const.
  75. N ln N N\ln N
  76. ( N n A ) = N ! n A ! n B ! {N\choose n_{A}}=\frac{N!}{n_{A}!n_{B}!}
  77. 2 E 1 \sqrt{2E_{1}}
  78. 2 E 2 \sqrt{2E_{2}}
  79. v 1 2 + v 2 2 = 2 E A v 3 2 = 2 E B v_{1}^{2}+v_{2}^{2}=2E_{A}\qquad v_{3}^{2}=2E_{B}
  80. v 2 2 + v 3 2 = 2 E A v 1 2 = 2 E B v_{2}^{2}+v_{3}^{2}=2E_{A}\qquad v_{1}^{2}=2E_{B}
  81. v 3 2 + v 1 2 = 2 E A v 2 2 = 2 E B v_{3}^{2}+v_{1}^{2}=2E_{A}\qquad v_{2}^{2}=2E_{B}
  82. ( 3 2 ) = 3. {3\choose 2}=3.

Gibbs–Helmholtz_equation.html

  1. ( ( Δ G / T ) T ) p = - Δ H T 2 \left(\frac{\partial(\Delta G^{\ominus}/T)}{\partial T}\right)_{p}=-\frac{% \Delta H}{T^{2}}
  2. Δ G ( T 2 ) T 2 - Δ G ( T 1 ) T 1 = Δ H ( p ) ( 1 T 2 - 1 T 1 ) \frac{\Delta G^{\ominus}(T_{2})}{T_{2}}-\frac{\Delta G^{\ominus}(T_{1})}{T_{1}% }=\Delta H^{\ominus}(p)\left(\frac{1}{T_{2}}-\frac{1}{T_{1}}\right)
  3. Δ G T = - R ln K \frac{\Delta G^{\ominus}}{T}=-R\ln K
  4. H = G + S T H=G+ST\,\!
  5. H = U + p V H=U+pV\,\!
  6. d U = T d S - p d V dU=TdS-pdV\,\!
  7. d G = - S d T + V d p dG=-SdT+Vdp\,\!
  8. d G = - S d T + V d p = G T d T + G p d p dG=-SdT+Vdp=\frac{\partial G}{\partial T}dT+\frac{\partial G}{\partial p}dp\,\!
  9. d G p = - S d T = ( G T ) p d T - S = ( G T ) p . dG_{p}=-SdT=\left(\frac{\partial G}{\partial T}\right)_{p}dT\quad\rightarrow% \quad-S=\left(\frac{\partial G}{\partial T}\right)_{p}.\,\!
  10. ( ( G / T ) T ) p = 1 T ( G T ) p + G ( T - 1 ) T = T ( G T ) p - G T 2 = - S T - G T 2 = - H T 2 \left(\frac{\partial(G/T)}{\partial T}\right)_{p}=\frac{1}{T}\left(\frac{% \partial G}{\partial T}\right)_{p}+G\frac{\partial(T^{-1})}{\partial T}=\dfrac% {T\left(\dfrac{\partial G}{\partial T}\right)_{p}-G}{T^{2}}=\frac{-ST-G}{T^{2}% }=-\frac{H}{T^{2}}\,\!

Global_field.html

  1. v | x | v = 1. \prod_{v}|x|_{v}=1.
  2. θ v : K v × / N L v / K v ( L v × ) G ab , \theta_{v}:K_{v}^{\times}/N_{L_{v}/K_{v}}(L_{v}^{\times})\to G^{\,\text{ab}},

Glossary_of_field_theory.html

  1. α n = b \alpha^{n}=b
  2. F = F 0 < F 1 < < F k = E F=F_{0}<F_{1}<\cdots<F_{k}=E
  3. F i / F i - 1 F_{i}/F_{i-1}

Glossary_of_group_theory.html

  1. ( a b b ) * ( b c a ) = a b b b c a . (abb)*(bca)=abbbca.
  2. p := lim n / p n \mathbb{Z}_{p}:=\underleftarrow{\lim}_{n}\mathbb{Z}/p^{n}
  3. ^ := lim n / n . \hat{\mathbb{Z}}:=\underleftarrow{\lim}_{n}\mathbb{Z}/n.
  4. a , b G , a * b = b * a . \forall a,b\in G,\ a*b=b*a\mbox{.}~{}
  5. [ a , b ] = : a - 1 b - 1 a b [a,b]=\,\!:\,a^{-1}b^{-1}ab
  6. n n
  7. n n
  8. F F

Glossary_of_ring_theory.html

  1. { a n x n + a n - 1 x n - 1 + a n - 2 x n - 2 + + a 1 x + a 0 | a n , a n - 1 , a n - 2 , , a 0 R } \{a_{n}x^{n}+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\ldots+a_{1}x+a_{0}|a_{n},a_{n-1},a% _{n-2},\ldots,a_{0}\in R\}
  2. ( i a i x i ) + ( i b i x i ) = i ( a i + b i ) x i \left(\sum_{i}a_{i}x^{i}\right)+\left(\sum_{i}b_{i}x^{i}\right)=\sum_{i}(a_{i}% +b_{i})x^{i}
  3. ( i a i x i ) ( j b j x j ) = k ( i , j : i + j = k a i b j ) x k \left(\sum_{i}a_{i}x^{i}\right)\cdot\left(\sum_{j}b_{j}x^{j}\right)=\sum_{k}% \left(\sum_{i,j:i+j=k}a_{i}b_{j}\right)x^{k}
  4. σ End ( R ) \sigma\in\textrm{End}(R)
  5. R [ x ; σ ] R[x;\sigma]
  6. { a n x n + a n - 1 x n - 1 + a n - 2 x n - 2 + + a 1 x + a 0 | a n , a n - 1 , a n - 2 , , a 0 R } \{a_{n}x^{n}+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\ldots+a_{1}x+a_{0}|a_{n},a_{n-1},a% _{n-2},\ldots,a_{0}\in R\}
  7. x a = σ ( a ) x a R xa=\sigma(a)x\;\forall a\in R

Glottochronology.html

  1. L = 2 ln ( r ) L=2\ln(r)
  2. t = ln ( c ) - L t=\frac{\ln(c)}{-L}
  3. t = ln ( c ) - L c t=\sqrt{\frac{\ln(c)}{-Lc}}

GNU_TeXmacs.html

  1. \Rightarrow
  2. \Uparrow

GNU_Units.html

  1. Δ P = 8 π 2 ρ f L Q 2 d 5 , \Delta P=\frac{8}{\pi^{2}}\rho fL\frac{Q^{2}}{d^{5}}\,,
  2. Δ P = A 1 ρ f L Q 2 d 5 \Delta P=A_{1}\,\rho fL\frac{Q^{2}}{d^{5}}

Goldbach's_weak_conjecture.html

  1. e 3100 2 × 10 1346 e^{3100}\approx 2\times 10^{1346}
  2. n > e 3100 2 × 10 1346 n>e^{3100}\approx 2\times 10^{1346}
  3. 𝔐 \mathfrak{M}
  4. ( a / q - c r 0 / q x , a / q + c r 0 / q x ) \left(a/q-cr_{0}/qx,a/q+cr_{0}/qx\right)
  5. a / q , q < r 0 a/q,q<r_{0}
  6. c c
  7. 𝔪 \mathfrak{m}
  8. 𝔪 = ( / ) 𝔐 \mathfrak{m}=(\mathbb{R}/\mathbb{Z})\setminus\mathfrak{M}

Golden_angle.html

  1. a + b a = a b . \frac{a+b}{a}=\frac{a}{b}.
  2. 360 ( 1 - 1 φ ) = 360 ( 2 - φ ) = 360 φ 2 = 180 ( 3 - 5 ) degrees 360\left(1-\frac{1}{\varphi}\right)=360(2-\varphi)=\frac{360}{\varphi^{2}}=180% (3-\sqrt{5})\,\text{ degrees}
  3. 2 π ( 1 - 1 φ ) = 2 π ( 2 - φ ) = 2 π φ 2 = π ( 3 - 5 ) radians , 2\pi\left(1-\frac{1}{\varphi}\right)=2\pi(2-\varphi)=\frac{2\pi}{\varphi^{2}}=% \pi(3-\sqrt{5})\,\text{ radians},
  4. f = b a + b = 1 1 + φ . f=\frac{b}{a+b}=\frac{1}{1+\varphi}.
  5. 1 + φ = φ 2 , {1+\varphi}=\varphi^{2},
  6. f = 1 φ 2 f=\frac{1}{\varphi^{2}}
  7. f 0.381966. f\approx 0.381966.\,
  8. g 360 × 0.381966 137.508 , g\approx 360\times 0.381966\approx 137.508^{\circ},\,
  9. g 2 π × 0.381966 2.39996. g\approx 2\pi\times 0.381966\approx 2.39996.\,

Golden_rectangle.html

  1. 1 : 1 + 5 2 1:\tfrac{1+\sqrt{5}}{2}
  2. 1 : φ 1:\varphi
  3. φ \varphi

Golden_spiral.html

  1. φ φ
  2. φ φ
  3. b b
  4. r = a e b θ r=ae^{b\theta}\,
  5. θ = 1 b ln ( r / a ) , \theta=\frac{1}{b}\ln(r/a),
  6. e e
  7. a a
  8. b b
  9. θ θ
  10. e b θ right = φ e^{b\theta_{\mathrm{right}}}\,=\varphi
  11. b b
  12. b = ln φ θ right . b={\ln{\varphi}\over\theta_{\mathrm{right}}}.
  13. b b
  14. π 2 \textstyle\frac{\pi}{2}
  15. b b
  16. b b
  17. | b | = ln φ 90 0.0053468 |b|={\ln{\varphi}\over 90}\doteq 0.0053468\,
  18. θ θ
  19. | b | = ln φ π / 2 0.3063489 |b|={\ln{\varphi}\over\pi/2}\doteq 0.3063489\,
  20. θ θ
  21. r = a c θ r=ac^{\theta}\,
  22. c c
  23. c = e b c=e^{b}\,
  24. c c
  25. c = φ 1 90 1.0053611 c=\varphi^{\frac{1}{90}}\doteq 1.0053611
  26. θ θ
  27. c = φ 2 π 1.358456. c=\varphi^{\frac{2}{\pi}}\doteq 1.358456.
  28. θ θ
  29. φ φ

Golomb_coding.html

  1. M M
  2. N N
  3. q q
  4. M M
  5. r r
  6. M = 1 M=1
  7. x x
  8. q = ( x - 1 ) M q=\left\lfloor\frac{\left(x-1\right)}{M}\right\rfloor
  9. r = x - q M - 1 r=x-qM-1
  10. ( q + 1 ) r \left(q+1\right)r
  11. r r
  12. b = log 2 ( M ) b=\lceil\log_{2}(M)\rceil
  13. 0 r < 2 b - M 0\leq r<2^{b}-M
  14. 2 b - M r < M 2^{b}-M\leq r<M
  15. b = log 2 ( M ) b=\log_{2}(M)
  16. p = P ( X = 0 ) p=P(X=0)
  17. M M
  18. ( 1 - p ) M + ( 1 - p ) M + 1 1 < ( 1 - p ) M - 1 + ( 1 - p ) M . (1-p)^{M}+(1-p)^{M+1}\leq 1<(1-p)^{M-1}+(1-p)^{M}.
  19. M = - 1 log 2 ( 1 - p ) M=\left\lfloor\frac{-1}{\log_{2}(1-p)}\right\rfloor
  20. x x
  21. x = 2 | x | = 2 x , x 0 x^{\prime}=2|x|=2x,x\geq 0
  22. y y
  23. y = 2 | y | - 1 = - 2 y - 1 , y < 0 y^{\prime}=2|y|-1=-2y-1,y<0
  24. log 2 ( M ) \log_{2}(M)
  25. b = log 2 ( M ) b=\lceil\log_{2}(M)\rceil
  26. r < 2 b - M r<2^{b}-M
  27. r 2 b - M r\geq 2^{b}-M
  28. r + 2 b - M r+2^{b}-M
  29. b = log 2 ( 10 ) = 4 b=\lceil\log_{2}(10)\rceil=4
  30. 2 b - M = 16 - 10 = 6 2^{b}-M=16-10=6
  31. \vdots
  32. \vdots
  33. 111 111 N 0 \underbrace{111\cdots 111}_{N}0
  34. - 1 log 2 p \frac{-1}{\log_{2}p}

Goodstein's_theorem.html

  1. m = a k n k + a k - 1 n k - 1 + + a 0 , m=a_{k}n^{k}+a_{k-1}n^{k-1}+\cdots+a_{0},
  2. 35 = 32 + 2 + 1 = 2 5 + 2 1 + 2 0 . 35=32+2+1=2^{5}+2^{1}+2^{0}.
  3. 100 = 81 + 18 + 1 = 3 4 + 2 3 2 + 3 0 . 100=81+18+1=3^{4}+2\cdot 3^{2}+3^{0}.
  4. 35 = 2 2 2 + 1 + 2 + 1 , 35=2^{2^{2}+1}+2+1,
  5. 100 = 3 3 + 1 + 2 3 2 + 1. 100=3^{3+1}+2\cdot 3^{2}+1.
  6. 2 1 + 1 2^{1}+1
  7. 3 1 + 1 - 1 = 3 1 3^{1}+1-1=3^{1}
  8. 4 1 - 1 = 3 4^{1}-1=3
  9. 3 - 1 = 2 3-1=2
  10. 2 - 1 = 1 2-1=1
  11. 1 - 1 = 0 1-1=0
  12. 2 2 2^{2}
  13. 3 3 - 1 = 2 3 2 + 2 3 + 2 3^{3}-1=2\cdot 3^{2}+2\cdot 3+2
  14. 2 4 2 + 2 4 + 1 2\cdot 4^{2}+2\cdot 4+1
  15. 2 5 2 + 2 5 2\cdot 5^{2}+2\cdot 5
  16. 2 6 2 + 2 6 - 1 = 2 6 2 + 6 + 5 2\cdot 6^{2}+2\cdot 6-1=2\cdot 6^{2}+6+5
  17. 2 7 2 + 7 + 4 2\cdot 7^{2}+7+4
  18. \vdots
  19. \vdots
  20. 2 11 2 + 11 2\cdot 11^{2}+11
  21. 2 12 2 + 12 - 1 = 2 12 2 + 11 2\cdot 12^{2}+12-1=2\cdot 12^{2}+11
  22. \vdots
  23. \vdots
  24. 3 2 402653209 3\cdot 2^{402653209}
  25. 3 2 402653210 - 1 3\cdot 2^{402653210}-1
  26. 3 2 402653209 3\cdot 2^{402653209}
  27. 3 2 402653211 - 1 3\cdot 2^{402653211}-1
  28. 3 2 402653211 - 1 = 402653184 2 402653184 - 1 3\cdot 2^{402653211}-1=402653184\cdot 2^{402653184}-1
  29. 2 2 2 + 2 + 1 2^{2^{2}}+2+1
  30. 3 3 3 + 3 3^{3^{3}}+3
  31. 4 4 4 + 3 4^{4^{4}}+3
  32. 1.3 × 10 154 \approx 1.3\times 10^{154}
  33. 5 5 5 + 2 5^{5^{5}}+2
  34. 1.8 × 10 2184 \approx 1.8\times 10^{2184}
  35. 6 6 6 + 1 6^{6^{6}}+1
  36. 2.6 × 10 36 , 305 \approx 2.6\times 10^{36,305}
  37. 7 7 7 7^{7^{7}}
  38. 3.8 × 10 695 , 974 \approx 3.8\times 10^{695,974}
  39. 8 8 8 - 1 = 7 8 ( 7 8 7 + 7 8 6 + 7 8 5 + 7 8 4 + 7 8 3 + 7 8 2 + 7 8 + 7 ) 8^{8^{8}}-1=7\cdot 8^{(7\cdot 8^{7}+7\cdot 8^{6}+7\cdot 8^{5}+7\cdot 8^{4}+7% \cdot 8^{3}+7\cdot 8^{2}+7\cdot 8+7)}
  40. + 7 8 ( 7 8 7 + 7 8 6 + 7 8 5 + 7 8 4 + 7 8 3 + 7 8 2 + 7 8 + 6 ) + +7\cdot 8^{(7\cdot 8^{7}+7\cdot 8^{6}+7\cdot 8^{5}+7\cdot 8^{4}+7\cdot 8^{3}+7% \cdot 8^{2}+7\cdot 8+6)}+\cdots
  41. + 7 8 ( 8 + 2 ) + 7 8 ( 8 + 1 ) + 7 8 8 +7\cdot 8^{(8+2)}+7\cdot 8^{(8+1)}+7\cdot 8^{8}
  42. + 7 8 7 + 7 8 6 + 7 8 5 + 7 8 4 +7\cdot 8^{7}+7\cdot 8^{6}+7\cdot 8^{5}+7\cdot 8^{4}
  43. + 7 8 3 + 7 8 2 + 7 8 + 7 +7\cdot 8^{3}+7\cdot 8^{2}+7\cdot 8+7
  44. 6 × 10 15 , 151 , 335 \approx 6\times 10^{15,151,335}
  45. 7 9 ( 7 9 7 + 7 9 6 + 7 9 5 + 7 9 4 + 7 9 3 + 7 9 2 + 7 9 + 7 ) 7\cdot 9^{(7\cdot 9^{7}+7\cdot 9^{6}+7\cdot 9^{5}+7\cdot 9^{4}+7\cdot 9^{3}+7% \cdot 9^{2}+7\cdot 9+7)}
  46. + 7 9 ( 7 9 7 + 7 9 6 + 7 9 5 + 7 9 4 + 7 9 3 + 7 9 2 + 7 9 + 6 ) + +7\cdot 9^{(7\cdot 9^{7}+7\cdot 9^{6}+7\cdot 9^{5}+7\cdot 9^{4}+7\cdot 9^{3}+7% \cdot 9^{2}+7\cdot 9+6)}+\cdots
  47. + 7 9 ( 9 + 2 ) + 7 9 ( 9 + 1 ) + 7 9 9 +7\cdot 9^{(9+2)}+7\cdot 9^{(9+1)}+7\cdot 9^{9}
  48. + 7 9 7 + 7 9 6 + 7 9 5 + 7 9 4 +7\cdot 9^{7}+7\cdot 9^{6}+7\cdot 9^{5}+7\cdot 9^{4}
  49. + 7 9 3 + 7 9 2 + 7 9 + 6 +7\cdot 9^{3}+7\cdot 9^{2}+7\cdot 9+6
  50. 4.3 × 10 369 , 693 , 099 \approx 4.3\times 10^{369,693,099}
  51. \vdots
  52. \vdots
  53. f ( 3 4 4 4 + 4 ) = 3 ω ω ω + ω = f ( 3 5 5 5 + 5 ) f(3\cdot 4^{4^{4}}+4)=3\omega^{\omega^{\omega}}+\omega=f(3\cdot 5^{5^{5}}+5)
  54. f ( 3 4 4 4 + 4 ) f(3\cdot 4^{4^{4}}+4)
  55. f ( ( 3 5 5 5 + 5 ) - 1 ) . f((3\cdot 5^{5^{5}}+5)-1).
  56. f ( 3 4 4 4 + 4 , 4 ) = 3 ω ω ω + ω = f ( 3 9 9 9 + 9 , 9 ) f(3\cdot 4^{4^{4}}+4,4)=3\omega^{\omega^{\omega}}+\omega=f(3\cdot 9^{9^{9}}+9,9)
  57. f ( 3 4 4 4 + 4 , 4 ) f(3\cdot 4^{4^{4}}+4,4)
  58. f ( ( 3 9 9 9 + 9 ) - 1 , 9 ) . f((3\cdot 9^{9^{9}}+9)-1,9).
  59. 𝒢 : \mathcal{G}:\mathbb{N}\to\mathbb{N}\,\!
  60. 𝒢 ( n ) \mathcal{G}(n)
  61. 𝒢 \mathcal{G}\,\!
  62. H α H_{\alpha}\,\!
  63. f α f_{\alpha}\,\!
  64. 𝒢 \mathcal{G}\,\!
  65. H ϵ 0 H_{\epsilon_{0}}\,\!
  66. f ϵ 0 f_{\epsilon_{0}}\,\!
  67. 𝒢 \mathcal{G}\,\!
  68. H α H_{\alpha}\,\!
  69. α < ϵ 0 \alpha<\epsilon_{0}\,\!
  70. H ϵ 0 H_{\epsilon_{0}}\,\!
  71. 𝒢 . \mathcal{G}\,\!.
  72. f , g : f,g:\mathbb{N}\to\mathbb{N}\,\!
  73. f f\,\!
  74. g g\,\!
  75. f ( n ) > g ( n ) f(n)>g(n)\,\!
  76. n n\,\!
  77. 𝒢 ( n ) = H R 2 ω ( n + 1 ) ( 1 ) - 1 , \mathcal{G}(n)=H_{R_{2}^{\omega}(n+1)}(1)-1,
  78. R 2 ω ( n ) R_{2}^{\omega}(n)
  79. n = 2 m 1 + 2 m 2 + + 2 m k n=2^{m_{1}}+2^{m_{2}}+\cdots+2^{m_{k}}
  80. m 1 > m 2 > > m k , m_{1}>m_{2}>\cdots>m_{k},
  81. 𝒢 ( n ) = f R 2 ω ( m 1 ) ( f R 2 ω ( m 2 ) ( ( f R 2 ω ( m k ) ( 3 ) ) ) ) - 2 \mathcal{G}(n)=f_{R_{2}^{\omega}(m_{1})}(f_{R_{2}^{\omega}(m_{2})}(\cdots(f_{R% _{2}^{\omega}(m_{k})}(3))\cdots))-2
  82. 𝒢 ( n ) \mathcal{G}(n)
  83. 2 0 2^{0}
  84. 2 - 1 2-1
  85. H ω ( 1 ) - 1 H_{\omega}(1)-1
  86. f 0 ( 3 ) - 2 f_{0}(3)-2
  87. 2 1 2^{1}
  88. 2 1 + 1 - 1 2^{1}+1-1
  89. H ω + 1 ( 1 ) - 1 H_{\omega+1}(1)-1
  90. f 1 ( 3 ) - 2 f_{1}(3)-2
  91. 2 1 + 2 0 2^{1}+2^{0}
  92. 2 2 - 1 2^{2}-1
  93. H ω ω ( 1 ) - 1 H_{\omega^{\omega}}(1)-1
  94. f 1 ( f 0 ( 3 ) ) - 2 f_{1}(f_{0}(3))-2
  95. 2 2 2^{2}
  96. 2 2 + 1 - 1 2^{2}+1-1
  97. H ω ω + 1 ( 1 ) - 1 H_{\omega^{\omega}+1}(1)-1
  98. f ω ( 3 ) - 2 f_{\omega}(3)-2
  99. 2 2 + 2 0 2^{2}+2^{0}
  100. 2 2 + 2 - 1 2^{2}+2-1
  101. H ω ω + ω ( 1 ) - 1 H_{\omega^{\omega}+\omega}(1)-1
  102. f ω ( f 0 ( 3 ) ) - 2 f_{\omega}(f_{0}(3))-2
  103. 2 2 + 2 1 2^{2}+2^{1}
  104. 2 2 + 2 + 1 - 1 2^{2}+2+1-1
  105. H ω ω + ω + 1 ( 1 ) - 1 H_{\omega^{\omega}+\omega+1}(1)-1
  106. f ω ( f 1 ( 3 ) ) - 2 f_{\omega}(f_{1}(3))-2
  107. 2 2 + 2 1 + 2 0 2^{2}+2^{1}+2^{0}
  108. 2 2 + 1 - 1 2^{2+1}-1
  109. H ω ω + 1 ( 1 ) - 1 H_{\omega^{\omega+1}}(1)-1
  110. f ω ( f 1 ( f 0 ( 3 ) ) ) - 2 f_{\omega}(f_{1}(f_{0}(3)))-2
  111. 2 2 + 1 2^{2+1}
  112. 2 2 + 1 + 1 - 1 2^{2+1}+1-1
  113. H ω ω + 1 + 1 ( 1 ) - 1 H_{\omega^{\omega+1}+1}(1)-1
  114. f ω + 1 ( 3 ) - 2 f_{\omega+1}(3)-2
  115. \vdots
  116. 2 2 + 1 + 2 2 2^{2+1}+2^{2}
  117. 2 2 + 1 + 2 2 + 1 - 1 2^{2+1}+2^{2}+1-1
  118. H ω ω + 1 + ω ω + 1 ( 1 ) - 1 H_{\omega^{\omega+1}+\omega^{\omega}+1}(1)-1
  119. f ω + 1 ( f ω ( 3 ) ) - 2 f_{\omega+1}(f_{\omega}(3))-2
  120. \vdots
  121. 2 2 2 + 2 1 + 2 0 2^{2^{2}}+2^{1}+2^{0}
  122. 2 2 2 + 2 2 - 1 2^{2^{2}}+2^{2}-1
  123. H ω ω ω + ω ω ( 1 ) - 1 H_{\omega^{\omega^{\omega}}+\omega^{\omega}}(1)-1
  124. f ω ω ( f 1 ( f 0 ( 3 ) ) ) - 2 f_{\omega^{\omega}}(f_{1}(f_{0}(3)))-2

Government_budget_balance.html

  1. t t
  2. G t G_{t}
  3. T t T_{t}
  4. Primary deficit = G t - T t . \,\text{Primary deficit}=G_{t}-T_{t}.\,
  5. D t - 1 D_{t-1}
  6. r r
  7. Total deficit t = r D t - 1 + G t - T t \,\text{Total deficit}_{t}=r\cdot D_{t-1}+G_{t}-T_{t}\,
  8. D t = ( 1 + r ) D t - 1 + G t - T t . {D_{t}}=(1+r)D_{t-1}+G_{t}-T_{t}.\,
  9. Y = C + I + G + ( X - M ) Y=C+I+G+(X-M)
  10. Y = C + S + T Y=C+S+T
  11. C + S + T = Y = C + I + G + ( X - M ) . C+S+T=Y=C+I+G+(X-M).
  12. S + T = I + G + ( X - M ) . S+T=I+G+(X-M).
  13. ( S - I ) = ( G - T ) + ( X - M ) . (S-I)=(G-T)+(X-M).
  14. ( G - T ) = ( S - I ) - N X (G-T)=(S-I)-NX

Gödel_numbering.html

  1. ( x 1 , x 2 , x 3 , , x n ) (x_{1},x_{2},x_{3},...,x_{n})
  2. enc ( x 1 , x 2 , x 3 , , x n ) = 2 x 1 3 x 2 5 x 3 p n x n . \mathrm{enc}(x_{1},x_{2},x_{3},\dots,x_{n})=2^{x_{1}}\cdot 3^{x_{2}}\cdot 5^{x% _{3}}\cdots p_{n}^{x_{n}}.\,
  3. s 1 s 2 s 3 s n s_{1}s_{2}s_{3}\dots s_{n}
  4. h ( s 1 ) × K ( n - 1 ) + h ( s 2 ) × K ( n - 2 ) + + h ( s n - 1 ) × K 1 + h ( s n ) × K 0 . h(s_{1})\times K^{(n-1)}+h(s_{2})\times K^{(n-2)}+\cdots+h(s_{n-1})\times K^{1% }+h(s_{n})\times K^{0}.
  5. A , B r C , A,B\vdash_{r}C,
  6. g r ( f ( A ) , f ( B ) ) = f ( C ) . g_{r}(f(A),f(B))=f(C).

Grade_(slope).html

  1. 100 rise run 100\frac{\,\text{rise}}{\,\text{run}}
  2. 1000 rise run 1000\frac{\,\text{rise}}{\,\text{run}}
  3. tan α = Δ h d \tan{\alpha}=\frac{\Delta h}{d}
  4. α = arctan Δ h d \alpha=\arctan{\frac{\Delta h}{d}}
  5. α = arctan % slope 100 \alpha=\arctan{\frac{\%\,\,\text{slope}}{100}}
  6. α = arctan 1 n \alpha=\arctan{\frac{1}{n}}

Graded_ring.html

  1. R i R_{i}
  2. R i R j R i + j R_{i}R_{j}\subset R_{i+j}
  3. A = n 0 A n = A 0 A 1 A 2 A=\bigoplus_{n\in\mathbb{N}_{0}}A_{n}=A_{0}\oplus A_{1}\oplus A_{2}\oplus\cdots
  4. A 0 A_{0}
  5. 0 \mathbb{N}_{0}
  6. A = 0 A i A=\oplus_{0}^{\infty}A_{i}
  7. A 0 A_{0}
  8. A 0 A_{0}
  9. A n A_{n}
  10. 𝔞 \mathfrak{a}
  11. 𝔞 \mathfrak{a}
  12. A / I A/I
  13. A / I = n ( A n + I ) / I . A/I=\bigoplus_{n\in\mathbb{N}}(A_{n}+I)/I.
  14. M = i 0 M i , M=\bigoplus_{i\in\mathbb{N}_{0}}M_{i},
  15. A i M j M i + j . A_{i}M_{j}\subseteq M_{i+j}.
  16. f : N M f:N\to M
  17. f ( N i ) M i f(N_{i})\subseteq M_{i}
  18. N i = N M i N_{i}=N\cap M_{i}
  19. M ( l ) M(l)
  20. M ( l ) n = M n + l M(l)_{n}=M_{n+l}
  21. f : M N f:M\to N
  22. f ( M n ) N n + d f(M_{n})\subset N_{n+d}
  23. P ( M , t ) [ [ t ] ] P(M,t)\in\mathbb{Z}[\![t]\!]
  24. P ( M , t ) = ( M n ) t n P(M,t)=\sum\ell(M_{n})t^{n}
  25. ( M n ) \ell(M_{n})
  26. k [ x 0 , , x n ] k[x_{0},\dots,x_{n}]
  27. n dim k M n n\mapsto\dim_{k}M_{n}
  28. A i R j A i + j A_{i}R_{j}\subseteq A_{i+j}
  29. R i A j A i + j R_{i}A_{j}\subseteq A_{i+j}
  30. A = i G A i A=\bigoplus_{i\in G}A_{i}
  31. A i A j A i j . A_{i}A_{j}\subseteq A_{i\cdot j}.
  32. A = k [ t 1 , , t n ] A=k[t_{1},\ldots,t_{n}]
  33. A i A_{i}

Gradian.html

  1. 1 / 400 {1}/{400}
  2. 9 / 10 {9}/{10}
  3. π / 200 {\pi}/{200}
  4. 1 / 360 {1}/{360}
  5. 1 ( 2 / π 1{(2}/{\pi}
  6. 1 / 3 {1}/{3}
  7. 2 / 3 {2}/{3}
  8. 1 / 24 {1}/{24}
  9. 2 / 3 {2}/{3}
  10. i n i^{n}
  11. 100 n 100n
  12. 1 24 \frac{1}{24}
  13. 1 12 \frac{1}{12}
  14. 1 10 \frac{1}{10}
  15. 1 8 \frac{1}{8}
  16. 1 ( 2 π ) \frac{1}{(2π)}
  17. 1 6 \frac{1}{6}
  18. 1 5 \frac{1}{5}
  19. 1 4 \frac{1}{4}
  20. 1 3 \frac{1}{3}
  21. 2 5 \frac{2}{5}
  22. 1 2 \frac{1}{2}
  23. 3 4 \frac{3}{4}
  24. π 12 \frac{\pi}{12}
  25. π 6 \frac{\pi}{6}
  26. π 5 \frac{\pi}{5}
  27. π 4 \frac{\pi}{4}
  28. π 3 \frac{\pi}{3}
  29. 2 π 5 2\frac{\pi}{5}
  30. π 2 \frac{\pi}{2}
  31. 2 π 3 2\frac{\pi}{3}
  32. 4 π 5 4\frac{\pi}{5}
  33. π \pi
  34. 3 π 2 3\frac{\pi}{2}
  35. π \pi
  36. 2 3 \frac{2}{3}
  37. 1 3 \frac{1}{3}
  38. 2 3 \frac{2}{3}
  39. 1 3 \frac{1}{3}

Gradient_descent.html

  1. F ( 𝐱 ) F(\mathbf{x})
  2. 𝐚 \mathbf{a}
  3. F ( 𝐱 ) F(\mathbf{x})
  4. 𝐚 \mathbf{a}
  5. F F
  6. 𝐚 \mathbf{a}
  7. - F ( 𝐚 ) -\nabla F(\mathbf{a})
  8. 𝐛 = 𝐚 - γ F ( 𝐚 ) \mathbf{b}=\mathbf{a}-\gamma\nabla F(\mathbf{a})
  9. γ \gamma
  10. F ( 𝐚 ) F ( 𝐛 ) F(\mathbf{a})\geq F(\mathbf{b})
  11. 𝐱 0 \mathbf{x}_{0}
  12. F F
  13. 𝐱 0 , 𝐱 1 , 𝐱 2 , \mathbf{x}_{0},\mathbf{x}_{1},\mathbf{x}_{2},\dots
  14. 𝐱 n + 1 = 𝐱 n - γ n F ( 𝐱 n ) , n 0. \mathbf{x}_{n+1}=\mathbf{x}_{n}-\gamma_{n}\nabla F(\mathbf{x}_{n}),\ n\geq 0.
  15. F ( 𝐱 0 ) F ( 𝐱 1 ) F ( 𝐱 2 ) , F(\mathbf{x}_{0})\geq F(\mathbf{x}_{1})\geq F(\mathbf{x}_{2})\geq\cdots,
  16. ( 𝐱 n ) (\mathbf{x}_{n})
  17. γ \gamma
  18. F F
  19. F F
  20. F \nabla F
  21. γ \gamma
  22. F F
  23. F F
  24. F F
  25. F F
  26. f ( x 1 , x 2 ) = ( 1 - x 1 ) 2 + 100 ( x 2 - x 1 2 ) 2 . f(x_{1},x_{2})=(1-x_{1})^{2}+100(x_{2}-x_{1}^{2})^{2}.\quad
  27. F ( x , y ) = sin ( 1 2 x 2 - 1 4 y 2 + 3 ) cos ( 2 x + 1 - e y ) F(x,y)=\sin\left(\frac{1}{2}x^{2}-\frac{1}{4}y^{2}+3\right)\cos(2x+1-e^{y})
  28. A 𝐱 - 𝐛 = 0 A\mathbf{x}-\mathbf{b}=0
  29. F ( x ) = A 𝐱 - 𝐛 2 . F(x)=\|A\mathbf{x}-\mathbf{b}\|^{2}.
  30. A A
  31. 𝐛 \mathbf{b}
  32. F ( 𝐱 ) = 2 A T ( A 𝐱 - 𝐛 ) . \nabla F(\mathbf{x})=2A^{T}(A\mathbf{x}-\mathbf{b}).
  33. γ \gamma
  34. γ \gamma
  35. A A
  36. { 3 x 1 - cos ( x 2 x 3 ) - 3 2 = 0 4 x 1 2 - 625 x 2 2 + 2 x 2 - 1 = 0 exp ( - x 1 x 2 ) + 20 x 3 + 10 π - 3 3 = 0 \begin{cases}3x_{1}-\cos(x_{2}x_{3})-\tfrac{3}{2}=0\\ 4x_{1}^{2}-625x_{2}^{2}+2x_{2}-1=0\\ \exp(-x_{1}x_{2})+20x_{3}+\tfrac{10\pi-3}{3}=0\end{cases}
  37. G ( 𝐱 ) = [ 3 x 1 - cos ( x 2 x 3 ) - 3 2 4 x 1 2 - 625 x 2 2 + 2 x 2 - 1 exp ( - x 1 x 2 ) + 20 x 3 + 10 π - 3 3 ] G(\mathbf{x})=\begin{bmatrix}3x_{1}-\cos(x_{2}x_{3})-\tfrac{3}{2}\\ 4x_{1}^{2}-625x_{2}^{2}+2x_{2}-1\\ \exp(-x_{1}x_{2})+20x_{3}+\tfrac{10\pi-3}{3}\\ \end{bmatrix}
  38. 𝐱 = [ x 1 x 2 x 3 ] \mathbf{x}=\begin{bmatrix}x_{1}\\ x_{2}\\ x_{3}\\ \end{bmatrix}
  39. F ( 𝐱 ) = 1 2 G T ( 𝐱 ) G ( 𝐱 ) F(\mathbf{x})=\tfrac{1}{2}G^{\mathrm{T}}(\mathbf{x})G(\mathbf{x})
  40. = 1 2 ( ( 3 x 1 - cos ( x 2 x 3 ) - 3 2 ) 2 + ( 4 x 1 2 - 625 x 2 2 + 2 x 2 - 1 ) 2 + ( exp ( - x 1 x 2 ) + 20 x 3 + 10 π - 3 3 ) 2 ) =\tfrac{1}{2}((3x_{1}-\cos(x_{2}x_{3})-\tfrac{3}{2})^{2}+(4x_{1}^{2}-625x_{2}^% {2}+2x_{2}-1)^{2}+(\exp(-x_{1}x_{2})+20x_{3}+\tfrac{10\pi-3}{3})^{2})
  41. 𝐱 ( 0 ) = [ x 1 x 2 x 3 ] = [ 0 0 0 ] \mathbf{x}^{(0)}=\begin{bmatrix}x_{1}\\ x_{2}\\ x_{3}\\ \end{bmatrix}=\begin{bmatrix}0\\ 0\\ 0\\ \end{bmatrix}
  42. 𝐱 ( 1 ) = 𝐱 ( 0 ) - γ 0 F ( x ( 0 ) ) \mathbf{x}^{(1)}=\mathbf{x}^{(0)}-\gamma_{0}\nabla F(x^{(0)})
  43. F ( 𝐱 ( 0 ) ) = J G ( 𝐱 ( 0 ) ) T G ( 𝐱 ( 0 ) ) \nabla F(\mathbf{x}^{(0)})=J_{G}(\mathbf{x}^{(0)})^{\mathrm{T}}G(\mathbf{x}^{(% 0)})
  44. J G ( 𝐱 ( 0 ) ) J_{G}(\mathbf{x}^{(0)})
  45. J G = [ 3 sin ( x 2 x 3 ) x 3 sin ( x 2 x 3 ) x 2 8 x 1 - 1250 x 2 + 2 0 - x 2 exp ( - x 1 x 2 ) - x 1 exp ( - x 1 x 2 ) 20 ] J_{G}=\begin{bmatrix}3&\sin(x_{2}x_{3})x_{3}&\sin(x_{2}x_{3})x_{2}\\ 8x_{1}&-1250x_{2}+2&0\\ -x_{2}\exp{(-x_{1}x_{2})}&-x_{1}\exp(-x_{1}x_{2})&20\\ \end{bmatrix}
  46. 𝐱 ( 0 ) \mathbf{x}^{(0)}
  47. J G ( 𝐱 ( 0 ) ) = [ 3 0 0 0 2 0 0 0 20 ] J_{G}\left(\mathbf{x}^{(0)}\right)=\begin{bmatrix}3&0&0\\ 0&2&0\\ 0&0&20\end{bmatrix}
  48. G ( 𝐱 ( 0 ) ) = [ - 2.5 - 1 10.472 ] G(\mathbf{x}^{(0)})=\begin{bmatrix}-2.5\\ -1\\ 10.472\end{bmatrix}
  49. 𝐱 ( 1 ) = 0 - γ 0 [ - 7.5 - 2 209.44 ] . \mathbf{x}^{(1)}=0-\gamma_{0}\begin{bmatrix}-7.5\\ -2\\ 209.44\end{bmatrix}.
  50. F ( 𝐱 ( 0 ) ) = 0.5 ( ( - 2.5 ) 2 + ( - 1 ) 2 + ( 10.472 ) 2 ) = 58.456 F\left(\mathbf{x}^{(0)}\right)=0.5((-2.5)^{2}+(-1)^{2}+(10.472)^{2})=58.456
  51. F ( 𝐱 ( n ) ) F(\mathbf{x}^{(n)})
  52. 𝐱 ( n ) \mathbf{x}^{(n)}
  53. γ 0 \gamma_{0}
  54. F ( 𝐱 ( 1 ) ) F ( 𝐱 ( 0 ) ) F(\mathbf{x}^{(1)})\leq F(\mathbf{x}^{(0)})
  55. γ 0 = 0.001 \gamma_{0}=0.001
  56. 𝐱 ( 1 ) = [ 0.0075 0.002 - 0.20944 ] \mathbf{x}^{(1)}=\begin{bmatrix}0.0075\\ 0.002\\ -0.20944\\ \end{bmatrix}
  57. F ( 𝐱 ( 1 ) ) = 0.5 ( ( - 2.48 ) 2 + ( - 1.00 ) 2 + ( 6.28 ) 2 ) = 23.306 F\left(\mathbf{x}^{(1)}\right)=0.5((-2.48)^{2}+(-1.00)^{2}+(6.28)^{2})=23.306
  58. F ( 𝐱 ( 0 ) ) = 58.456 F(\mathbf{x}^{(0)})=58.456
  59. F ( 𝐱 ( 1 ) ) = 23.306 F(\mathbf{x}^{(1)})=23.306
  60. γ \gamma
  61. γ \gamma
  62. γ . \gamma.
  63. x ( t ) = - f ( x ( t ) ) x^{\prime}(t)=-\nabla f(x(t))
  64. F F
  65. F \nabla F
  66. F F
  67. k k
  68. 𝒪 ( 1 / k ) \mathcal{O}(1/k)
  69. 𝒪 ( 1 / k 2 ) \mathcal{O}(1/k^{2})

Graham's_law.html

  1. Rate 1 Rate 2 = M 2 M 1 {\mbox{Rate}~{}_{1}\over\mbox{Rate}~{}_{2}}=\sqrt{M_{2}\over M_{1}}
  2. r α 1 d {\mbox{r}~{}}\ \alpha\ {\mbox{1}~{}\over\sqrt{d}}
  3. Rate H 2 Rate O 2 = 32 2 = 16 1 = 4 1 {\mbox{Rate H}~{}_{2}\over\mbox{Rate O}~{}_{2}}={\sqrt{32}\over\sqrt{2}}={% \sqrt{16}\over\sqrt{1}}=\frac{4}{1}
  4. M 2 = M 1 Rate 1 2 Rate 2 2 {M_{2}}={M_{1}\mbox{Rate}~{}_{1}^{2}\over\mbox{Rate}~{}_{2}^{2}}

Graham's_number.html

  1. a b c \scriptstyle a^{b^{c^{\cdot^{\cdot^{\cdot}}}}}
  2. F 7 ( 12 ) = F ( F ( F ( F ( F ( F ( F ( 12 ) ) ) ) ) ) ) \scriptstyle F^{7}(12)\;=\;F(F(F(F(F(F(F(12)))))))
  3. F ( n ) = 2 n 3 \scriptstyle F(n)\;=\;2\uparrow^{n}3
  4. N = 2 6 \scriptstyle N^{\prime}\;=\;2\;\uparrow\uparrow\uparrow\;6
  5. f 64 ( 4 ) \scriptstyle f^{64}(4)
  6. f ( n ) = 3 n 3 \scriptstyle f(n)\;=\;3\uparrow^{n}3
  7. G = 3 3 3 3 3 3 3 3 } 64 layers \left.\begin{matrix}G&=&3\underbrace{\uparrow\uparrow\cdots\cdots\cdots\cdots% \cdots\uparrow}3\\ &&3\underbrace{\uparrow\uparrow\cdots\cdots\cdots\cdots\uparrow}3\\ &&\underbrace{\qquad\;\;\vdots\qquad\;\;}\\ &&3\underbrace{\uparrow\uparrow\cdots\cdot\cdot\uparrow}3\\ &&3\uparrow\uparrow\uparrow\uparrow 3\end{matrix}\right\}\,\text{64 layers}
  8. G = g 64 , where g 1 = 3 3 , g n = 3 g n - 1 3 , G=g_{64},\,\text{ where }g_{1}=3\uparrow\uparrow\uparrow\uparrow 3,\ g_{n}=3% \uparrow^{g_{n-1}}3,
  9. G = f 64 ( 4 ) , where f ( n ) = 3 n 3 , G=f^{64}(4),\,\text{ where }f(n)=3\uparrow^{n}3,
  10. f 4 ( n ) = f ( f ( f ( f ( n ) ) ) ) \scriptstyle f^{4}(n)\;=\;f(f(f(f(n))))
  11. H 0 , H 1 , H 2 , \scriptstyle\,\text{H}_{0},\,\text{H}_{1},\,\text{H}_{2},\cdots
  12. f ( n ) = H n + 2 ( 3 , 3 ) \scriptstyle f(n)\;=\;\,\text{H}_{n+2}(3,3)
  13. f ( n ) > A ( n , n ) \scriptstyle f(n)\;>\;A(n,\,n)
  14. f ( n ) = 3 3 n \scriptstyle f(n)\;=\;3\rightarrow 3\rightarrow n
  15. 3 3 64 2 < G < 3 3 65 2. 3\rightarrow 3\rightarrow 64\rightarrow 2<G<3\rightarrow 3\rightarrow 65% \rightarrow 2.\,
  16. \scriptstyle\uparrow\uparrow
  17. g 1 = 3 3 = 3 ( 3 3 ) = 3 ( 3 ( 3 ( 3 3 ) ) ) g_{1}=3\uparrow\uparrow\uparrow\uparrow 3=3\uparrow\uparrow\uparrow(3\uparrow% \uparrow\uparrow 3)=3\uparrow\uparrow(3\uparrow\uparrow(3\uparrow\uparrow\ % \dots\ (3\uparrow\uparrow 3)\dots))
  18. 3 3 = 3 ( 3 3 ) . 3\uparrow\uparrow\uparrow 3\ =\ 3\uparrow\uparrow(3\uparrow\uparrow 3).
  19. \scriptstyle\uparrow\uparrow
  20. \scriptstyle\uparrow
  21. 3 X = 3 ( 3 ( 3 ( 3 3 ) ) ) = 3 3 3 where there are X 3s . 3\uparrow\uparrow X\ =\ 3\uparrow(3\uparrow(3\uparrow\dots(3\uparrow 3)\dots))% \ =\ 3^{3^{\cdot^{\cdot^{\cdot^{3}}}}}\quad\,\text{where there are X 3s}.
  22. g 1 = 3 ( 3 ( 3 ( 3 3 ) ) ) where the number of 3s is 3 ( 3 3 ) g_{1}=3\uparrow\uparrow(3\uparrow\uparrow(3\uparrow\uparrow\ \dots\ (3\uparrow% \uparrow 3)\dots))\quad\,\text{where the number of 3s is}\quad 3\uparrow% \uparrow(3\uparrow\uparrow 3)
  23. g 1 = 3 3 3 } 3 3 3 } 3 3 3 } 3 where the number of towers is 3 3 3 } 3 3 3 } 3 g_{1}=\left.\begin{matrix}3^{3^{\cdot^{\cdot^{\cdot^{\cdot^{3}}}}}}\end{matrix% }\right\}\left.\begin{matrix}3^{3^{\cdot^{\cdot^{\cdot^{3}}}}}\end{matrix}% \right\}\dots\left.\begin{matrix}3^{3^{3}}\end{matrix}\right\}3\quad\,\text{% where the number of towers is}\quad\left.\begin{matrix}3^{3^{\cdot^{\cdot^{% \cdot^{3}}}}}\end{matrix}\right\}\left.\begin{matrix}3^{3^{3}}\end{matrix}% \right\}3
  24. n = 3 3 3 3 \scriptstyle n\;=\;3\uparrow 3\uparrow 3\ \dots\ \uparrow 3
  25. 3 3 3 = 7625597484987 \scriptstyle 3\uparrow 3\uparrow 3\;=\;7625597484987
  26. \scriptstyle\uparrow

Gram–Schmidt_process.html

  1. proj 𝐮 ( 𝐯 ) = 𝐯 , 𝐮 𝐮 , 𝐮 𝐮 , \mathrm{proj}_{\mathbf{u}}\,(\mathbf{v})={\langle\mathbf{v},\mathbf{u}\rangle% \over\langle\mathbf{u},\mathbf{u}\rangle}\mathbf{u},
  2. 𝐯 , 𝐮 \langle\mathbf{v},\mathbf{u}\rangle
  3. proj 0 ( 𝐯 ) := 0 \mathrm{proj}_{0}\,(\mathbf{v}):=0
  4. proj 0 \mathrm{proj}_{0}
  5. 𝐮 1 \displaystyle\mathbf{u}_{1}
  6. ( v α ) α < λ (v_{\alpha})_{\alpha<\lambda}
  7. ( u α ) α < κ (u_{\alpha})_{\alpha<\kappa}
  8. κ λ \kappa\leq\lambda
  9. α λ \alpha\leq\lambda
  10. { u β : β < min ( α , κ ) } \{u_{\beta}:\beta<\min(\alpha,\kappa)\}
  11. { v β : β < α } \{v_{\beta}:\beta<\alpha\}
  12. κ < λ \kappa<\lambda
  13. ( u α ) α < κ (u_{\alpha})_{\alpha<\kappa}
  14. ( v α ) α < λ (v_{\alpha})_{\alpha<\lambda}
  15. S = { 𝐯 1 = ( 3 1 ) , 𝐯 2 = ( 2 2 ) } . S=\left\{\mathbf{v}_{1}=\begin{pmatrix}3\\ 1\end{pmatrix},\mathbf{v}_{2}=\begin{pmatrix}2\\ 2\end{pmatrix}\right\}.
  16. 𝐮 1 = 𝐯 1 = ( 3 1 ) \mathbf{u}_{1}=\mathbf{v}_{1}=\begin{pmatrix}3\\ 1\end{pmatrix}
  17. 𝐮 2 = 𝐯 2 - proj 𝐮 1 ( 𝐯 2 ) = ( 2 2 ) - proj ( 3 1 ) ( ( 2 2 ) ) = ( 2 2 ) - ( 4 / 5 ) ( 3 1 ) = ( - 2 / 5 6 / 5 ) . \mathbf{u}_{2}=\mathbf{v}_{2}-\mathrm{proj}_{\mathbf{u}_{1}}\,(\mathbf{v}_{2})% =\begin{pmatrix}2\\ 2\end{pmatrix}-\mathrm{proj}_{({3\atop 1})}\,({\begin{pmatrix}2\\ 2\end{pmatrix})}=\begin{pmatrix}2\\ 2\end{pmatrix}-\begin{pmatrix}4/5\end{pmatrix}\begin{pmatrix}3\\ 1\end{pmatrix}=\begin{pmatrix}-2/5\\ 6/5\end{pmatrix}.
  18. 𝐮 1 , 𝐮 2 = ( 3 1 ) , ( - 2 / 5 6 / 5 ) = - 6 5 + 6 5 = 0 , \langle\mathbf{u}_{1},\mathbf{u}_{2}\rangle=\left\langle\begin{pmatrix}3\\ 1\end{pmatrix},\begin{pmatrix}-2/5\\ 6/5\end{pmatrix}\right\rangle=-\frac{6}{5}+\frac{6}{5}=0,
  19. 𝐞 1 = 1 10 ( 3 1 ) \mathbf{e}_{1}={1\over\sqrt{10}}\begin{pmatrix}3\\ 1\end{pmatrix}
  20. 𝐞 2 = 1 40 25 ( - 2 / 5 6 / 5 ) = 1 10 ( - 1 3 ) . \mathbf{e}_{2}={1\over\sqrt{40\over 25}}\begin{pmatrix}-2/5\\ 6/5\end{pmatrix}={1\over\sqrt{10}}\begin{pmatrix}-1\\ 3\end{pmatrix}.
  21. 𝐮 k \mathbf{u}_{k}
  22. 𝐮 k = 𝐯 k - proj 𝐮 1 ( 𝐯 k ) - proj 𝐮 2 ( 𝐯 k ) - - proj 𝐮 k - 1 ( 𝐯 k ) , \mathbf{u}_{k}=\mathbf{v}_{k}-\mathrm{proj}_{\mathbf{u}_{1}}\,(\mathbf{v}_{k})% -\mathrm{proj}_{\mathbf{u}_{2}}\,(\mathbf{v}_{k})-\cdots-\mathrm{proj}_{% \mathbf{u}_{k-1}}\,(\mathbf{v}_{k}),
  23. 𝐮 k ( 1 ) = 𝐯 k - proj 𝐮 1 ( 𝐯 k ) , 𝐮 k ( 2 ) = 𝐮 k ( 1 ) - proj 𝐮 2 ( 𝐮 k ( 1 ) ) , 𝐮 k ( k - 2 ) = 𝐮 k ( k - 3 ) - proj 𝐮 k - 2 ( 𝐮 k ( k - 3 ) ) , 𝐮 k ( k - 1 ) = 𝐮 k ( k - 2 ) - proj 𝐮 k - 1 ( 𝐮 k ( k - 2 ) ) . \begin{aligned}\displaystyle\mathbf{u}_{k}^{(1)}&\displaystyle=\mathbf{v}_{k}-% \mathrm{proj}_{\mathbf{u}_{1}}\,(\mathbf{v}_{k}),\\ \displaystyle\mathbf{u}_{k}^{(2)}&\displaystyle=\mathbf{u}_{k}^{(1)}-\mathrm{% proj}_{\mathbf{u}_{2}}\,(\mathbf{u}_{k}^{(1)}),\\ &\displaystyle\,\,\,\vdots\\ \displaystyle\mathbf{u}_{k}^{(k-2)}&\displaystyle=\mathbf{u}_{k}^{(k-3)}-% \mathrm{proj}_{\mathbf{u}_{k-2}}\,(\mathbf{u}_{k}^{(k-3)}),\\ \displaystyle\mathbf{u}_{k}^{(k-1)}&\displaystyle=\mathbf{u}_{k}^{(k-2)}-% \mathrm{proj}_{\mathbf{u}_{k-1}}\,(\mathbf{u}_{k}^{(k-2)}).\end{aligned}
  24. 𝐮 k ( i ) \mathbf{u}_{k}^{(i)}
  25. 𝐮 k ( i - 1 ) \mathbf{u}_{k}^{(i-1)}
  26. 𝐮 k ( i ) \mathbf{u}_{k}^{(i)}
  27. 𝐮 k ( i - 1 ) \mathbf{u}_{k}^{(i-1)}
  28. 𝐯 i 𝐯 i 𝐯 i \mathbf{v}_{i}\leftarrow\frac{\mathbf{v}_{i}}{\|\mathbf{v}_{i}\|}
  29. 𝐯 j 𝐯 j - proj 𝐯 i ( 𝐯 j ) \mathbf{v}_{j}\leftarrow\mathbf{v}_{j}-\mathrm{proj}_{\mathbf{v}_{i}}\,(% \mathbf{v}_{j})
  30. 𝐞 j = 1 D j - 1 D j | 𝐯 1 , 𝐯 1 𝐯 2 , 𝐯 1 𝐯 j , 𝐯 1 𝐯 1 , 𝐯 2 𝐯 2 , 𝐯 2 𝐯 j , 𝐯 2 𝐯 1 , 𝐯 j - 1 𝐯 2 , 𝐯 j - 1 𝐯 j , 𝐯 j - 1 𝐯 1 𝐯 2 𝐯 j | \mathbf{e}_{j}=\frac{1}{\sqrt{D_{j-1}D_{j}}}\begin{vmatrix}\langle\mathbf{v}_{% 1},\mathbf{v}_{1}\rangle&\langle\mathbf{v}_{2},\mathbf{v}_{1}\rangle&\dots&% \langle\mathbf{v}_{j},\mathbf{v}_{1}\rangle\\ \langle\mathbf{v}_{1},\mathbf{v}_{2}\rangle&\langle\mathbf{v}_{2},\mathbf{v}_{% 2}\rangle&\dots&\langle\mathbf{v}_{j},\mathbf{v}_{2}\rangle\\ \vdots&\vdots&\ddots&\vdots\\ \langle\mathbf{v}_{1},\mathbf{v}_{j-1}\rangle&\langle\mathbf{v}_{2},\mathbf{v}% _{j-1}\rangle&\dots&\langle\mathbf{v}_{j},\mathbf{v}_{j-1}\rangle\\ \mathbf{v}_{1}&\mathbf{v}_{2}&\dots&\mathbf{v}_{j}\end{vmatrix}
  31. 𝐮 j = 1 D j - 1 | 𝐯 1 , 𝐯 1 𝐯 2 , 𝐯 1 𝐯 j , 𝐯 1 𝐯 1 , 𝐯 2 𝐯 2 , 𝐯 2 𝐯 j , 𝐯 2 𝐯 1 , 𝐯 j - 1 𝐯 2 , 𝐯 j - 1 𝐯 j , 𝐯 j - 1 𝐯 1 𝐯 2 𝐯 j | \mathbf{u}_{j}=\frac{1}{D_{j-1}}\begin{vmatrix}\langle\mathbf{v}_{1},\mathbf{v% }_{1}\rangle&\langle\mathbf{v}_{2},\mathbf{v}_{1}\rangle&\dots&\langle\mathbf{% v}_{j},\mathbf{v}_{1}\rangle\\ \langle\mathbf{v}_{1},\mathbf{v}_{2}\rangle&\langle\mathbf{v}_{2},\mathbf{v}_{% 2}\rangle&\dots&\langle\mathbf{v}_{j},\mathbf{v}_{2}\rangle\\ \vdots&\vdots&\ddots&\vdots\\ \langle\mathbf{v}_{1},\mathbf{v}_{j-1}\rangle&\langle\mathbf{v}_{2},\mathbf{v}% _{j-1}\rangle&\dots&\langle\mathbf{v}_{j},\mathbf{v}_{j-1}\rangle\\ \mathbf{v}_{1}&\mathbf{v}_{2}&\dots&\mathbf{v}_{j}\end{vmatrix}
  32. D j = | 𝐯 1 , 𝐯 1 𝐯 2 , 𝐯 1 𝐯 j , 𝐯 1 𝐯 1 , 𝐯 2 𝐯 2 , 𝐯 2 𝐯 j , 𝐯 2 𝐯 1 , 𝐯 j 𝐯 2 , 𝐯 j 𝐯 j , 𝐯 j | . D_{j}=\begin{vmatrix}\langle\mathbf{v}_{1},\mathbf{v}_{1}\rangle&\langle% \mathbf{v}_{2},\mathbf{v}_{1}\rangle&\dots&\langle\mathbf{v}_{j},\mathbf{v}_{1% }\rangle\\ \langle\mathbf{v}_{1},\mathbf{v}_{2}\rangle&\langle\mathbf{v}_{2},\mathbf{v}_{% 2}\rangle&\dots&\langle\mathbf{v}_{j},\mathbf{v}_{2}\rangle\\ \vdots&\vdots&\ddots&\vdots\\ \langle\mathbf{v}_{1},\mathbf{v}_{j}\rangle&\langle\mathbf{v}_{2},\mathbf{v}_{% j}\rangle&\dots&\langle\mathbf{v}_{j},\mathbf{v}_{j}\rangle\end{vmatrix}.
  33. j j
  34. j j
  35. 𝐕 \mathbf{V}
  36. 𝐕 * 𝐕 \mathbf{V}^{*}\mathbf{V}
  37. 𝐕 * 𝐕 = 𝐋𝐋 * , \mathbf{V}^{*}\mathbf{V}=\mathbf{L}\mathbf{L}^{*},
  38. 𝐋 \mathbf{L}
  39. 𝐔 = 𝐕 ( 𝐋 - 1 ) * \mathbf{U}=\mathbf{V}(\mathbf{L}^{-1})^{*}
  40. 𝐕 \mathbf{V}
  41. 𝐕 * 𝐕 \mathbf{V}^{*}\mathbf{V}

Grand_unification_energy.html

  1. Λ G U T \Lambda_{GUT}

Graph_isomorphism.html

  1. f : V ( G ) V ( H ) f\colon V(G)\to V(H)\,\!
  2. G H G\simeq H
  3. v V ( G ) v deg v \sum_{v\in V(G)}v\cdot\,\text{deg }v

Graph_of_a_function.html

  1. f ( x ) = { a , if x = 1 d , if x = 2 c , if x = 3. f(x)=\left\{\begin{matrix}a,&\mbox{if }~{}x=1\\ d,&\mbox{if }~{}x=2\\ c,&\mbox{if }~{}x=3.\end{matrix}\right.
  2. f ( x ) = x 3 - 9 x f(x)=x^{3}-9x
  3. x = x 1 , , x n x=x_{1},\ldots,x_{n}
  4. ( f , - 1 ) (\nabla f,-1)
  5. g ( x , z ) = f ( x ) - z g(x,z)=f(x)-z
  6. g \nabla g

Gravitational_binding_energy.html

  1. U = 3 G M 2 5 R U=\frac{3GM^{2}}{5R}
  2. R R
  3. ρ \rho
  4. m shell = 4 π r 2 ρ d r m_{\mathrm{shell}}=4\pi r^{2}\rho\,dr
  5. m interior = 4 3 π r 3 ρ m_{\mathrm{interior}}=\frac{4}{3}\pi r^{3}\rho
  6. 𝑑𝑈 = - G m shell m interior r {\it dU}=-G\frac{m_{\mathrm{shell}}m_{\mathrm{interior}}}{r}
  7. U = - G 0 R ( 4 π r 2 ρ ) ( 4 3 π r 3 ρ ) r d r = - G 16 3 π 2 ρ 2 0 R r 4 d r = - G 16 15 π 2 ρ 2 R 5 U=-G\int_{0}^{R}{\frac{(4\pi r^{2}\rho)(\tfrac{4}{3}\pi r^{3}\rho)}{r}}dr=-G{% \frac{16}{3}}\pi^{2}\rho^{2}\int_{0}^{R}{r^{4}}dr=-G{\frac{16}{15}}{\pi}^{2}{% \rho}^{2}R^{5}
  8. ρ \rho
  9. ρ = M 4 3 π R 3 \rho=\frac{M}{\frac{4}{3}\pi R^{3}}
  10. U = - G 16 15 π 2 R 5 ( M 4 3 π R 3 ) 2 = - 3 G M 2 5 R U=-G\frac{16}{15}\pi^{2}R^{5}\left(\frac{M}{\frac{4}{3}\pi R^{3}}\right)^{2}=-% \frac{3GM^{2}}{5R}
  11. B E = 0.60 β 1 - β 2 BE=\frac{0.60\,\beta}{1-\frac{\beta}{2}}
  12. β = G M / R c 2 \beta\ =G\,M/R\,{c}^{2}
  13. G = 6.6742 × 10 - 11 m 3 k g - 1 s e c - 2 G=6.6742\times 10^{-11}\,m^{3}kg^{-1}sec^{-2}
  14. c 2 = 8.98755 × 10 16 m 2 s e c - 2 c^{2}=8.98755\times 10^{16}\,m^{2}sec^{-2}
  15. M s o l a r = 1.98844 × 10 30 k g M_{solar}=1.98844\times 10^{30}\,kg
  16. M x = M M M_{x}=\frac{M}{M_{\odot}}
  17. B E = 885.975 M x R - 738.313 M x BE=\frac{885.975\,M_{x}}{R-738.313\,M_{x}}

Gravitational_field.html

  1. 𝐠 = 𝐅 m = - d 2 𝐑 d t 2 = - G M 𝐑 ^ | 𝐑 | 2 = - Φ , \mathbf{g}=\frac{\mathbf{F}}{m}=-\frac{{\rm d}^{2}\mathbf{R}}{{\rm d}t^{2}}=-% GM\frac{\mathbf{\hat{R}}}{|\mathbf{R}|^{2}}=-\nabla\Phi,
  2. 𝐑 ^ \mathbf{\hat{R}}
  3. - 𝐠 = 2 Φ = 4 π G ρ -\nabla\cdot\mathbf{g}=\nabla^{2}\Phi=4\pi G\rho\!
  4. 𝐠 j (net) = i j 𝐠 i = 1 m j i j 𝐅 i = - G i j m i 𝐑 ^ i j | 𝐑 i - 𝐑 j | 2 = - i j Φ i \mathbf{g}_{j}^{\,\text{(net)}}=\sum_{i\neq j}\mathbf{g}_{i}=\frac{1}{m_{j}}% \sum_{i\neq j}\mathbf{F}_{i}=-G\sum_{i\neq j}m_{i}\frac{\mathbf{\hat{R}}_{ij}}% {{|\mathbf{R}_{i}-\mathbf{R}_{j}}|^{2}}=-\sum_{i\neq j}\nabla\Phi_{i}
  5. 𝐑 ^ i j \mathbf{\hat{R}}_{ij}
  6. G = 8 π G c 4 T . {G}=\frac{8\pi G}{c^{4}}{T}.

Gravity_wave.html

  1. c \scriptstyle c
  2. k \scriptstyle k
  3. c = g k , c=\sqrt{\frac{g}{k}},
  4. c = g k + σ k ρ , c=\sqrt{\frac{g}{k}+\frac{\sigma k}{\rho}},
  5. ( u ( x , z , t ) , w ( x , z , t ) ) . \scriptstyle(u^{\prime}(x,z,t),w^{\prime}(x,z,t)).
  6. 𝐮 = ( u ( x , z , t ) , w ( x , z , t ) ) = ( ψ z , - ψ x ) , \,\textbf{u}^{\prime}=(u^{\prime}(x,z,t),w^{\prime}(x,z,t))=(\psi_{z},-\psi_{x% }),\,
  7. ( x , z ) \scriptstyle\left(x,z\right)
  8. × 𝐮 = 0. \scriptstyle\nabla\times\,\textbf{u}^{\prime}=0.\,
  9. 2 ψ = 0. \scriptstyle\nabla^{2}\psi=0.\,
  10. ψ ( x , z , t ) = e i k ( x - c t ) Ψ ( z ) , \psi\left(x,z,t\right)=e^{ik\left(x-ct\right)}\Psi\left(z\right),\,
  11. ( D 2 - k 2 ) Ψ = 0 , D = d d z . \left(D^{2}-k^{2}\right)\Psi=0,\,\,\,\ D=\frac{d}{dz}.
  12. z = - . \scriptstyle z=-\infty.
  13. z = 0 \scriptstyle z=0
  14. z = η , \scriptstyle z=\eta,
  15. η \scriptstyle\eta
  16. u = D Ψ = 0 , on z = - . u=D\Psi=0,\,\,\,\text{on}\,z=-\infty.
  17. Ψ = A e k z \scriptstyle\Psi=Ae^{kz}
  18. z ( - , η ) \scriptstyle z\in\left(-\infty,\eta\right)
  19. z = η ( x , t ) \scriptstyle z=\eta\left(x,t\right)\,
  20. η t + u η x = w ( η ) . \frac{\partial\eta}{\partial t}+u^{\prime}\frac{\partial\eta}{\partial x}=w^{% \prime}\left(\eta\right).\,
  21. η t = w ( 0 ) , \frac{\partial\eta}{\partial t}=w^{\prime}\left(0\right),\,
  22. w ( η ) \scriptstyle w^{\prime}\left(\eta\right)\,
  23. z = 0. \scriptstyle z=0.\,
  24. c η = Ψ \scriptstyle c\eta=\Psi\,
  25. z = η \scriptstyle z=\eta
  26. p ( z = η ) = - σ κ , p\left(z=\eta\right)=-\sigma\kappa,\,
  27. κ = 2 η = η x x . \kappa=\nabla^{2}\eta=\eta_{xx}.\,
  28. p ( z = η ) = - σ η x x . p\left(z=\eta\right)=-\sigma\eta_{xx}.\,
  29. [ P ( η ) + p ( 0 ) ] = - σ η x x . \left[P\left(\eta\right)+p^{\prime}\left(0\right)\right]=-\sigma\eta_{xx}.
  30. P = - ρ g z + Const. , \scriptstyle P=-\rho gz+\,\text{Const.},
  31. p = g η ρ - σ η x x , on z = 0. p=g\eta\rho-\sigma\eta_{xx},\qquad\,\text{on }z=0.\,
  32. u t = - 1 ρ p x \frac{\partial u^{\prime}}{\partial t}=-\frac{1}{\rho}\frac{\partial p^{\prime% }}{\partial x}\,
  33. p = ρ c D Ψ . \scriptstyle p^{\prime}=\rho cD\Psi.
  34. c ρ D Ψ = g η ρ - σ η x x . c\rho D\Psi=g\eta\rho-\sigma\eta_{xx}.\,
  35. c η = Ψ \scriptstyle c\eta=\Psi\,
  36. c 2 ρ D Ψ = g Ψ ρ + σ k 2 Ψ . \scriptstyle c^{2}\rho D\Psi=g\Psi\rho+\sigma k^{2}\Psi.
  37. Ψ = e k z \scriptstyle\Psi=e^{kz}
  38. c = g k + σ k ρ . c=\sqrt{\frac{g}{k}+\frac{\sigma k}{\rho}}.
  39. c = ω / k \scriptstyle c=\omega/k
  40. ω \scriptstyle\omega
  41. ω = g k . \omega=\sqrt{gk}.
  42. c g = d ω d k , c_{g}=\frac{d\omega}{dk},
  43. c g = 1 2 g k = 1 2 c . c_{g}=\frac{1}{2}\sqrt{\frac{g}{k}}=\frac{1}{2}c.
  44. c p = c g = g h . c_{p}=c_{g}=\sqrt{gh}.
  45. ( ω , k ) \scriptstyle\left(\omega,k\right)

Gray_(unit).html

  1. 1 Gy = 1 J kg = 1 m 2 s 2 1\ \mathrm{Gy}=1\ \frac{\mathrm{J}}{\mathrm{kg}}=1\ \frac{\mathrm{m}^{2}}{% \mathrm{s}^{2}}

Greek_numerals.html

  1. α ¯ \overline{α}
  2. β ¯ \overline{β}
  3. γ ¯ \overline{γ}
  4. χ ξ ϛ ¯ \overline{χξϛ}
  5. α ¯ \overline{α}
  6. ι ¯ \overline{ι}
  7. ρ ¯ \overline{ρ}
  8. β ¯ \overline{β}
  9. κ ¯ \overline{κ}
  10. σ ¯ \overline{σ}
  11. Γ ¯ \overline{Γ}
  12. λ ¯ \overline{λ}
  13. τ ¯ \overline{τ}
  14. Δ ¯ \overline{Δ}
  15. μ ¯ \overline{μ}
  16. υ ¯ \overline{υ}
  17. ε ¯ \overline{ε}
  18. ν ¯ \overline{ν}
  19. φ ¯ \overline{φ}
  20. ξ ¯ \overline{ξ}
  21. χ ¯ \overline{χ}
  22. ζ ¯ \overline{ζ}
  23. ο ¯ \overline{ο}
  24. ψ ¯ \overline{ψ}
  25. η ¯ \overline{η}
  26. π ¯ \overline{π}
  27. ω ¯ \overline{ω}
  28. θ ¯ \overline{θ}
  29. π ε ϱ ι φ ε ϱ ε ι ω ~ ν ε ν ϑ ε ι ω ~ ν ε ξ η κ \omicron σ τ ω ~ ν π δ π ε π ε π \stigma π \stigma π ζ π μ α γ π α δ ι ε π α κ ζ κ β π α ν κ δ π β ι γ ι ϑ π β λ \stigma ϑ μ \stigma κ ε μ \stigma ι δ μ \stigma γ μ ε ν β μ ε μ μ ε κ ϑ \begin{array}[]{ccc}\pi\varepsilon\varrho\iota\varphi\varepsilon\varrho% \varepsilon\iota\tilde{\omega}\nu&\varepsilon\overset{\,\text{'}}{\nu}% \vartheta\varepsilon\iota\tilde{\omega}\nu&\overset{\,\text{`}}{\varepsilon}% \xi\eta\kappa\omicron\sigma\tau\tilde{\omega}\nu\\ \begin{array}[]{|l|}\hline\pi\delta\angle^{\prime}\\ \pi\varepsilon\\ \pi\varepsilon\angle^{\prime}\\ \hline\pi\stigma\\ \pi\stigma\angle^{\prime}\\ \pi\zeta\\ \hline\end{array}&\begin{array}[]{|r|r|r|}\hline\pi&\mu\alpha&\gamma\\ \pi\alpha&\delta&\iota\varepsilon\\ \pi\alpha&\kappa\zeta&\kappa\beta\\ \hline\pi\alpha&\nu&\kappa\delta\\ \pi\beta&\iota\gamma&\iota\vartheta\\ \pi\beta&\lambda\stigma&\vartheta\\ \hline\end{array}&\begin{array}[]{|r|r|r|r|}\hline\circ&\circ&\mu\stigma&% \kappa\varepsilon\\ \circ&\circ&\mu\stigma&\iota\delta\\ \circ&\circ&\mu\stigma&\gamma\\ \hline\circ&\circ&\mu\varepsilon&\nu\beta\\ \circ&\circ&\mu\varepsilon&\mu\\ \circ&\circ&\mu\varepsilon&\kappa\vartheta\\ \hline\end{array}\end{array}

Greeks_(finance).html

  1. σ \sigma
  2. τ \tau
  3. Δ \Delta
  4. ν \nu
  5. Θ \Theta
  6. Δ \Delta
  7. Γ \Gamma
  8. ν \nu
  9. Γ \Gamma
  10. Δ = V S \Delta=\frac{\partial V}{\partial S}
  11. Δ \Delta
  12. V V
  13. S S
  14. ν = V σ \nu=\frac{\partial V}{\partial\sigma}
  15. ν \nu
  16. κ \kappa
  17. τ \tau
  18. Λ \Lambda
  19. Θ = - V τ \Theta=-\frac{\partial V}{\partial\tau}
  20. Θ \Theta
  21. ρ = V r \rho=\frac{\partial V}{\partial r}
  22. ρ \rho
  23. λ = V S × S V \lambda=\frac{\partial V}{\partial S}\times\frac{S}{V}
  24. λ \lambda
  25. Ω \Omega
  26. Γ = Δ S = 2 V S 2 \Gamma=\frac{\partial\Delta}{\partial S}=\frac{\partial^{2}V}{\partial S^{2}}
  27. Γ \Gamma
  28. Vanna : Δ σ \,\text{Vanna}:\frac{\partial\Delta}{\partial\sigma}
  29. ν S \frac{\partial\nu}{\partial S}
  30. 2 V S σ \frac{\partial^{2}V}{\partial S\partial\sigma}
  31. Vanna = Δ σ = ν S = 2 V S σ \,\text{Vanna}=\frac{\partial\Delta}{\partial\sigma}=\frac{\partial\nu}{% \partial S}=\frac{\partial^{2}V}{\partial S\partial\sigma}
  32. Vomma = ν σ = 2 V σ 2 \,\text{Vomma}=\frac{\partial\nu}{\partial\sigma}=\frac{\partial^{2}V}{% \partial\sigma^{2}}
  33. Charm = - Δ τ = Θ S = - 2 V S τ \,\text{Charm}=-\frac{\partial\Delta}{\partial\tau}=\frac{\partial\Theta}{% \partial S}=-\frac{\partial^{2}V}{\partial S\,\partial\tau}
  34. ν τ = 2 V σ τ \frac{\partial\nu}{\partial\tau}=\frac{\partial^{2}V}{\partial\sigma\,\partial\tau}
  35. ρ σ = 2 V σ r \frac{\partial\rho}{\partial\sigma}=\frac{\partial^{2}V}{\partial\sigma\,% \partial r}
  36. Color = Γ τ = 3 V S 2 τ \,\text{Color}=\frac{\partial\Gamma}{\partial\tau}=\frac{\partial^{3}V}{% \partial S^{2}\,\partial\tau}
  37. Speed = Γ S = 3 V S 3 \,\text{Speed}=\frac{\partial\Gamma}{\partial S}=\frac{\partial^{3}V}{\partial S% ^{3}}
  38. Ultima = vomma σ = 3 V σ 3 \,\text{Ultima}=\frac{\partial\,\text{vomma}}{\partial\sigma}=\frac{\partial^{% 3}V}{\partial\sigma^{3}}
  39. Zomma = Γ σ = vanna S = 3 V S 2 σ \,\text{Zomma}=\frac{\partial\Gamma}{\partial\sigma}=\frac{\partial\,\text{% vanna}}{\partial S}=\frac{\partial^{3}V}{\partial S^{2}\,\partial\sigma}
  40. ϕ \phi
  41. Φ \Phi
  42. S S\,
  43. K K\,
  44. r r\,
  45. q q\,
  46. τ = T - t \tau=T-t\,
  47. σ \sigma\,
  48. S e - q τ Φ ( d 1 ) - e - r τ K Φ ( d 2 ) Se^{-q\tau}\Phi(d_{1})-e^{-r\tau}K\Phi(d_{2})\,
  49. e - r τ K Φ ( - d 2 ) - S e - q τ Φ ( - d 1 ) e^{-r\tau}K\Phi(-d_{2})-Se^{-q\tau}\Phi(-d_{1})\,
  50. e - q τ Φ ( d 1 ) e^{-q\tau}\Phi(d_{1})\,
  51. - e - q τ Φ ( - d 1 ) -e^{-q\tau}\Phi(-d_{1})\,
  52. S e - q τ ϕ ( d 1 ) τ = K e - r τ ϕ ( d 2 ) τ Se^{-q\tau}\phi(d_{1})\sqrt{\tau}=Ke^{-r\tau}\phi(d_{2})\sqrt{\tau}\,
  53. - e - q τ S ϕ ( d 1 ) σ 2 τ - r K e - r τ Φ ( d 2 ) + q S e - q τ Φ ( d 1 ) -e^{-q\tau}\frac{S\phi(d_{1})\sigma}{2\sqrt{\tau}}-rKe^{-r\tau}\Phi(d_{2})+qSe% ^{-q\tau}\Phi(d_{1})\,
  54. - e - q τ S ϕ ( d 1 ) σ 2 τ + r K e - r τ Φ ( - d 2 ) - q S e - q τ Φ ( - d 1 ) -e^{-q\tau}\frac{S\phi(d_{1})\sigma}{2\sqrt{\tau}}+rKe^{-r\tau}\Phi(-d_{2})-% qSe^{-q\tau}\Phi(-d_{1})\,
  55. K τ e - r τ Φ ( d 2 ) K\tau e^{-r\tau}\Phi(d_{2})\,
  56. - K τ e - r τ Φ ( - d 2 ) -K\tau e^{-r\tau}\Phi(-d_{2})\,
  57. e - q τ ϕ ( d 1 ) S σ τ e^{-q\tau}\frac{\phi(d_{1})}{S\sigma\sqrt{\tau}}\,
  58. - e - q τ ϕ ( d 1 ) d 2 σ = ν S [ 1 - d 1 σ τ ] -e^{-q\tau}\phi(d_{1})\frac{d_{2}}{\sigma}\,=\frac{\nu}{S}\left[1-\frac{d_{1}}% {\sigma\sqrt{\tau}}\right]\,
  59. q e - q τ Φ ( d 1 ) - e - q τ ϕ ( d 1 ) 2 ( r - q ) τ - d 2 σ τ 2 τ σ τ qe^{-q\tau}\Phi(d_{1})-e^{-q\tau}\phi(d_{1})\frac{2(r-q)\tau-d_{2}\sigma\sqrt{% \tau}}{2\tau\sigma\sqrt{\tau}}\,
  60. - q e - q τ Φ ( - d 1 ) - e - q τ ϕ ( d 1 ) 2 ( r - q ) τ - d 2 σ τ 2 τ σ τ -qe^{-q\tau}\Phi(-d_{1})-e^{-q\tau}\phi(d_{1})\frac{2(r-q)\tau-d_{2}\sigma% \sqrt{\tau}}{2\tau\sigma\sqrt{\tau}}\,
  61. - e - q τ ϕ ( d 1 ) S 2 σ τ ( d 1 σ τ + 1 ) = - Γ S ( d 1 σ τ + 1 ) -e^{-q\tau}\frac{\phi(d_{1})}{S^{2}\sigma\sqrt{\tau}}\left(\frac{d_{1}}{\sigma% \sqrt{\tau}}+1\right)=-\frac{\Gamma}{S}\left(\frac{d_{1}}{\sigma\sqrt{\tau}}+1% \right)\,
  62. e - q τ ϕ ( d 1 ) ( d 1 d 2 - 1 ) S σ 2 τ = Γ ( d 1 d 2 - 1 σ ) e^{-q\tau}\frac{\phi(d_{1})\left(d_{1}d_{2}-1\right)}{S\sigma^{2}\sqrt{\tau}}=% \Gamma\cdot\left(\frac{d_{1}d_{2}-1}{\sigma}\right)\,
  63. - e - q τ ϕ ( d 1 ) 2 S τ σ τ [ 2 q τ + 1 + 2 ( r - q ) τ - d 2 σ τ σ τ d 1 ] -e^{-q\tau}\frac{\phi(d_{1})}{2S\tau\sigma\sqrt{\tau}}\left[2q\tau+1+\frac{2(r% -q)\tau-d_{2}\sigma\sqrt{\tau}}{\sigma\sqrt{\tau}}d_{1}\right]\,
  64. S e - q τ ϕ ( d 1 ) τ [ q + ( r - q ) d 1 σ τ - 1 + d 1 d 2 2 τ ] Se^{-q\tau}\phi(d_{1})\sqrt{\tau}\left[q+\frac{\left(r-q\right)d_{1}}{\sigma% \sqrt{\tau}}-\frac{1+d_{1}d_{2}}{2\tau}\right]\,
  65. S e - q τ ϕ ( d 1 ) τ d 1 d 2 σ = ν d 1 d 2 σ Se^{-q\tau}\phi(d_{1})\sqrt{\tau}\frac{d_{1}d_{2}}{\sigma}=\nu\frac{d_{1}d_{2}% }{\sigma}\,
  66. - ν σ 2 [ d 1 d 2 ( 1 - d 1 d 2 ) + d 1 2 + d 2 2 ] \frac{-\nu}{\sigma^{2}}\left[d_{1}d_{2}(1-d_{1}d_{2})+d_{1}^{2}+d_{2}^{2}\right]
  67. - e - r τ Φ ( d 2 ) -e^{-r\tau}\Phi(d_{2})\,
  68. e - r τ Φ ( - d 2 ) e^{-r\tau}\Phi(-d_{2})\,
  69. e - r τ ϕ ( d 2 ) K σ τ e^{-r\tau}\frac{\phi(d_{2})}{K\sigma\sqrt{\tau}}\,
  70. d 1 = ln ( S / K ) + ( r - q + σ 2 / 2 ) τ σ τ d_{1}=\frac{\ln(S/K)+(r-q+\sigma^{2}/2)\tau}{\sigma\sqrt{\tau}}
  71. d 2 = ln ( S / K ) + ( r - q - σ 2 / 2 ) τ σ τ = d 1 - σ τ d_{2}=\frac{\ln(S/K)+(r-q-\sigma^{2}/2)\tau}{\sigma\sqrt{\tau}}=d_{1}-\sigma% \sqrt{\tau}
  72. ϕ ( x ) = e - x 2 2 2 π \phi(x)=\frac{e^{-\frac{x^{2}}{2}}}{\sqrt{2\pi}}
  73. Φ ( x ) = 1 2 π - x e - y 2 2 d y = 1 - 1 2 π x e - y 2 2 d y \Phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-\frac{y^{2}}{2}}\,dy=1-% \frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}e^{-\frac{y^{2}}{2}}\,dy

Green's_function.html

  1. δ \delta
  2. L G ( x , s ) = - δ ( x - s ) . LG(x,s)=-\delta(x-s).
  3. G ( x , s ) = G ( x - s ) . G(x,s)=G(x-s).
  4. L G ( x , s ) f ( s ) d s = δ ( x - s ) f ( s ) d s = f ( x ) . \int LG(x,s)f(s)\,ds=\int\delta(x-s)f(s)\,ds=f(x).
  5. L u ( x ) = L G ( x , s ) f ( s ) d s . Lu(x)=\int LG(x,s)f(s)\,ds.
  6. L u ( x ) = L ( G ( x , s ) f ( s ) d s ) , Lu(x)=L\left(\int G(x,s)f(s)\,ds\right),
  7. L = d d x [ p ( x ) d d x ] + q ( x ) L=\dfrac{d}{dx}\left[p(x)\dfrac{d}{dx}\right]+q(x)
  8. D u = { α 1 u ( 0 ) + β 1 u ( 0 ) α 2 u ( l ) + β 2 u ( l ) . Du=\begin{cases}\alpha_{1}u^{\prime}(0)+\beta_{1}u(0)\\ \alpha_{2}u^{\prime}(l)+\beta_{2}u(l).\end{cases}
  9. L u = f D u = 0 \begin{aligned}\displaystyle Lu&\displaystyle=f\\ \displaystyle Du&\displaystyle=0\end{aligned}
  10. L u = f D u = 0 , \begin{aligned}\displaystyle Lu&\displaystyle=f\\ \displaystyle Du&\displaystyle=0,\end{aligned}
  11. u ( x ) = 0 f ( s ) G ( x , s ) d s , u(x)=\int_{0}^{\ell}f(s)G(x,s)\,ds,
  12. G ( x , s ) G(x,s)
  13. x x
  14. s s
  15. x s x\neq s
  16. L G ( x , s ) = 0 LG(x,s)=0
  17. s 0 s\neq 0
  18. D G ( x , s ) = 0 DG(x,s)=0
  19. G ( s + 0 , s ) - G ( s - 0 , s ) = 1 / p ( s ) G^{\prime}(s_{+0},s)-G^{\prime}(s_{-0},s)=1/p(s)
  20. G ( x , s ) = G ( s , x ) G(x,s)=G(s,x)
  21. Ψ n ( x ) \Psi_{n}(x)
  22. Ψ n \Psi_{n}
  23. λ n \lambda_{n}
  24. L Ψ n = λ n Ψ n L\Psi_{n}=\lambda_{n}\Psi_{n}
  25. { Ψ n } \left\{\Psi_{n}\right\}
  26. δ ( x - x ) = n = 0 Ψ n ( x ) Ψ n ( x ) . \delta(x-x^{\prime})=\sum_{n=0}^{\infty}\Psi_{n}^{\dagger}(x)\Psi_{n}(x^{% \prime}).
  27. G ( x , x ) = n = 0 Ψ n ( x ) Ψ n ( x ) λ n , G(x,x^{\prime})=\sum_{n=0}^{\infty}\dfrac{\Psi_{n}^{\dagger}(x)\Psi_{n}(x^{% \prime})}{\lambda_{n}},
  28. \dagger
  29. θ ( t ) \theta(t)
  30. r = x 2 + y 2 + z 2 r=\sqrt{x^{2}+y^{2}+z^{2}}
  31. ρ = x 2 + y 2 \rho=\sqrt{x^{2}+y^{2}}
  32. L L
  33. G G
  34. t + γ \partial_{t}+\gamma
  35. θ ( t ) e - γ t \theta(t)\mathrm{e}^{-\gamma t}
  36. ( t + γ ) 2 \left(\partial_{t}+\gamma\right)^{2}
  37. θ ( t ) t e - γ t \theta(t)t\mathrm{e}^{-\gamma t}
  38. t 2 + 2 γ t + ω 0 2 \partial_{t}^{2}+2\gamma\partial_{t}+\omega_{0}^{2}
  39. θ ( t ) e - γ t 1 ω sin ( ω t ) \theta(t)\mathrm{e}^{-\gamma t}\frac{1}{\omega}\sin(\omega t)
  40. ω = ω 0 2 - γ 2 \omega=\sqrt{\omega_{0}^{2}-\gamma^{2}}
  41. Δ 2D \Delta\text{2D}
  42. = x 2 + y 2 =\partial_{x}^{2}+\partial_{y}^{2}
  43. 1 2 π ln ρ \frac{1}{2\pi}\ln\rho
  44. 2 \nabla^{2}
  45. = x 2 + y 2 + z 2 = Δ =\partial_{x}^{2}+\partial_{y}^{2}+\partial_{z}^{2}=\Delta
  46. - 1 4 π r \frac{-1}{4\pi r}
  47. Δ + k 2 \Delta+k^{2}
  48. - e - i k r 4 π r \frac{-\mathrm{e}^{-ikr}}{4\pi r}
  49. = 1 c 2 t 2 - Δ \square=\frac{1}{c^{2}}\partial_{t}^{2}-\Delta
  50. δ ( t - r c ) 4 π r \frac{\delta(t-\frac{r}{c})}{4\pi r}
  51. t - k Δ \partial_{t}-k\Delta
  52. θ ( t ) ( 1 4 π k t ) 3 / 2 e - r 2 / 4 k t \theta(t)\left(\frac{1}{4\pi kt}\right)^{3/2}\mathrm{e}^{-r^{2}/4kt}
  53. V A d V = S A d σ ^ . \int_{V}\nabla\cdot\vec{A}\ dV=\int_{S}\vec{A}\cdot d\hat{\sigma}.
  54. A = ϕ ψ - ψ ϕ \vec{A}=\phi\nabla\psi-\psi\nabla\phi
  55. A \nabla\cdot\vec{A}
  56. \nabla
  57. A \displaystyle\nabla\cdot\vec{A}
  58. V ( ϕ 2 ψ - ψ 2 ϕ ) d V = S ( ϕ ψ - ψ ϕ ) d σ ^ . \int_{V}(\phi\nabla^{2}\psi-\psi\nabla^{2}\phi)dV=\int_{S}(\phi\nabla\psi-\psi% \nabla\phi)\cdot d\hat{\sigma}.
  59. 2 \nabla^{2}
  60. L G ( x , x ) = 2 G ( x , x ) = δ ( x - x ) . LG(x,x^{\prime})=\nabla^{2}G(x,x^{\prime})=\delta(x-x^{\prime}).
  61. ψ = G \psi=G
  62. V [ ϕ ( x ) δ ( x - x ) - G ( x , x ) 2 ϕ ( x ) ] d 3 x = S [ ϕ ( x ) G ( x , x ) - G ( x , x ) ϕ ( x ) ] d σ ^ . \int_{V}\left[\phi(x^{\prime})\delta(x-x^{\prime})-G(x,x^{\prime}){\nabla^{% \prime}}^{2}\phi(x^{\prime})\right]\ d^{3}x^{\prime}=\int_{S}\left[\phi(x^{% \prime})\nabla^{\prime}G(x,x^{\prime})-G(x,x^{\prime})\nabla^{\prime}\phi(x^{% \prime})\right]\cdot d\hat{\sigma}^{\prime}.
  63. 2 ϕ ( x ) = 0 \nabla^{2}\phi(x)=0
  64. 2 ϕ ( x ) = - ρ ( x ) \nabla^{2}\phi(x)=-\rho(x)
  65. ϕ ( x ) \phi(x)
  66. ϕ ( x ) \phi(x)
  67. ϕ ( x ) \phi(x)
  68. ϕ ( x ) \phi(x)
  69. V ϕ ( x ) δ ( x - x ) d 3 x \int\limits_{V}{\phi(x^{\prime})\delta(x-x^{\prime})\ d^{3}x^{\prime}}
  70. ϕ ( x ) \phi(x)
  71. ϕ ( x ) = V G ( x , x ) ρ ( x ) d 3 x + S [ ϕ ( x ) G ( x , x ) - G ( x , x ) ϕ ( x ) ] d σ ^ . \phi(x)=\int_{V}G(x,x^{\prime})\rho(x^{\prime})\ d^{3}x^{\prime}+\int_{S}\left% [\phi(x^{\prime})\nabla^{\prime}G(x,x^{\prime})-G(x,x^{\prime})\nabla^{\prime}% \phi(x^{\prime})\right]\cdot d\hat{\sigma}^{\prime}.
  72. ϕ ( x ) \phi(x)
  73. ρ ( x ) \rho(x)
  74. ϕ ( x ) d σ ^ \nabla\phi(x^{\prime})\cdot d\hat{\sigma}^{\prime}
  75. G ( x , x ) G(x,x^{\prime})
  76. S G ( x , x ) d σ ^ = V 2 G ( x , x ) d 3 x = V δ ( x - x ) d 3 x = 1 \int_{S}\nabla^{\prime}G(x,x^{\prime})\cdot d\hat{\sigma}^{\prime}=\int_{V}% \nabla^{\prime 2}G(x,x^{\prime})d^{3}x^{\prime}=\int_{V}\delta(x-x^{\prime})d^% {3}x^{\prime}=1
  77. G ( x , x ) G(x,x^{\prime})
  78. 1 / S 1/S
  79. S ϕ ( x ) G ( x , x ) d σ ^ = ϕ S \int_{S}\phi(x^{\prime})\nabla^{\prime}G(x,x^{\prime})\cdot d\hat{\sigma}^{% \prime}=\langle\phi\rangle_{S}
  80. ϕ S \langle\phi\rangle_{S}
  81. G ( x , x ) = 1 | x - x | . G(x,x^{\prime})=\dfrac{1}{|x-x^{\prime}|}.
  82. ϕ ( x ) = V ρ ( x ) | x - x | d 3 x . \phi(x)=\int_{V}\dfrac{\rho(x^{\prime})}{|x-x^{\prime}|}\,d^{3}x^{\prime}.
  83. L u \displaystyle Lu
  84. g ′′ ( x , s ) + k 2 g ( x , s ) = δ ( x - s ) . g^{\prime\prime}(x,s)+k^{2}g(x,s)=\delta(x-s).
  85. x s x\neq s
  86. g ( x , s ) = c 1 cos k x + c 2 sin k x . g(x,s)=c_{1}\cos kx+c_{2}\sin kx.
  87. x < s x<s
  88. x = 0 x=0
  89. g ( 0 , s ) = c 1 1 + c 2 0 = 0 , c 1 = 0 g(0,s)=c_{1}\cdot 1+c_{2}\cdot 0=0,\quad c_{1}=0
  90. x < s x<s
  91. s π 2 k s\neq\tfrac{\pi}{2k}
  92. x > s x>s
  93. x = π 2 k x=\tfrac{\pi}{2k}
  94. g ( π 2 k , s ) = c 3 0 + c 4 1 = 0 , c 4 = 0 g\left(\tfrac{\pi}{2k},s\right)=c_{3}\cdot 0+c_{4}\cdot 1=0,\quad c_{4}=0
  95. g ( 0 , s ) = 0 g(0,s)=0
  96. g ( x , s ) = { c 2 sin k x , for x < s c 3 cos k x , for s < x g(x,s)=\begin{cases}c_{2}\sin kx,&\,\text{for }x<s\\ c_{3}\cos kx,&\,\text{for }s<x\end{cases}
  97. c 2 c_{2}
  98. c 3 c_{3}
  99. x = s x=s
  100. c 2 sin k s = c 3 cos k s c_{2}\sin ks=c_{3}\cos ks
  101. x = s - ϵ x=s-\epsilon
  102. x = s + ϵ x=s+\epsilon
  103. ϵ \epsilon
  104. c 3 ( - k sin k s ) - c 2 ( k cos k s ) = 1 c_{3}\cdot\left(-k\sin ks\right)-c_{2}\cdot\left(k\cos ks\right)=1
  105. c 2 c_{2}
  106. c 3 c_{3}
  107. c 2 = - cos k s k ; c 3 = - sin k s k c_{2}=-\frac{\cos ks}{k}\quad;\quad c_{3}=-\frac{\sin ks}{k}
  108. g ( x , s ) = { - cos k s k sin k x , x < s - sin k s k cos k x , s < x g(x,s)=\begin{cases}-\frac{\cos ks}{k}\sin kx,&x<s\\ -\frac{\sin ks}{k}\cos kx,&s<x\end{cases}
  109. G ( x , y , x 0 , y 0 ) = 1 2 π \displaystyle G(x,y,x_{0},y_{0})=\dfrac{1}{2\pi}