wpmath0000001_8

Equivalent_noise_resistance.html

  1. R n = π W n k T 0 R_{n}=\frac{\pi W_{n}}{kT_{0}}
  2. W n W_{n}
  3. k k
  4. T 0 T_{0}
  5. k T 0 = 4.00 × 10 - 21 [ W s ] kT_{0}=4.00\times 10^{-21}\,[Ws]
  6. R n = e 2 4 k T 0 Δ f . R_{n}=\frac{e^{2}}{4kT_{0}\,\Delta f}.

Erlang_(unit).html

  1. E = λ h E=\lambda h
  2. P b = B ( E , m ) = E m m ! i = 0 m E i i ! P_{b}=B(E,m)=\frac{\frac{E^{m}}{m!}}{\sum_{i=0}^{m}\frac{E^{i}}{i!}}
  3. P b P_{b}
  4. B ( E , 0 ) = 1. B(E,0)=1.\,
  5. B ( E , j ) = E B ( E , j - 1 ) E B ( E , j - 1 ) + j j = 1 , 2 , , m . B(E,j)=\frac{EB(E,j-1)}{EB(E,j-1)+j}\ \forall{j}=1,2,\ldots,m.
  6. 1 B ( E , 0 ) = 1 \frac{1}{B(E,0)}=1
  7. 1 B ( E , j ) = 1 + j E 1 B ( E , j - 1 ) j = 1 , 2 , , m . \frac{1}{B(E,j)}=1+\frac{j}{E}\frac{1}{B(E,j-1)}\ \forall{j}=1,2,\ldots,m.
  8. P b = B ( E , m ) P_{b}=B(E,m)\,
  9. B e = E P b B_{e}=EP_{b}\,
  10. R R
  11. R f R_{f}
  12. R = B e R f R=B_{e}R_{f}\,
  13. E i + 1 = E 0 + R E_{i+1}=E_{0}+R\,
  14. E 0 E_{0}
  15. E E
  16. P W = A N N ! N N - A ( i = 0 N - 1 A i i ! ) + A N N ! N N - A P_{W}={{\frac{A^{N}}{N!}\frac{N}{N-A}}\over\left(\sum\limits_{i=0}^{N-1}\frac{% A^{i}}{i!}\right)+\frac{A^{N}}{N!}\frac{N}{N-A}}\,
  17. P b ( N , A , S ) = A N ( S N ) i = 0 N A i ( S i ) P_{b}(N,A,S)=\frac{A^{N}{\left(\begin{array}[]{c}S\\ N\end{array}\right)}}{\sum_{i=0}^{N}A^{i}{\left(\begin{array}[]{c}S\\ i\end{array}\right)}}
  18. P ( b ) = [ ( S - 1 ) ! N ! ( S - 1 - N ) ! ] M N X = 1 N [ ( S - 1 ) ! X ! ( S - 1 - X ) ! ] M X P(b)=\frac{\left[\frac{\left(S-1\right)!}{N!\cdot\left(S-1-N\right)!}\right]% \cdot M^{N}}{\sum_{X=1}^{N}\left[\frac{\left(S-1\right)!}{X!\cdot\left(S-1-X% \right)!}\right]\cdot M^{X}}
  19. M = A S - A ( 1 - P ( b ) ) . M=\frac{A}{S-A\cdot\left(1-P(b)\right)}.

Escape_velocity.html

  1. v e = 2 G M r , v_{e}=\sqrt{\frac{2GM}{r}},
  2. v e v_{e}
  3. ( K + U g ) i = ( K + U g ) f (K+U_{g})_{i}=(K+U_{g})_{f}\,
  4. 1 2 m v e 2 + - G M m r = 0 + 0 \frac{1}{2}mv_{e}^{2}+\frac{-GMm}{r}=0+0
  5. v e = 2 G M r v_{e}=\sqrt{\frac{2GM}{r}}
  6. m m
  7. v e = 2 G M r = 2 μ r = 2 g r v_{e}=\sqrt{\frac{2GM}{r}}=\sqrt{\frac{2\mu}{r}}=\sqrt{2gr\,}
  8. v e v_{e}
  9. 2 \sqrt{2}
  10. v e v_{e}
  11. v e 2.364 × 10 - 5 r ρ . v_{e}\approx 2.364\times 10^{-5}r\sqrt{\rho}.\,
  12. F = G M m r 2 . F=G\frac{Mm}{r^{2}}.
  13. d W = G M m r 2 d r . dW=G\frac{Mm}{r^{2}}\,dr.
  14. W = r 0 G M m r 2 d r = G M m r 0 . W=\int_{r_{0}}^{\infty}G\frac{Mm}{r^{2}}\,dr=G\frac{Mm}{r_{0}}.
  15. 1 2 m v 0 2 = G M m r 0 , \tfrac{1}{2}mv_{0}^{2}=G\frac{Mm}{r_{0}},
  16. v 0 = 2 G M r 0 . v_{0}=\sqrt{\frac{2GM}{r_{0}}}.
  17. 11.2 2 + 42.1 2 km / s = 43.56 km / s \scriptstyle\sqrt{11.2^{2}\ +\ 42.1^{2}}\ \mathrm{km}/\mathrm{s}=\ 43.56\ % \mathrm{km}/\mathrm{s}

Estimator.html

  1. θ ^ \scriptstyle\hat{\theta}
  2. θ \theta
  3. θ \theta
  4. θ ^ \widehat{\theta}
  5. θ ^ ( X ) \widehat{\theta}(X)
  6. θ ^ ( x ) \widehat{\theta}(x)
  7. θ ^ \widehat{\theta}
  8. x x
  9. θ ^ \widehat{\theta}
  10. e ( x ) = θ ^ ( x ) - θ , e(x)=\widehat{\theta}(x)-\theta,
  11. θ \theta
  12. θ ^ \widehat{\theta}
  13. MSE ( θ ^ ) = E [ ( θ ^ ( X ) - θ ) 2 ] . \operatorname{MSE}(\widehat{\theta})=\operatorname{E}[(\widehat{\theta}(X)-% \theta)^{2}].
  14. x x
  15. θ ^ \widehat{\theta}
  16. d ( x ) = θ ^ ( x ) - E ( θ ^ ( X ) ) = θ ^ ( x ) - E ( θ ^ ) , d(x)=\widehat{\theta}(x)-\operatorname{E}(\widehat{\theta}(X))=\widehat{\theta% }(x)-\operatorname{E}(\widehat{\theta}),
  17. E ( θ ^ ( X ) ) \operatorname{E}(\widehat{\theta}(X))
  18. θ ^ \widehat{\theta}
  19. var ( θ ^ ) = E [ ( θ ^ - E ( θ ^ ) ) 2 ] \operatorname{var}(\widehat{\theta})=\operatorname{E}[(\widehat{\theta}-% \operatorname{E}(\widehat{\theta}))^{2}]
  20. θ ^ \widehat{\theta}
  21. B ( θ ^ ) = E ( θ ^ ) - θ B(\widehat{\theta})=\operatorname{E}(\widehat{\theta})-\theta
  22. E ( θ ^ ) - θ = E ( θ ^ - θ ) \operatorname{E}(\widehat{\theta})-\theta=\operatorname{E}(\widehat{\theta}-\theta)
  23. θ ^ \widehat{\theta}
  24. θ \theta
  25. B ( θ ^ ) = 0 B(\widehat{\theta})=0
  26. MSE ( θ ^ ) = var ( θ ^ ) + ( B ( θ ^ ) ) 2 , \operatorname{MSE}(\widehat{\theta})=\operatorname{var}(\widehat{\theta})+(B(% \widehat{\theta}))^{2},
  27. lim n Pr { | t n - θ | < ϵ } = 1. \lim_{n\to\infty}\Pr\left\{\left|t_{n}-\theta\right|<\epsilon\right\}=1.
  28. 1 / n 1/\sqrt{n}
  29. 𝐷 \xrightarrow{D}
  30. n ( t n - θ ) 𝐷 N ( 0 , V ) , \sqrt{n}(t_{n}-\theta)\xrightarrow{D}N(0,V),
  31. θ \theta
  32. X ¯ \bar{X}

Euclidean_algorithm.html

  1. g c d ( a , b , c ) = g c d ( a , g c d ( b , c ) ) = g c d ( g c d ( a , b ) , c ) = g c d ( g c d ( a , c ) , b ) . gcd(a,b,c)=gcd(a, gcd(b,c))=gcd(gcd(a,b),c)=gcd(gcd(a,c),b).
  2. 1071 = 2 × 462 + 147. 1071=2×462+147.
  3. 462 = 3 × 147 + 21. 462=3×147+21.
  4. 147 = 7 × 21 + 0. 147=7×21+0.
  5. 1071 = q < s u b > 0462 + r 0 1071=q<sub>0462+r_{0}
  6. | r k + 1 r k | < 1 φ 0.618 , \left|\frac{r_{k+1}}{r_{k}}\right|<\frac{1}{\varphi}\sim 0.618,
  7. φ \varphi
  8. m g = m s a + m t b . mg=msa+mtb.
  9. ( a b ) = ( q 0 1 1 0 ) ( b r 0 ) = ( q 0 1 1 0 ) ( q 1 1 1 0 ) ( r 0 r 1 ) = = i = 0 N ( q i 1 1 0 ) ( r N - 1 0 ) . \begin{pmatrix}a\\ b\end{pmatrix}=\begin{pmatrix}q_{0}&1\\ 1&0\end{pmatrix}\begin{pmatrix}b\\ r_{0}\end{pmatrix}=\begin{pmatrix}q_{0}&1\\ 1&0\end{pmatrix}\begin{pmatrix}q_{1}&1\\ 1&0\end{pmatrix}\begin{pmatrix}r_{0}\\ r_{1}\end{pmatrix}=\cdots=\prod_{i=0}^{N}\begin{pmatrix}q_{i}&1\\ 1&0\end{pmatrix}\begin{pmatrix}r_{N-1}\\ 0\end{pmatrix}\,.
  10. 𝐌 = ( m 11 m 12 m 21 m 22 ) = i = 0 N ( q i 1 1 0 ) = ( q 0 1 1 0 ) ( q 1 1 1 0 ) ( q N 1 1 0 ) . \mathbf{M}=\begin{pmatrix}m_{11}&m_{12}\\ m_{21}&m_{22}\end{pmatrix}=\prod_{i=0}^{N}\begin{pmatrix}q_{i}&1\\ 1&0\end{pmatrix}=\begin{pmatrix}q_{0}&1\\ 1&0\end{pmatrix}\begin{pmatrix}q_{1}&1\\ 1&0\end{pmatrix}\cdots\begin{pmatrix}q_{N}&1\\ 1&0\end{pmatrix}\,.
  11. ( a b ) = 𝐌 ( r N - 1 0 ) = 𝐌 ( g 0 ) . \begin{pmatrix}a\\ b\end{pmatrix}=\mathbf{M}\begin{pmatrix}r_{N-1}\\ 0\end{pmatrix}=\mathbf{M}\begin{pmatrix}g\\ 0\end{pmatrix}\,.
  12. ( g 0 ) = 𝐌 - 1 ( a b ) = ( - 1 ) N + 1 ( m 22 - m 12 - m 21 m 11 ) ( a b ) . \begin{pmatrix}g\\ 0\end{pmatrix}=\mathbf{M}^{-1}\begin{pmatrix}a\\ b\end{pmatrix}=(-1)^{N+1}\begin{pmatrix}m_{22}&-m_{12}\\ -m_{21}&m_{11}\end{pmatrix}\begin{pmatrix}a\\ b\end{pmatrix}\,.
  13. 1 = s u + t w . 1=su+tw.
  14. v = s u v + t w v = s L + t w v . v=suv+twv=sL+twv.
  15. a x + b y = c ax+by=c
  16. a x c m o d b . ax≡cmodb.
  17. s a + t b = g sa+tb=g
  18. a x + m y = 1. ax+my=1.
  19. x 1 \displaystyle x_{1}
  20. M i = M m i . M_{i}=\frac{M}{m_{i}}.
  21. M i h i 1 mod m i . M_{i}h_{i}\equiv 1\bmod m_{i}\,.
  22. x ( x 1 M 1 h 1 + x 2 M 2 h 2 + + x N M N h N ) mod M . x\equiv(x_{1}M_{1}h_{1}+x_{2}M_{2}h_{2}+\cdots+x_{N}M_{N}h_{N})\bmod M\,.
  23. gcd ( 3 , 4 ) = gcd ( 3 , 1 ) = gcd ( 2 , 1 ) = gcd ( 1 , 1 ) . \begin{aligned}&\displaystyle\gcd(3,4)&\displaystyle\leftarrow\\ \displaystyle=&\displaystyle\gcd(3,1)&\displaystyle\rightarrow\\ \displaystyle=&\displaystyle\gcd(2,1)&\displaystyle\rightarrow\\ \displaystyle=&\displaystyle\gcd(1,1).\end{aligned}
  24. a b = q 0 + r 0 b b r 0 = q 1 + r 1 r 0 r 0 r 1 = q 2 + r 2 r 1 r k - 2 r k - 1 = q k + r k r k - 1 r N - 2 r N - 1 = q N . \begin{aligned}\displaystyle\frac{a}{b}&\displaystyle=q_{0}+\frac{r_{0}}{b}\\ \displaystyle\frac{b}{r_{0}}&\displaystyle=q_{1}+\frac{r_{1}}{r_{0}}\\ \displaystyle\frac{r_{0}}{r_{1}}&\displaystyle=q_{2}+\frac{r_{2}}{r_{1}}\\ &\displaystyle{}\ \vdots\\ \displaystyle\frac{r_{k-2}}{r_{k-1}}&\displaystyle=q_{k}+\frac{r_{k}}{r_{k-1}}% \\ &\displaystyle{}\ \vdots\\ \displaystyle\frac{r_{N-2}}{r_{N-1}}&\displaystyle=q_{N}\,.\end{aligned}
  25. a b = q 0 + 1 q 1 + r 1 r 0 . \frac{a}{b}=q_{0}+\cfrac{1}{q_{1}+\cfrac{r_{1}}{r_{0}}}\,.
  26. a b = q 0 + 1 q 1 + 1 q 2 + r 2 r 1 . \frac{a}{b}=q_{0}+\cfrac{1}{q_{1}+\cfrac{1}{q_{2}+\cfrac{r_{2}}{r_{1}}}}\,.
  27. a b = q 0 + 1 q 1 + 1 q 2 + 1 + 1 q N = [ q 0 ; q 1 , q 2 , , q N ] . \frac{a}{b}=q_{0}+\cfrac{1}{q_{1}+\cfrac{1}{q_{2}+\cfrac{1}{\ddots+\cfrac{1}{q% _{N}}}}}=[q_{0};q_{1},q_{2},\ldots,q_{N}]\,.
  28. 1071 462 = 2 + 1 3 + 1 7 = [ 2 ; 3 , 7 ] \frac{1071}{462}=2+\cfrac{1}{3+\cfrac{1}{7}}=[2;3,7]
  29. T ( a , b ) = T ( m , n ) T(a,b)=T(m,n)
  30. T ( a ) = 1 a 0 b < a T ( a , b ) . T(a)=\frac{1}{a}\sum_{0\leq b<a}T(a,b).
  31. τ ( a ) = 1 φ ( a ) 0 b < a gcd ( a , b ) = 1 T ( a , b ) . \tau(a)=\frac{1}{\varphi(a)}\sum_{\begin{smallmatrix}0\leq b<a\\ \gcd(a,b)=1\end{smallmatrix}}T(a,b).
  32. τ ( a ) = 12 π 2 ln 2 ln a + C + O ( a - 1 / 6 - ϵ ) \tau(a)=\frac{12}{\pi^{2}}\ln 2\ln a+C+O(a^{-1/6-\epsilon})
  33. C = - 1 2 + 6 ln 2 π 2 ( 4 γ - 24 π 2 ζ ( 2 ) + 3 ln 2 - 2 ) 1.467 C=-\frac{1}{2}+\frac{6\ln 2}{\pi^{2}}(4\gamma-24\pi^{2}\zeta^{\prime}(2)+3\ln 2% -2)\approx 1.467
  34. T ( a ) = 1 a d a φ ( d ) τ ( d ) T(a)=\frac{1}{a}\sum_{d\mid a}\varphi(d)\tau(d)
  35. T ( a ) C + 12 π 2 ln 2 ( ln a - d a Λ ( d ) d ) T(a)\approx C+\frac{12}{\pi^{2}}\ln 2\left(\ln a-\sum_{d\mid a}\frac{\Lambda(d% )}{d}\right)
  36. Y ( n ) = 1 n 2 a = 1 n b = 1 n T ( a , b ) = 1 n a = 1 n T ( a ) . Y(n)=\frac{1}{n^{2}}\sum_{a=1}^{n}\sum_{b=1}^{n}T(a,b)=\frac{1}{n}\sum_{a=1}^{% n}T(a).
  37. Y ( n ) 12 π 2 ln 2 ln n + 0.06. Y(n)\approx\frac{12}{\pi^{2}}\ln 2\ln n+0.06.
  38. O ( i < N h i ( h i - h i + 1 + 2 ) ) O ( h i < N ( h i - h i + 1 + 2 ) ) O ( h ( h 0 + 2 N ) ) O ( h 2 ) . O\Big(\sum_{i<N}h_{i}(h_{i}-h_{i+1}+2)\Big)\subseteq O\Big(h\sum_{i<N}(h_{i}-h% _{i+1}+2)\Big)\subseteq O(h(h_{0}+2N))\subseteq O(h^{2}).
  39. a / b = m g / n g = m / n a/b=mg/ng=m/n
  40. r < s u b > k 2 ( x ) = q k ( x ) r k 1 ( x ) + r k ( x ) r<sub>k−2(x)=q_{k}(x)r_{k−1}(x)+r_{k}(x)
  41. ω = D . \omega=\sqrt{D}.\,
  42. ω = 1 + D 2 . \omega=\frac{1+\sqrt{D}}{2}.
  43. ρ < s u b > 0 = α ψ 0 β = ( ξ ψ 0 η ) δ ρ<sub>0=α−ψ_{0}β=(ξ−ψ_{0}η)δ

Euclidean_distance.html

  1. 𝐩𝐪 ¯ \overline{\mathbf{p}\mathbf{q}}
  2. d ( 𝐩 , 𝐪 ) = d ( 𝐪 , 𝐩 ) \displaystyle\mathrm{d}(\mathbf{p},\mathbf{q})=\mathrm{d}(\mathbf{q},\mathbf{p})
  3. 𝐩 = p 1 2 + p 2 2 + + p n 2 = 𝐩 𝐩 , \left\|\mathbf{p}\right\|=\sqrt{p_{1}^{2}+p_{2}^{2}+\cdots+p_{n}^{2}}=\sqrt{% \mathbf{p}\cdot\mathbf{p}},
  4. 𝐪 - 𝐩 = ( q 1 - p 1 , q 2 - p 2 , , q n - p n ) \mathbf{q}-\mathbf{p}=(q_{1}-p_{1},q_{2}-p_{2},\cdots,q_{n}-p_{n})
  5. 𝐪 - 𝐩 = ( 𝐪 - 𝐩 ) ( 𝐪 - 𝐩 ) . \left\|\mathbf{q}-\mathbf{p}\right\|=\sqrt{(\mathbf{q}-\mathbf{p})\cdot(% \mathbf{q}-\mathbf{p})}.
  6. 𝐪 - 𝐩 = 𝐩 2 + 𝐪 2 - 2 𝐩 𝐪 . \left\|\mathbf{q}-\mathbf{p}\right\|=\sqrt{\left\|\mathbf{p}\right\|^{2}+\left% \|\mathbf{q}\right\|^{2}-2\mathbf{p}\cdot\mathbf{q}}.
  7. ( x - y ) 2 = | x - y | . \sqrt{(x-y)^{2}}=|x-y|.
  8. d ( 𝐩 , 𝐪 ) = ( q 1 - p 1 ) 2 + ( q 2 - p 2 ) 2 . \mathrm{d}(\mathbf{p},\mathbf{q})=\sqrt{(q_{1}-p_{1})^{2}+(q_{2}-p_{2})^{2}}.
  9. r 1 2 + r 2 2 - 2 r 1 r 2 cos ( θ 1 - θ 2 ) . \sqrt{r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}\cos(\theta_{1}-\theta_{2})}.
  10. d ( p , q ) = ( p 1 - q 1 ) 2 + ( p 2 - q 2 ) 2 + ( p 3 - q 3 ) 2 . d(p,q)=\sqrt{(p_{1}-q_{1})^{2}+(p_{2}-q_{2})^{2}+(p_{3}-q_{3})^{2}}.
  11. d ( p , q ) = ( p 1 - q 1 ) 2 + ( p 2 - q 2 ) 2 + + ( p i - q i ) 2 + + ( p n - q n ) 2 . d(p,q)=\sqrt{(p_{1}-q_{1})^{2}+(p_{2}-q_{2})^{2}+\cdots+(p_{i}-q_{i})^{2}+% \cdots+(p_{n}-q_{n})^{2}}.
  12. d 2 ( p , q ) = ( p 1 - q 1 ) 2 + ( p 2 - q 2 ) 2 + + ( p i - q i ) 2 + + ( p n - q n ) 2 . d^{2}(p,q)=(p_{1}-q_{1})^{2}+(p_{2}-q_{2})^{2}+\cdots+(p_{i}-q_{i})^{2}+\cdots% +(p_{n}-q_{n})^{2}.

Euclidean_domain.html

  1. f f
  2. R { 0 } R\setminus\{0\}
  3. a R { 0 } \scriptstyle a\in R\setminus\{0\}
  4. f ( a ) = min x R { 0 } g ( x a ) f(a)=\min_{x\in R\setminus\{0\}}g(xa)
  5. f ( x ) = i = 1 n v i ( x ) \scriptstyle f(x)=\sum_{i=1}^{n}v_{i}(x)
  6. v i \scriptstyle v_{i}
  7. ( - 19 ) , \mathbb{Q}(\sqrt{-19}),
  8. a + b - 19 2 \frac{a+b\sqrt{-19}}{2}
  9. a a
  10. b b
  11. [ X , Y ] / ( X 2 + Y 2 + 1 ) \mathbb{R}[X,Y]/(X^{2}+Y^{2}+1)
  12. ( d ) \mathbb{Q}(\sqrt{d})
  13. ( - 5 ) \mathbb{Q}(\sqrt{-5})
  14. ( - 19 ) \mathbb{Q}(\sqrt{-19})
  15. ( 69 ) \mathbb{Q}(\sqrt{69})
  16. ( - 1 ) \mathbb{Q}(\sqrt{-1})
  17. ( d ) \mathbb{Q}(\sqrt{d})

Euclidean_geometry.html

  1. A L 2 A\propto L^{2}
  2. V L 3 V\propto L^{3}
  3. | P Q | = ( p x - q x ) 2 + ( p y - q y ) 2 |PQ|=\sqrt{(p_{x}-q_{x})^{2}+(p_{y}-q_{y})^{2}}\,

Euclidean_space.html

  1. n n
  2. n n
  3. 𝐱 \mathbf{x}
  4. 𝐲 \mathbf{y}
  5. 𝐱 𝐲 = i = 1 n x i y i = x 1 y 1 + x 2 y 2 + + x n y n , \mathbf{x}\cdot\mathbf{y}=\sum_{i=1}^{n}x_{i}y_{i}=x_{1}y_{1}+x_{2}y_{2}+% \cdots+x_{n}y_{n},
  6. i i
  7. 𝐱 \mathbf{x}
  8. 𝐲 \mathbf{y}
  9. 𝐱 \mathbf{x}
  10. 𝐱 \mathbf{x}
  11. 𝐱 = 𝐱 𝐱 = i = 1 n ( x i ) 2 . \|\mathbf{x}\|=\sqrt{\mathbf{x}\cdot\mathbf{x}}=\sqrt{\sum_{i=1}^{n}(x_{i})^{2% }}.
  12. d ( 𝐱 , 𝐲 ) = 𝐱 - 𝐲 = i = 1 n ( x i - y i ) 2 . d(\mathbf{x},\mathbf{y})=\|\mathbf{x}-\mathbf{y}\|=\sqrt{\sum_{i=1}^{n}(x_{i}-% y_{i})^{2}}.
  13. 𝐑 \mathbf{R}
  14. θ θ
  15. 0 ° θ 180 ° 0°≤θ≤180°
  16. 𝐱 \mathbf{x}
  17. 𝐲 \mathbf{y}
  18. θ = arccos ( 𝐱 𝐲 𝐱 𝐲 ) \theta=\arccos\left(\frac{\mathbf{x}\cdot\mathbf{y}}{\|\mathbf{x}\|\|\mathbf{y% }\|}\right)
  19. a r c c o s arccos
  20. n > 1 n>1
  21. n = 2 n=2
  22. 2 π
  23. 𝐲𝐱 = 𝐱𝐲 ∠\mathbf{y}\mathbf{x}=−∠\mathbf{x}\mathbf{y}
  24. θ θ
  25. θ −θ
  26. 𝐱 \mathbf{x}
  27. 𝐲 \mathbf{y}
  28. Q Q
  29. 𝐱 \mathbf{x}
  30. 𝐲 \mathbf{y}
  31. Q 𝐱 Q 𝐲 = 𝐱 𝐲 Q\mathbf{x}\cdot Q\mathbf{y}=\mathbf{x}\cdot\mathbf{y}
  32. | Q 𝐱 | = | 𝐱 | . |Q\mathbf{x}|=|\mathbf{x}|.
  33. O ( n ) O(n)
  34. Q Q
  35. Q 𝖳 Q = Q Q 𝖳 = I , Q^{\mathsf{T}}Q=QQ^{\mathsf{T}}=I,
  36. Q Q
  37. Q Q
  38. I I
  39. S O ( n ) SO(n)
  40. n ( n 1 ) / 2 n(n−1)/2
  41. O ( n ) O(n)
  42. S O ( 1 ) SO(1)
  43. S O ( 2 ) SO(2)
  44. S < s u p > 1 S<sup>1
  45. S O ( n ) SO(n)
  46. n 4 n≤4
  47. n = 2 n=2
  48. O ( n ) O(n)
  49. n 2 n≤2
  50. E ( n ) E(n)
  51. I S O ( n ) ISO(n)
  52. I S O ( n ) ISO(n)
  53. 𝐄 < s u p > n \mathbf{E}<sup>n
  54. n = 1 n=1
  55. 𝐑 < s u p > n \mathbf{R}<sup>n

Euclidean_vector.html

  1. A B . \overrightarrow{AB}.
  2. A B \overrightarrow{AB}
  3. A B \overrightarrow{A^{\prime}B^{\prime}}
  4. A B \overrightarrow{AB}
  5. a \vec{a}
  6. a \underset{{}^{\sim}}{a}
  7. A B \overrightarrow{AB}
  8. 𝔞 \mathfrak{a}
  9. 𝐚 = ( 2 , 3 ) . \mathbf{a}=(2,3).
  10. O A \overrightarrow{OA}
  11. 𝐚 = ( a 1 , a 2 , a 3 ) . \mathbf{a}=(a_{1},a_{2},a_{3}).
  12. 𝐚 = ( a x , a y , a z ) . \mathbf{a}=(a\text{x},a\text{y},a\text{z}).
  13. 𝐚 = ( a 1 , a 2 , a 3 , , a n - 1 , a n ) . \mathbf{a}=(a_{1},a_{2},a_{3},\cdots,a_{n-1},a_{n}).
  14. 𝐚 = [ a 1 a 2 a 3 ] \mathbf{a}=\begin{bmatrix}a_{1}\\ a_{2}\\ a_{3}\\ \end{bmatrix}
  15. 𝐚 = [ a 1 a 2 a 3 ] . \mathbf{a}=[a_{1}\ a_{2}\ a_{3}].
  16. 𝐞 1 = ( 1 , 0 , 0 ) , 𝐞 2 = ( 0 , 1 , 0 ) , 𝐞 3 = ( 0 , 0 , 1 ) . {\mathbf{e}}_{1}=(1,0,0),\ {\mathbf{e}}_{2}=(0,1,0),\ {\mathbf{e}}_{3}=(0,0,1).
  17. 𝐚 = ( a 1 , a 2 , a 3 ) = a 1 ( 1 , 0 , 0 ) + a 2 ( 0 , 1 , 0 ) + a 3 ( 0 , 0 , 1 ) , \mathbf{a}=(a_{1},a_{2},a_{3})=a_{1}(1,0,0)+a_{2}(0,1,0)+a_{3}(0,0,1),
  18. 𝐚 = 𝐚 1 + 𝐚 2 + 𝐚 3 = a 1 𝐞 1 + a 2 𝐞 2 + a 3 𝐞 3 , \mathbf{a}=\mathbf{a}_{1}+\mathbf{a}_{2}+\mathbf{a}_{3}=a_{1}{\mathbf{e}}_{1}+% a_{2}{\mathbf{e}}_{2}+a_{3}{\mathbf{e}}_{3},
  19. 𝐢 , 𝐣 , 𝐤 \mathbf{i},\mathbf{j},\mathbf{k}
  20. 𝐱 ^ , 𝐲 ^ , 𝐳 ^ \mathbf{\hat{x}},\mathbf{\hat{y}},\mathbf{\hat{z}}
  21. 𝐚 = 𝐚 x + 𝐚 y + 𝐚 z = a x 𝐢 + a y 𝐣 + a z 𝐤 . \mathbf{a}=\mathbf{a}\text{x}+\mathbf{a}\text{y}+\mathbf{a}\text{z}=a\text{x}{% \mathbf{i}}+a\text{y}{\mathbf{j}}+a\text{z}{\mathbf{k}}.
  22. 𝐱 ^ , 𝐲 ^ , 𝐳 ^ \mathbf{\hat{x}},\mathbf{\hat{y}},\mathbf{\hat{z}}
  23. s y m b o l ρ ^ , s y m b o l ϕ ^ , 𝐳 ^ symbol{\hat{\rho}},symbol{\hat{\phi}},\mathbf{\hat{z}}
  24. 𝐫 ^ , s y m b o l θ ^ , s y m b o l ϕ ^ \mathbf{\hat{r}},symbol{\hat{\theta}},symbol{\hat{\phi}}
  25. 𝐞 1 = ( 1 , 0 , 0 ) , 𝐞 2 = ( 0 , 1 , 0 ) , 𝐞 3 = ( 0 , 0 , 1 ) {\mathbf{e}}_{1}=(1,0,0),\ {\mathbf{e}}_{2}=(0,1,0),\ {\mathbf{e}}_{3}=(0,0,1)
  26. 𝐚 = a 1 𝐞 1 + a 2 𝐞 2 + a 3 𝐞 3 . {\mathbf{a}}=a_{1}{\mathbf{e}}_{1}+a_{2}{\mathbf{e}}_{2}+a_{3}{\mathbf{e}}_{3}.
  27. 𝐚 = a 1 𝐞 1 + a 2 𝐞 2 + a 3 𝐞 3 {\mathbf{a}}=a_{1}{\mathbf{e}}_{1}+a_{2}{\mathbf{e}}_{2}+a_{3}{\mathbf{e}}_{3}
  28. 𝐛 = b 1 𝐞 1 + b 2 𝐞 2 + b 3 𝐞 3 {\mathbf{b}}=b_{1}{\mathbf{e}}_{1}+b_{2}{\mathbf{e}}_{2}+b_{3}{\mathbf{e}}_{3}
  29. a 1 = b 1 , a 2 = b 2 , a 3 = b 3 . a_{1}=b_{1},\quad a_{2}=b_{2},\quad a_{3}=b_{3}.\,
  30. 𝐚 = a 1 𝐞 1 + a 2 𝐞 2 + a 3 𝐞 3 {\mathbf{a}}=a_{1}{\mathbf{e}}_{1}+a_{2}{\mathbf{e}}_{2}+a_{3}{\mathbf{e}}_{3}
  31. 𝐛 = b 1 𝐞 1 + b 2 𝐞 2 + b 3 𝐞 3 {\mathbf{b}}=b_{1}{\mathbf{e}}_{1}+b_{2}{\mathbf{e}}_{2}+b_{3}{\mathbf{e}}_{3}
  32. a 1 = - b 1 , a 2 = - b 2 , a 3 = - b 3 . a_{1}=-b_{1},\quad a_{2}=-b_{2},\quad a_{3}=-b_{3}.\,
  33. 𝐚 + 𝐛 = ( a 1 + b 1 ) 𝐞 1 + ( a 2 + b 2 ) 𝐞 2 + ( a 3 + b 3 ) 𝐞 3 . \mathbf{a}+\mathbf{b}=(a_{1}+b_{1})\mathbf{e}_{1}+(a_{2}+b_{2})\mathbf{e}_{2}+% (a_{3}+b_{3})\mathbf{e}_{3}.
  34. 𝐚 - 𝐛 = ( a 1 - b 1 ) 𝐞 1 + ( a 2 - b 2 ) 𝐞 2 + ( a 3 - b 3 ) 𝐞 3 . \mathbf{a}-\mathbf{b}=(a_{1}-b_{1})\mathbf{e}_{1}+(a_{2}-b_{2})\mathbf{e}_{2}+% (a_{3}-b_{3})\mathbf{e}_{3}.
  35. r 𝐚 = ( r a 1 ) 𝐞 1 + ( r a 2 ) 𝐞 2 + ( r a 3 ) 𝐞 3 . r\mathbf{a}=(ra_{1})\mathbf{e}_{1}+(ra_{2})\mathbf{e}_{2}+(ra_{3})\mathbf{e}_{% 3}.
  36. 𝐚 = a 1 2 + a 2 2 + a 3 2 \left\|\mathbf{a}\right\|=\sqrt{{a_{1}}^{2}+{a_{2}}^{2}+{a_{3}}^{2}}
  37. 𝐚 = 𝐚 𝐚 . \left\|\mathbf{a}\right\|=\sqrt{\mathbf{a}\cdot\mathbf{a}}.
  38. 𝐚 ^ = 𝐚 𝐚 = a 1 𝐚 𝐞 1 + a 2 𝐚 𝐞 2 + a 3 𝐚 𝐞 3 \mathbf{\hat{a}}=\frac{\mathbf{a}}{\left\|\mathbf{a}\right\|}=\frac{a_{1}}{% \left\|\mathbf{a}\right\|}\mathbf{e}_{1}+\frac{a_{2}}{\left\|\mathbf{a}\right% \|}\mathbf{e}_{2}+\frac{a_{3}}{\left\|\mathbf{a}\right\|}\mathbf{e}_{3}
  39. 0 \vec{0}
  40. 𝐚 𝐛 = 𝐚 𝐛 cos θ \mathbf{a}\cdot\mathbf{b}=\left\|\mathbf{a}\right\|\left\|\mathbf{b}\right\|\cos\theta
  41. 𝐚 𝐛 = a 1 b 1 + a 2 b 2 + a 3 b 3 . \mathbf{a}\cdot\mathbf{b}=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}.
  42. 𝐚 × 𝐛 = 𝐚 𝐛 sin ( θ ) 𝐧 \mathbf{a}\times\mathbf{b}=\left\|\mathbf{a}\right\|\left\|\mathbf{b}\right\|% \sin(\theta)\,\mathbf{n}
  43. 𝐚 × 𝐛 = ( a 2 b 3 - a 3 b 2 ) 𝐞 1 + ( a 3 b 1 - a 1 b 3 ) 𝐞 2 + ( a 1 b 2 - a 2 b 1 ) 𝐞 3 . {\mathbf{a}}\times{\mathbf{b}}=(a_{2}b_{3}-a_{3}b_{2}){\mathbf{e}}_{1}+(a_{3}b% _{1}-a_{1}b_{3}){\mathbf{e}}_{2}+(a_{1}b_{2}-a_{2}b_{1}){\mathbf{e}}_{3}.
  44. ( 𝐚 𝐛 𝐜 ) = 𝐚 ( 𝐛 × 𝐜 ) . (\mathbf{a}\ \mathbf{b}\ \mathbf{c})=\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c% }).
  45. ( 𝐚 𝐛 𝐜 ) = | ( a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 ) | (\mathbf{a}\ \mathbf{b}\ \mathbf{c})=\left|\begin{pmatrix}a_{1}&a_{2}&a_{3}\\ b_{1}&b_{2}&b_{3}\\ c_{1}&c_{2}&c_{3}\\ \end{pmatrix}\right|
  46. ( 𝐚 𝐛 𝐜 ) = ( 𝐜 𝐚 𝐛 ) = ( 𝐛 𝐜 𝐚 ) = - ( 𝐚 𝐜 𝐛 ) = - ( 𝐛 𝐚 𝐜 ) = - ( 𝐜 𝐛 𝐚 ) . (\mathbf{a}\ \mathbf{b}\ \mathbf{c})=(\mathbf{c}\ \mathbf{a}\ \mathbf{b})=(% \mathbf{b}\ \mathbf{c}\ \mathbf{a})=-(\mathbf{a}\ \mathbf{c}\ \mathbf{b})=-(% \mathbf{b}\ \mathbf{a}\ \mathbf{c})=-(\mathbf{c}\ \mathbf{b}\ \mathbf{a}).
  47. 𝐚 = a 1 𝐞 1 + a 2 𝐞 2 + a 3 𝐞 3 = u 𝐧 1 + v 𝐧 2 + w 𝐧 3 \mathbf{a}=a_{1}\mathbf{e}_{1}+a_{2}\mathbf{e}_{2}+a_{3}\mathbf{e}_{3}=u% \mathbf{n}_{1}+v\mathbf{n}_{2}+w\mathbf{n}_{3}
  48. [ u v w ] = [ c 11 c 12 c 13 c 21 c 22 c 23 c 31 c 32 c 33 ] [ a 1 a 2 a 3 ] \begin{bmatrix}u\\ v\\ w\\ \end{bmatrix}=\begin{bmatrix}c_{11}&c_{12}&c_{13}\\ c_{21}&c_{22}&c_{23}\\ c_{31}&c_{32}&c_{33}\end{bmatrix}\begin{bmatrix}a_{1}\\ a_{2}\\ a_{3}\end{bmatrix}
  49. ( a 1 𝐞 1 + a 2 𝐞 2 ) + ( b 1 𝐞 1 + b 2 𝐞 2 ) = ( a 1 + b 1 ) 𝐞 1 + ( a 2 + b 2 ) 𝐞 2 (a_{1}{\mathbf{e}}_{1}+a_{2}{\mathbf{e}}_{2})+(b_{1}{\mathbf{e}}_{1}+b_{2}{% \mathbf{e}}_{2})=(a_{1}+b_{1}){\mathbf{e}}_{1}+(a_{2}+b_{2}){\mathbf{e}}_{2}
  50. ( a 1 𝐞 1 + a 2 𝐞 2 + a 3 𝐞 3 + a 4 𝐞 4 ) + ( b 1 𝐞 1 + b 2 𝐞 2 + b 3 𝐞 3 + b 4 𝐞 4 ) = ( a 1 + b 1 ) 𝐞 1 + ( a 2 + b 2 ) 𝐞 2 + ( a 3 + b 3 ) 𝐞 3 + ( a 4 + b 4 ) 𝐞 4 . \begin{aligned}\displaystyle(a_{1}{\mathbf{e}}_{1}+a_{2}{\mathbf{e}}_{2}+a_{3}% {\mathbf{e}}_{3}+a_{4}{\mathbf{e}}_{4})&\displaystyle+(b_{1}{\mathbf{e}}_{1}+b% _{2}{\mathbf{e}}_{2}+b_{3}{\mathbf{e}}_{3}+b_{4}{\mathbf{e}}_{4})=\\ \displaystyle(a_{1}+b_{1}){\mathbf{e}}_{1}+(a_{2}+b_{2}){\mathbf{e}}_{2}&% \displaystyle+(a_{3}+b_{3}){\mathbf{e}}_{3}+(a_{4}+b_{4}){\mathbf{e}}_{4}.\end% {aligned}
  51. ( a 1 𝐞 1 + a 2 𝐞 2 ) ( b 1 𝐞 1 + b 2 𝐞 2 ) = ( a 1 b 2 - a 2 b 1 ) 𝐞 1 𝐞 2 . (a_{1}{\mathbf{e}}_{1}+a_{2}{\mathbf{e}}_{2})\wedge(b_{1}{\mathbf{e}}_{1}+b_{2% }{\mathbf{e}}_{2})=(a_{1}b_{2}-a_{2}b_{1})\mathbf{e}_{1}\mathbf{e}_{2}.
  52. 𝐱 = x 1 𝐞 1 + x 2 𝐞 2 + x 3 𝐞 3 . {\mathbf{x}}=x_{1}{\mathbf{e}}_{1}+x_{2}{\mathbf{e}}_{2}+x_{3}{\mathbf{e}}_{3}.
  53. 𝐲 - 𝐱 = ( y 1 - x 1 ) 𝐞 1 + ( y 2 - x 2 ) 𝐞 2 + ( y 3 - x 3 ) 𝐞 3 . {\mathbf{y}}-{\mathbf{x}}=(y_{1}-x_{1}){\mathbf{e}}_{1}+(y_{2}-x_{2}){\mathbf{% e}}_{2}+(y_{3}-x_{3}){\mathbf{e}}_{3}.
  54. 𝐱 t = t 𝐯 + 𝐱 0 , {\mathbf{x}}_{t}=t{\mathbf{v}}+{\mathbf{x}}_{0},
  55. 𝐅 = m 𝐚 {\mathbf{F}}=m{\mathbf{a}}
  56. E = 𝐅 ( 𝐱 2 - 𝐱 1 ) . E={\mathbf{F}}\cdot({\mathbf{x}}_{2}-{\mathbf{x}}_{1}).
  57. f ( x α ) f(x^{\alpha})
  58. x α ( τ ) x^{\alpha}(\tau)
  59. f f
  60. d f d τ = α = 1 n d x α d τ f x α . \frac{df}{d\tau}=\sum_{\alpha=1}^{n}\frac{dx^{\alpha}}{d\tau}\frac{\partial f}% {\partial x^{\alpha}}.
  61. α \alpha
  62. x α ( τ ) x^{\alpha}(\tau)
  63. t α = d x α d τ . t^{\alpha}=\frac{dx^{\alpha}}{d\tau}.
  64. f f
  65. d d τ = α t α x α . \frac{d}{d\tau}=\sum_{\alpha}t^{\alpha}\frac{\partial}{\partial x^{\alpha}}.
  66. 𝐚 a α x α . \mathbf{a}\equiv a^{\alpha}\frac{\partial}{\partial x^{\alpha}}.

Eudoxus_of_Cnidus.html

  1. a / b a/b
  2. c / d c/d
  3. a / b = c / d a/b=c/d

Euler's_formula.html

  1. e i x = cos x + i sin x e^{ix}=\cos x+i\sin x
  2. 1 1 + x 2 = 1 2 ( 1 1 - i x + 1 1 + i x ) \dfrac{1}{1+x^{2}}=\dfrac{1}{2}\left(\dfrac{1}{1-ix}+\dfrac{1}{1+ix}\right)
  3. d x 1 + a x = 1 a ln ( 1 + a x ) + C \int\dfrac{dx}{1+ax}=\dfrac{1}{a}\ln\left(1+ax\right)+C
  4. i x = ln ( cos x + i sin x ) ix=\ln\left(\cos x+i\sin x\right)
  5. ln \ln
  6. z = x + i y = | z | ( cos ϕ + i sin ϕ ) = r e i ϕ z ¯ = x - i y = | z | ( cos ϕ - i sin ϕ ) = r e - i ϕ \begin{aligned}\displaystyle z&\displaystyle=x+iy&\displaystyle=|z|(\cos\phi+i% \sin\phi)&\displaystyle=re^{i\phi}\\ \displaystyle\bar{z}&\displaystyle=x-iy&\displaystyle=|z|(\cos\phi-i\sin\phi)&% \displaystyle=re^{-i\phi}\end{aligned}
  7. x = Re { z } x=\mathrm{Re}\{z\}\,
  8. y = Im { z } y=\mathrm{Im}\{z\}\,
  9. r = | z | = x 2 + y 2 r=|z|=\sqrt{x^{2}+y^{2}}
  10. ϕ = arg z = \phi=\arg z=\,
  11. a = e ln ( a ) a=e^{\ln(a)}
  12. e a e b = e a + b e^{a}e^{b}=e^{a+b}
  13. z = | z | e i ϕ = e ln | z | e i ϕ = e ln | z | + i ϕ z=|z|e^{i\phi}=e^{\ln|z|}e^{i\phi}=e^{\ln|z|+i\phi}
  14. ln z = ln | z | + i ϕ . \ln z=\ln|z|+i\phi\ .
  15. ( e a ) k = e a k , (e^{a})^{k}=e^{ak}\ ,
  16. cos x = Re { e i x } = e i x + e - i x 2 sin x = Im { e i x } = e i x - e - i x 2 i \begin{aligned}\displaystyle\cos x&\displaystyle=\mathrm{Re}\{e^{ix}\}={e^{ix}% +e^{-ix}\over 2}\\ \displaystyle\sin x&\displaystyle=\mathrm{Im}\{e^{ix}\}={e^{ix}-e^{-ix}\over 2% i}\end{aligned}
  17. e i x \displaystyle e^{ix}
  18. cos ( i y ) = e - y + e y 2 = cosh ( y ) sin ( i y ) = e - y - e y 2 i = - e y - e - y 2 i = i sinh ( y ) . \begin{aligned}\displaystyle\cos(iy)&\displaystyle={e^{-y}+e^{y}\over 2}=\cosh% (y)\\ \displaystyle\sin(iy)&\displaystyle={e^{-y}-e^{y}\over 2i}=-{e^{y}-e^{-y}\over 2% i}=i\sinh(y)\ .\end{aligned}
  19. cos x cos y \displaystyle\cos x\cdot\cos y
  20. cos ( n x ) \displaystyle\cos(nx)
  21. t e i t t\mapsto e^{it}
  22. 𝕊 1 \mathbb{S}^{1}
  23. 𝕊 1 \mathbb{S}^{1}
  24. τ \tau\mathbb{Z}
  25. τ = 2 π \tau=2\pi
  26. e z = 1 + z 1 ! + z 2 2 ! + z 3 3 ! + = n = 0 z n n ! . e^{z}=1+\frac{z}{1!}+\frac{z^{2}}{2!}+\frac{z^{3}}{3!}+\dots=\sum_{n=0}^{% \infty}\frac{z^{n}}{n!}.
  27. e z = lim n ( 1 + z n ) n . e^{z}=\lim_{n\rightarrow\infty}\left(1+\frac{z}{n}\right)^{n}~{}.
  28. e z = e r + j θ = e r e j θ = R e j θ e^{z}=e^{r+j\theta}=e^{r}e^{j\theta}=R^{\prime}e^{j\theta}
  29. i 0 \displaystyle i^{0}
  30. e i x \displaystyle e^{ix}
  31. e i x = r ( cos ( θ ) + i sin ( θ ) ) . e^{ix}=r(\cos(\theta)+i\sin(\theta))\,.
  32. i e i x = ( cos ( θ ) + i sin ( θ ) ) d r d x + r ( - sin ( θ ) + i cos ( θ ) ) d θ d x . ie^{ix}=(\cos(\theta)+i\sin(\theta))\frac{dr}{dx}+r(-\sin(\theta)+i\cos(\theta% ))\frac{d\theta}{dx}\,.
  33. r ( cos ( θ ) + i sin ( θ ) ) r(\cos(\theta)+i\sin(\theta))
  34. e i x e^{ix}
  35. d r d x = 0 \textstyle\frac{dr}{dx}=0
  36. d θ d x = 1 \textstyle\frac{d\theta}{dx}=1
  37. r ( 0 ) = 1 r(0)=1
  38. θ ( 0 ) = 0 \theta(0)=0
  39. e i 0 = 1 e^{i0}=1
  40. r = 1 r=1
  41. θ = x \theta=x
  42. e i x = 1 ( cos ( x ) + i sin ( x ) ) e^{ix}=1(\cos(x)+i\sin(x))

Euler's_identity.html

  1. e i π + 1 = 0 e^{i\pi}+1=0
  2. e e
  3. i i
  4. i i
  5. π \pi
  6. x x
  7. e i x = cos x + i sin x e^{ix}=\cos x+i\sin x
  8. x x
  9. π \pi
  10. e i π = cos π + i sin π . e^{i\pi}=\cos\pi+i\sin\pi.
  11. cos π = - 1 \cos\pi=-1\,\!
  12. sin π = 0 , \sin\pi=0,
  13. e i π = - 1 + 0 i , e^{i\pi}=-1+0i,
  14. e i π + 1 = 0. e^{i\pi}+1=0.
  15. π \pi
  16. π \pi
  17. e e
  18. e e
  19. i i
  20. π \pi
  21. k = 0 n - 1 e 2 π i k / n = 0. \sum_{k=0}^{n-1}e^{2\pi ik/n}=0.
  22. n n
  23. e ( i ± j ± k ) 3 π + 1 = 0. e^{\frac{(i\pm j\pm k)}{\sqrt{3}}\pi}+1=0.\,
  24. a 1 2 + a 2 2 + a 3 2 = 1 {a_{1}}^{2}+{a_{2}}^{2}+{a_{3}}^{2}=1
  25. e ( a 1 i + a 2 j + a 3 k ) π + 1 = 0. e^{(a_{1}i+a_{2}j+a_{3}k)\pi}+1=0.\,
  26. a 1 2 + a 2 2 + + a 7 2 = 1 {a_{1}}^{2}+{a_{2}}^{2}+\dots+{a_{7}}^{2}=1
  27. e ( a 1 i 1 + a 2 i 2 + + a 7 i 7 ) π + 1 = 0. e^{(a_{1}i_{1}+a_{2}i_{2}+\dots+a_{7}i_{7})\pi}+1=0.\,
  28. e e

Euler's_sum_of_powers_conjecture.html

  1. i = 1 n a i k = b k \sum_{i=1}^{n}a_{i}^{k}=b^{k}
  2. n > 1 n>1
  3. a 1 , a 2 , , a n , b a_{1},a_{2},\dots,a_{n},b
  4. n k n\geq k
  5. a 1 k + a 2 k = b k a_{1}^{k}+a_{2}^{k}=b^{k}
  6. 2 k 2\geq k
  7. 59 4 + 158 4 = 133 4 + 134 4 ; 59^{4}+158^{4}=133^{4}+134^{4};
  8. 3 3 + 4 3 + 5 3 = 6 3 3^{3}+4^{3}+5^{3}=6^{3}
  9. x 1 3 + x 2 3 = x 3 3 + x 4 3 x_{1}^{3}+x_{2}^{3}=x_{3}^{3}+x_{4}^{3}
  10. x 1 = 1 - ( a - 3 b ) ( a 2 + 3 b 2 ) , x 2 = ( a + 3 b ) ( a 2 + 3 b 2 ) - 1 x_{1}=1-(a-3b)(a^{2}+3b^{2}),x_{2}=(a+3b)(a^{2}+3b^{2})-1
  11. x 3 = ( a + 3 b ) - ( a 2 + 3 b 2 ) 2 , x 4 = ( a 2 + 3 b 2 ) 2 - ( a - 3 b ) x_{3}=(a+3b)-(a^{2}+3b^{2})^{2},x_{4}=(a^{2}+3b^{2})^{2}-(a-3b)
  12. a a
  13. b b
  14. i = 1 n a i k = j = 1 m b j k \sum_{i=1}^{n}a_{i}^{k}=\sum_{j=1}^{m}b_{j}^{k}
  15. i = 1 n a i k = b k \sum_{i=1}^{n}a_{i}^{k}=b^{k}

Euler's_theorem.html

  1. a φ ( n ) 1 ( mod n ) a^{\varphi(n)}\equiv 1\;\;(\mathop{{\rm mod}}n)
  2. a φ ( n ) = a k M = ( a k ) M 1 M = 1 ( mod n ) . a^{\varphi(n)}=a^{kM}=(a^{k})^{M}\equiv 1^{M}=1\;\;(\mathop{{\rm mod}}n).
  3. i = 1 φ ( n ) x i i = 1 φ ( n ) a x i a φ ( n ) i = 1 φ ( n ) x i ( mod n ) , \prod_{i=1}^{\varphi(n)}x_{i}\equiv\prod_{i=1}^{\varphi(n)}ax_{i}\equiv a^{% \varphi(n)}\prod_{i=1}^{\varphi(n)}x_{i}\;\;(\mathop{{\rm mod}}n),
  4. a φ ( n ) 1 ( mod n ) . a^{\varphi(n)}\equiv 1\;\;(\mathop{{\rm mod}}n).

Euler's_totient_function.html

  1. φ ( n ) φ(n)
  2. φ ( n ) φ(n)
  3. ϕ ( n ) ϕ(n)
  4. φ ( n ) φ(n)
  5. 1 k n 1≤k≤n
  6. g c d ( n , k ) = 1 gcd(n,k)=1
  7. φ ( m n ) = φ ( m ) φ ( n ) φ(mn)=φ(m)φ(n)
  8. n = 9 n=9
  9. g c d ( 9 , 3 ) = g c d ( 9 , 6 ) = 3 gcd(9,3)=gcd(9,6)=3
  10. g c d ( 9 , 9 ) = 9 gcd(9,9)=9
  11. 1 k 9 1≤k≤9
  12. φ ( 9 ) = 6 φ(9)=6
  13. φ ( 1 ) = 1 φ(1)=1
  14. g c d ( 1 , 1 ) = 1 gcd(1,1)=1
  15. φ ( n ) φ(n)
  16. φ ( n ) = n p n ( 1 - 1 p ) , \varphi(n)=n\prod_{p\mid n}\left(1-\frac{1}{p}\right),
  17. φ ( p k ) = p k - p k - 1 = p k - 1 ( p - 1 ) = p k ( 1 - 1 p ) . \varphi(p^{k})=p^{k}-p^{k-1}=p^{k-1}(p-1)=p^{k}\left(1-\frac{1}{p}\right).
  18. n = p 1 k 1 p r k r , n=p_{1}^{k_{1}}\cdots p_{r}^{k_{r}},
  19. φ ( n ) = φ ( p 1 k 1 ) φ ( p 2 k 2 ) φ ( p r k r ) = p 1 k 1 ( 1 - 1 p 1 ) p 2 k 2 ( 1 - 1 p 2 ) p r k r ( 1 - 1 p r ) = p 1 k 1 p 2 k 2 p r k r ( 1 - 1 p 1 ) ( 1 - 1 p 2 ) ( 1 - 1 p r ) = n ( 1 - 1 p 1 ) ( 1 - 1 p 2 ) ( 1 - 1 p r ) . \begin{aligned}\displaystyle\varphi(n)&\displaystyle=\varphi(p_{1}^{k_{1}})% \varphi(p_{2}^{k_{2}})\cdots\varphi(p_{r}^{k_{r}})\\ &\displaystyle=p_{1}^{k_{1}}\left(1-\frac{1}{p_{1}}\right)p_{2}^{k_{2}}\left(1% -\frac{1}{p_{2}}\right)\cdots p_{r}^{k_{r}}\left(1-\frac{1}{p_{r}}\right)\\ &\displaystyle=p_{1}^{k_{1}}p_{2}^{k_{2}}\cdots p_{r}^{k_{r}}\left(1-\frac{1}{% p_{1}}\right)\left(1-\frac{1}{p_{2}}\right)\cdots\left(1-\frac{1}{p_{r}}\right% )\\ &\displaystyle=n\left(1-\frac{1}{p_{1}}\right)\left(1-\frac{1}{p_{2}}\right)% \cdots\left(1-\frac{1}{p_{r}}\right).\end{aligned}
  20. φ ( 36 ) = φ ( 2 2 3 2 ) = 36 ( 1 - 1 2 ) ( 1 - 1 3 ) = 36 1 2 2 3 = 12. \varphi(36)=\varphi\left(2^{2}3^{2}\right)=36\left(1-\frac{1}{2}\right)\left(1% -\frac{1}{3}\right)=36\cdot\frac{1}{2}\cdot\frac{2}{3}=12.
  21. { 𝐱 } [ m ] = k = 1 n x k e - 2 π i m k n , 𝐱 𝐤 = { gcd ( k , n ) } for k { 1 n } φ ( n ) = { 𝐱 } [ 1 ] = k = 1 n gcd ( k , n ) e - 2 π i k n . \begin{aligned}\displaystyle\mathcal{F}\left\{\mathbf{x}\right\}[m]&% \displaystyle=\sum\limits_{k=1}^{n}x_{k}\cdot e^{{-2\pi i}\frac{mk}{n}},% \mathbf{x_{k}}=\left\{\gcd(k,n)\right\}\quad\,\text{for}\,k\in\left\{1\dots n% \right\}\\ \displaystyle\varphi(n)&\displaystyle=\mathcal{F}\left\{\mathbf{x}\right\}[1]=% \sum\limits_{k=1}^{n}\gcd(k,n)e^{{-2\pi i}\frac{k}{n}}.\end{aligned}
  22. φ ( n ) = k = 1 n gcd ( k , n ) cos 2 π k n . \varphi(n)=\sum\limits_{k=1}^{n}\gcd(k,n)\cos{2\pi\frac{k}{n}}.
  23. d n φ ( d ) = n , \sum_{d\mid n}\varphi(d)=n,
  24. 1 20 , 2 20 , 3 20 , 4 20 , 5 20 , 6 20 , 7 20 , 8 20 , 9 20 , 10 20 , 11 20 , 12 20 , 13 20 , 14 20 , 15 20 , 16 20 , 17 20 , 18 20 , 19 20 , 20 20 \tfrac{1}{20},\,\tfrac{2}{20},\,\tfrac{3}{20},\,\tfrac{4}{20},\,\tfrac{5}{20},% \,\tfrac{6}{20},\,\tfrac{7}{20},\,\tfrac{8}{20},\,\tfrac{9}{20},\,\tfrac{10}{2% 0},\,\tfrac{11}{20},\,\tfrac{12}{20},\,\tfrac{13}{20},\,\tfrac{14}{20},\,% \tfrac{15}{20},\,\tfrac{16}{20},\,\tfrac{17}{20},\,\tfrac{18}{20},\,\tfrac{19}% {20},\,\tfrac{20}{20}
  25. 1 20 , 1 10 , 3 20 , 1 5 , 1 4 , 3 10 , 7 20 , 2 5 , 9 20 , 1 2 , 11 20 , 3 5 , 13 20 , 7 10 , 3 4 , 4 5 , 17 20 , 9 10 , 19 20 , 1 1 \tfrac{1}{20},\,\tfrac{1}{10},\,\tfrac{3}{20},\,\tfrac{1}{5},\,\tfrac{1}{4},\,% \tfrac{3}{10},\,\tfrac{7}{20},\,\tfrac{2}{5},\,\tfrac{9}{20},\,\tfrac{1}{2},\,% \tfrac{11}{20},\,\tfrac{3}{5},\,\tfrac{13}{20},\,\tfrac{7}{10},\,\tfrac{3}{4},% \,\tfrac{4}{5},\,\tfrac{17}{20},\,\tfrac{9}{10},\,\tfrac{19}{20},\,\tfrac{1}{1}
  26. ( 1 20 , 3 20 , 7 20 , 9 20 , 11 20 , 13 20 , 17 20 , 19 20 ) . \left(\tfrac{1}{20},\,\tfrac{3}{20},\,\tfrac{7}{20},\,\tfrac{9}{20},\,\tfrac{1% 1}{20},\,\tfrac{13}{20},\,\tfrac{17}{20},\,\tfrac{19}{20}\right).
  27. ( 1 10 , 3 10 , 7 10 , 9 10 ) , \left(\tfrac{1}{10},\,\tfrac{3}{10},\,\tfrac{7}{10},\,\tfrac{9}{10}\right),
  28. ( 1 5 , 2 5 , 3 5 , 4 5 ) , \left(\tfrac{1}{5},\,\tfrac{2}{5},\,\tfrac{3}{5},\,\tfrac{4}{5}\right),
  29. φ ( n ) = d n d μ ( n d ) = n d n μ ( d ) d , \varphi(n)=\sum_{d\mid n}d\cdot\mu\left(\frac{n}{d}\right)=n\sum_{d\mid n}% \frac{\mu(d)}{d},
  30. p n ( 1 - 1 p ) \prod_{p\mid n}\left(1-\frac{1}{p}\right)
  31. d n μ ( d ) d . \sum_{d\mid n}\frac{\mu(d)}{d}.
  32. n > 1 n>1
  33. φ ( n ) = n lim s 1 ζ ( s ) d | n μ ( d ) ( e 1 / d ) ( s - 1 ) \varphi(n)=n\lim\limits_{s\rightarrow 1}\zeta(s)\sum\limits_{d|n}\mu(d)(e^{1/d% })^{(s-1)}
  34. ζ ( s ) \zeta(s)
  35. μ \mu
  36. e e
  37. d d
  38. φ ( n ) \varphi(n)
  39. a φ ( n ) 1 mod n . a^{\varphi(n)}\equiv 1\mod n.\,
  40. a a e mod n a\mapsto a^{e}\mod n
  41. e e
  42. b b d mod n b\mapsto b^{d}\mod n
  43. d d
  44. e e
  45. φ ( n ) \varphi(n)
  46. φ ( n ) \varphi(n)
  47. n n
  48. d d
  49. n n
  50. n n
  51. p p
  52. q q
  53. n n
  54. a b a\mid b
  55. φ ( a ) φ ( b ) . \varphi(a)\mid\varphi(b).
  56. n φ ( a n - 1 ) n\mid\varphi(a^{n}-1)
  57. φ ( m n ) = φ ( m ) φ ( n ) d φ ( d ) \varphi(mn)=\varphi(m)\varphi(n)\cdot\frac{d}{\varphi(d)}
  58. φ ( 2 m ) = { 2 φ ( m ) if m is even φ ( m ) if m is odd \varphi(2m)=\begin{cases}2\varphi(m)&\,\text{ if }m\,\text{ is even}\\ \varphi(m)&\,\text{ if }m\,\text{ is odd}\end{cases}
  59. φ ( n m ) = n m - 1 φ ( n ) . \;\varphi\left(n^{m}\right)=n^{m-1}\varphi(n).
  60. φ ( lcm ( m , n ) ) φ ( gcd ( m , n ) ) = φ ( m ) φ ( n ) . \varphi(\mathrm{lcm}(m,n))\cdot\varphi(\mathrm{gcd}(m,n))=\varphi(m)\cdot% \varphi(n).
  61. lcm ( m , n ) gcd ( m , n ) = m n . \mathrm{lcm}(m,n)\cdot\mathrm{gcd}(m,n)=m\cdot n.
  62. φ ( n ) \varphi(n)\;
  63. n 3. n\geq 3.
  64. 2 r φ ( n ) . 2^{r}\mid\varphi(n).
  65. 4 n 4\nmid n
  66. l 2 n l\geq 2n
  67. l φ ( a n - 1 ) l\mid\varphi(a^{n}-1)
  68. d n μ 2 ( d ) φ ( d ) = n φ ( n ) \sum_{d\mid n}\frac{\mu^{2}(d)}{\varphi(d)}=\frac{n}{\varphi(n)}
  69. 1 k n ( k , n ) = 1 k = 1 2 n φ ( n ) for n > 1 \sum_{1\leq k\leq n\atop(k,n)=1}\!\!k=\frac{1}{2}n\varphi(n)\,\text{ for }n>1
  70. k = 1 n φ ( k ) = 1 2 ( 1 + k = 1 n μ ( k ) n k 2 ) = 3 π 2 n 2 + O ( n ( log n ) 2 / 3 ( log log n ) 4 / 3 ) \sum_{k=1}^{n}\varphi(k)=\frac{1}{2}\left(1+\sum_{k=1}^{n}\mu(k)\left\lfloor% \frac{n}{k}\right\rfloor^{2}\right)=\frac{3}{\pi^{2}}n^{2}+O\left(n(\log n)^{2% /3}(\log\log n)^{4/3}\right)
  71. k = 1 n φ ( k ) k = k = 1 n μ ( k ) k n k = 6 π 2 n + O ( ( log n ) 2 / 3 ( log log n ) 4 / 3 ) \sum_{k=1}^{n}\frac{\varphi(k)}{k}=\sum_{k=1}^{n}\frac{\mu(k)}{k}\left\lfloor% \frac{n}{k}\right\rfloor=\frac{6}{\pi^{2}}n+O\left((\log n)^{2/3}(\log\log n)^% {4/3}\right)
  72. k = 1 n k φ ( k ) = 315 ζ ( 3 ) 2 π 4 n - log n 2 + O ( ( log n ) 2 / 3 ) \sum_{k=1}^{n}\frac{k}{\varphi(k)}=\frac{315\zeta(3)}{2\pi^{4}}n-\frac{\log n}% {2}+O\left((\log n)^{2/3}\right)
  73. k = 1 n 1 φ ( k ) = 315 ζ ( 3 ) 2 π 4 ( log n + γ - p prime log p p 2 - p + 1 ) + O ( ( log n ) 2 / 3 n ) \sum_{k=1}^{n}\frac{1}{\varphi(k)}=\frac{315\zeta(3)}{2\pi^{4}}\left(\log n+% \gamma-\sum_{p\,\text{ prime}}\frac{\log p}{p^{2}-p+1}\right)+O\left(\frac{(% \log n)^{2/3}}{n}\right)
  74. 1 k n ( k , m ) = 1 1 = n φ ( m ) m + O ( 2 ω ( m ) ) , \sum_{1\leq k\leq n\atop(k,m)=1}1=n\frac{\varphi(m)}{m}+O\left(2^{\omega(m)}% \right),
  75. gcd ( k , n ) = 1 1 k n gcd ( k - 1 , n ) = φ ( n ) d ( n ) , \sum_{\stackrel{1\leq k\leq n}{\gcd(k,n)=1}}\gcd(k-1,n)=\varphi(n)d(n),
  76. μ ( n ) \mu(n)
  77. φ ( n ) \varphi(n)
  78. ϕ = 1 + 5 2 = 1.618 \phi=\frac{1+\sqrt{5}}{2}=1.618\dots
  79. ϕ = - k = 1 φ ( k ) k log ( 1 - 1 ϕ k ) \phi=-\sum_{k=1}^{\infty}\frac{\varphi(k)}{k}\log\left(1-\frac{1}{\phi^{k}}\right)
  80. 1 ϕ = - k = 1 μ ( k ) k log ( 1 - 1 ϕ k ) . \frac{1}{\phi}=-\sum_{k=1}^{\infty}\frac{\mu(k)}{k}\log\left(1-\frac{1}{\phi^{% k}}\right).
  81. k = 1 μ ( k ) - φ ( k ) k log ( 1 - 1 ϕ k ) = 1. \sum_{k=1}^{\infty}\frac{\mu(k)-\varphi(k)}{k}\log\left(1-\frac{1}{\phi^{k}}% \right)=1.
  82. e e
  83. e = k = 1 ( 1 - 1 ϕ k ) μ ( k ) - φ ( k ) k . e=\prod_{k=1}^{\infty}\left(1-\frac{1}{\phi^{k}}\right)^{\frac{\mu(k)-\varphi(% k)}{k}}.
  84. k = 1 φ ( k ) k ( - log ( 1 - x k ) ) = x 1 - x \sum_{k=1}^{\infty}\frac{\varphi(k)}{k}(-\log(1-x^{k}))=\frac{x}{1-x}
  85. k = 1 μ ( k ) k ( - log ( 1 - x k ) ) = x , \sum_{k=1}^{\infty}\frac{\mu(k)}{k}(-\log(1-x^{k}))=x,
  86. n = 1 φ ( n ) n s = ζ ( s - 1 ) ζ ( s ) . \sum_{n=1}^{\infty}\frac{\varphi(n)}{n^{s}}=\frac{\zeta(s-1)}{\zeta(s)}.
  87. n = 1 φ ( n ) q n 1 - q n = q ( 1 - q ) 2 \sum_{n=1}^{\infty}\frac{\varphi(n)q^{n}}{1-q^{n}}=\frac{q}{(1-q)^{2}}
  88. lim sup φ ( n ) n = 1 , \lim\sup\frac{\varphi(n)}{n}=1,
  89. φ ( n ) n 1 - δ . \frac{\varphi(n)}{n^{1-\delta}}\rightarrow\infty.
  90. 6 π 2 < φ ( n ) σ ( n ) n 2 < 1 , \frac{6}{\pi^{2}}<\frac{\varphi(n)\sigma(n)}{n^{2}}<1,
  91. lim inf φ ( n ) n log log n = e - γ . \lim\inf\frac{\varphi(n)}{n}\log\log n=e^{-\gamma}.
  92. lim inf φ ( n ) n = 0. \lim\inf\frac{\varphi(n)}{n}=0.
  93. φ ( n ) > n e γ log log n + 3 log log n \varphi(n)>\frac{n}{e^{\gamma}\;\log\log n+\frac{3}{\log\log n}}
  94. φ ( n ) < n e γ log log n \varphi(n)<\frac{n}{e^{\gamma}\log\log n}
  95. φ ( 1 ) + φ ( 2 ) + + φ ( n ) = 3 n 2 π 2 + O ( n ( log n ) 2 / 3 ( log log n ) 4 / 3 ) ( n ) , \varphi(1)+\varphi(2)+\cdots+\varphi(n)=\frac{3n^{2}}{\pi^{2}}+O\left(n(\log n% )^{2/3}(\log\log n)^{4/3}\right)\ \ (n\rightarrow\infty),
  96. 6 π 2 . \tfrac{6}{\pi^{2}}.
  97. lim inf φ ( n + 1 ) φ ( n ) = 0 \lim\inf\frac{\varphi(n+1)}{\varphi(n)}=0
  98. lim sup φ ( n + 1 ) φ ( n ) = . \lim\sup\frac{\varphi(n+1)}{\varphi(n)}=\infty.
  99. { φ ( n + 1 ) φ ( n ) , n = 1 , 2 , } \left\{\frac{\varphi(n+1)}{\varphi(n)},\;\;n=1,2,\cdots\right\}
  100. { φ ( n ) n , n = 1 , 2 , } \left\{\frac{\varphi(n)}{n},\;\;n=1,2,\cdots\right\}
  101. x log x exp ( ( C + o ( 1 ) ) ( log log log x ) 2 ) \frac{x}{\log x}\exp\left({(C+o(1))(\log\log\log x)^{2}}\right)
  102. | { n : ϕ ( n ) x } | = ζ ( 2 ) ζ ( 3 ) ζ ( 6 ) x + R ( x ) |\{n:\phi(n)\leq x\}|=\frac{\zeta(2)\zeta(3)}{\zeta(6)}\cdot x+R(x)
  103. x / ( log x ) k x/(\log x)^{k}
  104. [ 1 , n ] [1,n]
  105. e e
  106. φ ( n ) \varphi(n)\;
  107. ϕ ( n ) \phi(n)\;
  108. n = 0 n=0
  109. φ ( n ) φ(n)
  110. n = 0 n=0

Euler_number.html

  1. 1 cosh t = 2 e t + e - t = n = 0 E n n ! t n \frac{1}{\cosh t}=\frac{2}{e^{t}+e^{-t}}=\sum_{n=0}^{\infty}\frac{E_{n}}{n!}% \cdot t^{n}\!
  2. E 2 n = i k = 1 2 n + 1 j = 0 k ( k j ) ( - 1 ) j ( k - 2 j ) 2 n + 1 2 k i k k E_{2n}=i\sum_{k=1}^{2n+1}\sum_{j=0}^{k}{k\choose j}\frac{(-1)^{j}(k-2j)^{2n+1}% }{2^{k}i^{k}k}
  3. E 2 n = ( 2 n ) ! 0 k 1 , , k n n ( K k 1 , , k n ) δ n , m k m ( - 1 2 ! ) k 1 ( - 1 4 ! ) k 2 ( - 1 ( 2 n ) ! ) k n , E_{2n}=(2n)!\sum_{0\leq k_{1},\ldots,k_{n}\leq n}~{}\left(\begin{array}[]{c}K% \\ k_{1},\ldots,k_{n}\end{array}\right)\delta_{n,\sum mk_{m}}\left(\frac{-1~{}}{2% !}\right)^{k_{1}}\left(\frac{-1~{}}{4!}\right)^{k_{2}}\cdots\left(\frac{-1~{}}% {(2n)!}\right)^{k_{n}},
  4. E 2 n = ( - 1 ) n - 1 ( 2 n - 1 ) ! 0 k 1 , , k n 2 n - 1 ( K k 1 , , k n ) δ 2 n - 1 , ( 2 m - 1 ) k m ( - 1 1 ! ) k 1 ( 1 3 ! ) k 2 ( ( - 1 ) n ( 2 n - 1 ) ! ) k n , E_{2n}=(-1)^{n-1}(2n-1)!\sum_{0\leq k_{1},\ldots,k_{n}\leq 2n-1}\left(\begin{% array}[]{c}K\\ k_{1},\ldots,k_{n}\end{array}\right)\delta_{2n-1,\sum(2m-1)k_{m}}\left(\frac{-% 1~{}}{1!}\right)^{k_{1}}\left(\frac{1}{3!}\right)^{k_{2}}\cdots\left(\frac{(-1% )^{n}}{(2n-1)!}\right)^{k_{n}},
  5. K = k 1 + + k n K=k_{1}+\cdots+k_{n}
  6. ( K k 1 , , k n ) K ! k 1 ! k n ! \left(\begin{array}[]{c}K\\ k_{1},\ldots,k_{n}\end{array}\right)\equiv\frac{K!}{k_{1}!\cdots k_{n}!}
  7. 2 k 1 + 4 k 2 + + 2 n k n = 2 n 2k_{1}+4k_{2}+\cdots+2nk_{n}=2n
  8. k 1 + 3 k 2 + + ( 2 n - 1 ) k n = 2 n - 1 k_{1}+3k_{2}+\cdots+(2n-1)k_{n}=2n-1
  9. E 10 = 10 ! ( - 1 10 ! + 2 2 ! 8 ! + 2 4 ! 6 ! - 3 2 ! 2 6 ! - 3 2 ! 4 ! 2 + 4 2 ! 3 4 ! - 1 2 ! 5 ) = 9 ! ( - 1 9 ! + 3 1 ! 2 7 ! + 6 1 ! 3 ! 5 ! + 1 3 ! 3 - 5 1 ! 4 5 ! - 10 1 ! 3 3 ! 2 + 7 1 ! 6 3 ! - 1 1 ! 9 ) = - 50 , 521. \begin{aligned}\displaystyle E_{10}&\displaystyle=10!\left(-\frac{1}{10!}+% \frac{2}{2!8!}+\frac{2}{4!6!}-\frac{3}{2!^{2}6!}-\frac{3}{2!4!^{2}}+\frac{4}{2% !^{3}4!}-\frac{1}{2!^{5}}\right)\\ &\displaystyle=9!\left(-\frac{1}{9!}+\frac{3}{1!^{2}7!}+\frac{6}{1!3!5!}+\frac% {1}{3!^{3}}-\frac{5}{1!^{4}5!}-\frac{10}{1!^{3}3!^{2}}+\frac{7}{1!^{6}3!}-% \frac{1}{1!^{9}}\right)\\ &\displaystyle=-50,521.\end{aligned}
  10. E 2 n = ( - 1 ) n ( 2 n ) ! | 1 2 ! 1 1 4 ! 1 2 ! 1 1 ( 2 n - 2 ) ! 1 ( 2 n - 4 ) ! 1 2 ! 1 1 ( 2 n ) ! 1 ( 2 n - 2 ) ! 1 4 ! 1 2 ! | . \begin{aligned}\displaystyle E_{2n}&\displaystyle=(-1)^{n}(2n)!~{}\begin{% vmatrix}\frac{1}{2!}&1&&&\\ \frac{1}{4!}&\frac{1}{2!}&1&&\\ \vdots&&\ddots&\ddots&\\ \frac{1}{(2n-2)!}&\frac{1}{(2n-4)!}&&\frac{1}{2!}&1\\ \frac{1}{(2n)!}&\frac{1}{(2n-2)!}&\cdots&\frac{1}{4!}&\frac{1}{2!}\end{vmatrix% }.\end{aligned}
  11. | E 2 n | > 8 n π ( 4 n π e ) 2 n . |E_{2n}|>8\sqrt{\frac{n}{\pi}}\left(\frac{4n}{\pi e}\right)^{2n}.
  12. sec x + tan x \sec x+\tan x
  13. n = 0 A n n ! x n \sum_{n=0}^{\infty}\frac{A_{n}}{n!}x^{n}
  14. A n A_{n}
  15. A n A_{n}
  16. ( - 1 ) n 2 E n (-1)^{\frac{n}{2}}E_{n}
  17. E n E_{n}
  18. A n A_{n}
  19. ( - 1 ) n - 1 2 2 n + 1 ( 2 n + 1 - 1 ) B n + 1 n + 1 (-1)^{\frac{n-1}{2}}\frac{2^{n+1}(2^{n+1}-1)B_{n+1}}{n+1}
  20. B n B_{n}

Euler–Maclaurin_formula.html

  1. m m
  2. n n
  3. f ( x ) f(x)
  4. x x
  5. [ m , n ] [m,n]
  6. I = m n f ( x ) d x I=\int_{m}^{n}f(x)\,dx
  7. S = f ( m + 1 ) + + f ( n - 1 ) + f ( n ) S=f\left(m+1\right)+\cdots+f\left(n-1\right)+f(n)
  8. x = m x=m
  9. n n
  10. p p
  11. f ( x ) f(x)
  12. p p
  13. [ m , n ] [m,n]
  14. S - I = k = 1 p B k k ! ( f ( k - 1 ) ( n ) - f ( k - 1 ) ( m ) ) + R S-I=\sum_{k=1}^{p}{\frac{B_{k}}{k!}\left(f^{(k-1)}(n)-f^{(k-1)}(m)\right)}+R
  15. k k
  16. R R
  17. p p
  18. n , m , p n,m,p
  19. f f
  20. i = m + 1 n f ( i ) = m n f ( x ) d x + B 1 ( f ( n ) - f ( m ) ) + k = 1 p B 2 k ( 2 k ) ! ( f ( 2 k - 1 ) ( n ) - f ( 2 k - 1 ) ( m ) ) + R . \sum_{i=m+1}^{n}f(i)=\int^{n}_{m}f(x)\,dx+B_{1}\left(f(n)-f(m)\right)+\sum_{k=% 1}^{p}\frac{B_{2k}}{(2k)!}\left(f^{(2k-1)}(n)-f^{(2k-1)}(m)\right)+R.
  21. [ r , r + 1 ] [r,r+1]
  22. r = m , m + 1 , , n - 1 r=m,m+1,\cdots,n-1
  23. P k ( x ) P_{k}(x)
  24. B k ( x ) B_{k}(x)
  25. k = 0 , 1 , 2 k=0,1,2\cdots
  26. B 0 ( x ) = 1 B n ( x ) = n B n - 1 ( x ) and 0 1 B n ( x ) d x = 0 for n 1 \begin{aligned}\displaystyle B_{0}(x)&\displaystyle=1\\ \displaystyle B_{n}^{\prime}(x)&\displaystyle=nB_{n-1}(x)\,\text{ and }\int_{0% }^{1}B_{n}(x)\,dx=0\,\text{ for }n\geq 1\end{aligned}
  27. P n ( x ) = B n ( x - x ) P_{n}(x)=B_{n}\left(x-\lfloor x\rfloor\right)
  28. x ⌊x⌋
  29. x x
  30. x - x x-⌊x⌋
  31. 0 , 1 0,1
  32. B k ( 1 ) = B k ( 0 ) B_{k}(1)=B_{k}(0)
  33. k 1 k\neq 1
  34. P 1 ( x ) P_{1}(x)
  35. P k ( x ) P_{k}(x)
  36. B ~ k ( x ) \tilde{B}_{k}(x)
  37. R R
  38. R = m n f ( 2 p ) ( x ) P 2 p ( x ) ( 2 p ) ! d x R=\int_{m}^{n}f^{(2p)}(x){P_{2p}(x)\over(2p)!}\,dx
  39. R = - m n f ( 2 p + 1 ) ( x ) P ( 2 p + 1 ) ( x ) ( 2 p + 1 ) ! d x p > 0 R=-\int_{m}^{n}f^{(2p+1)}(x){P_{(2p+1)}(x)\over(2p+1)!}\,dx\quad p>0
  40. | B n ( x ) | 2 n ! ( 2 π ) n ζ ( n ) \left|B_{n}\left(x\right)\right|\leq\frac{2\cdot n!}{\left(2\pi\right)^{n}}% \zeta\left(n\right)
  41. ζ ζ
  42. n n
  43. x x
  44. ζ ( n ) ζ(n)
  45. n n
  46. | R | 2 ζ ( 2 p ) ( 2 π ) 2 p m n | f ( 2 p ) ( x ) | d x \left|R\right|\leq\frac{2\zeta(2p)}{(2\pi)^{2p}}\int_{m}^{n}\left|f^{(2p)}(x)% \right|\ \,dx
  47. I = x 0 x N f ( x ) d x = h ( f 0 2 + f 1 + f 2 + f N - 1 + f N 2 ) + h 2 12 [ f 0 - f N ] - h 4 720 [ f 0 ′′′ - f N ′′′ ] + I=\int_{x_{0}}^{x_{N}}f(x)\,dx=h\left(\frac{f_{0}}{2}+f_{1}+f_{2}...+f_{N-1}+% \frac{f_{N}}{2}\right)+\frac{h^{2}}{12}[f^{\prime}_{0}-f^{\prime}_{N}]-\frac{h% ^{4}}{720}[f^{\prime\prime\prime}_{0}-f^{\prime\prime\prime}_{N}]+...
  48. N N
  49. x 0 x_{0}
  50. x N x_{N}
  51. h h
  52. h = ( x N - x 0 ) / N h=(x_{N}-x_{0})/N
  53. 1 + 1 4 + 1 9 + 1 16 + 1 25 + = n = 1 1 n 2 1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}+\cdots=\sum_{n=1}^{\infty}% \frac{1}{n^{2}}
  54. i = 0 n i 3 = ( n ( n + 1 ) 2 ) 2 \sum_{i=0}^{n}i^{3}=\left(\frac{n(n+1)}{2}\right)^{2}
  55. n = a b f ( n ) a b f ( x ) d x + f ( b ) + f ( a ) 2 + k = 1 B 2 k ( 2 k ) ! ( f ( 2 k - 1 ) ( b ) - f ( 2 k - 1 ) ( a ) ) \sum_{n=a}^{b}f(n)\sim\int_{a}^{b}f(x)\,dx+\frac{f(b)+f(a)}{2}+\sum_{k=1}^{% \infty}\,\frac{B_{2k}}{(2k)!}\left(f^{(2k-1)}(b)-f^{(2k-1)}(a)\right)
  56. a - {\scriptstyle a\to-\infty}
  57. b + {\scriptstyle b\to+\infty}
  58. k = 0 1 ( z + k ) 2 0 1 ( z + k ) 2 d k = 1 z + 1 2 z 2 + t = 1 B 2 t z 2 t + 1 \sum_{k=0}^{\infty}\frac{1}{(z+k)^{2}}\sim\underbrace{\int_{0}^{\infty}\frac{1% }{(z+k)^{2}}\,dk}_{=\frac{1}{z}}+\frac{1}{2z^{2}}+\sum_{t=1}^{\infty}\frac{B_{% 2t}}{z^{2t+1}}
  59. ψ ( 1 ) ( z ) {\scriptstyle\psi^{(1)}(z)}
  60. ψ ( 1 ) ( z ) = d 2 d z 2 ln Γ ( z ) {\scriptstyle\psi^{(1)}(z)=\frac{d^{2}}{dz^{2}}\ln\Gamma(z)}
  61. Γ ( z ) {\scriptstyle\Gamma(z)}
  62. ( z - 1 ) ! {\scriptstyle(z-1)!}
  63. z {\scriptstyle z}
  64. ψ ( 1 ) ( z ) {\scriptstyle\psi^{(1)}(z)}
  65. k = 1 n 1 k s = 1 n s - 1 + s 1 n x x s + 1 d x with s \R { 1 } \sum_{k=1}^{n}\frac{1}{k^{s}}=\frac{1}{n^{s-1}}+s\int_{1}^{n}\frac{\lfloor x% \rfloor}{x^{s+1}}dx\qquad\,\text{with }\quad s\in\R\setminus\{1\}
  66. k = 1 n 1 k = log n + 1 2 + 1 2 n - 1 n x - x x 2 d x \sum_{k=1}^{n}\frac{1}{k}=\log n+\frac{1}{2}+\frac{1}{2n}-\int_{1}^{n}\frac{x-% \lfloor x\rfloor}{x^{2}}dx
  67. n = 0 , 1 , 2 , n=0,1,2,...
  68. B 1 ( x ) = x - 1 2 B 2 ( x ) = x 2 - x + 1 6 B 3 ( x ) = x 3 - 3 2 x 2 + 1 2 x B 4 ( x ) = x 4 - 2 x 3 + x 2 - 1 30 \begin{aligned}\displaystyle B_{1}(x)&\displaystyle=x-\frac{1}{2}\\ \displaystyle B_{2}(x)&\displaystyle=x^{2}-x+\frac{1}{6}\\ \displaystyle B_{3}(x)&\displaystyle=x^{3}-\frac{3}{2}x^{2}+\frac{1}{2}x\\ \displaystyle B_{4}(x)&\displaystyle=x^{4}-2x^{3}+x^{2}-\frac{1}{30}\\ &\displaystyle\vdots\end{aligned}
  69. n 1 n≠ 1
  70. B n ( 0 ) = B n ( 1 ) = B n ( n th Bernoulli number ) B_{n}(0)=B_{n}(1)=B_{n}\quad(n\,\text{th Bernoulli number})
  71. n = 1 n=1
  72. B 1 ( 0 ) = - B 1 ( 1 ) = B 1 B_{1}(0)=-B_{1}(1)=B_{1}
  73. 0 , 1 0, 1
  74. n = 1 n=1
  75. P n ( 0 ) = P n ( 1 ) = B n , n 1 P_{n}(0)=P_{n}(1)=B_{n},n\neq 1
  76. k k
  77. k k + 1 f ( x ) d x = k k + 1 u d v \int_{k}^{k+1}f(x)\,dx=\int_{k}^{k+1}u\,dv
  78. u = f ( x ) d u = f ( x ) d x d v = P 0 ( x ) d x since P 0 ( x ) = 1 v = P 1 ( x ) \begin{aligned}\displaystyle u&\displaystyle=f(x)\\ \displaystyle du&\displaystyle=f^{\prime}(x)\,dx\\ \displaystyle dv&\displaystyle=P_{0}(x)\,dx&&\displaystyle\,\text{since }P_{0}% (x)=1\\ \displaystyle v&\displaystyle=P_{1}(x)\end{aligned}
  79. k k + 1 f ( x ) d x = [ u v ] k k + 1 - k k + 1 v d u = [ f ( x ) P 1 ( x ) ] k k + 1 - k k + 1 f ( x ) P 1 ( x ) d x = B 1 ( 1 ) f ( k + 1 ) - B 1 ( 0 ) f ( k ) - k k + 1 f ( x ) P 1 ( x ) d x \begin{aligned}\displaystyle\int_{k}^{k+1}f(x)\,dx&\displaystyle=\Big[uv\Big]_% {k}^{k+1}-\int_{k}^{k+1}v\,du\\ &\displaystyle=\Big[f(x)P_{1}(x)\Big]_{k}^{k+1}-\int_{k}^{k+1}f^{\prime}(x)P_{% 1}(x)\,dx\\ &\displaystyle=B_{1}(1)f(k+1)-B_{1}(0)f(k)-\int_{k}^{k+1}f^{\prime}(x)P_{1}(x)% \,dx\end{aligned}
  80. 0 1 f ( x ) d x + + n - 1 n f ( x ) d x = 0 n f ( x ) d x = f ( 0 ) 2 + f ( 1 ) + + f ( n - 1 ) + f ( n ) 2 - 0 n f ( x ) P 1 ( x ) d x \begin{aligned}\displaystyle\int_{0}^{1}f(x)\,dx+\cdots+\int_{n-1}^{n}f(x)\,dx% &\displaystyle=\int_{0}^{n}f(x)\,dx\\ &\displaystyle=\frac{f(0)}{2}+f(1)+\cdots+f(n-1)+{f(n)\over 2}-\int_{0}^{n}f^{% \prime}(x)P_{1}(x)\,dx\end{aligned}
  81. k = 1 n f ( k ) = 0 n f ( x ) d x + f ( n ) - f ( 0 ) 2 + 0 n f ( x ) P 1 ( x ) d x ( 1 ) \sum_{k=1}^{n}f(k)=\int_{0}^{n}f(x)\,dx+{f(n)-f(0)\over 2}+\int_{0}^{n}f^{% \prime}(x)P_{1}(x)\,dx\qquad(1)
  82. k k + 1 f ( x ) P 1 ( x ) d x = k k + 1 u d v \int_{k}^{k+1}f^{\prime}(x)P_{1}(x)\,dx=\int_{k}^{k+1}u\,dv
  83. u = f ( x ) d u = f ′′ ( x ) d x d v = P 1 ( x ) d x v = 1 2 P 2 ( x ) \begin{aligned}\displaystyle u&\displaystyle=f^{\prime}(x)\\ \displaystyle du&\displaystyle=f^{\prime\prime}(x)\,dx\\ \displaystyle dv&\displaystyle=P_{1}(x)\,dx\\ \displaystyle v&\displaystyle=\frac{1}{2}P_{2}(x)\end{aligned}
  84. [ u v ] k k + 1 - k k + 1 v d u = [ f ( x ) P 2 ( x ) 2 ] k k + 1 - 1 2 k k + 1 f ′′ ( x ) P 2 ( x ) d x = B 2 2 ( f ( k + 1 ) - f ( k ) ) - 1 2 k k + 1 f ′′ ( x ) P 2 ( x ) d x \begin{aligned}\displaystyle\Big[uv\Big]_{k}^{k+1}-\int_{k}^{k+1}v\,du&% \displaystyle=\left[{f^{\prime}(x)P_{2}(x)\over 2}\right]_{k}^{k+1}-{1\over 2}% \int_{k}^{k+1}f^{\prime\prime}(x)P_{2}(x)\,dx\\ &\displaystyle={B_{2}\over 2}(f^{\prime}(k+1)-f^{\prime}(k))-{1\over 2}\int_{k% }^{k+1}f^{\prime\prime}(x)P_{2}(x)\,dx\end{aligned}
  85. k = 1 n f ( k ) = 0 n f ( x ) d x + f ( n ) - f ( 0 ) 2 + B 2 2 ( f ( n ) - f ( 0 ) ) - 1 2 0 n f ′′ ( x ) P 2 ( x ) d x . \sum_{k=1}^{n}f(k)=\int_{0}^{n}f(x)\,dx+{f(n)-f(0)\over 2}+\frac{B_{2}}{2}(f^{% \prime}(n)-f^{\prime}(0))-{1\over 2}\int_{0}^{n}f^{\prime\prime}(x)P_{2}(x)\,dx.

Europa_(moon).html

  1. × 10 2 1 \times 10^{2}1
  2. × 10 2 1 \times 10^{2}1

Evaporation.html

  1. ln ( P 2 P 1 ) = - Δ H v a p R ( 1 T 2 - 1 T 1 ) \ln\left(\frac{P_{2}}{P_{1}}\right)=-\frac{\Delta H_{vap}}{R}\left(\frac{1}{T_% {2}}-\frac{1}{T_{1}}\right)

Event_(probability_theory).html

  1. P ( A ) = | A | | Ω | ( alternatively: Pr ( A ) = | A | | Ω | ) \mathrm{P}(A)=\frac{|A|}{|\Omega|}\,\ \left(\,\text{alternatively:}\ \Pr(A)=% \frac{|A|}{|\Omega|}\right)
  2. { ω Ω u < X ( ω ) v } \{\omega\in\Omega\mid u<X(\omega)\leq v\}\,
  3. u < X v . u<X\leq v\,.
  4. Pr ( u < X v ) = F ( v ) - F ( u ) . \Pr(u<X\leq v)=F(v)-F(u)\,.
  5. u < X ( ω ) v u<X(\omega)\leq v

Evolutionary_psychology.html

  1. r b > c rb>c
  2. c c
  3. b b
  4. r r

Examples_of_differential_equations.html

  1. d y d x = f ( x ) g ( y ) \frac{dy}{dx}=f(x)g(y)
  2. d y g ( y ) = f ( x ) d x \frac{dy}{g(y)}=f(x)dx
  3. d y g ( y ) = f ( x ) d x \int\frac{dy}{g(y)}=\int f(x)dx
  4. g ( y ) g(y)
  5. y = c o n s t y=const
  6. g ( y ) = 0 g(y)=0
  7. d y d t + f ( t ) y = 0 \frac{dy}{dt}+f(t)y=0
  8. f ( t ) f(t)
  9. d y y = - f ( t ) d t \frac{dy}{y}=-f(t)\,dt
  10. ln | y | = ( - f ( t ) d t ) + C \ln|y|=\left(-\int f(t)\,dt\right)+C\,
  11. y = ± e ( - f ( t ) d t ) + C = ± e C e - f ( t ) d t y=\pm e^{\left(-\int f(t)\,dt\right)+C}=\pm e^{C}e^{-\int f(t)\,dt}
  12. e C > 0 e^{C}>0
  13. ± e C 0 \pm e^{C}\neq 0
  14. y = A e - f ( t ) d t y=Ae^{-\int f(t)\,dt}
  15. d y d t + f ( t ) y = - f ( t ) A e - f ( t ) d t + f ( t ) A e - f ( t ) d t = 0 \frac{dy}{dt}+f(t)y=-f(t)\cdot Ae^{-\int f(t)\,dt}+f(t)\cdot Ae^{-\int f(t)\,% dt}=0
  16. f ( t ) = α f(t)=\alpha
  17. y = A e - α t y=Ae^{-\alpha t}
  18. α > 0 \alpha>0
  19. α \alpha
  20. d y d t + α y = 0 , y ( 1 ) = 2 , y ( 2 ) = 1 \frac{dy}{dt}+\alpha y=0,y(1)=2,y(2)=1
  21. α = l n ( 2 ) \alpha=ln(2)
  22. y = 4 e - l n ( 2 ) t = 2 2 - t y=4e^{-ln(2)t}=2^{2-t}
  23. d y d x + p ( x ) y = q ( x ) \frac{dy}{dx}+p(x)y=q(x)
  24. μ = e x 0 x p ( t ) d t \mu=e^{\int_{x_{0}}^{x}p(t)\,dt}
  25. d μ d x = e x 0 x p ( t ) d t p ( x ) = μ p ( x ) \frac{d{\mu}}{dx}=e^{\int_{x_{0}}^{x}p(t)\,dt}\cdot p(x)=\mu p(x)
  26. μ d y d x + μ p ( x ) y = μ q ( x ) \mu{\frac{dy}{dx}}+\mu{p(x)y}=\mu{q(x)}
  27. μ d y d x + y d μ d x = μ q ( x ) \mu{\frac{dy}{dx}}+y{\frac{d{\mu}}{dx}}=\mu{q(x)}
  28. d d x ( μ y ) = μ q ( x ) \frac{d}{dx}{(\mu{y})}=\mu{q(x)}
  29. μ y = ( μ q ( x ) d x ) + C \mu{y}=\left(\int\mu q(x)\,dx\right)+C
  30. μ \mu
  31. y = ( μ q ( x ) d x ) + C μ y=\frac{\left(\int\mu q(x)\,dx\right)+C}{\mu}
  32. m d 2 x d t 2 + k x = 0 , m\frac{d^{2}x}{dt^{2}}+kx=0,
  33. C e λ t Ce^{\lambda t}
  34. λ 2 + 1 = 0 \lambda^{2}+1=0
  35. λ \lambda
  36. i i
  37. - i -i
  38. x ( t ) = A cos t + B sin t x(t)=A\cos t+B\sin t
  39. x ( 0 ) = A cos 0 + B sin 0 = A = 1 , x(0)=A\cos 0+B\sin 0=A=1,\,
  40. x ( 0 ) = - A sin 0 + B cos 0 = B = 0 , x^{\prime}(0)=-A\sin 0+B\cos 0=B=0,\,
  41. m d 2 x d t 2 + c d x d t + k x = 0 , m\frac{d^{2}x}{dt^{2}}+c\frac{dx}{dt}+kx=0,
  42. c c
  43. C e λ t Ce^{\lambda t}
  44. m λ 2 + c λ + k = 0. m\lambda^{2}+c\lambda+k=0.\,
  45. c 2 < 4 k m c^{2}<4km
  46. x ( t ) = e a t ( cos b t - a b sin b t ) x(t)=e^{at}\left(\cos bt-\frac{a}{b}\sin bt\right)
  47. m = 1 m=1
  48. 0 < c = - 2 a 0<c=-2a
  49. k = a 2 + b 2 k=a^{2}+b^{2}
  50. y 1 = y 1 + 2 y 2 + t y_{1}^{\prime}=y_{1}+2y_{2}+t
  51. y 2 = 2 y 1 - 2 y 2 + sin ( t ) y_{2}^{\prime}=2y_{1}-2y_{2}+\sin(t)

Exciton.html

  1. f ex f_{\rm ex}
  2. Φ ex 2 ( s y m b o l r e , s y m b o l r e ) \Phi^{2}_{\rm ex}(symbol{r}_{e},symbol{r}_{e})
  3. Φ ex ( s y m b o l r e , s y m b o l r h ) \Phi_{\rm ex}(symbol{r}_{e},symbol{r}_{h})
  4. s y m b o l r e symbol{r}_{e}
  5. s y m b o l r h symbol{r}_{h}
  6. f ex f_{\rm ex}
  7. f i f_{i}
  8. f ex f_{\rm ex}
  9. Φ ex ( s y m b o l r e , s y m b o l r h ) \Phi_{\rm ex}(symbol{r}_{e},symbol{r}_{h})
  10. f i f_{i}
  11. f i = 8 ( μ m E ex E i ) 3 / 2 π a ex 3 v f ex f_{i}=8\left(\frac{\mu}{m}\frac{E_{\rm ex}}{E_{i}}\right)^{3/2}\frac{\pi a^{3}% _{\rm ex}}{v}f_{\rm ex}
  12. m = m e + m h m=m_{e}+m_{h}
  13. μ = ( m e - 1 + m h - 1 ) - 1 \mu=(m_{e}^{-1}+m_{h}^{-1})^{-1}
  14. E ex E_{\rm ex}
  15. E i E_{i}
  16. a ex a_{\rm ex}
  17. v v
  18. f i f_{i}
  19. f ex f_{\rm ex}
  20. π a ex 3 v \pi a^{3}_{\rm ex}\gg v

Expander_graph.html

  1. h ( G ) = min 0 < | S | n 2 | S | | S | , h(G)=\min_{0<|S|\leq\frac{n}{2}}\frac{|\partial S|}{|S|},
  2. h out ( G ) h_{\,\text{out}}(G)
  3. h out ( G ) = min 0 < | S | n 2 | out ( S ) | | S | , h_{\,\text{out}}(G)=\min_{0<|S|\leq\frac{n}{2}}\frac{|\partial_{\,\text{out}}(% S)|}{|S|},
  4. out ( S ) \partial_{\,\text{out}}(S)
  5. V ( G ) S V(G)\setminus S
  6. out ( S ) \partial_{\,\text{out}}(S)
  7. h in ( G ) h_{\,\text{in}}(G)
  8. h in ( G ) = min 0 < | S | n 2 | in ( S ) | | S | , h_{\,\text{in}}(G)=\min_{0<|S|\leq\frac{n}{2}}\frac{|\partial_{\,\text{in}}(S)% |}{|S|},
  9. in ( S ) \partial_{\,\text{in}}(S)
  10. V ( G ) S V(G)\setminus S
  11. A i j A_{ij}
  12. λ 1 λ 2 λ n \lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}
  13. u \R n u\in\R^{n}
  14. u i = 1 / n u_{i}=1/n
  15. λ = max { | λ 2 | , | λ n | } \lambda=\max\{|\lambda_{2}|,|\lambda_{n}|\}
  16. λ = max v u , v 0 A v 2 v 2 , \lambda=\max_{v\perp u,v\neq 0}\frac{\|Av\|_{2}}{\|v\|_{2}},
  17. v 2 = ( i = 1 n v i 2 ) 1 / 2 \|v\|_{2}=\left(\sum_{i=1}^{n}v_{i}^{2}\right)^{1/2}
  18. v \R n v\in\R^{n}
  19. 1 d A \tfrac{1}{d}A
  20. h out ( G ) h ( G ) d h out ( G ) . h_{\,\text{out}}(G)\leq h(G)\leq d\cdot h_{\,\text{out}}(G).
  21. 1 2 ( d - λ 2 ) h ( G ) 2 d ( d - λ 2 ) . \tfrac{1}{2}(d-\lambda_{2})\leq h(G)\leq\sqrt{2d(d-\lambda_{2})}.
  22. h out ( G ) ( 4 ( d - λ 2 ) + 1 ) 2 - 1 h_{\,\text{out}}(G)\leq\left(\sqrt{4(d-\lambda_{2})}+1\right)^{2}-1
  23. h in ( G ) 8 ( d - λ 2 ) . h_{\,\text{in}}(G)\leq\sqrt{8(d-\lambda_{2})}.
  24. h 2 d \frac{h^{2}}{d}
  25. h out h_{\,\text{out}}
  26. h in 2 h_{\,\text{in}}^{2}
  27. O ( d - λ 2 ) O(d-\lambda_{2})
  28. n × n \mathbb{Z}_{n}\times\mathbb{Z}_{n}
  29. n = / n \mathbb{Z}_{n}=\mathbb{Z}/n\mathbb{Z}
  30. ( x , y ) n × n (x,y)\in\mathbb{Z}_{n}\times\mathbb{Z}_{n}
  31. ( x ± 2 y , y ) , ( x ± ( 2 y + 1 ) , y ) , ( x , y ± 2 x ) , ( x , y ± ( 2 x + 1 ) ) . (x\pm 2y,y),(x\pm(2y+1),y),(x,y\pm 2x),(x,y\pm(2x+1)).
  32. λ ( G ) 5 2 \lambda(G)\leq 5\sqrt{2}
  33. λ 2 d - 1 - o ( 1 ) \lambda\geq 2\sqrt{d-1}-o(1)
  34. λ 2 d - 1 \lambda\leq 2\sqrt{d-1}
  35. λ 2 d - 1 + ϵ \lambda\leq 2\sqrt{d-1}+\epsilon
  36. 1 - o ( 1 ) 1-o(1)
  37. λ = max { | λ 2 | , | λ n | } \lambda=\max\{|\lambda_{2}|,|\lambda_{n}|\}
  38. d n | S | | T | \tfrac{d}{n}\cdot|S|\cdot|T|
  39. E ( S , T ) = 2 | E ( G [ S T ] ) | + E ( S T , T ) + E ( S , T S ) E(S,T)=2|E(G[S\cap T])|+E(S\setminus T,T)+E(S,T\setminus S)
  40. | E ( S , T ) - d | S | | T | n | d λ | S | | T | , \left|E(S,T)-\frac{d\cdot|S|\cdot|T|}{n}\right|\leq d\lambda\sqrt{|S|\cdot|T|},

Expected_value.html

  1. E [ X ] = x 1 p 1 + x 2 p 2 + + x k p k . \operatorname{E}[X]=x_{1}p_{1}+x_{2}p_{2}+\cdots+x_{k}p_{k}\;.
  2. E [ X ] = x 1 p 1 + x 2 p 2 + + x k p k 1 = x 1 p 1 + x 2 p 2 + + x k p k p 1 + p 2 + + p k . \operatorname{E}[X]=\frac{x_{1}p_{1}+x_{2}p_{2}+\cdots+x_{k}p_{k}}{1}=\frac{x_% {1}p_{1}+x_{2}p_{2}+\cdots+x_{k}p_{k}}{p_{1}+p_{2}+\cdots+p_{k}}\;.
  3. E [ X ] = 1 1 6 + 2 1 6 + 3 1 6 + 4 1 6 + 5 1 6 + 6 1 6 = 3.5. \operatorname{E}[X]=1\cdot\frac{1}{6}+2\cdot\frac{1}{6}+3\cdot\frac{1}{6}+4% \cdot\frac{1}{6}+5\cdot\frac{1}{6}+6\cdot\frac{1}{6}=3.5.
  4. E [ gain from $ 1 bet ] = - $ 1 37 38 + $ 35 ( = 36 - 1 ) 1 38 = - $ 0.0526. \operatorname{E}[\,\,\text{gain from }\$1\,\text{ bet}\,]=-\$1\cdot\frac{37}{3% 8}\ +\ \$35(=36-1)\cdot\frac{1}{38}=-\$0.0526.
  5. E [ X ] = i = 1 x i p i , \operatorname{E}[X]=\sum_{i=1}^{\infty}x_{i}\,p_{i},
  6. i = 1 x i p i = c ( 1 - 1 2 + 1 3 - 1 4 + ) \sum_{i=1}^{\infty}x_{i}\,p_{i}=c\,\bigg(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}% +\cdots\bigg)
  7. X X
  8. f ( x ) f(x)
  9. E [ X ] = - x f ( x ) d x . \operatorname{E}[X]=\int_{-\infty}^{\infty}xf(x)\,\mathrm{d}x.
  10. E [ X ] = Ω X d P = Ω X ( ω ) P ( d ω ) \operatorname{E}[X]=\int_{\Omega}X\,\mathrm{d}P=\int_{\Omega}X(\omega)P(% \mathrm{d}\omega)
  11. E [ g ( X ) ] = - g ( x ) f ( x ) d x . \operatorname{E}[g(X)]=\int_{-\infty}^{\infty}g(x)f(x)\,\mathrm{d}x.
  12. E [ g ( X ) ] = - g ( x ) dP ( X x ) = { g ( a ) + a g ( x ) P ( X > x ) d x if P ( g ( X ) g ( a ) ) = 1 g ( b ) - - b g ( x ) P ( X x ) d x if P ( g ( X ) g ( b ) ) = 1. \operatorname{E}[g(X)]=\int_{-\infty}^{\infty}g(x)\,\mathrm{d}\mathrm{P}(X\leq x% )=\begin{cases}g(a)+\int_{a}^{\infty}g^{\prime}(x)\mathrm{P}(X>x)\,\mathrm{d}x% &\mathrm{if}\ \mathrm{P}(g(X)\geq g(a))=1\\ g(b)-\int_{-\infty}^{b}g^{\prime}(x)\mathrm{P}(X\leq x)\,\mathrm{d}x&\mathrm{% if}\ \mathrm{P}(g(X)\leq g(b))=1.\end{cases}
  13. E [ | X | α ] = α 0 t α - 1 P ( | X | > t ) d t . \operatorname{E}\left[\left|X\right|^{\alpha}\right]=\alpha\int_{0}^{\infty}t^% {\alpha-1}\mathrm{P}(\left|X\right|>t)\,\mathrm{d}t.
  14. E [ | X | ] = E [ X ] = 0 { 1 - F ( t ) } d t , \operatorname{E}[|X|]=\operatorname{E}[X]=\int_{0}^{\infty}\{1-F(t)\}\,\mathrm% {d}t,
  15. E [ g ( X 1 , , X n ) ] = - - g ( x 1 , , x n ) f ( x 1 , , x n ) d x 1 d x n . \operatorname{E}[g(X_{1},\dots,X_{n})]=\int_{-\infty}^{\infty}\cdots\int_{-% \infty}^{\infty}g(x_{1},\ldots,x_{n})~{}f(x_{1},\ldots,x_{n})~{}\mathrm{d}x_{1% }\cdots\mathrm{d}x_{n}.
  16. E [ X + c ] = E [ X ] + c E [ X + Y ] = E [ X ] + E [ Y ] E [ a X ] = a E [ X ] \begin{aligned}\displaystyle\operatorname{E}[X+c]&\displaystyle=\operatorname{% E}[X]+c\\ \displaystyle\operatorname{E}[X+Y]&\displaystyle=\operatorname{E}[X]+% \operatorname{E}[Y]\\ \displaystyle\operatorname{E}[aX]&\displaystyle=a\operatorname{E}[X]\end{aligned}
  17. E [ a X + b Y + c ] = a E [ X ] + b E [ Y ] + c \operatorname{E}[aX+bY+c]=a\operatorname{E}[X]+b\operatorname{E}[Y]+c\,
  18. E [ X | Y = y ] = x x P ( X = x | Y = y ) . \operatorname{E}[X|Y=y]=\sum\limits_{x}x\cdot\operatorname{P}(X=x|Y=y).
  19. E [ X ] = E [ E [ X | Y ] ] . \operatorname{E}[X]=\operatorname{E}\left[\operatorname{E}[X|Y]\right].
  20. E [ E [ X | Y ] ] \displaystyle\operatorname{E}\left[\operatorname{E}[X|Y]\right]
  21. E [ X ] = E [ E [ X | Y ] ] \operatorname{E}[X]=\operatorname{E}[\operatorname{E}[X|Y]]
  22. | E [ X ] | E [ | X | ] |\operatorname{E}[X]|\leq\operatorname{E}[|X|]
  23. E [ X Y ] = x y j ( x , y ) d x d y . \operatorname{E}[XY]=\iint xy\,j(x,y)\,\mathrm{d}x\,\mathrm{d}y.
  24. Cov ( X , Y ) = E [ X Y ] - E [ X ] E [ Y ] . \operatorname{Cov}(X,Y)=\operatorname{E}[XY]-\operatorname{E}[X]\operatorname{% E}[Y].
  25. E [ X Y ] = x y j ( x , y ) d x d y = x y f ( x ) g ( y ) d y d x = [ x f ( x ) d x ] [ y g ( y ) d y ] = E [ X ] E [ Y ] \begin{aligned}\displaystyle\operatorname{E}[XY]&\displaystyle=\iint xy\,j(x,y% )\,\mathrm{d}x\,\mathrm{d}y=\iint xyf(x)g(y)\,\mathrm{d}y\,\mathrm{d}x\\ &\displaystyle=\left[\int xf(x)\,\mathrm{d}x\right]\left[\int yg(y)\,\mathrm{d% }y\right]=\operatorname{E}[X]\operatorname{E}[Y]\end{aligned}
  26. E [ g ( X ) ] = Ω g ( X ) dP g ( E [ X ] ) , \operatorname{E}[g(X)]=\int_{\Omega}g(X)\,\mathrm{d}\mathrm{P}\neq g(% \operatorname{E}[X]),
  27. P ( X 𝒜 ) = E [ I 𝒜 ( X ) ] \operatorname{P}({X\in\mathcal{A}})=\operatorname{E}[I_{\mathcal{A}}(X)]
  28. I 𝒜 ( X ) I_{\mathcal{A}}(X)
  29. 𝒜 \mathcal{A}
  30. X 𝒜 I 𝒜 ( X ) = 1 , X 𝒜 I 𝒜 ( X ) = 0 X\in\mathcal{A}\rightarrow I_{\mathcal{A}}(X)=1,X\not\in\mathcal{A}\rightarrow I% _{\mathcal{A}}(X)=0
  31. Var ( X ) = E [ X 2 ] - ( E [ X ] ) 2 . \operatorname{Var}(X)=\operatorname{E}[X^{2}]-(\operatorname{E}[X])^{2}.
  32. A ^ \hat{A}
  33. | ψ |\psi\rangle
  34. A ^ = ψ | A | ψ \langle\hat{A}\rangle=\langle\psi|A|\psi\rangle
  35. A ^ \hat{A}
  36. ( Δ A ) 2 = A ^ 2 - A ^ 2 (\Delta A)^{2}=\langle\hat{A}^{2}\rangle-\langle\hat{A}\rangle^{2}
  37. E [ X ] = E [ ( x 1 , 1 x 1 , 2 x 1 , n x 2 , 1 x 2 , 2 x 2 , n x m , 1 x m , 2 x m , n ) ] = ( E [ x 1 , 1 ] E [ x 1 , 2 ] E [ x 1 , n ] E [ x 2 , 1 ] E [ x 2 , 2 ] E [ x 2 , n ] E [ x m , 1 ] E [ x m , 2 ] E [ x m , n ] ) . \operatorname{E}[X]=\operatorname{E}\left[\begin{pmatrix}x_{1,1}&x_{1,2}&% \cdots&x_{1,n}\\ x_{2,1}&x_{2,2}&\cdots&x_{2,n}\\ \vdots&\vdots&\ddots&\vdots\\ x_{m,1}&x_{m,2}&\cdots&x_{m,n}\end{pmatrix}\right]=\begin{pmatrix}% \operatorname{E}[x_{1,1}]&\operatorname{E}[x_{1,2}]&\cdots&\operatorname{E}[x_% {1,n}]\\ \operatorname{E}[x_{2,1}]&\operatorname{E}[x_{2,2}]&\cdots&\operatorname{E}[x_% {2,n}]\\ \vdots&\vdots&\ddots&\vdots\\ \operatorname{E}[x_{m,1}]&\operatorname{E}[x_{m,2}]&\cdots&\operatorname{E}[x_% {m,n}]\end{pmatrix}.
  38. E [ X ] = i = 1 P ( X i ) . \operatorname{E}[X]=\sum\limits_{i=1}^{\infty}P(X\geq i).
  39. i = 1 P ( X i ) = i = 1 j = i P ( X = j ) . \sum\limits_{i=1}^{\infty}\mathrm{P}(X\geq i)=\sum\limits_{i=1}^{\infty}\sum% \limits_{j=i}^{\infty}P(X=j).
  40. i = 1 j = i P ( X = j ) = j = 1 i = 1 j P ( X = j ) = j = 1 j P ( X = j ) = E [ X ] . \begin{aligned}\displaystyle\sum\limits_{i=1}^{\infty}\sum\limits_{j=i}^{% \infty}P(X=j)&\displaystyle=\sum\limits_{j=1}^{\infty}\sum\limits_{i=1}^{j}P(X% =j)\\ &\displaystyle=\sum\limits_{j=1}^{\infty}j\,P(X=j)\\ &\displaystyle=\operatorname{E}[X].\end{aligned}
  41. i = 1 ( 1 - p ) i = 1 p - 1 \sum_{i=1}^{\infty}(1-p)^{i}=\frac{1}{p}-1
  42. i = 1 r i = i = 0 r i - 1 = 1 1 - r - 1 , \sum_{i=1}^{\infty}r^{i}=\sum_{i=0}^{\infty}r^{i}-1=\frac{1}{1-r}-1,
  43. r = 1 - p < 1 r=1-p<1
  44. E [ X ] = 0 P ( X x ) d x \operatorname{E}[X]=\int_{0}^{\infty}P(X\geq x)\;\mathrm{d}x
  45. E [ X ] = 0 ( - x ) ( - f X ( x ) ) d x = [ - x ( 1 - F ( x ) ) ] 0 + 0 ( 1 - F ( x ) ) d x \operatorname{E}[X]=\int_{0}^{\infty}(-x)(-f_{X}(x))\;\mathrm{d}x=\left[-x(1-F% (x))\right]_{0}^{\infty}+\int_{0}^{\infty}(1-F(x))\;\mathrm{d}x
  46. 1 - F ( x ) = o ( 1 x ) 1-F(x)=o\left(\frac{1}{x}\right)
  47. x . x\rightarrow\infty.
  48. 0 P ( X x ) d x = 0 x f X ( t ) d t d x = 0 0 t f X ( t ) d x d t = 0 t f X ( t ) d t = E [ X ] \int_{0}^{\infty}\!\mathrm{P}(X\geq x)\;\mathrm{d}x=\int_{0}^{\infty}\int_{x}^% {\infty}f_{X}(t)\;\mathrm{d}t\;\mathrm{d}x=\int_{0}^{\infty}\int_{0}^{t}f_{X}(% t)\;\mathrm{d}x\;\mathrm{d}t=\int_{0}^{\infty}tf_{X}(t)\;\mathrm{d}t=% \operatorname{E}[X]
  49. E [ X ] = 0 0 x d t d F ( x ) = 0 t d F ( x ) d t = 0 ( 1 - F ( t ) ) d t . \operatorname{E}[X]=\int_{0}^{\infty}\int_{0}^{x}\!\mathrm{d}t\,\mathrm{d}F(x)% =\int_{0}^{\infty}\int_{t}^{\infty}\!\mathrm{d}F(x)\mathrm{d}t=\int_{0}^{% \infty}\!(1-F(t))\,\mathrm{d}t.

Expert_system.html

  1. R 1 : M a n ( x ) = > M o r t a l ( x ) R1:Man(x)=>Mortal(x)

Exponential_distribution.html

  1. f ( x ; λ ) = { λ e - λ x x 0 , 0 x < 0. f(x;\lambda)=\begin{cases}\lambda e^{-\lambda x}&x\geq 0,\\ 0&x<0.\end{cases}
  2. f ( x ; λ ) = λ e - λ x H ( x ) f(x;\lambda)=\mathrm{\lambda}e^{-\lambda x}H(x)
  3. F ( x ; λ ) = { 1 - e - λ x x 0 , 0 x < 0. F(x;\lambda)=\begin{cases}1-e^{-\lambda x}&x\geq 0,\\ 0&x<0.\end{cases}
  4. F ( x ; λ ) = ( 1 - e - λ x ) H ( x ) F(x;\lambda)=\mathrm{(}1-e^{-\lambda x})H(x)
  5. f ( x ; β ) = { 1 β e - x β x 0 , 0 x < 0. f(x;\beta)=\begin{cases}\frac{1}{\beta}e^{-\frac{x}{\beta}}&x\geq 0,\\ 0&x<0.\end{cases}
  6. E [ X ] = 1 λ = β \mathrm{E}[X]=\frac{1}{\lambda}=\beta
  7. Var [ X ] = 1 λ 2 , \mathrm{Var}[X]=\frac{1}{\lambda^{2}},
  8. E [ X n ] = n ! λ n . \mathrm{E}\left[X^{n}\right]=\frac{n!}{\lambda^{n}}.
  9. m [ X ] = ln ( 2 ) λ < E [ X ] , \,\text{m}[X]=\frac{\ln(2)}{\lambda}<\mathrm{E}[X],
  10. | E [ X ] - m [ X ] | = 1 - ln ( 2 ) λ < 1 λ = standard deviation , |\,\text{E}[X]-\,\text{m}[X]|=\frac{1-\ln(2)}{\lambda}<\frac{1}{\lambda}=\,% \text{standard deviation},
  11. Pr ( T > s + t | T > s ) = Pr ( T > t ) , s , t 0. \Pr\left(T>s+t|T>s\right)=\Pr(T>t),\qquad\forall s,t\geq 0.
  12. F - 1 ( p ; λ ) = - ln ( 1 - p ) λ , 0 p < 1 F^{-1}(p;\lambda)=\frac{-\ln(1-p)}{\lambda},\qquad 0\leq p<1
  13. e λ e^{\lambda}
  14. e λ 0 e^{\lambda_{0}}
  15. Δ ( λ 0 | | λ ) = log ( λ 0 ) - log ( λ ) + λ λ 0 - 1. \Delta(\lambda_{0}||\lambda)=\log(\lambda_{0})-\log(\lambda)+\frac{\lambda}{% \lambda_{0}}-1.
  16. [ 0 , ) [0,\infty)
  17. μ \mu
  18. λ = 1 μ \lambda=\tfrac{1}{\mu}
  19. X X
  20. E [ X ] E[X]
  21. min { X 1 , , X n } \min\left\{X_{1},\dots,X_{n}\right\}
  22. λ = λ 1 + + λ n . \lambda=\lambda_{1}+\cdots+\lambda_{n}.
  23. Pr ( min { X 1 , , X n } > x ) \displaystyle\Pr\left(\min\{X_{1},\dots,X_{n}\}>x\right)
  24. Pr ( X k = min { X 1 , , X n } ) = λ k λ 1 + + λ n . \Pr\left(X_{k}=\min\{X_{1},\dots,X_{n}\}\right)=\frac{\lambda_{k}}{\lambda_{1}% +\cdots+\lambda_{n}}.
  25. max { X 1 , , X n } \max\{X_{1},\dots,X_{n}\}
  26. 1 / ( [ sample mean | X ¯ ] ) 1/([\,\text{sample mean}|\bar{X}])
  27. ( n - 1 ) / ( X ¯ * n ) (n-1)/(\bar{X}*n)
  28. ( n - 2 ) / ( X ¯ * n ) (n-2)/(\bar{X}*n)
  29. L ( λ ) = i = 1 n λ exp ( - λ x i ) = λ n exp ( - λ i = 1 n x i ) = λ n exp ( - λ n x ¯ ) , L(\lambda)=\prod_{i=1}^{n}\lambda\exp(-\lambda x_{i})=\lambda^{n}\exp\left(-% \lambda\sum_{i=1}^{n}x_{i}\right)=\lambda^{n}\exp\left(-\lambda n\overline{x}% \right),
  30. x ¯ = 1 n i = 1 n x i \overline{x}={1\over n}\sum_{i=1}^{n}x_{i}
  31. d d λ ln ( L ( λ ) ) = d d λ ( n ln ( λ ) - λ n x ¯ ) = n λ - n x ¯ { > 0 , 0 < λ < 1 x ¯ , = 0 , λ = 1 x ¯ , < 0 , λ > 1 x ¯ . \frac{\mathrm{d}}{\mathrm{d}\lambda}\ln(L(\lambda))=\frac{\mathrm{d}}{\mathrm{% d}\lambda}\left(n\ln(\lambda)-\lambda n\overline{x}\right)=\frac{n}{\lambda}-n% \overline{x}\ \begin{cases}>0,&0<\lambda<\frac{1}{\overline{x}},\\ =0,&\lambda=\frac{1}{\overline{x}},\\ <0,&\lambda>\frac{1}{\overline{x}}.\end{cases}
  32. λ ^ = 1 x ¯ . \widehat{\lambda}=\frac{1}{\overline{x}}.
  33. λ \lambda
  34. x ¯ \overline{x}
  35. 1 / λ = β , 1/\lambda=\beta,
  36. β \beta
  37. 2 n λ ^ χ 1 - α 2 , 2 n 2 < 1 λ < 2 n λ ^ χ α 2 , 2 n 2 \frac{2n}{\widehat{\lambda}\chi^{2}_{1-\frac{\alpha}{2},2n}}<\frac{1}{\lambda}% <\frac{2n}{\widehat{\lambda}\chi^{2}_{\frac{\alpha}{2},2n}}
  38. 2 n x ¯ χ 1 - α 2 , 2 n 2 < 1 λ < 2 n x ¯ χ α 2 , 2 n 2 \frac{2n\overline{x}}{\chi^{2}_{1-\frac{\alpha}{2},2n}}<\frac{1}{\lambda}<% \frac{2n\overline{x}}{\chi^{2}_{\frac{\alpha}{2},2n}}
  39. 100 ( p ) 100(p)
  40. λ l o w = λ ^ ( 1 - 1.96 n ) \lambda_{low}=\widehat{\lambda}\left(1-\frac{1.96}{\sqrt{n}}\right)
  41. λ u p p = λ ^ ( 1 + 1.96 n ) \lambda_{upp}=\widehat{\lambda}\left(1+\frac{1.96}{\sqrt{n}}\right)
  42. Gamma ( λ ; α , β ) = β α Γ ( α ) λ α - 1 exp ( - λ β ) . \mathrm{Gamma}(\lambda;\alpha,\beta)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}% \lambda^{\alpha-1}\exp(-\lambda\beta).
  43. p ( λ ) \displaystyle p(\lambda)
  44. p ( λ ) = Gamma ( λ ; α + n , β + n x ¯ ) . p(\lambda)=\mathrm{Gamma}(\lambda;\alpha+n,\beta+n\overline{x}).
  45. α + n β + n x ¯ . \frac{\alpha+n}{\beta+n\overline{x}}.
  46. T = F - 1 ( U ) T=F^{-1}(U)
  47. F - 1 ( p ) = - ln ( 1 - p ) λ . F^{-1}(p)=\frac{-\ln(1-p)}{\lambda}.
  48. T = - ln ( U ) λ . T=\frac{-\ln(U)}{\lambda}.
  49. lim n n Beta ( 1 , n ) = Exp ( 1 ) . \lim_{n\to\infty}n{\rm Beta}(1,n)={\rm Exp}(1).
  50. i X i \sum_{i}X_{i}
  51. μ - β log ( e - X 1 - e - X ) Logistic ( μ , β ) \mu-\beta\log\left(\tfrac{e^{-X}}{1-e^{-X}}\right)\sim\mathrm{Logistic}(\mu,\beta)
  52. μ - β log ( X Y ) Logistic ( μ , β ) \mu-\beta\log\left(\tfrac{X}{Y}\right)\sim\mathrm{Logistic}(\mu,\beta)
  53. e - X k PowerLaw ( k , λ ) \tfrac{e^{-X}}{k}\sim\mathrm{PowerLaw}(k,\lambda)
  54. X Weibull ( 1 λ , 1 ) X\sim\mathrm{Weibull}(\tfrac{1}{\lambda},1)
  55. lim n n min ( X 1 , , X n ) Exp ( 1 ) \lim_{n\to\infty}n\min\left(X_{1},\ldots,X_{n}\right)\sim\textrm{Exp}(1)
  56. Y Geometric ( 1 1 + λ ) Y\sim\mathrm{Geometric}(\tfrac{1}{1+\lambda})
  57. Y Γ ( α , β α ) Y\sim\Gamma(\alpha,\tfrac{\beta}{\alpha})
  58. X Y K ( α , β ) \sqrt{XY}\sim\mathrm{K}(\alpha,\beta)
  59. X Y Pareto ( 1 , n ) \frac{X}{Y}\sim\mathrm{Pareto}(1,n)
  60. Y Γ ( n , 1 λ ) Y\sim\Gamma(n,\tfrac{1}{\lambda})
  61. X Y Pareto ( 1 , n ) \tfrac{X}{Y}\sim\mathrm{Pareto}(1,n)
  62. Z = λ X X λ Y Y Z=\frac{\lambda_{X}X}{\lambda_{Y}Y}
  63. f Z ( z ) = 1 ( z + 1 ) 2 f_{Z}(z)=\frac{1}{(z+1)^{2}}
  64. λ X λ Y \frac{\lambda_{X}}{\lambda_{Y}}
  65. p ML ( x n + 1 x 1 , , x n ) = ( 1 x ¯ ) exp ( - x n + 1 x ¯ ) p_{\rm ML}(x_{n+1}\mid x_{1},\ldots,x_{n})=\left(\frac{1}{\overline{x}}\right)% \exp\left(-\frac{x_{n+1}}{\overline{x}}\right)
  66. p CNML ( x n + 1 x 1 , , x n ) = n n + 1 ( x ¯ ) n ( n x ¯ + x n + 1 ) n + 1 , p_{\rm CNML}(x_{n+1}\mid x_{1},\ldots,x_{n})=\frac{n^{n+1}\left(\overline{x}% \right)^{n}}{\left(n\overline{x}+x_{n+1}\right)^{n+1}},
  67. x n + 1 / x ¯ {x_{n+1}}/{\overline{x}}
  68. E λ 0 [ Δ ( λ 0 p ML ) ] = ψ ( n ) + 1 n - 1 - log ( n ) E λ 0 [ Δ ( λ 0 p CNML ) ] = ψ ( n ) + 1 n - log ( n ) \begin{aligned}\displaystyle{\rm E}_{\lambda_{0}}\left[\Delta(\lambda_{0}\mid% \mid p_{\rm ML})\right]&\displaystyle=\psi(n)+\frac{1}{n-1}-\log(n)\\ \displaystyle{\rm E}_{\lambda_{0}}\left[\Delta(\lambda_{0}\mid\mid p_{\rm CNML% })\right]&\displaystyle=\psi(n)+\frac{1}{n}-\log(n)\end{aligned}

Exponential_function.html

  1. y = e x y=e^{x}
  2. e x e^{x}\,
  3. ln x \ln x\,
  4. e x e^{x}\,
  5. e x + C e^{x}+C\,
  6. e x = n = 0 x n n ! = 1 + x + x 2 2 ! + x 3 3 ! + x 4 4 ! + e^{x}=\sum_{n=0}^{\infty}{x^{n}\over n!}=1+x+{x^{2}\over 2!}+{x^{3}\over 3!}+{% x^{4}\over 4!}+\cdots
  7. x = 1 y d t t x=\int_{1}^{y}{\mathrm{d}t\over t}
  8. e x = lim n ( 1 + x n ) n e^{x}=\lim_{n\rightarrow\infty}\left(1+\frac{x}{n}\right)^{n}
  9. lim n ( 1 + 1 n ) n \lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}
  10. exp ( x ) = lim n ( 1 + x n ) n \exp(x)=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^{n}
  11. exp ( x + y ) = exp ( x ) exp ( y ) \exp(x+y)=\exp(x)\cdot\exp(y)
  12. d d x e x = e x \frac{\mathrm{d}}{\mathrm{d}x}e^{x}=e^{x}
  13. e x \displaystyle e^{x}
  14. d d x e f ( x ) = f ( x ) e f ( x ) {\mathrm{d}\over\mathrm{d}x}e^{f(x)}=f^{\prime}(x)e^{f(x)}
  15. e x = 1 + x 1 - x x + 2 - 2 x x + 3 - 3 x x + 4 - e^{x}=1+\cfrac{x}{1-\cfrac{x}{x+2-\cfrac{2x}{x+3-\cfrac{3x}{x+4-\ddots}}}}
  16. e z = 1 + 2 z 2 - z + z 2 6 + z 2 10 + z 2 14 + e^{z}=1+\cfrac{2z}{2-z+\cfrac{z^{2}}{6+\cfrac{z^{2}}{10+\cfrac{z^{2}}{14+% \ddots}}}}
  17. x / y {x}/{y}
  18. e x y = 1 + 2 x 2 y - x + x 2 6 y + x 2 10 y + x 2 14 y + e^{\frac{x}{y}}=1+\cfrac{2x}{2y-x+\cfrac{x^{2}}{6y+\cfrac{x^{2}}{10y+\cfrac{x^% {2}}{14y+\ddots}}}}
  19. e 2 = 1 + 4 0 + 2 2 6 + 2 2 10 + 2 2 14 + = 7 + 2 5 + 1 7 + 1 9 + 1 11 + e^{2}=1+\cfrac{4}{0+\cfrac{2^{2}}{6+\cfrac{2^{2}}{10+\cfrac{2^{2}}{14+\ddots\,% }}}}=7+\cfrac{2}{5+\cfrac{1}{7+\cfrac{1}{9+\cfrac{1}{11+\ddots\,}}}}
  20. e 3 = 1 + 6 - 1 + 3 2 6 + 3 2 10 + 3 2 14 + = 13 + 54 7 + 9 14 + 9 18 + 9 22 + e^{3}=1+\cfrac{6}{-1+\cfrac{3^{2}}{6+\cfrac{3^{2}}{10+\cfrac{3^{2}}{14+\ddots% \,}}}}=13+\cfrac{54}{7+\cfrac{9}{14+\cfrac{9}{18+\cfrac{9}{22+\ddots\,}}}}
  21. e z = n = 0 z n n ! e^{z}=\sum_{n=0}^{\infty}\frac{z^{n}}{n!}
  22. 2 π i 2\pi i
  23. e a + b i = e a ( cos b + i sin b ) e^{a+bi}=e^{a}(\cos b+i\sin b)
  24. e z + w = e z e w e^{z+w}=e^{z}e^{w}\,
  25. e 0 = 1 e^{0}=1\,
  26. e z 0 e^{z}\neq 0
  27. d d z e z = e z {\mathrm{d}\over\mathrm{d}z}e^{z}=e^{z}
  28. ( e z ) n = e n z , n \,(e^{z})^{n}=e^{nz},n\in\mathbb{Z}
  29. z w = e w log z z^{w}=e^{w\log z}
  30. ( e z ) w e z w (e^{z})^{w}\neq e^{zw}
  31. ( e z ) w = e ( z + 2 π i n ) w (e^{z})^{w}=e^{(z+2\pi in)w}\,
  32. a b = ( r e θ i ) b = ( e ln ( r ) + θ i ) b = e ( ln ( r ) + θ i ) b a^{b}=(re^{\theta i})^{b}=(e^{\ln(r)+\theta i})^{b}=e^{(\ln(r)+\theta i)b}
  33. lim n ( 1 + x n ) n \textstyle\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^{n}
  34. 𝔤 \mathfrak{g}
  35. 𝔤 G \mathfrak{g}\to G
  36. F ( m ) = 2 2 m + 1 \,F(m)=2^{2^{m}}+1
  37. M M ( p ) = 2 2 p - 1 - 1 \,MM(p)=2^{2^{p}-1}-1

Exponentiation_by_squaring.html

  1. x n = { x ( x 2 ) n - 1 2 , if n is odd ( x 2 ) n 2 , if n is even . x^{n}=\begin{cases}x\,(x^{2})^{\frac{n-1}{2}},&\mbox{if }~{}n\mbox{ is odd}\\ (x^{2})^{\frac{n}{2}},&\mbox{if }~{}n\mbox{ is even}~{}.\end{cases}
  2. log 2 n \lfloor\log_{2}n\rfloor
  3. log 2 n \lfloor\log_{2}n\rfloor
  4. \lfloor\;\rfloor
  5. i = 0 O ( log ( n ) ) ( 2 i O ( log ( x ) ) ) k = O ( ( n log ( x ) ) k ) \sum\limits_{i=0}^{O(\log(n))}(2^{i}O(\log(x)))^{k}=O((n\log(x))^{k})
  6. x 3 , x 5 , , x 2 k - 1 x^{3},x^{5},...,x^{2^{k}-1}
  7. log ( n ) < k ( k + 1 ) 2 2 k 2 k + 1 - k - 2 + 1. \log(n)<\frac{k(k+1)\cdot 2^{2k}}{2^{k+1}-k-2}+1.
  8. x 3 , x 5 , , x 2 k - 1 x^{3},x^{5},...,x^{2^{k}-1}
  9. n = i = 0 w - 1 n i b i n=\sum_{i=0}^{w-1}n_{i}b_{i}
  10. 0 n i < h 0\leqslant n_{i}<h
  11. i [ 0 , w - 1 ] i\in[0,w-1]
  12. x n = i = 0 w - 1 x i n i = j = 1 h - 1 [ n i = j x i ] j x^{n}=\prod_{i=0}^{w-1}{x_{i}}^{n_{i}}=\prod_{j=1}^{h-1}{\bigg[\prod_{n_{i}=j}% x_{i}\bigg]}^{j}
  13. x n x^{n}
  14. 𝐆 \mathbf{G}
  15. n n
  16. x 0 n 0 x 1 n 1 = ( x 0 x 1 q ) n 0 x 1 n 1 mod n 0 {x_{0}}^{n_{0}}\cdot{x_{1}}^{n_{1}}={\left(x_{0}\cdot{x_{1}}^{q}\right)}^{n_{0% }}\cdot{x_{1}}^{n_{1}\mod{n_{0}}}
  17. q = n 1 / n 0 q=\left\lfloor{n_{1}}/{n_{0}}\right\rfloor
  18. q q
  19. n 1 mod n 0 n_{1}\mod n_{0}
  20. x x
  21. 𝐆 \mathbf{G}
  22. n n
  23. x n x^{n}
  24. l l
  25. x b 0 , , x b l i x^{b_{0}},...,x^{b_{l_{i}}}
  26. M [ 0 , l - 1 ] M\in\left[0,l-1\right]
  27. i [ 0 , l - 1 ] , n M n i \forall i\in\left[0,l-1\right],{n_{M}}\geq{n_{i}}
  28. N ( [ 0 , l - 1 ] - M ) N\in\left(\left[0,l-1\right]-M\right)
  29. i ( [ 0 , l - 1 ] - M ) , n N n i \forall i\in\left(\left[0,l-1\right]-M\right),{n_{N}}\geq{n_{i}}
  30. n N = 0 {n_{N}}=0
  31. q = n M / n N q=\left\lfloor{n_{M}}/{n_{N}}\right\rfloor
  32. n N = ( n M mod n N ) {n_{N}}=\left({n_{M}}\mod{n_{N}}\right)
  33. x M q {x_{M}}^{q}
  34. x N = x N x M q {x_{N}}={x_{N}}\cdot{x_{M}}^{q}
  35. x n = x M n M x^{n}={x_{M}}^{n_{M}}
  36. q q
  37. 2 log 2 ( 722340 ) 40 2\log_{2}(722340)\leq 40
  38. n n
  39. b b
  40. n = i = 0 l - 1 n i b i with | n i | < b n=\sum_{i=0}^{l-1}n_{i}b^{i}\,\text{ with }|n_{i}|<b
  41. b = 2 b=2
  42. n i { - 1 , 0 , 1 } n_{i}\in\{-1,0,1\}
  43. ( n l - 1 n 0 ) s (n_{l-1}\dots n_{0})_{s}
  44. n = 478 n=478
  45. ( 10 1 ¯ 1100 1 ¯ 10 ) s (10\bar{1}1100\bar{1}10)_{s}
  46. ( 100 1 ¯ 1000 1 ¯ 0 ) s (100\bar{1}1000\bar{1}0)_{s}
  47. 1 ¯ \bar{1}
  48. - 1 -1
  49. n n
  50. n i n i + 1 = 0 for all i 0 n_{i}n_{i+1}=0\,\text{ for all }i\geqslant 0
  51. ( n l - 1 n 0 ) NAF (n_{l-1}\dots n_{0})_{\,\text{NAF}}
  52. ( 1000 1 ¯ 000 1 ¯ 0 ) NAF (1000\bar{1}000\bar{1}0)_{\,\text{NAF}}
  53. n = ( n l n l - 1 n 0 ) 2 n=(n_{l}n_{l-1}\dots n_{0})_{2}
  54. n l = n l - 1 = 0 n_{l}=n_{l-1}=0
  55. c 0 = 0 c_{0}=0
  56. i = 0 i=0
  57. l - 1 l-1
  58. c i + 1 = 1 2 ( c i + n i + n i + 1 ) c_{i+1}=\left\lfloor\frac{1}{2}(c_{i}+n_{i}+n_{i+1})\right\rfloor
  59. n i = c i + n i - 2 c i + 1 n_{i}^{\prime}=c_{i}+n_{i}-2c_{i+1}
  60. ( n l - 1 n 0 ) NAF (n_{l-1}^{\prime}\dots n_{0}^{\prime})_{\,\text{NAF}}
  61. n i = n i + 1 = 0 n_{i}=n_{i+1}=0
  62. a 15 = x × ( x × [ x × x 2 ] 2 ) 2 a^{15}=x\times(x\times[x\times x^{2}]^{2})^{2}\!
  63. a 15 = x 3 × ( [ x 3 ] 2 ) 2 a^{15}=x^{3}\times([x^{3}]^{2})^{2}\!
  64. x 2 k - 1 x^{2^{k}-1}

EXPSPACE.html

  1. EXPSPACE = k DSPACE ( 2 n k ) = k NSPACE ( 2 n k ) \mbox{EXPSPACE}~{}=\bigcup_{k\in\mathbb{N}}\mbox{DSPACE}~{}(2^{n^{k}})=\bigcup% _{k\in\mathbb{N}}\mbox{NSPACE}~{}(2^{n^{k}})

EXPTIME.html

  1. EXPTIME = k DTIME ( 2 n k ) . \mbox{EXPTIME}~{}=\bigcup_{k\in\mathbb{N}}\mbox{ DTIME }~{}\left(2^{n^{k}}% \right).
  2. \subseteq
  3. \subseteq
  4. \subseteq
  5. \subseteq
  6. \subseteq
  7. \subsetneq
  8. \subsetneq
  9. \subsetneq
  10. \subseteq
  11. 2 2 n 2^{2^{n}}

Extended_periodic_table.html

  1. Es 99 254 + 20 48 Ca 119 302 Uue * \,{}^{254}_{99}\mathrm{Es}+\,^{48}_{20}\mathrm{Ca}\to\,^{302}_{119}\mathrm{Uue% }^{*}
  2. Bk 97 249 + 22 50 Ti 119 296 Uue + 3 0 1 n \,{}^{249}_{97}\mathrm{Bk}+\,^{50}_{22}\mathrm{Ti}\to\,^{296}_{119}\mathrm{Uue% }\,+3\,^{1}_{0}\mathrm{n}
  3. Bk 97 249 + 22 50 Ti 119 295 Uue + 4 0 1 n \,{}^{249}_{97}\mathrm{Bk}+\,^{50}_{22}\mathrm{Ti}\to\,^{295}_{119}\mathrm{Uue% }\,+4\,^{1}_{0}\mathrm{n}
  4. Pu 94 244 + 26 58 Fe 120 302 Ubn * 𝑓𝑖𝑠𝑠𝑖𝑜𝑛 𝑜𝑛𝑙𝑦 \,{}^{244}_{94}\mathrm{Pu}+\,^{58}_{26}\mathrm{Fe}\to\,^{302}_{120}\mathrm{Ubn% }^{*}\to\ \mathit{fission\ only}
  5. U 92 238 + 28 64 Ni 120 302 Ubn * 𝑓𝑖𝑠𝑠𝑖𝑜𝑛 𝑜𝑛𝑙𝑦 \,{}^{238}_{92}\mathrm{U}+\,^{64}_{28}\mathrm{Ni}\to\,^{302}_{120}\mathrm{Ubn}% ^{*}\to\ \mathit{fission\ only}
  6. Cm 96 248 + 24 54 Cr 120 302 Ubn * \,{}^{248}_{96}\mathrm{Cm}+\,^{54}_{24}\mathrm{Cr}\to\,^{302}_{120}\mathrm{Ubn% }^{*}
  7. Cf 98 249 + 22 50 Ti 120 299 Ubn * \,{}^{249}_{98}\mathrm{Cf}+\,^{50}_{22}\mathrm{Ti}\to\,^{299}_{120}\mathrm{Ubn% }^{*}
  8. U 92 238 + 28 n a t Ni 296 , 298 , 299 , 300 , 302 Ubn * 𝑓𝑖𝑠𝑠𝑖𝑜𝑛 . \,{}^{238}_{92}\mathrm{U}+\,^{nat}_{28}\mathrm{Ni}\to\,^{296,298,299,300,302}% \mathrm{Ubn}^{*}\to\ \mathit{fission}.
  9. U 92 238 + 30 66 Zn 122 304 Ubb * no atoms . \,{}^{238}_{92}\mathrm{U}+\,^{66}_{30}\mathrm{Zn}\to\,^{304}_{122}\mathrm{Ubb}% ^{*}\to\ \mbox{no atoms}~{}.
  10. U 92 238 + 30 70 Zn 122 308 Ubb * no atoms . \,{}^{238}_{92}\mathrm{U}+\,^{70}_{30}\mathrm{Zn}\to\,^{308}_{122}\mathrm{Ubb}% ^{*}\to\ \mbox{no atoms}~{}.
  11. Er 68 n a t + 54 136 Xe 298 , 300 , 302 , 303 , 304 , 306 Ubb * no atoms . \,{}^{nat}_{68}\mathrm{Er}+\,^{136}_{54}\mathrm{Xe}\to\,^{298,300,302,303,304,% 306}\mathrm{Ubb}^{*}\to\ \mbox{no atoms}~{}.
  12. U 92 238 + 32 n a t Ge 308 , 310 , 311 , 312 , 314 Ubq * f i s s i o n . \,{}^{238}_{92}\mathrm{U}+\,^{nat}_{32}\mathrm{Ge}\to\,^{308,310,311,312,314}% \mathrm{Ubq}^{*}\to\ fission.
  13. Ta 73 n a t + 54 136 Xe 316 , 317 Ubs * no atoms . \,{}^{nat}_{73}\mathrm{Ta}+\,^{136}_{54}\mathrm{Xe}\to\,^{316,317}\mathrm{Ubs}% ^{*}\to\mbox{no atoms}~{}.
  14. v = Z α c Z c 137.036 v=Z\alpha c\approx\frac{Zc}{137.036}
  15. E = m c 2 1 - v 2 c 2 = m c 2 1 - Z 2 α 2 , E=\frac{mc^{2}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}=\frac{mc^{2}}{\sqrt{1-Z^{2}% \alpha^{2}}},

Extended_real_number_line.html

  1. f ( x ) = x - 2 . f(x)=x^{-2}.
  2. 1 d x x \int_{1}^{\infty}\frac{dx}{x}
  3. f n ( x ) = { 2 n ( 1 - n x ) , if 0 x 1 n 0 , if 1 n < x 1 f_{n}(x)=\begin{cases}2n(1-nx),&\mbox{if }~{}0\leq x\leq\frac{1}{n}\\ 0,&\mbox{if }~{}\frac{1}{n}<x\leq 1\end{cases}
  4. a + = + + a \displaystyle a+\infty=+\infty+a
  5. lim x - f ( x ) \lim_{x\to-\infty}{f(x)}
  6. lim x + f ( x ) \lim_{x\to+\infty}{f(x)}

Extinction_ratio.html

  1. r e = P 1 P 0 r_{e}=\frac{P_{1}}{P_{0}}

Extractor_(mathematics).html

  1. ( N , M , D , K , ϵ ) (N,M,D,K,\epsilon)
  2. N N
  3. M M
  4. D D
  5. A A
  6. K K
  7. A A
  8. ϵ \epsilon
  9. E : [ N ] × [ D ] [ M ] E:[N]\times[D]\rightarrow[M]
  10. X X
  11. n n
  12. log K \log K
  13. E ( X , U D ) E(X,U_{D})
  14. ϵ \epsilon
  15. U M U_{M}
  16. U T U_{T}
  17. [ T ] [T]
  18. K , D , ϵ K,D,\epsilon
  19. N N
  20. M M
  21. K D KD

Extraterrestrial_life.html

  1. N = R f p n e f f i f c L N=R_{\ast}\cdot f_{p}\cdot n_{e}\cdot f_{\ell}\cdot f_{i}\cdot f_{c}\cdot L

Extreme_value_theory.html

  1. X 1 , , X n X_{1},\dots,X_{n}
  2. M n = max ( X 1 , , X n ) M_{n}=\max(X_{1},\dots,X_{n})
  3. Pr ( M n z ) \displaystyle\Pr(M_{n}\leq z)
  4. I n = I ( M n > z ) I_{n}=I(M_{n}>z)
  5. p ( z ) = ( 1 - ( F ( z ) ) n ) p(z)=(1-(F(z))^{n})
  6. z z
  7. n n
  8. O ( 1 / p ( z ) ) O(1/p(z))
  9. F F
  10. a n > 0 a_{n}>0
  11. b n b_{n}\in\mathbb{R}
  12. Pr { ( M n - b n ) / a n z } G ( z ) \Pr\{(M_{n}-b_{n})/a_{n}\leq z\}\rightarrow G(z)
  13. n n\rightarrow\infty
  14. G ( z ) exp [ - ( 1 + ζ z ) - 1 / ζ ] G(z)\propto\exp\left[-(1+\zeta z)^{-1/\zeta}\right]
  15. ζ \zeta
  16. G ( z ) = { exp { - ( - ( z - b a ) ) α } z < b 1 z b G(z)=\begin{cases}\exp\left\{-\left(-\left(\frac{z-b}{a}\right)\right)^{\alpha% }\right\}&z<b\\ 1&z\geq b\end{cases}
  17. M n M_{n}
  18. G ( z ) = exp { - exp ( - ( z - b a ) ) } for z . G(z)=\exp\left\{-\exp\left(-\left(\frac{z-b}{a}\right)\right)\right\}\,\text{ % for }z\in\mathbb{R}.
  19. M n M_{n}
  20. G ( z ) = { 0 z b exp { - ( z - b a ) - α } z > b . G(z)=\begin{cases}0&z\leq b\\ \exp\left\{-\left(\frac{z-b}{a}\right)^{-\alpha}\right\}&z>b.\end{cases}
  21. M n M_{n}
  22. α > 0 \alpha>0

Factorial.html

  1. 5 ! = 5 × 4 × 3 × 2 × 1 = 120. 5!=5\times 4\times 3\times 2\times 1=120.
  2. n ! = k = 1 n k n!=\prod_{k=1}^{n}k\!
  3. n ! = { 1 if n = 0 , ( n - 1 ) ! × n if n > 0 n!=\begin{cases}1&\,\text{if }n=0,\\ (n-1)!\times n&\,\text{if }n>0\end{cases}
  4. n ! = D n x n n!=D^{n}x^{n}\;
  5. 0 ! = 1 , 0!=1,
  6. e x = n = 0 x n n ! . e^{x}=\sum_{n=0}^{\infty}\frac{x^{n}}{n!}.
  7. ( 0 0 ) = 0 ! 0 ! 0 ! = 1 {\textstyle\left({{0}\atop{0}}\right)}=\tfrac{0!}{0!0!}=1
  8. ( n n ) = n ! n ! 0 ! = 1 {\textstyle\left({{n}\atop{n}}\right)}=\tfrac{n!}{n!0!}=1
  9. n k ¯ = n ( n - 1 ) ( n - 2 ) ( n - k + 1 ) n^{\underline{k}}=n(n-1)(n-2)\cdots(n-k+1)
  10. n k ¯ k ! = n ( n - 1 ) ( n - 2 ) ( n - k + 1 ) k ( k - 1 ) ( k - 2 ) 1 . \frac{n^{\underline{k}}}{k!}=\frac{n(n-1)(n-2)\cdots(n-k+1)}{k(k-1)(k-2)\cdots 1}.
  11. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  12. n k ¯ = n ! ( n - k ) ! ; n^{\underline{k}}=\frac{n!}{(n-k)!};
  13. ( n k ) = n k ¯ k ! = n ! ( n - k ) ! k ! = n n - k ¯ ( n - k ) ! = ( n n - k ) . {\left({{n}\atop{k}}\right)}=\frac{n^{\underline{k}}}{k!}=\frac{n!}{(n-k)!k!}=% \frac{n^{\underline{n-k}}}{(n-k)!}={\left({{n}\atop{n-k}}\right)}.
  14. ( n - 1 ) ! 0 ( mod n ) . (n-1)!\ \equiv\ 0\;\;(\mathop{{\rm mod}}n).
  15. ( p - 1 ) ! - 1 ( mod p ) (p-1)!\ \equiv\ -1\;\;(\mathop{{\rm mod}}p)
  16. n ! n!
  17. i = 1 n p i \sum_{i=1}^{\infty}\left\lfloor\frac{n}{p^{i}}\right\rfloor
  18. n - s p ( n ) p - 1 \frac{n-s_{p}(n)}{p-1}
  19. s p ( n ) s_{p}(n)
  20. n = 0 1 n ! = 1 1 + 1 1 + 1 2 + 1 6 + 1 24 + 1 120 + = e . \sum_{n=0}^{\infty}\frac{1}{n!}=\frac{1}{1}+\frac{1}{1}+\frac{1}{2}+\frac{1}{6% }+\frac{1}{24}+\frac{1}{120}+\ldots=e\,.
  21. n = 0 1 ( n + 2 ) n ! = 1 2 + 1 3 + 1 8 + 1 30 + 1 144 = 1 . \sum_{n=0}^{\infty}\frac{1}{(n+2)n!}=\frac{1}{2}+\frac{1}{3}+\frac{1}{8}+\frac% {1}{30}+\frac{1}{144}\ldots=1\,.
  22. log n ! = x = 1 n log x . \log n!=\sum_{x=1}^{n}\log x.
  23. 1 n log x d x x = 1 n log x 0 n log ( x + 1 ) d x \int_{1}^{n}\log x\,dx\leq\sum_{x=1}^{n}\log x\leq\int_{0}^{n}\log(x+1)\,dx
  24. n log ( n e ) + 1 log n ! ( n + 1 ) log ( n + 1 e ) + 1. n\log\left(\frac{n}{e}\right)+1\leq\log n!\leq(n+1)\log\left(\frac{n+1}{e}% \right)+1.
  25. e ( n e ) n n ! e ( n + 1 e ) n + 1 . e\left(\frac{n}{e}\right)^{n}\leq n!\leq e\left(\frac{n+1}{e}\right)^{n+1}.
  26. ( n / 3 ) n < n ! (n/3)^{n}<n!
  27. n ! < ( n / 2 ) n n!<(n/2)^{n}
  28. n ! 2 π n ( n e ) n . n!\approx\sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}.
  29. n ! > 2 π n ( n e ) n . n!>\sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}.
  30. log n ! n log n - n + log ( n ( 1 + 4 n ( 1 + 2 n ) ) ) 6 + log ( π ) 2 \log n!\approx n\log n-n+\frac{\log(n(1+4n(1+2n)))}{6}+\frac{\log(\pi)}{2}
  31. = n log n - n + log ( 1 + 1 / ( 2 n ) + 1 / ( 8 n 2 ) ) 6 + log ( 2 n ) 2 + log ( π ) 2 . =n\log n-n+\frac{\log(1+1/(2n)+1/(8n^{2}))}{6}+\frac{\log(2n)}{2}+\frac{\log(% \pi)}{2}.
  32. 1 12 n \tfrac{1}{12n}
  33. × 10 9 97 \times 10^{9}97
  34. ( ( 1 × 2 ) × 3 ) × 4 ((1\times 2)\times 3)\times 4\dots
  35. π \sqrt{π}
  36. π 2 \frac{\sqrt{π}}{2}
  37. Γ ( z ) = 0 t z - 1 e - t d t . \Gamma(z)=\int_{0}^{\infty}t^{z-1}e^{-t}\,\mathrm{d}t.\!
  38. n ! = Γ ( n + 1 ) . n!=\Gamma(n+1).\,
  39. Γ ( z ) = lim n n z n ! k = 0 n ( z + k ) . \Gamma(z)=\lim_{n\to\infty}\frac{n^{z}n!}{\displaystyle\prod_{k=0}^{n}(z+k)}.\!
  40. Π ( z ) = 0 t z e - t d t . \Pi(z)=\int_{0}^{\infty}t^{z}e^{-t}\,\mathrm{d}t\,.
  41. Π ( z ) = Γ ( z + 1 ) . \Pi(z)=\Gamma(z+1)\,.
  42. Π ( n ) = n ! for n 𝐍 . \Pi(n)=n!\,\text{ for }n\in\mathbf{N}\,.
  43. Π ( z ) = z Π ( z - 1 ) . \Pi(z)=z\Pi(z-1)\,.
  44. Γ ( n + 1 ) = n Γ ( n ) . \Gamma(n+1)=n\Gamma(n)\,.
  45. z ! = Π ( z ) z!=\Pi(z)\,
  46. Γ ( 1 2 ) = ( - 1 2 ) ! = Π ( - 1 2 ) = π , \Gamma\left(\frac{1}{2}\right)=\left(-\frac{1}{2}\right)!=\Pi\left(-\frac{1}{2% }\right)=\sqrt{\pi},
  47. Γ ( 1 2 + n ) = ( - 1 2 + n ) ! = Π ( - 1 2 + n ) = π k = 1 n 2 k - 1 2 = ( 2 n ) ! 4 n n ! π = ( 2 n - 1 ) ! 2 2 n - 1 ( n - 1 ) ! π . \Gamma\left(\frac{1}{2}+n\right)=\left(-\frac{1}{2}+n\right)!=\Pi\left(-\frac{% 1}{2}+n\right)=\sqrt{\pi}\prod_{k=1}^{n}{2k-1\over 2}={(2n)!\over 4^{n}n!}% \sqrt{\pi}={(2n-1)!\over 2^{2n-1}(n-1)!}\sqrt{\pi}.
  48. Γ ( 4.5 ) = 3.5 ! = Π ( 3.5 ) = 1 2 3 2 5 2 7 2 π = 8 ! 4 4 4 ! π = 7 ! 2 7 3 ! π = 105 16 π 11.63. \Gamma\left(4.5\right)=3.5!=\Pi\left(3.5\right)={1\over 2}\cdot{3\over 2}\cdot% {5\over 2}\cdot{7\over 2}\sqrt{\pi}={8!\over 4^{4}4!}\sqrt{\pi}={7!\over 2^{7}% 3!}\sqrt{\pi}={105\over 16}\sqrt{\pi}\approx 11.63.
  49. Γ ( 1 2 - n ) = ( - 1 2 - n ) ! = Π ( - 1 2 - n ) = π k = 1 n 2 1 - 2 k = ( - 4 ) n n ! ( 2 n ) ! π . \Gamma\left(\frac{1}{2}-n\right)=\left(-\frac{1}{2}-n\right)!=\Pi\left(-\frac{% 1}{2}-n\right)=\sqrt{\pi}\prod_{k=1}^{n}{2\over 1-2k}={(-4)^{n}n!\over(2n)!}% \sqrt{\pi}.
  50. Γ ( - 2.5 ) = ( - 3.5 ) ! = Π ( - 3.5 ) = 2 - 1 2 - 3 2 - 5 π = ( - 4 ) 3 3 ! 6 ! π = - 8 15 π - 0.9453. \Gamma\left(-2.5\right)=(-3.5)!=\Pi\left(-3.5\right)={2\over-1}\cdot{2\over-3}% \cdot{2\over-5}\sqrt{\pi}={(-4)^{3}3!\over 6!}\sqrt{\pi}=-{8\over 15}\sqrt{\pi% }\approx-0.9453.
  51. n ! = Π ( n ) \displaystyle n!=\Pi(n)
  52. V n = π n / 2 Γ ( ( n / 2 ) + 1 ) R n . V_{n}=\frac{\pi^{n/2}}{\Gamma((n/2)+1)}R^{n}.
  53. f = ρ exp ( i φ ) = ( x + i y ) ! = Γ ( x + i y + 1 ) \ f=\rho\exp({\rm i}\varphi)=(x+{\rm i}y)!=\Gamma(x+{\rm i}y+1)
  54. ρ = const \rho=\rm const
  55. φ = const \varphi=\rm const
  56. - 3 x 3 ~{}-3\leq x\leq 3~{}
  57. - 2 y 2 ~{}-2\leq y\leq 2~{}
  58. φ = ± π \varphi=\pm\pi
  59. x + i y ( negative integers ) x+{\rm i}y\in\rm(negative~{}integers)
  60. | z | < 1 |z|<1
  61. z ! = n = 0 g n z n . z!=\sum_{n=0}^{\infty}g_{n}z^{n}.
  62. n n
  63. g n g_{n}
  64. 1 1
  65. 1 1
  66. - γ -\gamma
  67. - 0.5772156649 -0.5772156649
  68. π 2 12 + γ 2 2 \frac{\pi^{2}}{12}+\frac{\gamma^{2}}{2}
  69. 0.9890559955 0.9890559955
  70. - ζ ( 3 ) 3 - π 2 γ 12 - γ 3 6 -\frac{\zeta(3)}{3}-\frac{\pi^{2}\gamma}{12}-\frac{\gamma^{3}}{6}
  71. - 0.9074790760 -0.9074790760
  72. γ \gamma
  73. ζ \zeta
  74. P ( z ) = p ( z ) + log ( 2 π ) / 2 - z + ( z + 1 2 ) log ( z ) , P(z)=p(z)+\log(2\pi)/2-z+\left(z+\frac{1}{2}\right)\log(z),
  75. p ( z ) = a 0 z + a 1 z + a 2 z + a 3 z + p(z)=\cfrac{a_{0}}{z+\cfrac{a_{1}}{z+\cfrac{a_{2}}{z+\cfrac{a_{3}}{z+\ddots}}}}
  76. log ( z ! ) = P ( z ) \displaystyle\log(z!)=P(z)
  77. log ( Γ ( z + 1 ) ) = P ( z ) \log(\Gamma(z\!+\!1))=P(z)
  78. | ( Γ ( z + 1 ) ) | < π |\Im(\Gamma(z\!+\!1))|<\pi
  79. | ( z ) | > 2 |\Im(z)|>2
  80. ( z ) > 2 \Re(z)>2
  81. ( n - 1 ) ! = n ! n . (n-1)!=\frac{n!}{n}.
  82. ( 2 k - 1 ) ! ! = i = 1 k ( 2 i - 1 ) = ( 2 k ) ! 2 k k ! = P k 2 k 2 k = ( 2 k ) k ¯ 2 k . (2k-1)!!=\prod_{i=1}^{k}(2i-1)=\frac{(2k)!}{2^{k}k!}=\frac{{}_{2k}P_{k}}{2^{k}% }=\frac{{(2k)}^{\underline{k}}}{2^{k}}.
  83. n ! ! n!!
  84. n ! ! ! n!!!
  85. n ! ! ! n!!!
  86. n ! ( k ) n!^{(k)}
  87. n ! ( k ) = { n if 0 < n k n ( ( n - k ) ! ( k ) ) if n > k n!^{(k)}=\begin{cases}n&\,\text{if }0<n\leq k\\ n((n-k)!^{(k)})&\,\text{if }n>k\end{cases}
  88. n ! ( k ) = 1 if - k < n 0 n!^{(k)}=1\ \,\text{if }-k<n\leq 0
  89. n ! 2 n!_{2}
  90. n ! k n!_{k}
  91. n ! n!
  92. n ! ! n!!
  93. n ! ( k ) n!^{(k)}
  94. k k
  95. z ! ( k ) = z ( z - k ) ( k + 1 ) = k ( z - 1 ) / k ( z k ) ( z - k k ) ( k + 1 k ) = k ( z - 1 ) / k Γ ( z k + 1 ) Γ ( 1 k + 1 ) . z!^{(k)}=z(z-k)\cdots(k+1)=k^{(z-1)/k}\left(\frac{z}{k}\right)\left(\frac{z-k}% {k}\right)\cdots\left(\frac{k+1}{k}\right)=k^{(z-1)/k}\frac{\Gamma\left(\frac{% z}{k}+1\right)}{\Gamma\left(\frac{1}{k}+1\right)}\,.
  96. 2 n ( 2 n ) ! n ! 2 n \displaystyle 2^{n}\frac{(2n)!}{n!2^{n}}
  97. n n
  98. sf ( 4 ) = 1 ! × 2 ! × 3 ! × 4 ! = 288. \mathrm{sf}(4)=1!\times 2!\times 3!\times 4!=288.\,
  99. sf ( n ) = k = 1 n k ! = k = 1 n k n - k + 1 = 1 n 2 n - 1 3 n - 2 ( n - 1 ) 2 n 1 . \mathrm{sf}(n)=\prod_{k=1}^{n}k!=\prod_{k=1}^{n}k^{n-k+1}=1^{n}\cdot 2^{n-1}% \cdot 3^{n-2}\cdots(n-1)^{2}\cdot n^{1}.
  100. sf ( n ) = 0 i < j n ( j - i ) \mathrm{sf}(n)=\prod_{0\leq i<j\leq n}(j-i)
  101. n = 0 n=0
  102. n $ n ! n ! n ! n ! , n\$\equiv\begin{matrix}\underbrace{n!^{{n!}^{{\cdot}^{{\cdot}^{{\cdot}^{n!}}}}% }}\\ n!\end{matrix},\,
  103. n $ = n ! [ 4 ] n ! n\$=n![4]n!\,
  104. n $ = ( n ! ) ( n ! ) . n\$=(n!)\uparrow\uparrow(n!).\,
  105. 1 $ = 1 1\$=1\,
  106. 2 $ = 2 2 = 4 2\$=2^{2}=4\,
  107. 3 $ = 6 [ 4 ] 6 = 6 6 = 6 6 6 6 6 6 . 3\$=6[4]6={{}^{6}}6=6^{6^{6^{6^{6^{6}}}}}.
  108. a b c = a ( b c ) . a^{b^{c}}=a^{(b^{c})}.\,
  109. H ( n ) = k = 1 n k k = 1 1 2 2 3 3 ( n - 1 ) n - 1 n n . H(n)=\prod_{k=1}^{n}k^{k}=1^{1}\cdot 2^{2}\cdot 3^{3}\cdots(n-1)^{n-1}\cdot n^% {n}.
  110. H ( n ) A n ( 6 n 2 + 6 n + 1 ) / 12 e - n 2 / 4 H(n)\sim An^{(6n^{2}+6n+1)/12}e^{-n^{2}/4}

Fall_time.html

  1. t f \scriptstyle t_{f}\,

Falsifiability.html

  1. U ¬ O U\rightarrow\neg O
  2. O \ \ O
  3. ¬ U \neg U

Faraday_constant.html

  1. 𝐅 \mathbf{F}
  2. F = e N A F\,=\,eN_{A}

Fast_Fourier_transform.html

  1. O ( n 2 ) O(n^{2})
  2. O ( n log n ) O(n\log n)
  3. n n
  4. e - 2 π i N e^{-{2\pi i\over N}}
  5. O ( n 2 ) O(n^{2})
  6. O ( n log n ) O(n\log n)
  7. X k = n = 0 N - 1 x n e - i 2 π k n N k = 0 , , N - 1. X_{k}=\sum_{n=0}^{N-1}x_{n}e^{-{i2\pi k\frac{n}{N}}}\qquad k=0,\dots,N-1.
  8. n k = - ( k - n ) 2 / 2 + n 2 / 2 + k 2 / 2 nk=-(k-n)^{2}/2+n^{2}/2+k^{2}/2
  9. X N - k = X k * X_{N-k}=X_{k}^{*}
  10. 4 N - 2 log 2 2 N - 2 log 2 N - 4 4N-2\log_{2}^{2}N-2\log_{2}N-4
  11. N = 2 m N=2^{m}
  12. N log 2 N N\log_{2}N
  13. 2 N log 2 N 2N\log_{2}N
  14. N log 2 N N\log_{2}N
  15. 4 N log 2 N - 6 N + 8 4N\log_{2}N-6N+8
  16. 34 9 N log 2 N \sim\frac{34}{9}N\log_{2}N
  17. X 𝐤 = 𝐧 = 0 𝐍 - 1 e - 2 π i 𝐤 ( 𝐧 / 𝐍 ) x 𝐧 X_{\mathbf{k}}=\sum_{\mathbf{n}=0}^{\mathbf{N}-1}e^{-2\pi i\mathbf{k}\cdot(% \mathbf{n}/\mathbf{N})}x_{\mathbf{n}}
  18. 𝐧 = ( n 1 , , n d ) \mathbf{n}=(n_{1},\ldots,n_{d})
  19. n j = 0 N j - 1 n_{j}=0\ldots N_{j}-1
  20. 𝐧 / 𝐍 = ( n 1 / N 1 , , n d / N d ) \mathbf{n}/\mathbf{N}=(n_{1}/N_{1},\ldots,n_{d}/N_{d})
  21. N = N 1 N 2 N d N=N_{1}\cdot N_{2}\cdot\ldots\cdot N_{d}
  22. N N 1 O ( N 1 log N 1 ) + + N N d O ( N d log N d ) \displaystyle{}\qquad\frac{N}{N_{1}}O(N_{1}\log N_{1})+\cdots+\frac{N}{N_{d}}O% (N_{d}\log N_{d})
  23. n 1 × n 2 n_{1}\times n_{2}
  24. n 1 × n 2 n_{1}\times n_{2}
  25. ( n 1 , , n d / 2 ) (n_{1},\ldots,n_{d/2})
  26. ( n d / 2 + 1 , , n d ) (n_{d/2+1},\ldots,n_{d})
  27. 𝐫 = ( r 1 , r 2 , , r d ) \mathbf{r}=(r_{1},r_{2},\ldots,r_{d})
  28. 𝐫 = ( 1 , , 1 , r , 1 , , 1 ) \mathbf{r}=(1,\ldots,1,r,1,\ldots,1)
  29. N N
  30. n = 0 , 1 , . , N - 1 n=0,1,.......,N-1
  31. Δ t \Delta t
  32. T T
  33. T = N . Δ t T=N.\Delta t
  34. Δ f \Delta f
  35. + F m a x +F_{max}
  36. ± F m a x \pm F_{max}
  37. F m a x = N 2 . Δ f F_{max}=\frac{N}{2}.\Delta f
  38. Δ t = 1 2. F m a x \Delta t=\frac{1}{2.F_{max}}
  39. Δ f = 1 T \Delta f=\frac{1}{T}

Faster-than-light.html

  1. β r e l = β + β 1 + β 2 = 2 β 1 + β 2 1. \beta_{rel}={\beta+\beta\over 1+\beta^{2}}={2\beta\over 1+\beta^{2}}\leq 1.

Felix_Hausdorff.html

  1. = 2 0 \aleph=2^{\aleph_{0}}
  2. α \aleph_{\alpha}
  3. card ( T ( 0 ) ) = \mathrm{card}(T(\aleph_{0}))=\aleph
  4. = 1 \aleph=\aleph_{1}
  5. 1 \aleph\geq\aleph_{1}
  6. 1 \aleph_{1}
  7. \aleph
  8. μ \mu
  9. μ α = μ μ - 1 α . \aleph_{\mu}^{\aleph_{\alpha}}=\aleph_{\mu}\;\aleph_{\mu-1}^{\aleph_{\alpha}}.
  10. A A
  11. ω ξ , ω η \omega_{\xi},\omega_{\eta}
  12. A A
  13. ω ξ \omega_{\xi}
  14. ω η * \omega_{\eta}^{*}
  15. W W
  16. W W
  17. W W
  18. η α \eta_{\alpha}
  19. η α \eta_{\alpha}
  20. δ \delta
  21. σ \sigma
  22. n \mathbb{R}^{n}
  23. ρ \rho
  24. n \mathbb{R}^{n}
  25. n 3 n\geq 3
  26. n \mathbb{R}^{n}
  27. G δ G_{\delta}
  28. G δ σ δ G_{\delta\sigma\delta}
  29. φ ( x ) \varphi(x)
  30. φ ( x ) L p [ 0 , 1 ] \varphi(x)\in L^{p}[0,1]
  31. L p L^{p}
  32. F F
  33. E E
  34. E E
  35. F F
  36. E E
  37. E E
  38. F E F\subset E
  39. F F
  40. η α \eta_{\alpha}

Fermat's_little_theorem.html

  1. a p a ( mod p ) . a^{p}\equiv a\;\;(\mathop{{\rm mod}}p).
  2. a p - 1 1 ( mod p ) . a^{p-1}\equiv 1\;\;(\mathop{{\rm mod}}p).
  3. 2 p 2 ( mod p ) 2^{p}\equiv 2\;\;(\mathop{{\rm mod}}p)\,
  4. 2 341 2 ( mod 341 ) 2^{341}\equiv 2\;\;(\mathop{{\rm mod}}341)\,
  5. m n ( mod p - 1 ) m\equiv n\;\;(\mathop{{\rm mod}}p-1)
  6. a m a n ( mod p ) a^{m}\equiv a^{n}\;\;(\mathop{{\rm mod}}p)
  7. b ( p - 1 ) + n b(p-1)+n
  8. a m = a b ( p - 1 ) a n ( a p - 1 ) b a n 1 b a n a n ( mod p ) . a^{m}=a^{b(p-1)}\cdot a^{n}\equiv\left({a^{p-1}}\right)^{b}\cdot a^{n}\equiv 1% ^{b}\cdot a^{n}\equiv a^{n}\;\;(\mathop{{\rm mod}}p).
  9. a φ ( n ) 1 ( mod n ) a^{\varphi(n)}\equiv 1\;\;(\mathop{{\rm mod}}n)
  10. gcd ( a = 1 p - 1 a p - 1 , p ) = 1 \gcd\left(\sum_{a=1}^{p-1}a^{p-1},p\right)=1
  11. a p - 1 1 ( mod p ) a^{p-1}\equiv 1\;\;(\mathop{{\rm mod}}p)
  12. a ( p - 1 ) / q 1 ( mod p ) a^{(p-1)/q}\not\equiv 1\;\;(\mathop{{\rm mod}}p)

Fermat's_principle.html

  1. T = 𝐭 𝟎 𝐭 𝟏 d t = 1 c 𝐭 𝟎 𝐭 𝟏 c v d s d t d t = 1 c 𝐀 𝐁 n d s T=\int_{\mathbf{t_{0}}}^{\mathbf{t_{1}}}\,dt=\frac{1}{c}\int_{\mathbf{t_{0}}}^% {\mathbf{t_{1}}}\frac{c}{v}\frac{ds}{dt}\,dt=\frac{1}{c}\int_{\mathbf{A}}^{% \mathbf{B}}n\,ds
  2. t 0 t_{0}
  3. t 1 t_{1}
  4. S = 𝐀 𝐁 n d s S=\int_{\mathbf{A}}^{\mathbf{B}}n\,ds
  5. δ S = δ 𝐀 𝐁 n d s = 0 \delta S=\delta\int_{\mathbf{A}}^{\mathbf{B}}n\,ds=0
  6. n = n ( x 1 , x 2 , x 3 ) n=n\left(x_{1},x_{2},x_{3}\right)
  7. s = ( x 1 ( x 3 ) , x 2 ( x 3 ) , x 3 ) s=\left(x_{1}\left(x_{3}\right),x_{2}\left(x_{3}\right),x_{3}\right)
  8. n d s = n d x 1 2 + d x 2 2 + d x 3 2 d x 3 d x 3 = n 1 + x ˙ 1 2 + x ˙ 2 2 d x 3 nds=n\frac{\sqrt{dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}}}{dx_{3}}dx_{3}=n\sqrt{1+% \dot{x}_{1}^{2}+\dot{x}_{2}^{2}}\ dx_{3}
  9. x ˙ k = d x k / d x 3 \dot{x}_{k}=dx_{k}/dx_{3}
  10. δ S = δ x 3 A x 3 B n ( x 1 , x 2 , x 3 ) 1 + x ˙ 1 2 + x ˙ 2 2 d x 3 \delta S=\delta\int_{x_{3A}}^{x_{3B}}n\left(x_{1},x_{2},x_{3}\right)\sqrt{1+% \dot{x}_{1}^{2}+\dot{x}_{2}^{2}}\,dx_{3}
  11. = δ x 3 A x 3 B L ( x 1 ( x 3 ) , x 2 ( x 3 ) , x ˙ 1 ( x 3 ) , x ˙ 2 ( x 3 ) , x 3 ) d x 3 = 0 =\delta\int_{x_{3A}}^{x_{3B}}L\left(x_{1}\left(x_{3}\right),x_{2}\left(x_{3}% \right),\dot{x}_{1}\left(x_{3}\right),\dot{x}_{2}\left(x_{3}\right),x_{3}% \right)\,dx_{3}=0
  12. L ( x 1 , x 2 , x ˙ 1 , x ˙ 2 , x 3 ) L\left(x_{1},x_{2},\dot{x}_{1},\dot{x}_{2},x_{3}\right)

Fermat_pseudoprime.html

  1. a q - 1 1 ( mod q ) a^{q-1}\equiv 1\;\;(\mathop{{\rm mod}}q)
  2. a = q - 1 a=q-1
  3. a q - 1 1 ( mod q ) a^{q-1}\equiv 1\;\;(\mathop{{\rm mod}}q)
  4. 2 a q - 2. 2\leq a\leq q-2.
  5. φ \varphi

Fermi_paradox.html

  1. N ( r ) N(r)
  2. r r
  3. R g R_{g}
  4. 0 R g N ( r ) 4 π r 2 d r + R g N ( r ) 4 π r 2 d r , \int_{0}^{R_{g}}N(r)4\pi r^{2}\,dr+\int_{R_{g}}^{\infty}N(r)4\pi r^{2}\,dr,

Fermium.html

  1. U 92 238 + 15 n , 7 β - 99 253 Es \mathrm{{}^{238}_{\ 92}U\ \xrightarrow{+\ 15n,7\beta^{-}}\ ^{253}_{\ 99}Es}
  2. × 10 1 5 \times 10^{1}5
  3. × 10 - 14 \times 10^{-}14
  4. × 10 9 \times 10^{9}
  5. × 10 4 \times 10^{−}4

Ferroelectricity.html

  1. Δ E = 1 2 α 0 ( T - T 0 ) ( P x 2 + P y 2 + P z 2 ) + 1 4 α 11 ( P x 4 + P y 4 + P z 4 ) + 1 2 α 12 ( P x 2 P y 2 + P y 2 P z 2 + P z 2 P x 2 ) + 1 6 α 111 ( P x 6 + P y 6 + P z 6 ) + 1 2 α 112 [ P x 4 ( P y 2 + P z 2 ) + P y 4 ( P x 2 + P z 2 ) + P z 4 ( P x 2 + P y 2 ) ] + 1 2 α 123 P x 2 P y 2 P z 2 \begin{array}[]{ll}\Delta E=&\frac{1}{2}\alpha_{0}\left(T-T_{0}\right)\left(P_% {x}^{2}+P_{y}^{2}+P_{z}^{2}\right)+\frac{1}{4}\alpha_{11}\left(P_{x}^{4}+P_{y}% ^{4}+P_{z}^{4}\right)\\ &+\frac{1}{2}\alpha_{12}\left(P_{x}^{2}P_{y}^{2}+P_{y}^{2}P_{z}^{2}+P_{z}^{2}P% _{x}^{2}\right)\\ &+\frac{1}{6}\alpha_{111}\left(P_{x}^{6}+P_{y}^{6}+P_{z}^{6}\right)\\ &+\frac{1}{2}\alpha_{112}\left[P_{x}^{4}\left(P_{y}^{2}+P_{z}^{2}\right)+P_{y}% ^{4}\left(P_{x}^{2}+P_{z}^{2}\right)+P_{z}^{4}\left(P_{x}^{2}+P_{y}^{2}\right)% \right]\\ &+\frac{1}{2}\alpha_{123}P_{x}^{2}P_{y}^{2}P_{z}^{2}\end{array}
  2. α i , α i j , α i j k \alpha_{i},\alpha_{ij},\alpha_{ijk}
  3. α 0 > 0 \alpha_{0}>0
  4. α 111 > 0 \alpha_{111}>0
  5. α 11 < 0 \alpha_{11}<0
  6. α 11 > 0 \alpha_{11}>0
  7. Δ E = 1 2 α 0 ( T - T 0 ) P x 2 + 1 4 α 11 P x 4 + 1 6 α 111 P x 6 \Delta E=\frac{1}{2}\alpha_{0}\left(T-T_{0}\right)P_{x}^{2}+\frac{1}{4}\alpha_% {11}P_{x}^{4}+\frac{1}{6}\alpha_{111}P_{x}^{6}
  8. P = ± P s P=\pm P_{s}
  9. Δ E P x = α 0 ( T - T 0 ) P x + α 11 P x 3 + α 111 P x 5 = 0 \frac{\partial\Delta E}{\partial P_{x}}=\alpha_{0}\left(T-T_{0}\right)P_{x}+% \alpha_{11}P_{x}^{3}+\alpha_{111}P_{x}^{5}=0
  10. P x [ α 0 ( T - T 0 ) + α 11 P x 2 + α 111 P x 4 ] = 0 P_{x}\left[\alpha_{0}\left(T-T_{0}\right)+\alpha_{11}P_{x}^{2}+\alpha_{111}P_{% x}^{4}\right]=0
  11. α 0 ( T - T 0 ) + α 11 P x 2 + α 111 P x 4 = 0 \alpha_{0}\left(T-T_{0}\right)+\alpha_{11}P_{x}^{2}+\alpha_{111}P_{x}^{4}=0
  12. P s 2 = 1 2 α 111 [ - α 11 ± α 11 2 - 4 α 0 α 111 ( T - T 0 ) ] P_{s}^{2}=\frac{1}{2\alpha_{111}}\left[-\alpha_{11}\pm\sqrt{\alpha_{11}^{2}-4% \alpha_{0}\alpha_{111}\left(T-T_{0}\right)}\right]
  13. P s = 1 2 α 111 [ - α 11 + α 11 2 - 4 α 0 α 111 ( T - T 0 ) ] P_{s}=\sqrt{\frac{1}{2\alpha_{111}}\left[-\alpha_{11}+\sqrt{\alpha_{11}^{2}-4% \alpha_{0}\alpha_{111}\left(T-T_{0}\right)}\right]}
  14. α 11 = 0 \alpha_{11}=0
  15. P s = - α 0 ( T - T 0 ) α 111 P_{s}=\sqrt{-\frac{\alpha_{0}\left(T-T_{0}\right)}{\alpha_{111}}}
  16. Δ E = 1 2 α 0 ( T - T 0 ) P x 2 + 1 4 α 11 P x 4 + 1 6 α 111 P x 6 - E x P x \Delta E=\frac{1}{2}\alpha_{0}\left(T-T_{0}\right)P_{x}^{2}+\frac{1}{4}\alpha_% {11}P_{x}^{4}+\frac{1}{6}\alpha_{111}P_{x}^{6}-E_{x}P_{x}
  17. Δ E P x = α 0 ( T - T 0 ) P x + α 11 P x 3 + α 111 P x 5 - E x = 0 \frac{\partial\Delta E}{\partial P_{x}}=\alpha_{0}\left(T-T_{0}\right)P_{x}+% \alpha_{11}P_{x}^{3}+\alpha_{111}P_{x}^{5}-E_{x}=0
  18. E x = α 0 ( T - T 0 ) P x + α 11 P x 3 + α 111 P x 5 E_{x}=\alpha_{0}\left(T-T_{0}\right)P_{x}+\alpha_{11}P_{x}^{3}+\alpha_{111}P_{% x}^{5}
  19. 2 Δ E P x 2 < 0 \frac{\partial^{2}\Delta E}{\partial P_{x}^{2}}<0

Ferromagnetism.html

  1. c a \scriptstyle\frac{c}{a}
  2. c a - 1 = - ( 120 ± 5 ) × 10 - 4 \frac{c}{a}-1=-(120\pm 5)\times 10^{-4}

Feynman_diagram.html

  1. e + e - 2 γ e^{+}e^{-}\to 2\gamma
  2. | i |i\rangle
  3. | f |f\rangle
  4. S f i = f | S | i , S_{fi}=\langle f|S|i\rangle\;,
  5. S S
  6. S = n = 0 i n n ! j = 1 n d 4 x j T j = 1 n L v ( x j ) n = 0 S ( n ) , S=\sum_{n=0}^{\infty}{i^{n}\over n!}\int\prod_{j=1}^{n}d^{4}x_{j}T\prod_{j=1}^% {n}L_{v}(x_{j})\equiv\sum_{n=0}^{\infty}S^{(n)}\;,
  7. L v L_{v}
  8. T T
  9. n n
  10. S ( n ) S^{(n)}
  11. T j = 1 n L v ( x j ) = all possible contractions ( ± ) N j = 1 n L v ( x j ) , T\prod_{j=1}^{n}L_{v}(x_{j})=\sum_{\mathrm{all\;possible\;contractions}}(\pm)N% \prod_{j=1}^{n}L_{v}(x_{j})\;,
  12. N N
  13. ( ± ) (\pm)
  14. L v = - g ψ ¯ γ μ ψ A μ L_{v}=-g\bar{\psi}\gamma^{\mu}\psi A_{\mu}
  15. ψ \psi
  16. A μ A_{\mu}
  17. x j x_{j}
  18. A μ ( x i ) A_{\mu}(x_{i})
  19. x i x_{i}
  20. ψ ( x i ) \psi(x_{i})
  21. x i x_{i}
  22. ψ ¯ ( x i ) \bar{\psi}(x_{i})
  23. x i x_{i}
  24. S ( 2 ) = ( i e ) 2 2 ! d 4 x d 4 x T ψ ¯ ( x ) γ μ ψ ( x ) A μ ( x ) ψ ¯ ( x ) γ ν ψ ( x ) A ν ( x ) . S^{(2)}={(ie)^{2}\over 2!}\int d^{4}x\,d^{4}x^{\prime}\,T\bar{\psi}(x)\,\gamma% ^{\mu}\,\psi(x)\,A_{\mu}(x)\,\bar{\psi}(x^{\prime})\,\gamma^{\nu}\,\psi(x^{% \prime})\,A_{\nu}(x^{\prime}).\;
  25. N ψ ¯ ( x ) i e γ μ ψ ( x ) ψ ¯ ( x ) i e γ ν ψ ( x ) A μ ( x ) A ν ( x ) ¯ N\bar{\psi}(x)ie\gamma^{\mu}\psi(x)\bar{\psi}(x^{\prime})ie\gamma^{\nu}\psi(x^% {\prime})\underline{A_{\mu}(x)A_{\nu}(x^{\prime})}
  26. N ψ ¯ ( x ) γ μ ψ ( x ) ψ ¯ ( x ) γ ν ψ ( x ) A μ ( x ) A ν ( x ) ¯ , N\bar{\psi}(x)\gamma^{\mu}\psi(x)\bar{\psi}(x^{\prime})\gamma^{\nu}\psi(x^{% \prime})\underline{A_{\mu}(x)A_{\nu}(x^{\prime})}\;,
  27. A μ ( x ) A ν ( x ) ¯ = d 4 k ( 2 π ) 4 - i g μ ν k 2 + i 0 e - i k ( x - x ) \underline{A_{\mu}(x)A_{\nu}(x^{\prime})}=\int{d^{4}k\over(2\pi)^{4}}{-ig_{\mu% \nu}\over k^{2}+i0}e^{-ik(x-x^{\prime})}
  28. e - e - e^{-}e^{-}
  29. e + e + e^{+}e^{+}
  30. e - e + e^{-}e^{+}
  31. e - e + e^{-}e^{+}
  32. N ψ ¯ ( x ) γ μ ψ ( x ) ψ ¯ ( x ) ¯ γ ν ψ ( x ) A μ ( x ) A ν ( x ) , N\bar{\psi}(x)\,\gamma^{\mu}\,\underline{\psi(x)\,\bar{\psi}(x^{\prime})}\,% \gamma^{\nu}\,\psi(x^{\prime})\,A_{\mu}(x)\,A_{\nu}(x^{\prime})\;,
  33. ψ ( x ) ψ ¯ ( x ) ¯ = d 4 p ( 2 π ) 4 i γ p - m + i 0 e - i p ( x - x ) \underline{\psi(x)\bar{\psi}(x^{\prime})}=\int{d^{4}p\over(2\pi)^{4}}{i\over% \gamma p-m+i0}e^{-ip(x-x^{\prime})}
  34. S = 1 2 μ ϕ μ ϕ d d x . S=\int{1\over 2}\partial_{\mu}\phi\partial^{\mu}\phi d^{d}x\,.
  35. A B e i S D ϕ , \int_{A}^{B}e^{iS}D\phi\,,
  36. ϕ ( A ) \,\phi(A)
  37. ϕ ( B ) \,\phi(B)
  38. A B e i S ϕ ( x 1 ) ϕ ( x n ) D ϕ = A | ϕ ( x 1 ) ϕ ( x n ) | B , \int_{A}^{B}e^{iS}\phi(x_{1})...\phi(x_{n})D\phi=\langle A|\phi(x_{1})...\phi(% x_{n})|B\rangle\,,
  39. e i S ϕ ( x 1 ) ϕ ( x n ) D ϕ e i S D ϕ = 0 | ϕ ( x 1 ) . ϕ ( x n ) | 0 . {\int e^{iS}\phi(x_{1})...\phi(x_{n})D\phi\over\int e^{iS}D\phi}=\langle 0|% \phi(x_{1})....\phi(x_{n})|0\rangle\,.
  40. a a
  41. a 0 a\rightarrow 0
  42. ϕ ( x ) = d k ( 2 π ) d ϕ ( k ) e i k x = k ϕ ( k ) e i k x . \phi(x)=\int{dk\over(2\pi)^{d}}\phi(k)e^{ik\cdot x}=\int_{k}\phi(k)e^{ikx}\,.
  43. 2 π / a 2\pi/a
  44. 2 π 2\pi
  45. a 0 a\rightarrow 0
  46. S = < x , y > 1 2 ( ϕ ( x ) - ϕ ( y ) ) 2 , S=\sum_{<x,y>}{1\over 2}(\phi(x)-\phi(y))^{2}\,,
  47. < x , y > <x,y>
  48. x x
  49. y y
  50. μ ϕ \partial_{\mu}\phi
  51. S = k ( ( 1 - cos ( k 1 ) ) + ( 1 - cos ( k 2 ) ) + + ( 1 - cos ( k d ) ) ) ϕ k * ϕ k . S=\int_{k}((1-\cos(k_{1}))+(1-\cos(k_{2}))+...+(1-\cos(k_{d})))\phi^{*}_{k}% \phi^{k}\,.
  52. S = k 1 2 k 2 | ϕ ( k ) | 2 . S=\int_{k}{1\over 2}k^{2}|\phi(k)|^{2}\,.
  53. d d k d^{d}k
  54. ( 2 π / V ) d (2\pi/V)^{d}
  55. ϕ \,\phi
  56. ϕ ( k ) * = ϕ ( - k ) . \phi(k)^{*}=\phi(-k)\,.
  57. ϕ ( k ) \,\phi(k)
  58. S = k 1 2 k 2 ϕ ( k ) ϕ ( - k ) S=\int_{k}{1\over 2}k^{2}\phi(k)\phi(-k)
  59. S = 1 2 μ ϕ * μ ϕ d d x S=\int{1\over 2}\partial_{\mu}\phi^{*}\partial^{\mu}\phi d^{d}x
  60. S = k 1 2 k 2 | ϕ ( k ) | 2 S=\int_{k}{1\over 2}k^{2}|\phi(k)|^{2}
  61. ϕ ( x ) \,\phi(x)
  62. y i = A i j x j , y_{i}=A_{ij}x_{j}\,,
  63. det ( A ) d x 1 d x 2 d x n = d y 1 d y 2 d y n . \det(A)\int dx_{1}dx_{2}...dx_{n}=\int dy_{1}dy_{2}...dy_{n}\,.
  64. A T A = I A^{T}A=I\,
  65. det A = ± 1 \det A=\pm 1
  66. ϕ ( x ) \,\phi(x)
  67. ϕ ( k ) \,\phi(k)
  68. A k x = e i k x A_{kx}=e^{ikx}\,
  69. A k x - 1 = e - i k x A^{-1}_{kx}=e^{-ikx}\,
  70. 2 π 2\pi
  71. det A = 1 \det A=1\,
  72. exp ( i 2 k k 2 ϕ * ( k ) ϕ ( k ) ) D ϕ = k ϕ k e i 2 k 2 | ϕ k | 2 d d k \int\exp\biggl({i\over 2}\sum_{k}k^{2}\phi^{*}(k)\phi(k)\biggr)D\phi=\prod_{k}% \int_{\phi_{k}}e^{{i\over 2}k^{2}|\phi_{k}|^{2}d^{d}k}\,
  73. d d k \scriptstyle d^{d}k
  74. d d k = 1 / L d \scriptstyle d^{d}k={1/L}^{d}
  75. ϕ k \phi_{k}
  76. e k - 1 2 k 2 ϕ k * ϕ k = k e - k 2 | ϕ k | 2 d d k e^{\int_{k}-{1\over 2}k^{2}\phi^{*}_{k}\phi_{k}}=\prod_{k}e^{-k^{2}|\phi_{k}|^% {2}d^{d}k}
  77. ϕ ( x 1 ) ϕ ( x n ) = e - S ϕ ( x 1 ) ϕ ( x n ) D ϕ e - S D ϕ \langle\phi(x_{1})...\phi(x_{n})\rangle={\int e^{-S}\phi(x_{1})...\phi(x_{n})D% \phi\over\int e^{-S}D\phi}
  78. ϕ k \,\phi_{k}
  79. ϕ ( k ) \,\phi(k)
  80. V / k 2 V/k^{2}
  81. 1 / k 2 1/k^{2}
  82. ϕ C ( k ) \,\phi_{C}(k)
  83. ϕ C ( x ) \,\phi_{C}(x)
  84. ϕ ( k ) , ϕ ( - k ) \,\phi(k),\phi(-k)
  85. ϕ ( x 1 ) ϕ ( x n ) = lim | C | C ϕ C ( x 1 ) ϕ C ( x n ) | C | \langle\phi(x_{1})...\phi(x_{n})\rangle=\lim_{|C|\rightarrow\infty}{\sum_{C}% \phi_{C}(x_{1})...\phi_{C}(x_{n})\over|C|}
  86. | C | |C|
  87. ϕ k ϕ k = 0 \langle\phi_{k}\phi_{k^{\prime}}\rangle=0\,
  88. k k k\neq k^{\prime}
  89. ϕ k ϕ k = V k 2 \langle\phi_{k}\phi_{k}\rangle={V\over k^{2}}
  90. ϕ ( k ) ϕ ( k ) = δ ( k - k ) 1 k 2 \langle\phi(k)\phi(k^{\prime})\rangle=\delta(k-k^{\prime}){1\over k^{2}}
  91. ϕ ( k ) ϕ ( k ) = δ ( k - k ) 1 2 ( d - cos ( k 1 ) + cos ( k 2 ) + cos ( k d ) ) \langle\phi(k)\phi(k^{\prime})\rangle=\delta(k-k^{\prime}){1\over 2(d-\cos(k_{% 1})+\cos(k_{2})...+\cos(k_{d}))}
  92. 2 π 2\pi
  93. 2 π 2\pi
  94. δ ( k ) = ( 2 π ) d δ D ( k 1 ) δ D ( k 2 ) δ D ( k d ) \delta(k)=(2\pi)^{d}\delta_{D}(k_{1})\delta_{D}(k_{2})...\delta_{D}(k_{d})\,
  95. δ D ( k ) \delta_{D}(k)
  96. 2 π 2\pi
  97. μ μ ϕ = 0 \partial_{\mu}\partial^{\mu}\phi=0\,
  98. μ μ ϕ ( x ) ϕ ( y ) = 0 \partial_{\mu}\partial^{\mu}\langle\phi(x)\phi(y)\rangle=0
  99. Δ \Delta
  100. 2 Δ ( x ) = i δ ( x ) \partial^{2}\Delta(x)=i\delta(x)\,
  101. Δ ( k ) = i k 2 \Delta(k)={i\over k^{2}}
  102. ϕ ( k 1 ) ϕ ( k 2 ) ϕ ( k n ) \langle\phi(k_{1})\phi(k_{2})...\phi(k_{n})\rangle
  103. ϕ \phi
  104. ϕ \phi
  105. ϕ ( k 1 ) ϕ ( k 2 n ) = i , j δ ( k i - k j ) k i 2 \langle\phi(k_{1})...\phi(k_{2n})\rangle=\sum\prod_{i,j}{\delta(k_{i}-k_{j})% \over k_{i}^{2}}
  106. ϕ ( k 1 ) ϕ ( k 2 ) ϕ ( k 3 ) ϕ ( k 4 ) = δ ( k 1 - k 2 ) k 1 2 δ ( k 3 - k 4 ) k 3 2 + δ ( k 1 - k 3 ) k 3 2 δ ( k 2 - k 4 ) k 2 2 + δ ( k 1 - k 4 ) k 1 2 δ ( k 2 - k 3 ) k 2 2 \langle\phi(k_{1})\phi(k_{2})\phi(k_{3})\phi(k_{4})\rangle={\delta(k_{1}-k_{2}% )\over k_{1}^{2}}{\delta(k_{3}-k_{4})\over k_{3}^{2}}+{\delta(k_{1}-k_{3})% \over k_{3}^{2}}{\delta(k_{2}-k_{4})\over k_{2}^{2}}+{\delta(k_{1}-k_{4})\over k% _{1}^{2}}{\delta(k_{2}-k_{3})\over k_{2}^{2}}
  107. I = e - a x 2 / 2 d x = 2 π a I=\int e^{-ax^{2}/2}dx=\sqrt{2\pi\over a}
  108. n a n I = x 2 n 2 n e - a x 2 / 2 d x = 1 3 5... ( 2 n - 1 ) 2 2 2... 2 2 π a - 2 n + 1 2 {\partial^{n}\over\partial a^{n}}I=\int{x^{2n}\over 2^{n}}e^{-ax^{2}/2}dx={1% \cdot 3\cdot 5...\cdot(2n-1)\over 2\cdot 2\cdot 2...\;\;\;\;\;\cdot 2\;\;\;\;% \;\;}\sqrt{2\pi}a^{-{2n+1\over 2}}
  109. x 2 n = x 2 n e - a x 2 / 2 e - a x 2 / 2 = 1 3 5... ( 2 n - 1 ) 1 a n \langle x^{2n}\rangle={\int x^{2n}e^{-ax^{2}/2}\over\int e^{-ax^{2}/2}}=1\cdot 3% \cdot 5...\cdot(2n-1){1\over a^{n}}
  110. x 2 = 1 a \langle x^{2}\rangle={1\over a}
  111. x 1 x 2 x 3 x 2 n \langle x_{1}x_{2}x_{3}...x_{2n}\rangle
  112. x 2 n x^{2n}
  113. x 2 n = ( 2 n - 1 ) ( 2 n - 3 ) . 5 3 1 ( x 2 ) n \langle x^{2n}\rangle=(2n-1)\cdot(2n-3)....\cdot 5\cdot 3\cdot 1(\langle x^{2}% \rangle)^{n}
  114. S = μ ϕ μ ϕ + λ 4 ! ϕ 4 . S=\int\partial^{\mu}\phi\partial_{\mu}\phi+{\lambda\over 4!}\phi^{4}.
  115. S = k k 2 | ϕ ( k ) | 2 + k 1 k 2 k 3 k 4 ϕ ( k 1 ) ϕ ( k 2 ) ϕ ( k 3 ) ϕ ( k 4 ) δ ( k 1 + k 2 + k 3 + k 4 ) = S F + X . S=\int_{k}k^{2}|\phi(k)|^{2}+\int_{k_{1}k_{2}k_{3}k_{4}}\phi(k_{1})\phi(k_{2})% \phi(k_{3})\phi(k_{4})\delta(k_{1}+k_{2}+k_{3}+k_{4})=S_{F}+X.
  116. S F S_{F}
  117. λ \lambda
  118. e - S = e - S F ( 1 + X + 1 2 ! X X + 1 3 ! X X X + ) e^{-S}=e^{-S_{F}}(1+X+{1\over 2!}XX+{1\over 3!}XXX+...)
  119. ϕ ( k ) \phi(k)
  120. ϕ ( k 1 ) ϕ ( k 2 ) ϕ ( k 3 ) ϕ ( k 4 ) = e - S ϕ ( k 1 ) ϕ ( k 2 ) ϕ ( k 3 ) ϕ ( k 4 ) D ϕ Z \langle\phi(k_{1})\phi(k_{2})\phi(k_{3})\phi(k_{4})\rangle={\int e^{-S}\phi(k_% {1})\phi(k_{2})\phi(k_{3})\phi(k_{4})D\phi\over Z}
  121. ϕ ( k i ) \phi(k_{i})
  122. e - S F X e^{-S_{F}}X
  123. ϕ ( k ) \phi(k)
  124. λ 1 k 1 2 1 k 2 2 1 k 3 2 1 k 4 2 . \lambda{1\over k_{1}^{2}}{1\over k_{2}^{2}}{1\over k_{3}^{2}}{1\over k_{4}^{2}}.
  125. ϕ ( k ) \phi(k)
  126. ϕ ( k ) \phi(k)
  127. δ ( k 1 + k 2 ) k 1 2 \delta(k_{1}+k_{2})\over k_{1}^{2}
  128. 1 k 2 1\over k^{2}
  129. Z [ J ] Z[J]
  130. i W [ J ] ln Z [ J ] . iW[J]\equiv\ln Z[J].
  131. Z [ J ] k D k Z[J]\propto\sum_{k}{D_{k}}
  132. D k D_{k}
  133. C i C_{i}
  134. n i n_{i}
  135. C i C_{i}
  136. D k D_{k}
  137. n i ! n_{i}!
  138. D k D_{k}
  139. i C i n i n i ! \prod_{i}{C_{i}^{n_{i}}\over n_{i}!}
  140. i i
  141. D k D_{k}
  142. Z [ J ] Z[J]
  143. ( 1 0 ! + C 1 1 ! + C 1 2 2 ! + ) ( 1 + C 2 + 1 2 C 2 2 + ) \left(\frac{1}{0!}+\frac{C_{1}}{1!}+\frac{C^{2}_{1}}{2!}+\dots\right)\left(1+C% _{2}+\frac{1}{2}C^{2}_{2}+\dots\right)\dots
  144. Z [ J ] i n i = 0 C i n i n i ! = exp i C i exp W [ J ] . Z[J]\propto\prod_{i}{\sum^{\infty}_{n_{i}=0}{\frac{C_{i}^{n_{i}}}{n_{i}!}}}=% \exp{\sum_{i}{C_{i}}}\propto\exp{W[J]}.
  145. Z 0 = exp W [ 0 ] = 1 Z_{0}=\exp{W[0]}=1
  146. J J
  147. ϕ 1 ( x 1 ) ϕ n ( x n ) = e - S ϕ 1 ( x 1 ) ϕ n ( x n ) D ϕ e - S D ϕ . \langle\phi_{1}(x_{1})...\phi_{n}(x_{n})\rangle={\int e^{-S}\phi_{1}(x_{1})...% \phi_{n}(x_{n})D\phi\over\int e^{-S}D\phi}.
  148. e - S ϕ 1 ( x 1 ) ϕ n ( x n ) D ϕ = ( E i ) ( exp ( i C i ) ) . \int e^{-S}\phi_{1}(x_{1})...\phi_{n}(x_{n})D\phi=(\sum E_{i})(\exp(\sum_{i}C_% {i})).
  149. Z = e - S D ϕ = e - H T = e - ρ V Z=\int e^{-S}D\phi=e^{-HT}=e^{-\rho V}
  150. ρ \rho
  151. δ ( k ) \delta(k)
  152. δ ( 0 ) \delta(0)
  153. h ( x ) ϕ ( x ) d d x = h ( k ) ϕ ( k ) d d k \int h(x)\phi(x)d^{d}x=\int h(k)\phi(k)d^{d}k\,
  154. log ( Z [ h ] ) = n , C h ( k 1 ) h ( k 2 ) h ( k n ) C ( k 1 , , k n ) \log(Z[h])=\sum_{n,C}h(k_{1})h(k_{2})...h(k_{n})C(k_{1},...,k_{n})\,
  155. C ( k 1 , . , k n ) C(k_{1},....,k_{n})
  156. Z [ h ] = e i S + i h ϕ D ϕ Z[h]=\int e^{iS+i\int h\phi}D\phi\,
  157. ρ ( y ) e i k y d n y = e i k y = i = 1 n e i h i y i \int\rho(y)e^{iky}d^{n}y=\langle e^{iky}\rangle=\langle\prod_{i=1}^{n}e^{ih_{i% }y_{i}}\rangle\,
  158. Z [ h ] = e i S e i x h ( x ) ϕ ( x ) D ϕ = e i h ϕ Z[h]=\int e^{iS}e^{i\int_{x}h(x)\phi(x)}D\phi=\langle e^{ih\phi}\rangle
  159. x e i h x ϕ x \langle\prod_{x}e^{ih_{x}\phi_{x}}\rangle
  160. δ ( x - y ) = e i k ( x - y ) d k \delta(x-y)=\int e^{ik(x-y)}dk
  161. ϕ \phi
  162. η \eta
  163. δ ( ϕ - η ) = e i h ( x ) ( ϕ ( x ) - η ( x ) d d x D h \delta(\phi-\eta)=\int e^{ih(x)(\phi(x)-\eta(x)d^{d}x}Dh
  164. ϕ ( x ) = 1 Z h ( x ) Z [ h ] = h ( x ) log ( Z [ h ] ) . \langle\phi(x)\rangle={1\over Z}{\partial\over\partial h(x)}Z[h]={\partial% \over\partial h(x)}\log(Z[h]).
  165. e M i j ψ ¯ i ψ j D ψ ¯ D ψ = Det ( M ) \int e^{M_{ij}{\bar{\psi}}^{i}\psi^{j}}D\bar{\psi}D\psi=\mathrm{Det}(M)
  166. ψ , ψ ¯ \scriptstyle\psi,\bar{\psi}
  167. e 1 2 A i j ψ i ψ j D ψ = Pfaff ( A ) \int e^{{1\over 2}A_{ij}\psi^{i}\psi^{j}}D\psi=\mathrm{Pfaff}(A)
  168. ψ \scriptstyle\psi
  169. ψ i ψ j = - ψ j ψ i \scriptstyle\psi^{i}\psi^{j}=-\psi^{j}\psi^{i}
  170. ψ ¯ \scriptstyle\bar{\psi}
  171. η ¯ \scriptstyle\bar{\eta}
  172. η \scriptstyle\eta
  173. ψ \scriptstyle\psi
  174. Z = e ψ ¯ M ψ + η ¯ ψ + ψ ¯ η D ψ ¯ D ψ = e ( ψ ¯ + η ¯ M - 1 ) M ( ψ + M - 1 η ) - η ¯ M - 1 η D ψ ¯ D ψ = Det ( M ) e - η ¯ M - 1 η Z=\int e^{\bar{\psi}M\psi+\bar{\eta}\psi+\bar{\psi}\eta}D\bar{\psi}D\psi=\int e% ^{(\bar{\psi}+\bar{\eta}M^{-1})M(\psi+M^{-1}\eta)-\bar{\eta}M^{-1}\eta}D\bar{% \psi}D\psi=\mathrm{Det}(M)e^{-\bar{\eta}M^{-1}\eta}
  175. η \eta
  176. ψ \psi
  177. η \eta
  178. ψ ¯ \scriptstyle\bar{\psi}
  179. ψ ¯ ψ = 1 Z η η ¯ Z | η = η ¯ = 0 = M - 1 \langle\bar{\psi}\psi\rangle={1\over Z}{\partial\over\partial\eta}{\partial% \over\partial\bar{\eta}}Z|_{\eta=\bar{\eta}=0}=M^{-1}
  180. η \eta
  181. η ¯ \scriptstyle\bar{\eta}
  182. M - 1 \scriptstyle M^{-1}
  183. e ϕ * M ϕ + h * ϕ + ϕ * h D ϕ * D ϕ = e h * M - 1 h Det ( M ) \int e^{\phi^{*}M\phi+h^{*}\phi+\phi^{*}h}D\phi^{*}D\phi={e^{h^{*}M^{-1}h}% \over\mathrm{Det}(M)}
  184. ϕ * ϕ = 1 Z h h * Z | h = h * = 0 = M - 1 \langle\phi^{*}\phi\rangle={1\over Z}{\partial\over\partial h}{\partial\over% \partial h^{*}}Z|_{h=h^{*}=0}=M^{-1}
  185. ψ ¯ ( γ μ μ - m ) ψ \int\bar{\psi}(\gamma^{\mu}\partial_{\mu}-m)\psi
  186. S = k ψ ¯ ( i γ μ k μ - m ) ψ . S=\int_{k}\bar{\psi}(i\gamma^{\mu}k_{\mu}-m)\psi.
  187. ψ ( k ) \psi(k)
  188. ψ ¯ ( k ) \scriptstyle\bar{\psi}(k)
  189. ψ ¯ ( k ) ψ ( k ) = δ ( k + k ) 1 γ k - m = δ ( k + k ) γ k + m k 2 - m 2 \langle\bar{\psi}(k^{\prime})\psi(k)\rangle=\delta(k+k^{\prime}){1\over{\gamma% \cdot k-m}}=\delta(k+k^{\prime}){\gamma\cdot k+m\over k^{2}-m^{2}}
  190. ψ ¯ ( k 1 ) ψ ¯ ( k 2 ) ψ ¯ ( k n ) ψ ( k 1 ) ψ ( k n ) = pairings ( - 1 ) S pairs i , j δ ( k i - k j ) 1 γ k i - m \langle\bar{\psi}(k_{1})\bar{\psi}(k_{2})...\bar{\psi}(k_{n})\psi(k^{\prime}_{% 1})...\psi(k_{n})\rangle=\sum_{\mathrm{pairings}}(-1)^{S}\prod_{\mathrm{pairs}% \;i,j}\delta(k_{i}-k_{j}){1\over\gamma\cdot k_{i}-m}
  191. S = 1 4 F μ ν F μ ν = - 1 2 ( μ A ν μ A ν - μ A μ ν A ν ) . S=\int{1\over 4}F^{\mu\nu}F_{\mu\nu}=\int-{1\over 2}(\partial^{\mu}A_{\nu}% \partial_{\mu}A^{\nu}-\partial^{\mu}A_{\mu}\partial_{\nu}A^{\nu}).\,
  192. δ ( μ A μ - f ) e - f 2 2 D f . \int\delta(\partial_{\mu}A^{\mu}-f)e^{-{f^{2}\over 2}}Df.
  193. e - ( μ A μ ) 2 2 . e^{-{(\partial_{\mu}A_{\mu})^{2}\over 2}}.
  194. S = μ A ν μ A ν S=\int\partial^{\mu}A^{\nu}\partial_{\mu}A_{\nu}
  195. A μ ( k ) A ν ( k ) = δ ( k + k ) g μ ν k 2 . \langle A_{\mu}(k)A_{\nu}(k^{\prime})\rangle=\delta(k+k^{\prime}){g_{\mu\nu}% \over k^{2}}.
  196. λ \lambda
  197. S = 1 2 ( μ A ν μ A ν - λ ( μ A μ ) 2 ) S=\int{1\over 2}(\partial^{\mu}A^{\nu}\partial_{\mu}A_{\nu}-\lambda(\partial_{% \mu}A^{\mu})^{2})
  198. A μ ( k ) A ν ( k ) = δ ( k + k ) g μ ν - λ k μ k ν k 2 k 2 . \langle A_{\mu}(k)A_{\nu}(k^{\prime})\rangle=\delta(k+k^{\prime}){g_{\mu\nu}-% \lambda{k_{\mu}k_{\nu}\over k^{2}}\over k^{2}}.
  199. δ ( μ A μ - f ) e - f 2 2 Det M \delta(\partial_{\mu}A_{\mu}-f)e^{-{f^{2}\over 2}}\mathrm{Det}{M}
  200. θ \scriptstyle\theta
  201. f ( r ) d x d y = f ( r ) d θ δ ( y ) | d y d θ | d x d y \int f(r)dxdy=\int f(r)\int d\theta\delta(y)|{dy\over d\theta}|dxdy
  202. θ \scriptstyle\theta
  203. f ( r ) d x d y = d θ f ( r ) δ ( y ) | d y d θ | d x d y \int f(r)dxdy=\int d\theta\int f(r)\delta(y)|{dy\over d\theta}|dxdy
  204. f ( r ) d x d y = d θ 0 f ( x ) | d y d θ | d x . \int f(r)dxdy=\int d\theta_{0}\int f(x)|{dy\over d\theta}|dx\,.
  205. θ \scriptstyle\theta
  206. 2 π \scriptstyle 2\pi
  207. y \scriptstyle y
  208. θ \theta
  209. f ( r ) d x d y = 2 π f ( x ) x d x \int f(r)dxdy=2\pi\int f(x)xdx
  210. D A δ ( F ( A ) ) Det ( F G ) D G e i S = D G δ ( F ( A ) ) Det ( F G ) e i S \int DA\int\delta(F(A))\mathrm{Det}({\partial F\over\partial G})DGe^{iS}=\int DG% \int\delta(F(A))\mathrm{Det}({\partial F\over\partial G})e^{iS}\,
  211. Det ( F G ) e i S G F D A \int\mathrm{Det}({\partial F\over\partial G})e^{iS_{GF}}DA\,
  212. μ A μ = f \partial_{\mu}A^{\mu}=f\,
  213. μ D μ α \partial_{\mu}D_{\mu}\alpha\,
  214. α \scriptstyle\alpha
  215. D e t ( μ D μ ) Det(\partial_{\mu}D_{\mu})\,
  216. e η ¯ μ D μ η D η ¯ D η \int e^{\bar{\eta}\partial_{\mu}D^{\mu}\eta}D\bar{\eta}D\eta\,
  217. S = T r μ A ν μ A ν + f j k i ν A i μ A μ j A ν k + f j r i f k l r A i A j A k A l + T r μ η ¯ μ η + η ¯ A j η S=\int Tr\partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}+f^{i}_{jk}\partial^{\nu}A_% {i}^{\mu}A^{j}_{\mu}A^{k}_{\nu}+f^{i}_{jr}f^{r}_{kl}A_{i}A_{j}A^{k}A^{l}+Tr% \partial_{\mu}\bar{\eta}\partial^{\mu}\eta+\bar{\eta}A_{j}\eta\,
  218. 1 p 2 + m 2 = 0 e - τ ( p 2 + m 2 ) d τ {1\over p^{2}+m^{2}}=\int_{0}^{\infty}e^{-\tau(p^{2}+m^{2})}d\tau
  219. Δ ( x ) = 0 d τ e - m 2 τ 1 ( 4 π τ ) d / 2 e - x 2 4 τ \Delta(x)=\int_{0}^{\infty}d\tau e^{-m^{2}\tau}{1\over({4\pi\tau})^{d/2}}e^{-x% ^{2}\over 4\tau}
  220. τ \tau
  221. τ \scriptstyle\sqrt{\tau}
  222. τ \tau
  223. τ \tau
  224. Δ ( x ) = 0 d τ D X e - 0 τ ( x ˙ 2 / 2 + m 2 ) d τ \Delta(x)=\int_{0}^{\infty}d\tau\int DXe^{-\int_{0}^{\tau}(\dot{x}^{2}/2+m^{2}% )d\tau^{\prime}}
  225. k 1 ( k 2 + m 2 ) 1 ( ( k + p ) 2 + m 2 ) . \int_{k}{1\over(k^{2}+m^{2})}{1\over((k+p)^{2}+m^{2})}\,.
  226. t , t e - t ( k 2 + m 2 ) - t ( ( k + p ) 2 + m 2 ) d t d t . \int_{t,t^{\prime}}e^{-t(k^{2}+m^{2})-t^{\prime}((k+p)^{2}+m^{2})}dtdt^{\prime% }\,.
  227. t , t e - ( t + t ) ( k 2 + m 2 ) - t 2 p k - t p 2 , \int_{t,t^{\prime}}e^{-(t+t^{\prime})(k^{2}+m^{2})-t^{\prime}2p\cdot k-t^{% \prime}p^{2}}\,,
  228. v \scriptstyle v
  229. v \scriptstyle v
  230. v \scriptstyle v
  231. d ( u v ) = d t d u = d t + d t , d(uv)=dt^{\prime}\;\;\;du=dt+dt^{\prime}\,,
  232. u d u d v = d t d t udu\wedge dv=dt\wedge dt^{\prime}\,
  233. u , v u e - u ( k 2 + m 2 + v 2 p k + v p 2 ) = 1 ( k 2 + m 2 + v 2 p k - v p 2 ) 2 d v \int_{u,v}ue^{-u(k^{2}+m^{2}+v2p\cdot k+vp^{2})}=\int{1\over(k^{2}+m^{2}+v2p% \cdot k-vp^{2})^{2}}dv
  234. v \,v
  235. 1 A B = 0 1 1 ( v A + ( 1 - v ) B ) 2 d v {1\over AB}=\int_{0}^{1}{1\over(vA+(1-v)B)^{2}}dv
  236. v \,v
  237. v \,v
  238. k = k + v p k^{\prime}=k+vp
  239. 0 1 1 ( k 2 + m 2 + v 2 p k + v p 2 ) 2 d k d v = 0 1 1 ( k 2 + m 2 + v ( 1 - v ) p 2 ) 2 d k d v \int_{0}^{1}\int{1\over(k^{2}+m^{2}+v2p\cdot k+vp^{2})^{2}}dkdv=\int_{0}^{1}% \int{1\over(k^{\prime 2}+m^{2}+v(1-v)p^{2})^{2}}dk^{\prime}dv
  240. d k 1 ( k 2 + m 2 ) 1 ( ( k + p 1 ) 2 + m 2 ) 1 ( ( k + p n ) 2 + m 2 ) \int dk{1\over(k^{2}+m^{2})}{1\over((k+p_{1})^{2}+m^{2})}...{1\over((k+p_{n})^% {2}+m^{2})}
  241. 1 D 0 D 1 D n = 0 0 e - u 0 D 0 - u n D n d u 0 d u n . {1\over D_{0}D_{1}...D_{n}}=\int_{0}^{\infty}...\int_{0}^{\infty}e^{-u_{0}D_{0% }...-u_{n}D_{n}}du_{0}...du_{n}\,.
  242. u i \scriptstyle u_{i}
  243. u = u 0 + u 1 + u n \scriptstyle u=u_{0}+u_{1}...+u_{n}
  244. v i = u i / u \scriptstyle v_{i}=u_{i}/u
  245. i { 1 , 2 , , n } \scriptstyle i\in\{1,2,...,n\}
  246. d u = d u 0 + d u 1 + d u n du=du_{0}+du_{1}...+du_{n}\,
  247. d ( u v i ) = d u i . d(uv_{i})=du_{i}\,.
  248. u n d u d v 1 d v 2 d v n = d u 0 d u 1 d u n . u^{n}du\wedge dv_{1}\wedge dv_{2}...\wedge dv_{n}=du_{0}\wedge du_{1}...\wedge du% _{n}\,.
  249. 0 simplex u n e - u ( v 0 D 0 + v 1 D 1 + v 2 D 2 + v n D n ) d v 1 d v n d u , \int_{0}^{\infty}\int_{\mathrm{simplex}}u^{n}e^{-u(v_{0}D_{0}+v_{1}D_{1}+v_{2}% D_{2}...+v_{n}D_{n})}dv_{1}...dv_{n}du\,,
  250. v i > 0 \scriptstyle v_{i}>0
  251. i = 1 n v i < 1 \scriptstyle\sum_{i=1}^{n}v_{i}<1
  252. v 0 = 1 - i = 1 n v i \scriptstyle v_{0}=1-\sum_{i=1}^{n}v_{i}
  253. 1 D 0 D n = n ! simplex 1 ( v 0 D 0 + v 1 D 1 + v n D n ) n + 1 d v 1 d v 2 d v n {1\over D_{0}...D_{n}}=n!\int_{\mathrm{simplex}}{1\over(v_{0}D_{0}+v_{1}D_{1}.% ..+v_{n}D_{n})^{n+1}}dv_{1}dv_{2}...dv_{n}
  254. v i \scriptstyle v_{i}
  255. ϕ ( x ) \phi(x)
  256. λ ϕ 4 \lambda\phi^{4}
  257. λ \lambda
  258. ϕ ( k 1 ) ϕ ( k 2 ) ϕ ( k 3 ) ϕ ( k 4 ) = i k 1 2 i k 2 2 i k 3 2 i k 4 2 i λ \langle\phi(k_{1})\phi(k_{2})\phi(k_{3})\phi(k_{4})\rangle={i\over k_{1}^{2}}{% i\over k_{2}^{2}}{i\over k_{3}^{2}}{i\over k_{4}^{2}}i\lambda\,
  259. i / k 2 i/k^{2}
  260. M = i λ M=i\lambda\,
  261. | M | 2 |M|^{2}
  262. d k | k k | \int dk|k\rangle\langle k|\,
  263. E 2 - k 2 = m 2 E^{2}-k^{2}=m^{2}\,
  264. δ ( E 2 - k 2 - m 2 ) | E , k E , k | d E d k = d k 2 E | k k | \int\delta(E^{2}-k^{2}-m^{2})|E,k\rangle\langle E,k|dEdk=\int{dk\over 2E}|k% \rangle\langle k|
  265. E = ( k 2 - m 2 ) 1 4 \sqrt{E}=(k^{2}-m^{2})^{1\over 4}
  266. m \sqrt{m}
  267. ϕ 4 \phi^{4}

Fibonacci_coding.html

  1. N N\!
  2. d ( 0 ) , d ( 1 ) , , d ( k - 1 ) , d ( k ) d(0),d(1),\ldots,d(k-1),d(k)\!
  3. N N\!
  4. N = i = 0 k - 1 d ( i ) F ( i + 2 ) , and d ( k - 1 ) = d ( k ) = 1. N=\sum_{i=0}^{k-1}d(i)F(i+2),\,\text{ and }d(k-1)=d(k)=1.\!
  5. F ( i ) F(i)
  6. i i
  7. F ( i + 2 ) F(i+2)
  8. i i
  9. 1 , 2 , 3 , 5 , 8 , 13 , 1,2,3,5,8,13,\ldots
  10. d ( k ) d(k)
  11. F ( 2 ) F(2)
  12. F ( 3 ) F(3)
  13. F ( 4 ) F(4)
  14. F ( 2 ) + F ( 4 ) F(2)+F(4)
  15. F ( 5 ) F(5)
  16. F ( 2 ) + F ( 5 ) F(2)+F(5)
  17. F ( 3 ) + F ( 5 ) F(3)+F(5)
  18. F ( 6 ) F(6)
  19. F ( 2 ) + F ( 6 ) F(2)+F(6)
  20. F ( 3 ) + F ( 6 ) F(3)+F(6)
  21. F ( 4 ) + F ( 6 ) F(4)+F(6)
  22. F ( 2 ) + F ( 4 ) + F ( 6 ) F(2)+F(4)+F(6)
  23. F ( 7 ) F(7)
  24. F ( 2 ) + F ( 7 ) F(2)+F(7)
  25. F ( 0 ) F(0)
  26. F ( 1 ) F(1)
  27. F ( 2 ) F(2)
  28. F ( 3 ) F(3)
  29. F ( 4 ) F(4)
  30. F ( 5 ) F(5)
  31. F ( 6 ) F(6)
  32. F ( 7 ) F(7)
  33. F ( 8 ) F(8)
  34. F ( 9 ) F(9)
  35. F ( 10 ) F(10)

Fibonacci_number.html

  1. 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 , 144 , 1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\;\ldots\;
  2. 0 , 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 , 144 , 0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\;\ldots\;
  3. F n = F n - 1 + F n - 2 , F_{n}=F_{n-1}+F_{n-2},\!\,
  4. F 1 = 1 , F 2 = 1 F_{1}=1,\;F_{2}=1
  5. F 0 = 0 , F 1 = 1. F_{0}=0,\;F_{1}=1.
  6. F n - 2 = F n - F n - 1 , F_{n-2}=F_{n}-F_{n-1},
  7. F - n = ( - 1 ) n + 1 F n . F_{-n}=(-1)^{n+1}F_{n}.
  8. F n = k = 0 n - 1 2 ( n - k - 1 k ) F_{n}=\sum_{k=0}^{\lfloor\frac{n-1}{2}\rfloor}{\textstyle\left({{n-k-1}\atop{k% }}\right)}
  9. F n = φ n - ψ n φ - ψ = φ n - ψ n 5 F_{n}=\frac{\varphi^{n}-\psi^{n}}{\varphi-\psi}=\frac{\varphi^{n}-\psi^{n}}{% \sqrt{5}}
  10. φ = 1 + 5 2 1.61803 39887 \varphi=\frac{1+\sqrt{5}}{2}\approx 1.61803\,39887\cdots\,
  11. ψ = 1 - 5 2 = 1 - φ = - 1 φ - 0.61803 39887 \psi=\frac{1-\sqrt{5}}{2}=1-\varphi=-{1\over\varphi}\approx-0.61803\,39887\cdots
  12. ψ = - 1 φ \psi=-\frac{1}{\varphi}
  13. F n = φ n - ( - φ ) - n 5 F_{n}=\frac{\varphi^{n}-(-\varphi)^{-n}}{\sqrt{5}}
  14. x 2 = x + 1 , x n = x n - 1 + x n - 2 , x^{2}=x+1,\,x^{n}=x^{n-1}+x^{n-2},\,
  15. φ n = φ n - 1 + φ n - 2 \varphi^{n}=\varphi^{n-1}+\varphi^{n-2}\,
  16. ψ n = ψ n - 1 + ψ n - 2 . \psi^{n}=\psi^{n-1}+\psi^{n-2}\,.
  17. U n = a φ n + b ψ n U_{n}=a\varphi^{n}+b\psi^{n}\,
  18. U n = a φ n - 1 + b ψ n - 1 + a φ n - 2 + b ψ n - 2 = U n - 1 + U n - 2 . U_{n}=a\varphi^{n-1}+b\psi^{n-1}+a\varphi^{n-2}+b\psi^{n-2}=U_{n-1}+U_{n-2}.\,
  19. { a + b = 0 φ a + ψ b = 1 \left\{\begin{array}[]{l}a+b=0\\ \varphi a+\psi b=1\end{array}\right.
  20. a = 1 φ - ψ = 1 5 , b = - a a=\frac{1}{\varphi-\psi}=\frac{1}{\sqrt{5}},\,b=-a
  21. | ψ | n 5 < 1 2 \frac{|\psi|^{n}}{\sqrt{5}}<\frac{1}{2}
  22. φ n 5 . \frac{\varphi^{n}}{\sqrt{5}}\,.
  23. F n = [ φ n 5 ] , n 0 , F_{n}=\bigg[\frac{\varphi^{n}}{\sqrt{5}}\bigg],\ n\geq 0,
  24. F n = φ n 5 + 1 2 , n 0. F_{n}=\bigg\lfloor\frac{\varphi^{n}}{\sqrt{5}}+\frac{1}{2}\bigg\rfloor,\ n\geq 0.
  25. n ( F ) = log φ ( F 5 + 1 2 ) n(F)=\bigg\lfloor\log_{\varphi}\left(F\cdot\sqrt{5}+\frac{1}{2}\right)\bigg\rfloor
  26. φ \varphi
  27. lim n F n + 1 F n = φ \lim_{n\to\infty}\frac{F_{n+1}}{F_{n}}=\varphi
  28. lim n F n + α F n = φ α \lim_{n\to\infty}\frac{F_{n+\alpha}}{F_{n}}=\varphi^{\alpha}
  29. φ 2 = φ + 1 , \varphi^{2}=\varphi+1,\,
  30. φ n \varphi^{n}
  31. φ \varphi
  32. φ n = F n φ + F n - 1 . \varphi^{n}=F_{n}\varphi+F_{n-1}.
  33. F n = F n - 1 + F n - 2 . F_{n}=F_{n-1}+F_{n-2}.
  34. ( F k + 2 F k + 1 ) = ( 1 1 1 0 ) ( F k + 1 F k ) F k + 1 = 𝐀 F k , \begin{aligned}\displaystyle{F_{k+2}\choose F_{k+1}}&\displaystyle=\begin{% pmatrix}1&1\\ 1&0\end{pmatrix}{F_{k+1}\choose F_{k}}\\ \displaystyle\vec{F}_{k+1}&\displaystyle=\mathbf{A}\vec{F}_{k}~{},\end{aligned}
  35. F n = 𝐀 n F 0 \vec{F}_{n}=\mathbf{A}^{n}\vec{F}_{0}
  36. φ = 1 2 ( 1 + 5 ) \varphi=~{}\scriptstyle\frac{1}{2}(1+\sqrt{5})\,\!
  37. - φ - 1 = 1 2 ( 1 - 5 ) -\varphi^{-1}=~{}\scriptstyle\frac{1}{2}(1-\sqrt{5})
  38. μ = ( φ 1 ) \vec{\mu}=\scriptstyle{\varphi\choose 1}
  39. ν = ( - φ - 1 1 ) \vec{\nu}=\scriptstyle{-\varphi^{-1}\choose 1}
  40. F 0 = ( 1 0 ) = 1 5 μ - 1 5 ν \scriptstyle\vec{F}_{0}={1\choose 0}=\frac{1}{\sqrt{5}}\vec{\mu}-\frac{1}{% \sqrt{5}}\vec{\nu}
  41. n n
  42. F n = 1 5 A n μ - 1 5 A n ν = 1 5 φ n μ - 1 5 ( - φ ) - n ν = 1 5 ( 1 + 5 2 ) n ( φ 1 ) - 1 5 ( 1 - 5 2 ) n ( - φ - 1 1 ) , \begin{aligned}\displaystyle\vec{F}_{n}&\displaystyle=\frac{1}{\sqrt{5}}A^{n}% \vec{\mu}-\frac{1}{\sqrt{5}}A^{n}\vec{\nu}\\ &\displaystyle=\frac{1}{\sqrt{5}}\varphi^{n}\vec{\mu}-\frac{1}{\sqrt{5}}(-% \varphi)^{-n}\vec{\nu}\\ &\displaystyle=\cfrac{1}{\sqrt{5}}\cdot\left(\cfrac{1+\sqrt{5}}{2}\right)^{n}{% \varphi\choose 1}-\cfrac{1}{\sqrt{5}}\cdot\left(\cfrac{1-\sqrt{5}}{2}\right)^{% n}{-\varphi^{-1}\choose 1}~{},\end{aligned}
  43. n n
  44. n n
  45. F n = 1 5 ( 1 + 5 2 ) n - 1 5 ( 1 - 5 2 ) n . F_{n}=\cfrac{1}{\sqrt{5}}\cdot\left(\cfrac{1+\sqrt{5}}{2}\right)^{n}-\cfrac{1}% {\sqrt{5}}\cdot\left(\cfrac{1-\sqrt{5}}{2}\right)^{n}~{}.
  46. A \displaystyle A
  47. Λ = ( φ 0 0 - φ - 1 ) \Lambda=\begin{pmatrix}\varphi&0\\ 0&-\varphi^{-1}\end{pmatrix}
  48. S = ( φ - φ - 1 1 1 ) S=\begin{pmatrix}\varphi&-\varphi^{-1}\\ 1&1\end{pmatrix}
  49. n n
  50. ( F n + 1 F n ) = A n ( F 1 F 0 ) = S Λ n S - 1 ( F 1 F 0 ) = S ( φ n 0 0 ( - φ ) - n ) S - 1 ( F 1 F 0 ) = ( φ - φ - 1 1 1 ) ( φ n 0 0 ( - φ ) - n ) 1 5 ( 1 φ - 1 - 1 φ ) ( 1 0 ) , \begin{aligned}\displaystyle{F_{n+1}\choose F_{n}}&\displaystyle=A^{n}{F_{1}% \choose F_{0}}\\ &\displaystyle=S\Lambda^{n}S^{-1}{F_{1}\choose F_{0}}\\ &\displaystyle=S\begin{pmatrix}\varphi^{n}&0\\ 0&(-\varphi)^{-n}\end{pmatrix}S^{-1}{F_{1}\choose F_{0}}\\ &\displaystyle=\begin{pmatrix}\varphi&-\varphi^{-1}\\ 1&1\end{pmatrix}\begin{pmatrix}\varphi^{n}&0\\ 0&(-\varphi)^{-n}\end{pmatrix}\frac{1}{\sqrt{5}}\begin{pmatrix}1&\varphi^{-1}% \\ -1&\varphi\end{pmatrix}{1\choose 0},\end{aligned}
  51. F n = φ n - ( - φ ) - n 5 . F_{n}=\cfrac{\varphi^{n}-(-\varphi)^{-n}}{\sqrt{5}}.
  52. φ = 1 + 1 1 + 1 1 + 1 1 + \varphi=1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\;\;\ddots\,}}}
  53. φ φ
  54. ( 1 1 1 0 ) n = ( F n + 1 F n F n F n - 1 ) . \begin{pmatrix}1&1\\ 1&0\end{pmatrix}^{n}=\begin{pmatrix}F_{n+1}&F_{n}\\ F_{n}&F_{n-1}\end{pmatrix}.
  55. ( - 1 ) n = F n + 1 F n - 1 - F n 2 . (-1)^{n}=F_{n+1}F_{n-1}-F_{n}^{2}\,.
  56. F m F n + F m - 1 F n - 1 = F m + n - 1 F m F n + 1 + F m - 1 F n = F m + n . \begin{aligned}\displaystyle{F_{m}}{F_{n}}+{F_{m-1}}{F_{n-1}}&\displaystyle=F_% {m+n-1}\\ \displaystyle F_{m}F_{n+1}+F_{m-1}F_{n}&\displaystyle=F_{m+n}~{}.\end{aligned}
  57. F 2 n - 1 = F n 2 + F n - 1 2 F 2 n = ( F n - 1 + F n + 1 ) F n = ( 2 F n - 1 + F n ) F n . \begin{aligned}\displaystyle F_{2n-1}&\displaystyle=F_{n}^{2}+F_{n-1}^{2}\\ \displaystyle F_{2n}&\displaystyle=(F_{n-1}+F_{n+1})F_{n}\\ &\displaystyle=(2F_{n-1}+F_{n})F_{n}~{}.\end{aligned}
  58. O ( l o g ( n ) ) O(log(n))
  59. O ( M ( n ) l o g ( n ) ) O(M(n) log(n))
  60. M ( n ) M(n)
  61. 5 x 2 + 4 5x^{2}+4
  62. 5 x 2 - 4 5x^{2}-4
  63. n = log φ ( F n 5 + 5 F n 2 ± 4 2 ) n=\log_{\varphi}\left(\frac{F_{n}\sqrt{5}+\sqrt{5F_{n}^{2}\pm 4}}{2}\right)
  64. F n = F n - 1 + F n - 2 , F_{n}=F_{n-1}+F_{n-2},\,
  65. i = 1 n F i = F n + 2 - 1 \sum_{i=1}^{n}F_{i}=F_{n+2}-1
  66. i = 0 n - 1 F 2 i + 1 = F 2 n \sum_{i=0}^{n-1}F_{2i+1}=F_{2n}
  67. i = 1 n F 2 i = F 2 n + 1 - 1. \sum_{i=1}^{n}F_{2i}=F_{2n+1}-1.
  68. i = 1 n F i 2 = F n F n + 1 , \sum_{i=1}^{n}{F_{i}}^{2}=F_{n}F_{n+1},
  69. F n 2 - F n + 1 F n - 1 = ( - 1 ) n - 1 F_{n}^{2}-F_{n+1}F_{n-1}=(-1)^{n-1}
  70. F n 2 - F n + r F n - r = ( - 1 ) n - r F r 2 F_{n}^{2}-F_{n+r}F_{n-r}=(-1)^{n-r}F_{r}^{2}
  71. F m F n + 1 - F m + 1 F n = ( - 1 ) n F m - n F_{m}F_{n+1}-F_{m+1}F_{n}=(-1)^{n}F_{m-n}
  72. F 2 n = F n + 1 2 - F n - 1 2 = F n ( F n + 1 + F n - 1 ) = F n L n F_{2n}=F_{n+1}^{2}-F_{n-1}^{2}=F_{n}\left(F_{n+1}+F_{n-1}\right)=F_{n}L_{n}
  73. F 3 n = 2 F n 3 + 3 F n F n + 1 F n - 1 = 5 F n 3 + 3 ( - 1 ) n F n F_{3n}=2F_{n}^{3}+3F_{n}F_{n+1}F_{n-1}=5F_{n}^{3}+3(-1)^{n}F_{n}
  74. F 3 n + 1 = F n + 1 3 + 3 F n + 1 F n 2 - F n 3 F_{3n+1}=F_{n+1}^{3}+3F_{n+1}F_{n}^{2}-F_{n}^{3}
  75. F 3 n + 2 = F n + 1 3 + 3 F n + 1 2 F n + F n 3 F_{3n+2}=F_{n+1}^{3}+3F_{n+1}^{2}F_{n}+F_{n}^{3}
  76. F 4 n = 4 F n F n + 1 ( F n + 1 2 + 2 F n 2 ) - 3 F n 2 ( F n 2 + 2 F n + 1 2 ) F_{4n}=4F_{n}F_{n+1}\left(F_{n+1}^{2}+2F_{n}^{2}\right)-3F_{n}^{2}\left(F_{n}^% {2}+2F_{n+1}^{2}\right)
  77. F k n + c = i = 0 k ( k i ) F c - i F n i F n + 1 k - i . F_{kn+c}=\sum_{i=0}^{k}{k\choose i}F_{c-i}F_{n}^{i}F_{n+1}^{k-i}.
  78. k = 2 k=2
  79. s ( x ) = k = 0 F k x k . s(x)=\sum_{k=0}^{\infty}F_{k}x^{k}.
  80. | x | < 1 φ , |x|<\frac{1}{\varphi},
  81. s ( x ) = x 1 - x - x 2 s(x)=\frac{x}{1-x-x^{2}}
  82. s ( x ) \displaystyle s(x)
  83. s ( x ) = x + x s ( x ) + x 2 s ( x ) s(x)=x+xs(x)+x^{2}s(x)
  84. x x
  85. n = 0 F n k n = k k 2 - k - 1 . \sum_{n=0}^{\infty}\,\frac{F_{n}}{k^{n}}\,=\,\frac{k}{k^{2}-k-1}.
  86. n = 1 F n 10 m ( n + 1 ) = 1 10 2 m - 10 m - 1 \sum_{n=1}^{\infty}{\frac{F_{n}}{10^{m(n+1)}}}=\frac{1}{10^{2m}-10^{m}-1}
  87. s ( 1 10 ) 10 = 1 89 = .011235 . \frac{s(\frac{1}{10})}{10}=\frac{1}{89}=.011235\ldots.
  88. s ( 1 100 ) 100 = 1 9899 = .00010102030508132134 . \frac{s(\frac{1}{100})}{100}=\frac{1}{9899}=.00010102030508132134\ldots.
  89. k = 0 1 F 2 k + 1 = 5 4 ϑ 2 2 ( 0 , 3 - 5 2 ) , \sum_{k=0}^{\infty}\frac{1}{F_{2k+1}}=\frac{\sqrt{5}}{4}\vartheta_{2}^{2}\left% (0,\frac{3-\sqrt{5}}{2}\right),
  90. k = 1 1 F k 2 = 5 24 ( ϑ 2 4 ( 0 , 3 - 5 2 ) - ϑ 4 4 ( 0 , 3 - 5 2 ) + 1 ) . \sum_{k=1}^{\infty}\frac{1}{F_{k}^{2}}=\frac{5}{24}\left(\vartheta_{2}^{4}% \left(0,\frac{3-\sqrt{5}}{2}\right)-\vartheta_{4}^{4}\left(0,\frac{3-\sqrt{5}}% {2}\right)+1\right).
  91. k = 0 1 1 + F 2 k + 1 = 5 2 , \sum_{k=0}^{\infty}\frac{1}{1+F_{2k+1}}=\frac{\sqrt{5}}{2},
  92. k = 1 ( - 1 ) k + 1 j = 1 k F j 2 = 5 - 1 2 . \sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{\sum_{j=1}^{k}{F_{j}}^{2}}=\frac{\sqrt{5}% -1}{2}.
  93. ψ = k = 1 1 F k = 3.359885666243 \psi=\sum_{k=1}^{\infty}\frac{1}{F_{k}}=3.359885666243\dots
  94. n = 0 1 F 2 n = 7 - 5 2 , \sum_{n=0}^{\infty}\frac{1}{F_{2^{n}}}=\frac{7-\sqrt{5}}{2}\,,
  95. n = 0 N 1 F 2 n = 3 - F 2 N - 1 F 2 N . \sum_{n=0}^{N}\frac{1}{F_{2^{n}}}=3-\frac{F_{2^{N}-1}}{F_{2^{N}}}.
  96. gcd ( F m , F n ) = F gcd ( m , n ) . \gcd(F_{m},F_{n})=F_{\gcd(m,n)}.
  97. p F p - ( 5 p ) . p\mid F_{p-\left(\frac{5}{p}\right)}.
  98. n F n - ( 5 n ) n\mid F_{n-\left(\frac{5}{n}\right)}
  99. ( F m + 1 F m F m F m - 1 ) \begin{pmatrix}F_{m+1}&F_{m}\\ F_{m}&F_{m-1}\end{pmatrix}
  100. ( 1 1 1 0 ) m \begin{pmatrix}1&1\\ 1&0\end{pmatrix}^{m}
  101. ( p 5 ) \left(\tfrac{p}{5}\right)
  102. ( p 5 ) = { 0 if p = 5 1 if p ± 1 ( mod 5 ) - 1 if p ± 2 ( mod 5 ) . \left(\frac{p}{5}\right)=\begin{cases}0&\textrm{if}\;p=5\\ 1&\textrm{if}\;p\equiv\pm 1\;\;(\mathop{{\rm mod}}5)\\ -1&\textrm{if}\;p\equiv\pm 2\;\;(\mathop{{\rm mod}}5).\end{cases}
  103. F p ( p 5 ) ( mod p ) and F p - ( p 5 ) 0 ( mod p ) . F_{p}\equiv\left(\frac{p}{5}\right)\;\;(\mathop{{\rm mod}}p)\quad\,\text{and}% \quad F_{p-\left(\frac{p}{5}\right)}\equiv 0\;\;(\mathop{{\rm mod}}p).
  104. ( 2 5 ) = - 1 , F 3 = 2 , F 2 = 1 , ( 3 5 ) = - 1 , F 4 = 3 , F 3 = 2 , ( 5 5 ) = 0 , F 5 = 5 , ( 7 5 ) = - 1 , F 8 = 21 , F 7 = 13 , ( 11 5 ) = + 1 , F 10 = 55 , F 11 = 89. \begin{aligned}\displaystyle(\tfrac{2}{5})&\displaystyle=-1,&\displaystyle F_{% 3}&\displaystyle=2,&\displaystyle F_{2}&\displaystyle=1,\\ \displaystyle(\tfrac{3}{5})&\displaystyle=-1,&\displaystyle F_{4}&% \displaystyle=3,&\displaystyle F_{3}&\displaystyle=2,\\ \displaystyle(\tfrac{5}{5})&\displaystyle=0,&\displaystyle F_{5}&\displaystyle% =5,\\ \displaystyle(\tfrac{7}{5})&\displaystyle=-1,&\displaystyle F_{8}&% \displaystyle=21,&\displaystyle F_{7}&\displaystyle=13,\\ \displaystyle(\tfrac{11}{5})&\displaystyle=+1,&\displaystyle F_{10}&% \displaystyle=55,&\displaystyle F_{11}&\displaystyle=89.\end{aligned}
  105. F p - ( p 5 ) 0 ( mod p 2 ) . F_{p-\left(\frac{p}{5}\right)}\equiv 0\;\;(\mathop{{\rm mod}}p^{2}).
  106. 5 F p ± 1 2 2 { 1 2 ( 5 ( p 5 ) ± 5 ) ( mod p ) if p 1 ( mod 4 ) 1 2 ( 5 ( p 5 ) 3 ) ( mod p ) if p 3 ( mod 4 ) . 5F^{2}_{\frac{p\pm 1}{2}}\equiv\begin{cases}\tfrac{1}{2}\left(5\left(\frac{p}{% 5}\right)\pm 5\right)\;\;(\mathop{{\rm mod}}p)&\textrm{if}\;p\equiv 1\;\;(% \mathop{{\rm mod}}4)\\ \tfrac{1}{2}\left(5\left(\frac{p}{5}\right)\mp 3\right)\;\;(\mathop{{\rm mod}}% p)&\textrm{if}\;p\equiv 3\;\;(\mathop{{\rm mod}}4).\end{cases}
  107. ( 7 5 ) = - 1 : 1 2 ( 5 ( 7 5 ) + 3 ) = - 1 , 1 2 ( 5 ( 7 5 ) - 3 ) = - 4. (\tfrac{7}{5})=-1:\qquad\tfrac{1}{2}\left(5(\tfrac{7}{5})+3\right)=-1,\quad% \tfrac{1}{2}\left(5(\tfrac{7}{5})-3\right)=-4.
  108. F 3 = 2 and F 4 = 3. F_{3}=2\,\text{ and }F_{4}=3.
  109. 5 F 3 2 = 20 - 1 ( mod 7 ) and 5 F 4 2 = 45 - 4 ( mod 7 ) 5F_{3}^{2}=20\equiv-1\;\;(\mathop{{\rm mod}}7)\;\;\,\text{ and }\;\;5F_{4}^{2}% =45\equiv-4\;\;(\mathop{{\rm mod}}7)
  110. ( 11 5 ) = + 1 : 1 2 ( 5 ( 11 5 ) + 3 ) = 4 , 1 2 ( 5 ( 11 5 ) - 3 ) = 1. (\tfrac{11}{5})=+1:\qquad\tfrac{1}{2}\left(5(\tfrac{11}{5})+3\right)=4,\quad% \tfrac{1}{2}\left(5(\tfrac{11}{5})-3\right)=1.
  111. F 5 = 5 and F 6 = 8. F_{5}=5\,\text{ and }F_{6}=8.
  112. 5 F 5 2 = 125 4 ( mod 11 ) and 5 F 6 2 = 320 1 ( mod 11 ) 5F_{5}^{2}=125\equiv 4\;\;(\mathop{{\rm mod}}11)\;\;\,\text{ and }\;\;5F_{6}^{% 2}=320\equiv 1\;\;(\mathop{{\rm mod}}11)
  113. ( 13 5 ) = - 1 : 1 2 ( 5 ( 13 5 ) - 5 ) = - 5 , 1 2 ( 5 ( 13 5 ) + 5 ) = 0. (\tfrac{13}{5})=-1:\qquad\tfrac{1}{2}\left(5(\tfrac{13}{5})-5\right)=-5,\quad% \tfrac{1}{2}\left(5(\tfrac{13}{5})+5\right)=0.
  114. F 6 = 8 and F 7 = 13. F_{6}=8\,\text{ and }F_{7}=13.
  115. 5 F 6 2 = 320 - 5 ( mod 13 ) and 5 F 7 2 = 845 0 ( mod 13 ) 5F_{6}^{2}=320\equiv-5\;\;(\mathop{{\rm mod}}13)\;\;\,\text{ and }\;\;5F_{7}^{% 2}=845\equiv 0\;\;(\mathop{{\rm mod}}13)
  116. ( 29 5 ) = + 1 : 1 2 ( 5 ( 29 5 ) - 5 ) = 0 , 1 2 ( 5 ( 29 5 ) + 5 ) = 5. (\tfrac{29}{5})=+1:\qquad\tfrac{1}{2}\left(5(\tfrac{29}{5})-5\right)=0,\quad% \tfrac{1}{2}\left(5(\tfrac{29}{5})+5\right)=5.
  117. F 14 = 377 and F 15 = 610. F_{14}=377\,\text{ and }F_{15}=610.
  118. 5 F 14 2 = 710645 0 ( mod 29 ) and 5 F 15 2 = 1860500 5 ( mod 29 ) 5F_{14}^{2}=710645\equiv 0\;\;(\mathop{{\rm mod}}29)\;\;\,\text{ and }\;\;5F_{% 15}^{2}=1860500\equiv 5\;\;(\mathop{{\rm mod}}29)
  119. F 1 = 1 , F 3 = 2 , F 5 = 5 , F 7 = 13 , F 9 = 34 = 2 17 , F 11 = 89 , F 13 = 233 , F 15 = 610 = 2 5 61. F_{1}=1,F_{3}=2,F_{5}=5,F_{7}=13,F_{9}=34=2\cdot 17,F_{11}=89,F_{13}=233,F_{15% }=610=2\cdot 5\cdot 61.
  120. a n = F 2 n - 1 \displaystyle a_{n}=F_{2n-1}
  121. b n = 2 F n F n - 1 \displaystyle b_{n}=2F_{n}F_{n-1}
  122. c n = F n 2 - F n - 1 2 . \displaystyle c_{n}=F_{n}^{2}-F_{n-1}^{2}.
  123. a n 2 = b n 2 + c n 2 a_{n}^{2}=b_{n}^{2}+c_{n}^{2}
  124. a = F n F n + 3 ; b = 2 F n + 1 F n + 2 ; c = F n + 1 2 + F n + 2 2 ; a 2 + b 2 = c 2 . a=F_{n}F_{n+3}\,;\,b=2F_{n+1}F_{n+2}\,;\,c=F_{n+1}^{2}+F_{n+2}^{2}\,;\,a^{2}+b% ^{2}=c^{2}\,.
  125. a = 1 × 5 = 5 \displaystyle a=1\times 5=5
  126. b = 2 × 2 × 3 = 12 \displaystyle b=2\times 2\times 3=12
  127. c = 2 2 + 3 2 = 13 \displaystyle c=2^{2}+3^{2}=13\,
  128. 5 2 + 12 2 = 13 2 . \displaystyle 5^{2}+12^{2}=13^{2}\,.
  129. φ n / 5 \varphi^{n}/\sqrt{5}
  130. n log 10 φ 0.2090 n n\,\log_{10}\varphi\approx 0.2090\,n
  131. n log b φ n\,\log_{b}\varphi
  132. θ = 2 π ϕ 2 n , r = c n \theta=\frac{2\pi}{\phi^{2}}n,\ r=c\sqrt{n}

Fick's_laws_of_diffusion.html

  1. . J = - D ϕ x . \bigg.J=-D\frac{\partial\phi}{\partial x}\bigg.
  2. J J
  3. ( mol m 2 s ) (\tfrac{\mathrm{mol}}{\mathrm{m}^{2}\cdot\mathrm{s}})
  4. J J
  5. D \,D
  6. ( m 2 s ) (\tfrac{\mathrm{m}^{2}}{\mathrm{s}})
  7. ϕ \,\phi
  8. ( mol m 3 ) (\tfrac{\mathrm{mol}}{\mathrm{m}^{3}})
  9. x \,x
  10. m \,\mathrm{m}
  11. D \,D
  12. \nabla
  13. 𝐉 = - D ϕ \mathbf{J}=-D\nabla\phi
  14. - ϕ x -\frac{\partial\phi}{\partial x}
  15. J i = - D c i R T μ i x J_{i}=-\frac{Dc_{i}}{RT}\frac{\partial\mu_{i}}{\partial x}
  16. y i y_{i}
  17. k g k g \tfrac{\mathrm{k}g}{\mathrm{k}g}
  18. J i = - ρ D y i J_{i}=-\rho D\nabla y_{i}
  19. ρ \rho
  20. k g m 3 \tfrac{\mathrm{k}g}{\mathrm{m}^{3}}
  21. ϕ t = D 2 ϕ x 2 \frac{\partial\phi}{\partial t}=D\,\frac{\partial^{2}\phi}{\partial x^{2}}\,\!
  22. ϕ \,\phi
  23. ( mol m 3 ) (\tfrac{\mathrm{mol}}{m^{3}})
  24. ϕ = ϕ ( x , t ) \,\phi=\phi(x,t)
  25. x x
  26. t t
  27. t \,t
  28. D \,D
  29. ( m 2 s ) (\tfrac{m^{2}}{s})
  30. x \,x
  31. m \,m
  32. Δ = 2 \Delta=\nabla^{2}
  33. ϕ t = D Δ ϕ \frac{\partial\phi}{\partial t}=D\,\Delta\phi
  34. x = 0 x=0
  35. n 0 n_{0}
  36. n ( x , t ) = n 0 erfc ( x 2 D t ) n\left(x,t\right)=n_{0}\mathrm{erfc}\left(\frac{x}{2\sqrt{Dt}}\right)
  37. n ( x , 0 ) = 0 , x > 0 n\left(x,0\right)=0,x>0
  38. n ( x , 0 ) = n 0 , x 0 n\left(x,0\right)=n_{0},x\leq 0
  39. 2 D t 2\sqrt{Dt}
  40. n ( x , t ) = n 0 [ 1 - 2 ( x 2 D t π ) ] n\left(x,t\right)=n_{0}\left[1-2\left(\frac{x}{2\sqrt{Dt\pi}}\right)\right]
  41. D D
  42. 2 0 t D ( t ) d t 2\sqrt{\int_{0}^{t}D(t^{\prime})dt^{\prime}}
  43. D = D ( x ) D=D(x)
  44. ϕ ( x , t ) t = ( D ( x ) ϕ ( x , t ) ) = D ( x ) Δ ϕ ( x , t ) + i = 1 3 D ( x ) x i ϕ ( x , t ) x i \frac{\partial\phi(x,t)}{\partial t}=\nabla\cdot(D(x)\nabla\phi(x,t))=D(x)% \Delta\phi(x,t)+\sum_{i=1}^{3}\frac{\partial D(x)}{\partial x_{i}}\frac{% \partial\phi(x,t)}{\partial x_{i}}
  45. D = D i j D=D_{ij}
  46. J = - D ϕ J=-D\nabla\phi
  47. J i = - j = 1 3 D i j ϕ x j . \;\;J_{i}=-\sum_{j=1}^{3}D_{ij}\frac{\partial\phi}{\partial x_{j}}\ .
  48. ϕ ( x , t ) t = ( D ϕ ( x , t ) ) = j = 1 3 D i j 2 ϕ ( x , t ) x i x j . \frac{\partial\phi(x,t)}{\partial t}=\nabla\cdot(D\nabla\phi(x,t))=\sum_{j=1}^% {3}D_{ij}\frac{\partial^{2}\phi(x,t)}{\partial x_{i}\partial x_{j}}\ .
  49. D i j D_{ij}
  50. ϕ ( x , t ) t = ( D ( x ) ϕ ( x , t ) ) = i , j = 1 3 ( D i j ( x ) 2 ϕ ( x , t ) x i x j + D i j ( x ) x i ϕ ( x , t ) x j ) . \frac{\partial\phi(x,t)}{\partial t}=\nabla\cdot(D(x)\nabla\phi(x,t))=\sum_{i,% j=1}^{3}\left(D_{ij}(x)\frac{\partial^{2}\phi(x,t)}{\partial x_{i}\partial x_{% j}}+\frac{\partial D_{ij}(x)}{\partial x_{i}}\frac{\partial\phi(x,t)}{\partial x% _{j}}\right)\ .
  51. ϕ i t = j ( D i j ϕ i ϕ j ϕ j ) . \frac{\partial\phi_{i}}{\partial t}=\sum_{j}\nabla\cdot\left(D_{ij}\frac{\phi_% {i}}{\phi_{j}}\nabla\,\phi_{j}\right)\,.
  52. ϕ i \phi_{i}
  53. D i j D_{ij}
  54. t ϕ i = j D i j Δ ϕ j \partial_{t}\phi_{i}=\sum_{j}D_{ij}\Delta\phi_{j}
  55. D i j α β D_{ij\,\alpha\beta}
  56. Flux = - P ( c 2 - c 1 ) \,\text{Flux}={-P(c_{2}-c_{1})}\,\!
  57. P \,P
  58. c 2 - c 1 \,c_{2}-c_{1}
  59. c 1 c_{1}
  60. c 2 c_{2}
  61. Δ x \Delta x
  62. Δ t \Delta t
  63. N ( x , t ) N(x,t)
  64. x x
  65. t t
  66. x x
  67. x + Δ x x+\Delta x
  68. - 1 2 [ N ( x + Δ x , t ) - N ( x , t ) ] -\frac{1}{2}\left[N(x+\Delta x,t)-N(x,t)\right]
  69. Δ t \Delta t
  70. J = - 1 2 [ N ( x + Δ x , t ) a Δ t - N ( x , t ) a Δ t ] J=-\frac{1}{2}\left[\frac{N(x+\Delta x,t)}{a\Delta t}-\frac{N(x,t)}{a\Delta t}\right]
  71. ( Δ x ) 2 (\Delta x)^{2}
  72. J = - ( Δ x ) 2 2 Δ t [ N ( x + Δ x , t ) a ( Δ x ) 2 - N ( x , t ) a ( Δ x ) 2 ] J=-\frac{\left(\Delta x\right)^{2}}{2\Delta t}\left[\frac{N(x+\Delta x,t)}{a(% \Delta x)^{2}}-\frac{N(x,t)}{a(\Delta x)^{2}}\right]
  73. ϕ ( x , t ) = N ( x , t ) a Δ x \phi(x,t)=\frac{N(x,t)}{a\Delta x}
  74. ( Δ x ) 2 2 Δ t \tfrac{\left(\Delta x\right)^{2}}{2\Delta t}
  75. D D
  76. J = - D [ ϕ ( x + Δ x , t ) Δ x - ϕ ( x , t ) Δ x ] J=-D\left[\frac{\phi(x+\Delta x,t)}{\Delta x}-\frac{\phi(x,t)}{\Delta x}\right]
  77. Δ x \Delta x
  78. . J = - D ϕ x . \bigg.J=-D\frac{\partial\phi}{\partial x}\bigg.
  79. ϕ t + x J = 0 ϕ t - x ( D x ϕ ) = 0 \frac{\partial\phi}{\partial t}+\,\frac{\partial}{\partial x}\,J=0\Rightarrow% \frac{\partial\phi}{\partial t}-\frac{\partial}{\partial x}\bigg(\,D\,\frac{% \partial}{\partial x}\phi\,\bigg)\,=0\!
  80. x ( D x ϕ ) = D x x ϕ = D 2 ϕ x 2 \frac{\partial}{\partial x}\bigg(\,D\,\frac{\partial}{\partial x}\phi\,\bigg)=% D\,\frac{\partial}{\partial x}\frac{\partial}{\partial x}\,\phi=D\,\frac{% \partial^{2}\phi}{\partial x^{2}}
  81. ϕ t = D 2 ϕ \frac{\partial\phi}{\partial t}=D\,\nabla^{2}\,\phi\,\!
  82. ϕ t = ( D ϕ ) \frac{\partial\phi}{\partial t}=\nabla\cdot(\,D\,\nabla\,\phi\,)\,\!
  83. ϕ \,\phi
  84. D \,D
  85. x \,x
  86. 2 ϕ = 0 \nabla^{2}\,\phi=0\!

Field_(mathematics).html

  1. b a a b = b a a b = 1. \frac{b}{a}\cdot\frac{a}{b}=\frac{ba}{ab}=1.
  2. a b ( c d + e f ) \frac{a}{b}\cdot\left(\frac{c}{d}+\frac{e}{f}\right)
  3. = a b ( c d f f + e f d d ) =\frac{a}{b}\cdot\left(\frac{c}{d}\cdot\frac{f}{f}+\frac{e}{f}\cdot\frac{d}{d}\right)
  4. = a b ( c f d f + e d f d ) = a b c f + e d d f =\frac{a}{b}\cdot\left(\frac{cf}{df}+\frac{ed}{fd}\right)=\frac{a}{b}\cdot% \frac{cf+ed}{df}
  5. = a ( c f + e d ) b d f = a c f b d f + a e d b d f = a c b d + a e b f =\frac{a(cf+ed)}{bdf}=\frac{acf}{bdf}+\frac{aed}{bdf}=\frac{ac}{bd}+\frac{ae}{bf}
  6. = a b c d + a b e f , =\frac{a}{b}\cdot\frac{c}{d}+\frac{a}{b}\cdot\frac{e}{f}\,\text{,}
  7. p ( X ) q ( X ) , \frac{p(X)}{q(X)},
  8. E \scriptstyle E
  9. E ( X ) \scriptstyle E(X)
  10. E [ [ X ] ] \scriptstyle E[[X]]
  11. E [ [ X ] ] \scriptstyle E[[X]]
  12. E [ [ X ] ] \scriptstyle E[[X]]
  13. E ( ( X ) ) \scriptstyle E((X))
  14. R [ X ] / ( X 2 + 1 ) \scriptstyle R[X]/(X^{2}\,+\,1)
  15. a + b X ¯ a + i b \scriptstyle\overline{a\,+\,bX}\;\rightarrow\;a\,+\,ib
  16. a b \scriptstyle a^{b}
  17. ± 2 \scriptstyle\pm 2
  18. ( - 1 ) 1 / 2 = - 1 \scriptstyle(-1)^{1/2}\;=\;\sqrt{-1}
  19. + : F × F F , \scriptstyle+\colon\,F\,\times\,F\;\to\;F,\,
  20. a , b a + b . \scriptstyle a,\,b\;\mapsto\;a\,+\,b.
  21. - : F F \scriptstyle-\colon\,F\;\to\;F
  22. a - a \scriptstyle a\;\mapsto\;-a
  23. - a \scriptstyle-a
  24. a + b = 0 \scriptstyle a\,+\,b\;=\;0
  25. - : F × F F , \scriptstyle-\colon F\,\times\,F\;\to\;F,\,
  26. a , b a - b := a + ( - b ) \scriptstyle a,\,b\;\mapsto\;a\,-\,b\;:=\;a\,+\,(-b)

Field_extension.html

  1. i i
  2. 𝔠 \mathfrak{c}

Field_of_fractions.html

  1. R R
  2. a b \frac{a}{b}
  3. a a
  4. b b
  5. R R
  6. b 0 b\neq 0
  7. R R
  8. Quot ( R ) \mathrm{Quot}(R)
  9. Frac ( R ) \mathrm{Frac}(R)
  10. \Q = Quot ( \Z ) \Q=\mathrm{Quot}(\Z)
  11. R := { a + b i | a , b \Z } R:=\{a+b\mathrm{i}|a,b\in\Z\}
  12. Quot ( R ) = { c + d i | c , d \Q } \mathrm{Quot}(R)=\{c+d\mathrm{i}|c,d\in\Q\}
  13. K K
  14. K [ X ] K[X]
  15. K ( X ) K(X)
  16. R R
  17. n , d R n,d\in R
  18. d 0 d\neq 0
  19. n d \frac{n}{d}
  20. ( n , d ) (n,d)
  21. ( n , d ) (n,d)
  22. ( m , b ) (m,b)
  23. n b = m d nb=md
  24. n d = m b \frac{n}{d}=\frac{m}{b}
  25. n b = m d nb=md
  26. Quot ( R ) \mathrm{Quot}(R)
  27. n d \frac{n}{d}
  28. n d \frac{n}{d}
  29. m b \frac{m}{b}
  30. n b + m d d b \frac{nb+md}{db}
  31. n d \frac{n}{d}
  32. m b \frac{m}{b}
  33. n m d b \frac{nm}{db}
  34. R R
  35. Quot ( R ) \mathrm{Quot}(R)
  36. n n
  37. R R
  38. e n e \frac{en}{e}
  39. e R e\in R
  40. e e
  41. n 1 = n \frac{n}{1}=n
  42. R R
  43. R R
  44. e n e = n 1 \frac{en}{e}=\frac{n}{1}
  45. R R
  46. h : R F h:R\rightarrow F
  47. R R
  48. F F
  49. g : Quot ( R ) F g:\mathrm{Quot}(R)\rightarrow F
  50. h h
  51. C C
  52. C C
  53. C C
  54. R R
  55. S S
  56. R R
  57. S S
  58. - 1 -1
  59. R R
  60. r s \frac{r}{s}
  61. r R r\in R
  62. s S s\in S
  63. ( r , s ) (r,s)
  64. ( r , s ) (r^{\prime},s^{\prime})
  65. t S t\in S
  66. t ( r s - r s ) = 0 t(rs^{\prime}-r^{\prime}s)=0
  67. S S
  68. P P
  69. S S
  70. - 1 -1
  71. R R
  72. R P R_{P}
  73. R R
  74. P P
  75. R P R_{P}
  76. R R
  77. S S
  78. R R
  79. S S
  80. - 1 -1
  81. R R

Figure-eight_knot_(mathematics).html

  1. x = ( 2 + cos ( 2 t ) ) cos ( 3 t ) y = ( 2 + cos ( 2 t ) ) sin ( 3 t ) z = sin ( 4 t ) \begin{aligned}\displaystyle x&\displaystyle=\left(2+\cos{(2t)}\right)\cos{(3t% )}\\ \displaystyle y&\displaystyle=\left(2+\cos{(2t)}\right)\sin{(3t)}\\ \displaystyle z&\displaystyle=\sin{(4t)}\end{aligned}
  2. F ( x , y , z , t ) = G ( x , y , z 2 - t 2 , 2 z t ) , F(x,y,z,t)=G(x,y,z^{2}-t^{2},2zt),\,\!
  3. G ( x , y , z , t ) = \displaystyle G(x,y,z,t)=
  4. Δ ( t ) = - t + 3 - t - 1 , \Delta(t)=-t+3-t^{-1},
  5. ( z ) = 1 - z 2 , \nabla(z)=1-z^{2},
  6. V ( q ) = q 2 - q + 1 - q - 1 + q - 2 . V(q)=q^{2}-q+1-q^{-1}+q^{-2}.
  7. q q
  8. q - 1 q^{-1}
  9. σ i \sigma_{i}

Figured_bass.html

  1. 3 5 {3}_{5}
  2. 3 6 {3}_{6}
  3. 4 6 {4}_{6}
  4. 5 6 {5}_{6}
  5. 3 4 {3}_{4}
  6. 2 4 {2}_{4}
  7. C 4 6 \mbox{C}~{}_{4}^{6}
  8. C 6 \mbox{C}~{}_{6}

Filter_(mathematics).html

  1. p \uparrow p
  2. { C } \{C\}
  3. { { N , N + 1 , N + 2 , } : N { 1 , 2 , 3 , } } \{\{N,N+1,N+2,\dots\}:N\in\{1,2,3,\dots\}\}
  4. ( 1 , 2 , 3 , ) (1,2,3,\dots)
  5. ( x α ) α A (x_{\alpha})_{\alpha\in A}
  6. { { x α : α A , α 0 α } : α 0 A } \{\{x_{\alpha}:\alpha\in A,\alpha_{0}\leq\alpha\}:\alpha_{0}\in A\}\,
  7. m ( A ) = { 1 if A F 0 if S A F undefined otherwise m(A)=\begin{cases}1&\,\text{if }A\in F\\ 0&\,\text{if }S\setminus A\in F\\ \,\text{undefined}&\,\text{otherwise}\end{cases}
  8. { x S : φ ( x ) } F \left\{\,x\in S:\varphi(x)\,\right\}\in F
  9. F = { U Y | U N x } F=\{U\cap Y\ |\ U\in N_{x}\}
  10. B N x B\cup N_{x}
  11. X X
  12. Y Y
  13. B B
  14. X X
  15. f : X Y f\colon X\to Y
  16. B B
  17. f f
  18. f [ B ] f[B]
  19. { f [ x ] : x B } \{f[x]:x\in B\}
  20. f [ B ] f[B]
  21. Y Y
  22. f f
  23. x x
  24. B x B\to x
  25. f [ B ] f ( x ) f[B]\to f(x)
  26. ( X , d ) (X,d)

Financial_economics.html

  1. V t + 1 2 σ 2 S 2 2 V S 2 + r S V S - r V = 0 \frac{\partial V}{\partial t}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}V}{% \partial S^{2}}+rS\frac{\partial V}{\partial S}-rV=0

Fine-structure_constant.html

  1. α = 1 4 π ε 0 e 2 c = μ 0 4 π e 2 c = k e e 2 c = c μ 0 2 R K \alpha=\frac{1}{4\pi\varepsilon_{0}}\frac{e^{2}}{\hbar c}=\frac{\mu_{0}}{4\pi}% \frac{e^{2}c}{\hbar}=\frac{k\text{e}e^{2}}{\hbar c}=\frac{c\mu_{0}}{2R\text{K}}
  2. α = e 2 c . \alpha=\frac{e^{2}}{\hbar c}.
  3. α = e 2 4 π . \alpha=\frac{e^{2}}{4\pi}.
  4. α = e 2 ( 4 π ε 0 ) c = 7.297 352 5698 ( 24 ) × 10 - 3 . \alpha=\frac{e^{2}}{(4\pi\varepsilon_{0})\hbar c}=7.297\,352\,5698(24)\times 1% 0^{-3}.
  5. α - 1 = 137.035 999 074 ( 44 ) . \alpha^{-1}=137.035\,999\,074(44).
  6. α - 1 = 137.035 999 173 ( 35 ) . \alpha^{-1}=137.035\,999\,173(35).
  7. α = ( e q P ) 2 . \alpha=\left(\frac{e}{q\text{P}}\right)^{2}.
  8. λ = 2 π d \lambda=2\pi d
  9. α = e 2 4 π ε 0 d / h c λ = e 2 4 π ε 0 d × 2 π d h c = e 2 4 π ε 0 d × d c = e 2 4 π ε 0 c . \alpha=\frac{e^{2}}{4\pi\varepsilon_{0}d}\left/\frac{hc}{\lambda}\right.=\frac% {e^{2}}{4\pi\varepsilon_{0}d}\times{\frac{2\pi d}{hc}}=\frac{e^{2}}{4\pi% \varepsilon_{0}d}\times{\frac{d}{\hbar c}}=\frac{e^{2}}{4\pi\varepsilon_{0}% \hbar c}.
  10. = =
  11. r e r\text{e}
  12. λ e \lambda\text{e}
  13. a 0 a_{0}
  14. r e = α λ e 2 π = α 2 a 0 r\text{e}={\alpha\lambda\text{e}\over 2\pi}=\alpha^{2}a_{0}
  15. α = 1 4 Z 0 G 0 \alpha=\tfrac{1}{4}Z_{0}G_{0}
  16. Δ α α em = ( - 0.6 ± 0.6 ) × 10 - 6 . \frac{\Delta\alpha}{\alpha_{\mathrm{em}}}=\left(-0.6\pm 0.6\right)\times 10^{-% 6}.
  17. Δ α / α \Delta\alpha/\alpha
  18. Δ α / α \Delta\alpha/\alpha
  19. Δ α / α \Delta\alpha/\alpha

Finitary_relation.html

  1. k k
  2. k k
  3. k k
  4. k k
  5. X X
  6. Y Y
  7. Z Z
  8. ( X , Y , Z ) \left(X,Y,Z\right)
  9. k k
  10. k k
  11. k k
  12. k k
  13. k k
  14. = =
  15. < <
  16. 5 < 12 5<12
  17. \mid
  18. 13 143 13\mid 143
  19. \in
  20. 1 1\in\mathbb{N}
  21. k k
  22. \varnothing
  23. 𝐱 \mathbf{x}
  24. 𝐱 \mathbf{x}

Finite-state_machine.html

  1. S 1 S_{1}
  2. ( Σ , S , s 0 , δ , F ) (\Sigma,S,s_{0},\delta,F)
  3. Σ \Sigma
  4. S S
  5. s 0 s_{0}
  6. S S
  7. δ \delta
  8. δ : S × Σ S \delta:S\times\Sigma\rightarrow S
  9. δ : S × Σ 𝒫 ( S ) \delta:S\times\Sigma\rightarrow\mathcal{P}(S)
  10. δ \delta
  11. F F
  12. S S
  13. δ \delta
  14. δ ( q , x ) \delta(q,x)
  15. q \isin S q\isin S
  16. x \isin Σ x\isin\Sigma
  17. M M
  18. q q
  19. x x
  20. δ ( q , x ) \delta(q,x)
  21. M M
  22. ( Σ , Γ , S , s 0 , δ , ω ) (\Sigma,\Gamma,S,s_{0},\delta,\omega)
  23. Σ \Sigma
  24. Γ \Gamma
  25. S S
  26. s 0 s_{0}
  27. S S
  28. s 0 s_{0}
  29. δ \delta
  30. δ : S × Σ S \delta:S\times\Sigma\rightarrow S
  31. ω \omega
  32. ω : S × Σ Γ \omega:S\times\Sigma\rightarrow\Gamma
  33. ω : S Γ \omega:S\rightarrow\Gamma
  34. ω ( s 0 ) \omega(s_{0})

Finite_difference.html

  1. f ( x + b ) f ( x + a ) f(x+b) −f(x+a)
  2. b a b−a
  3. Δ h [ f ] ( x ) = f ( x + h ) - f ( x ) . \Delta_{h}[f](x)=f(x+h)-f(x).
  4. Δ [ f ] ( x ) = Δ 1 [ f ] ( x ) \Delta[f](x)=\Delta_{1}[f](x)
  5. h [ f ] ( x ) = f ( x ) - f ( x - h ) . \nabla_{h}[f](x)=f(x)-f(x-h).
  6. δ h [ f ] ( x ) = f ( x + 1 2 h ) - f ( x - 1 2 h ) . \delta_{h}[f](x)=f(x+\tfrac{1}{2}h)-f(x-\tfrac{1}{2}h)~{}.
  7. f f
  8. x x
  9. f ( x ) = lim h 0 f ( x + h ) - f ( x ) h . f^{\prime}(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}.
  10. h h
  11. f ( x + h ) - f ( x ) h = Δ h [ f ] ( x ) h . \frac{f(x+h)-f(x)}{h}=\frac{\Delta_{h}[f](x)}{h}.
  12. h h
  13. h h
  14. f f
  15. Δ h [ f ] ( x ) h - f ( x ) = O ( h ) 0 as ( h 0 ) . \frac{\Delta_{h}[f](x)}{h}-f^{\prime}(x)=O(h)\to 0\quad\,\text{as }(h\to 0).
  16. h [ f ] ( x ) h - f ( x ) = O ( h ) 0 as ( h 0 ) . \frac{\nabla_{h}[f](x)}{h}-f^{\prime}(x)=O(h)\to 0\quad\,\text{as }(h\to 0).
  17. f f
  18. δ h [ f ] ( x ) h - f ( x ) = O ( h 2 ) . \frac{\delta_{h}[f](x)}{h}-f^{\prime}(x)=O(h^{2}).\!
  19. f ( n h ) = 1 f(nh)=1
  20. n n
  21. f ( n h ) = 2 f(nh)=2
  22. n n
  23. f ( n h ) = 0 f(nh)=0
  24. f f
  25. f ( x + h / 2 ) f(x+h/2)
  26. f ( x h / 2 ) f(x−h/2)
  27. f f
  28. x x
  29. f f
  30. f ′′ ( x ) δ h 2 [ f ] ( x ) h 2 = f ( x + h ) - 2 f ( x ) + f ( x - h ) h 2 . f^{\prime\prime}(x)\approx\frac{\delta_{h}^{2}[f](x)}{h^{2}}=\frac{f(x+h)-2f(x% )+f(x-h)}{h^{2}}.
  31. f ′′ ( x ) Δ h 2 [ f ] ( x ) h 2 = f ( x + 2 h ) - 2 f ( x + h ) + f ( x ) h 2 . f^{\prime\prime}(x)\approx\frac{\Delta_{h}^{2}[f](x)}{h^{2}}=\frac{f(x+2h)-2f(% x+h)+f(x)}{h^{2}}.
  32. n n
  33. Δ h n [ f ] ( x ) = i = 0 n ( - 1 ) i ( n i ) f ( x + ( n - i ) h ) , \Delta^{n}_{h}[f](x)=\sum_{i=0}^{n}(-1)^{i}{\left({{n}\atop{i}}\right)}f(x+(n-% i)h),
  34. Δ n [ f ] ( x ) = k = 0 n ( n k ) ( - 1 ) n - k f ( x + k ) \Delta^{n}[f](x)=\sum_{k=0}^{n}{\left({{n}\atop{k}}\right)}(-1)^{n-k}f(x+k)
  35. h n [ f ] ( x ) = i = 0 n ( - 1 ) i ( n i ) f ( x - i h ) , \nabla^{n}_{h}[f](x)=\sum_{i=0}^{n}(-1)^{i}{\left({{n}\atop{i}}\right)}f(x-ih),
  36. δ h n [ f ] ( x ) = i = 0 n ( - 1 ) i ( n i ) f ( x + ( n 2 - i ) h ) . \delta^{n}_{h}[f](x)=\sum_{i=0}^{n}(-1)^{i}{\left({{n}\atop{i}}\right)}f\left(% x+\left(\frac{n}{2}-i\right)h\right).
  37. ( n i ) \ {\left({{n}\atop{i}}\right)}
  38. n n
  39. h h
  40. δ n [ f ] ( x - h / 2 ) \delta^{n}[f](x-h/2)
  41. δ n [ f ] ( x + h / 2 ) \delta^{n}[f](x+h/2)
  42. n n
  43. d n f d x n ( x ) = Δ h n [ f ] ( x ) h n + O ( h ) = h n [ f ] ( x ) h n + O ( h ) = δ h n [ f ] ( x ) h n + O ( h 2 ) . \frac{d^{n}f}{dx^{n}}(x)=\frac{\Delta_{h}^{n}[f](x)}{h^{n}}+O(h)=\frac{\nabla_% {h}^{n}[f](x)}{h^{n}}+O(h)=\frac{\delta_{h}^{n}[f](x)}{h^{n}}+O(h^{2}).
  44. h h
  45. Δ h [ f ] ( x ) - 1 2 Δ h 2 [ f ] ( x ) h = - f ( x + 2 h ) - 4 f ( x + h ) + 3 f ( x ) 2 h \frac{\Delta_{h}[f](x)-\frac{1}{2}\Delta_{h}^{2}[f](x)}{h}=-\frac{f(x+2h)-4f(x% +h)+3f(x)}{2h}
  46. Δ k h n ( f , x ) = i 1 = 0 k - 1 i 2 = 0 k - 1 i n = 0 k - 1 Δ h n ( f , x + i 1 h + i 2 h + + i n h ) . \Delta^{n}_{kh}(f,x)=\sum\limits_{i_{1}=0}^{k-1}\sum\limits_{i_{2}=0}^{k-1}% \cdots\sum\limits_{i_{n}=0}^{k-1}\Delta^{n}_{h}(f,x+i_{1}h+i_{2}h+\cdots+i_{n}% h).
  47. Δ h n ( f g , x ) = k = 0 n ( n k ) Δ h k ( f , x ) Δ h n - k ( g , x + k h ) . \Delta^{n}_{h}(fg,x)=\sum\limits_{k=0}^{n}{\left({{n}\atop{k}}\right)}\Delta^{% k}_{h}(f,x)\Delta^{n-k}_{h}(g,x+kh).
  48. ( x k ) = ( x ) k k ! {x\choose k}=\frac{(x)_{k}}{k!}
  49. ( x ) k = x ( x - 1 ) ( x - 2 ) ( x - k + 1 ) (x)_{k}=x(x-1)(x-2)\cdots(x-k+1)
  50. ( x + y ) n = k = 0 n ( n k ) ( x ) n - k ( y ) k , (x+y)_{n}=\sum_{k=0}^{n}{n\choose k}(x)_{n-k}~{}(y)_{k}~{},
  51. f f
  52. x f = Δ 0 Δ 1 Δ 2 1 2 ¯ 0 ¯ 2 2 2 ¯ 2 3 4 f ( x ) = Δ 0 1 + Δ 1 ( x - x 0 ) 1 1 ! + Δ 2 ( x - x 0 ) 2 2 ! ( x 0 = 1 ) = 2 1 + 0 x - 1 1 + 2 ( x - 1 ) ( x - 2 ) 2 = 2 + ( x - 1 ) ( x - 2 ) \begin{matrix}\begin{array}[]{|c| |c|c|c|}\hline x&f=\Delta^{0}&\Delta^{1}&% \Delta^{2}\\ \hline 1&\underline{2}&&\\ &&\underline{0}&\\ 2&2&&\underline{2}\\ &&2&\\ 3&4&&\\ \hline\end{array}&\quad\begin{aligned}\displaystyle f(x)&\displaystyle=\Delta^% {0}\cdot 1+\Delta^{1}\cdot\dfrac{(x-x_{0})_{1}}{1!}+\Delta^{2}\cdot\dfrac{(x-x% _{0})_{2}}{2!}\quad(x_{0}=1)\\ \\ &\displaystyle=2\cdot 1+0\cdot\dfrac{x-1}{1}+2\cdot\dfrac{(x-1)(x-2)}{2}\\ \\ &\displaystyle=2+(x-1)(x-2)\\ \end{aligned}\end{matrix}
  53. Δ j , 0 = y j , Δ j , k = Δ j + 1 , k - 1 - Δ j , k - 1 x j + k - x j { k > 0 , j max ( j ) - k } , Δ 0 k = Δ 0 , k \Delta_{j,0}=y_{j},\quad\quad\Delta_{j,k}=\frac{\Delta_{j+1,k-1}-\Delta_{j,k-1% }}{x_{j+k}-x_{j}}\quad\ni\quad\left\{k>0,\ \ j\leq\max\left(j\right)-k\right\}% ,\quad\quad\Delta 0_{k}=\Delta_{0,k}
  54. P 0 = 1 , P k + 1 = P k ( ξ - x k ) , {P_{0}}=1,\quad\quad P_{k+1}=P_{k}\cdot\left(\xi-x_{k}\right)~{},
  55. f ( ξ ) = Δ 0 P ( ξ ) f(\xi)=\Delta 0\cdot P\left(\xi\right)
  56. f ( x ) = k = 0 ( x - a h k ) j = 0 k ( - 1 ) k - j ( k j ) f ( a + j h ) . f(x)=\sum_{k=0}{\frac{x-a}{h}\choose k}\sum_{j=0}^{k}(-1)^{k-j}{k\choose j}f(a% +jh).
  57. f f
  58. Δ h = T h - I , \Delta_{h}=T_{h}-I,\,
  59. I I
  60. Δ h = h D + 1 2 h 2 D 2 + 1 3 ! h 3 D 3 + = e h D - I , \Delta_{h}=hD+\frac{1}{2}h^{2}D^{2}+\frac{1}{3!}h^{3}D^{3}+\cdots=\mathrm{e}^{% hD}-I~{},
  61. h D = log ( 1 + Δ h ) = Δ h - 1 2 Δ h 2 + 1 3 Δ h 3 + . hD=\log(1+\Delta_{h})=\Delta_{h}-\tfrac{1}{2}\Delta_{h}^{2}+\tfrac{1}{3}\Delta% _{h}^{3}+\cdots.\,
  62. f ( x ) f’(x)
  63. h D = - log ( 1 - h ) and h D = 2 arsinh ( 1 2 δ h ) . hD=-\log(1-\nabla_{h})\quad\,\text{and}\quad hD=2\,\operatorname{arsinh}(% \tfrac{1}{2}\delta_{h}).
  64. h 0 h→0
  65. f ( x ) f(x)
  66. ( x ) n ( x T h - 1 ) n = x ( x - h ) ( x - 2 h ) ( x - ( n - 1 ) h ) ~{}(x)_{n}\equiv(xT_{h}^{-1})^{n}=x(x-h)(x-2h)\cdots(x-(n-1)h)
  67. Δ h h ( x ) n = n ( x ) n - 1 , \frac{\Delta_{h}}{h}~{}(x)_{n}=n~{}(x)_{n-1}~{},
  68. sin ( x T h - 1 ) = x - ( x ) 3 3 ! + ( x ) 5 5 ! - ( x ) 7 7 ! + . \sin(x\,T_{h}^{-1})=x-\frac{(x)_{3}}{3!}+\frac{(x)_{5}}{5!}-\frac{(x)_{7}}{7!}% +\cdots.
  69. Δ h h ( 1 + λ h ) x / h = Δ h h e ln ( 1 + λ h ) x / h = λ e ln ( 1 + λ h ) x / h , \frac{\Delta_{h}}{h}~{}(1+\lambda h)^{x/h}=\frac{\Delta_{h}}{h}~{}e^{\ln(1+% \lambda h)~{}x/h}=\lambda~{}e^{\ln(1+\lambda h)~{}x/h}~{},
  70. δ ( x ) sin [ π 2 ( 1 + x / h ) ] π ( x + h ) , \delta(x)\mapsto\frac{\sin\bigl[\frac{\pi}{2}(1+x/h)\bigr]}{\pi(x+h)}~{},
  71. Δ c = 0 \Delta c=0{\,}
  72. Δ ( a f + b g ) = a Δ f + b Δ g \Delta(af+bg)=a\,\Delta f+b\,\Delta g
  73. \nabla
  74. Δ \Delta
  75. Δ ( f g ) = f Δ g + g Δ f + Δ f Δ g \Delta(fg)=f\,\Delta g+g\,\Delta f+\Delta f\,\Delta g
  76. ( f g ) = f g + g f - f g \nabla(fg)=f\,\nabla g+g\,\nabla f-\nabla f\,\nabla g
  77. ( f g ) = 1 g det [ f g f g ] ( det [ g g 1 1 ] ) - 1 \nabla\left(\frac{f}{g}\right)=\frac{1}{g}\det\begin{bmatrix}\nabla f&\nabla g% \\ f&g\end{bmatrix}\left(\det{\begin{bmatrix}g&\nabla g\\ 1&1\end{bmatrix}}\right)^{-1}
  78. ( f g ) = g f - f g g ( g - g ) \nabla\left(\frac{f}{g}\right)=\frac{g\,\nabla f-f\,\nabla g}{g\cdot(g-\nabla g)}
  79. Δ ( f g ) = g Δ f - f Δ g g ( g + Δ g ) \Delta\left(\frac{f}{g}\right)=\frac{g\,\Delta f-f\,\Delta g}{g\cdot(g+\Delta g)}
  80. n = a b Δ f ( n ) = f ( b + 1 ) - f ( a ) \sum_{n=a}^{b}\Delta f(n)=f(b+1)-f(a)
  81. n = a b f ( n ) = f ( b ) - f ( a - 1 ) \sum_{n=a}^{b}\nabla f(n)=f(b)-f(a-1)
  82. Δ h μ [ f ] ( x ) = k = 0 N μ k f ( x + k h ) , \Delta_{h}^{\mu}[f](x)=\sum_{k=0}^{N}\mu_{k}f(x+kh),
  83. μ = ( μ 0 , , μ N ) \mu=(\mu_{0},\ldots,\mu_{N})
  84. μ k \mu_{k}
  85. x x
  86. μ k = μ k ( x ) \mu_{k}=\mu_{k}(x)
  87. h h
  88. x x
  89. h = h ( x ) h=h(x)
  90. R [ T h ] R[T_{h}]
  91. f x ( x , y ) f ( x + h , y ) - f ( x - h , y ) 2 h f_{x}(x,y)\approx\frac{f(x+h,y)-f(x-h,y)}{2h}
  92. f y ( x , y ) f ( x , y + k ) - f ( x , y - k ) 2 k f_{y}(x,y)\approx\frac{f(x,y+k)-f(x,y-k)}{2k}
  93. f x x ( x , y ) f ( x + h , y ) - 2 f ( x , y ) + f ( x - h , y ) h 2 f_{xx}(x,y)\approx\frac{f(x+h,y)-2f(x,y)+f(x-h,y)}{h^{2}}
  94. f y y ( x , y ) f ( x , y + k ) - 2 f ( x , y ) + f ( x , y - k ) k 2 f_{yy}(x,y)\approx\frac{f(x,y+k)-2f(x,y)+f(x,y-k)}{k^{2}}
  95. f x y ( x , y ) f ( x + h , y + k ) - f ( x + h , y - k ) - f ( x - h , y + k ) + f ( x - h , y - k ) 4 h k . f_{xy}(x,y)\approx\frac{f(x+h,y+k)-f(x+h,y-k)-f(x-h,y+k)+f(x-h,y-k)}{4hk}~{}.
  96. f f
  97. f x y ( x , y ) f ( x + h , y + k ) - f ( x + h , y ) - f ( x , y + k ) + 2 f ( x , y ) - f ( x - h , y ) - f ( x , y - k ) + f ( x - h , y - k ) 2 h k , f_{xy}(x,y)\approx\frac{f(x+h,y+k)-f(x+h,y)-f(x,y+k)+2f(x,y)-f(x-h,y)-f(x,y-k)% +f(x-h,y-k)}{2hk}~{},
  98. f ( x + h , y + k ) f(x+h,y+k)
  99. f ( x h , y k ) f(x−h,y−k)

Finite_field.html

  1. q q
  2. q q
  3. p p
  4. k k
  5. p p
  6. p p
  7. q q
  8. p p
  9. G F ( p ) GF(p)
  10. 𝐙 / p 𝐙 \mathbf{Z}/p\mathbf{Z}
  11. 𝔽 p \mathbb{F}_{p}
  12. p p
  13. p p
  14. 0 , , p 1 0,...,p−1
  15. p p
  16. F F
  17. x x
  18. F F
  19. n n
  20. n x n⋅x
  21. n n
  22. x x
  23. n n
  24. n 1 = 0 n⋅1=0
  25. F F
  26. p p
  27. ( k , x ) k x (k,x)\mapsto k\cdot x
  28. F F
  29. G F ( p ) GF(p)
  30. F F
  31. p p
  32. n n
  33. 𝔽 p n \mathbb{F}_{p^{n}}
  34. ( x + y ) p = x p + y p (x+y)^{p}=x^{p}+y^{p}
  35. x x
  36. y y
  37. p p
  38. ( x + y ) p (x+y)^{p}
  39. p p
  40. x x
  41. G F ( p ) GF(p)
  42. x = 0 x=0
  43. x = 1 x=1
  44. G F ( p ) GF(p)
  45. x x
  46. 1 1
  47. x x
  48. 1 , 2 , , p 1 1,2,...,p−1
  49. p p
  50. X p - X = a GF ( p ) ( X - a ) X^{p}-X=\prod_{a\in{\rm GF}(p)}(X-a)
  51. G F ( p ) GF(p)
  52. F F
  53. P = X q - X P=X^{q}-X
  54. G F ( p ) GF(p)
  55. F F
  56. P P
  57. q q
  58. P P
  59. 1 −1
  60. P P
  61. P P
  62. P P
  63. P P
  64. q q
  65. F F
  66. q q
  67. q q
  68. q q
  69. x q = x , x^{q}=x,
  70. X q - X X^{q}-X
  71. X q - X = a F ( X - a ) . X^{q}-X=\prod_{a\in F}(X-a).
  72. m m
  73. n n
  74. X p m - X X^{p^{m}}-X
  75. X p n - X X^{p^{n}}-X
  76. m m
  77. n n
  78. p p
  79. n > 1 n>1
  80. G F ( q ) GF(q)
  81. P P
  82. G F ( p ) X X GF(p)XX
  83. n n
  84. GF ( q ) = GF ( p ) [ X ] / ( P ) {\rm GF}(q)={\rm GF}(p)[X]/(P)
  85. G F ( p ) X X GF(p)XX
  86. P P
  87. q q
  88. G F ( q ) GF(q)
  89. G F ( p ) GF(p)
  90. n n
  91. G F ( p ) GF(p)
  92. P P
  93. G F ( p ) X X GF(p)XX
  94. G F ( 4 ) GF(4)
  95. P P
  96. P P
  97. X n + a X + b , X^{n}+aX+b,
  98. X n + a X + b X^{n}+aX+b
  99. k k
  100. G F ( 2 ) GF(2)
  101. X 2 + X + 1 X^{2}+X+1
  102. G F ( 4 ) GF(4)
  103. GF ( 4 ) = GF ( 2 ) [ X ] / ( X 2 + X + 1 ) . {\rm GF}(4)={\rm GF}(2)[X]/(X^{2}+X+1).
  104. a a
  105. G F ( 4 ) GF(4)
  106. G F ( 4 ) GF(4)
  107. x x
  108. y y
  109. x x
  110. y y
  111. p = 2 p=2
  112. p p
  113. r r
  114. G F ( p ) GF(p)
  115. G F ( p ) GF(p)
  116. r r
  117. p p
  118. p - 1 2 \frac{p-1}{2}
  119. p p
  120. 2 2
  121. p = 3 , 5 , 11 , 13 , p=3,5,11,13,...
  122. 3 3
  123. p = 5 , 7 , 17 , p=5,7,17,...
  124. p 3 m o d 4 p≡3mod4
  125. p = 3 , 7 , 11 , 19 , p=3,7,11,19,...
  126. 1 p 1 −1≡p−1
  127. r r
  128. α α
  129. r r
  130. i i
  131. 1 −1
  132. a + b α , a+b\alpha,
  133. a a
  134. b b
  135. G F ( p ) GF(p)
  136. G F ( p ) GF(p)
  137. G F ( p ) GF(p)
  138. - ( a + b α ) \displaystyle-(a+b\alpha)
  139. X 3 - X - 1 X^{3}-X-1
  140. G F ( 2 ) GF(2)
  141. G F ( 3 ) GF(3)
  142. 2 2
  143. 3 3
  144. G F ( 2 ) GF(2)
  145. G F ( 3 ) GF(3)
  146. G F ( 8 ) GF(8)
  147. G F ( 27 ) GF(27)
  148. a + b α + c α 2 , a+b\alpha+c\alpha^{2},
  149. a , b , c a,b,c
  150. G F ( 2 ) GF(2)
  151. G F ( 3 ) GF(3)
  152. α \alpha
  153. α 3 = α + 1. \alpha^{3}=\alpha+1.
  154. G F ( 8 ) GF(8)
  155. G F ( 27 ) GF(27)
  156. G F ( 2 ) GF(2)
  157. G F ( 3 ) GF(3)
  158. G F ( 2 ) GF(2)
  159. G F ( 3 ) GF(3)
  160. - ( a + b α + c α 2 ) \displaystyle-(a+b\alpha+c\alpha^{2})
  161. X 4 + X + 1 X^{4}+X+1
  162. G F ( 2 ) GF(2)
  163. 2 2
  164. G F ( 16 ) GF(16)
  165. a + b α + c α 2 + d α 3 , a+b\alpha+c\alpha^{2}+d\alpha^{3},
  166. a , b , c , d a,b,c,d
  167. G F ( 2 ) GF(2)
  168. α \alpha
  169. α 4 = α + 1. \alpha^{4}=\alpha+1.
  170. G F ( 2 ) GF(2)
  171. 2 2
  172. G F ( 16 ) GF(16)
  173. G F ( 2 ) GF(2)
  174. G F ( 2 ) GF(2)
  175. ( a + b α + c α 2 + d α 3 ) + ( e + f α + g α 2 + h α 3 ) = ( a + e ) + ( b + f ) α + ( c + g ) α 2 + ( d + h ) α 3 ( a + b α + c α 2 + d α 3 ) ( e + f α + g α 2 + h α 3 ) = ( a e + b h + c g + d f ) + ( a f + b e + b h + c g + d f + c h + d g ) α + ( a g + b f + c e + c h + d g + d h ) α 2 + ( a h + b g + c f + d e + d h ) α 3 \begin{aligned}\displaystyle(a+b\alpha+c\alpha^{2}+d\alpha^{3})+(e+f\alpha+g% \alpha^{2}+h\alpha^{3})&\displaystyle=(a+e)+(b+f)\alpha+(c+g)\alpha^{2}+(d+h)% \alpha^{3}\\ \displaystyle(a+b\alpha+c\alpha^{2}+d\alpha^{3})(e+f\alpha+g\alpha^{2}+h\alpha% ^{3})&\displaystyle=(ae+bh+cg+df)+(af+be+bh+cg+df+ch+dg)\alpha\;+\\ &\displaystyle\quad\;(ag+bf+ce+ch+dg+dh)\alpha^{2}+(ah+bg+cf+de+dh)\alpha^{3}% \end{aligned}
  176. G F ( q ) GF(q)
  177. q 1 q–1
  178. k k
  179. q 1 q–1
  180. x x
  181. G F ( q ) GF(q)
  182. k k
  183. q 1 q–1
  184. k k
  185. G F ( q ) GF(q)
  186. a a
  187. q 1 q–1
  188. G F ( q ) GF(q)
  189. a a
  190. q = 2 , 3 q=2,3
  191. φ ( q 1 ) φ(q−1)
  192. φ φ
  193. x x
  194. G F ( q ) GF(q)
  195. q q
  196. a a
  197. G F ( q ) GF(q)
  198. x x
  199. F F
  200. n n
  201. 0 n q 2 0≤n≤q−2
  202. n n
  203. x x
  204. a a
  205. G F ( q ) GF(q)
  206. q 1 q–1
  207. 1 = n = 0 , , q 2 1=n=0,...,q−2
  208. −∞
  209. G F ( q ) GF(q)
  210. n n
  211. n n
  212. a a
  213. n n
  214. F F
  215. F F
  216. n n
  217. G F ( q ) GF(q)
  218. n n
  219. n n
  220. q 1 q−1
  221. n n
  222. q 1 q−1
  223. n n
  224. G F ( q ) GF(q)
  225. φ ( n ) φ(n)
  226. n n
  227. G F ( q ) GF(q)
  228. g c d ( n , q 1 ) gcd(n,q−1)
  229. p p
  230. ( n p ) (np)
  231. n n
  232. ( n p ) (np)
  233. p p
  234. n n
  235. p p
  236. n n
  237. p p
  238. 1 1
  239. n n
  240. G F ( p ) GF(p)
  241. d d
  242. p p
  243. n n
  244. G F ( 64 ) GF(64)
  245. 6 6
  246. G F ( 2 ) GF(2)
  247. 6 6
  248. 1 , 2 , 3 , 6 1,2,3,6
  249. G F ( 64 ) GF(64)
  250. G F ( 2 ) GF(2)
  251. G F ( 64 ) GF(64)
  252. 2 2
  253. 3 3
  254. G F ( 4 ) GF(4)
  255. G F ( 8 ) GF(8)
  256. G F ( 64 ) GF(64)
  257. G F ( 2 ) GF(2)
  258. G F ( 4 ) GF(4)
  259. G F ( 8 ) GF(8)
  260. 10 10
  261. 54 54
  262. G F ( 64 ) GF(64)
  263. G F ( 64 ) GF(64)
  264. 6 6
  265. G F ( 2 ) GF(2)
  266. G F ( 2 ) GF(2)
  267. 9 = 54 6 9=\frac{54}{6}
  268. 6 6
  269. G F ( 2 ) GF(2)
  270. G F ( 64 ) GF(64)
  271. n n
  272. n n
  273. 63 63
  274. G F ( 4 ) GF(4)
  275. G F ( 8 ) GF(8)
  276. 54 54
  277. n n
  278. n n
  279. 6 6
  280. 9 9
  281. 12 12
  282. 21 21
  283. 36 36
  284. 63 63
  285. 54 54
  286. G F ( 2 ) GF(2)
  287. 9 9
  288. X 6 + X 3 + 1 , X^{6}+X^{3}+1,
  289. 21 21
  290. ( X 6 + X 4 + X 2 + X + 1 ) ( X 6 + X 5 + X 4 + X 2 + 1 ) . (X^{6}+X^{4}+X^{2}+X+1)(X^{6}+X^{5}+X^{4}+X^{2}+1).
  291. 36 36
  292. G F ( 64 ) GF(64)
  293. ( X 6 + X 4 + X 3 + X + 1 ) ( X 6 + X + 1 ) ( X 6 + X 5 + 1 ) ( X 6 + X 5 + X 3 + X 2 + 1 ) ( X 6 + X 5 + X 2 + X + 1 ) ( X 6 + X 5 + X 4 + X + 1 ) , (X^{6}+X^{4}+X^{3}+X+1)(X^{6}+X+1)(X^{6}+X^{5}+1)(X^{6}+X^{5}+X^{3}+X^{2}+1)(X% ^{6}+X^{5}+X^{2}+X+1)(X^{6}+X^{5}+X^{4}+X+1),
  294. G F ( 64 ) GF(64)
  295. p p
  296. p p
  297. G F ( q ) GF(q)
  298. ( x + y ) p = x p + y p (x+y)^{p}=x^{p}+y^{p}
  299. φ : x x p \varphi:x\mapsto x^{p}
  300. G F ( p ) GF(p)
  301. G F ( q ) GF(q)
  302. G F ( p ) GF(p)
  303. φ k \varphi^{k}
  304. φ \varphi
  305. k k
  306. φ k : x x p k . \varphi^{k}:x\mapsto x^{p^{k}}.
  307. φ n \varphi^{n}
  308. μ μ
  309. n n
  310. G F ( q ) GF(q)
  311. ( q 1 ) N ( q , n ) (q−1)N(q,n)
  312. N ( q , n ) N(q,n)
  313. N ( q , n ) 1 n ( q n - p | n , p prime q n p ) . N(q,n)\geq\frac{1}{n}\left(q^{n}-\sum_{p|n,\ p\,\text{ prime}}q^{\frac{n}{p}}% \right).
  314. q q
  315. n n
  316. n n
  317. G F ( q ) GF(q)
  318. q = n = 2 q=n=2
  319. 𝐅 \mathbf{F}
  320. f ( T ) = 1 + α 𝐅 ( T - α ) , f(T)=1+\prod_{\alpha\in\mathbf{F}}(T-\alpha),
  321. 𝐅 \mathbf{F}
  322. f ( α ) = 1 f(α)=1
  323. α α
  324. 𝐅 \mathbf{F}
  325. 𝐅 p ¯ \overline{\mathbf{F}_{p}}
  326. 𝐅 p ¯ \overline{\mathbf{F}_{p}}
  327. 𝐅 p ¯ \overline{\mathbf{F}_{p}}
  328. 𝐅 p ¯ \overline{\mathbf{F}_{p}}
  329. 𝐙 \mathbf{Z}
  330. p p
  331. p p
  332. 𝐅 < s u b > p < s u p > n \mathbf{F}<sub>p<sup>n

Finite_set.html

  1. { 2 , 4 , 6 , 8 , 10 } \{2,4,6,8,10\}\,\!
  2. { 1 , 2 , 3 , } . \{1,2,3,\ldots\}.
  3. f : S { 1 , , n } f\colon S\rightarrow\{1,\ldots,n\}
  4. S = { x 1 , x 2 , , x n } . S=\{x_{1},x_{2},\ldots,x_{n}\}.
  5. | S T | | S | + | T | . |S\cup T|\leq|S|+|T|.
  6. | S T | = | S | + | T | - | S T | . |S\cup T|=|S|+|T|-|S\cap T|.
  7. | S × T | = | S | × | T | . |S\times T|=|S|\times|T|.
  8. n {}^{n}
  9. n {}^{n}
  10. { x | x < n } \{x|x<n\}
  11. f : S S f:S\rightarrow S
  12. x , f ( x ) , f ( f ( x ) ) , x,f(x),f(f(x)),...
  13. x 1 , x 2 , x 3 , x_{1},x_{2},x_{3},...
  14. f ( x i ) = x i + 1 f(x_{i})=x_{i+1}

Finitely_generated_abelian_group.html

  1. ( , + ) \left(\mathbb{Z},+\right)
  2. n n
  3. ( n , + ) \left(\mathbb{Z}_{n},+\right)
  4. ( , + ) \left(\mathbb{Q},+\right)
  5. x 1 , , x n x_{1},\ldots,x_{n}
  6. k k
  7. 1 / k 1/k
  8. x 1 , , x n x_{1},\ldots,x_{n}
  9. ( * , ) \left(\mathbb{Q}^{*},\cdot\right)
  10. n q 1 q t , \mathbb{Z}^{n}\oplus\mathbb{Z}_{q_{1}}\oplus\cdots\oplus\mathbb{Z}_{q_{t}},
  11. n k 1 k u , \mathbb{Z}^{n}\oplus\mathbb{Z}_{k_{1}}\oplus\cdots\oplus\mathbb{Z}_{k_{u}},
  12. m j k \mathbb{Z}_{m}\simeq\mathbb{Z}_{j}\oplus\mathbb{Z}_{k}
  13. \mathbb{Q}
  14. \mathbb{Q}
  15. 2 \mathbb{Z}_{2}

Firearm.html

  1. E k = 1 2 m v 2 E\text{k}=\tfrac{1}{2}mv^{2}
  2. m m
  3. v v

First-order_logic.html

  1. \land
  2. ¬ \lnot
  3. \rightarrow
  4. x φ \forall x\varphi
  5. φ \varphi
  6. x φ \exists x\varphi
  7. φ \varphi
  8. x y ( P ( f ( x ) ) ¬ ( P ( x ) Q ( f ( y ) , x , z ) ) ) \forall x\forall y(P(f(x))\rightarrow\neg(P(x)\rightarrow Q(f(y),x,z)))
  9. x x \forall x\,x\rightarrow
  10. ¬ \lnot
  11. \land
  12. \lor
  13. \to
  14. ( ¬ x P ( x ) x ¬ P ( x ) ) (\lnot\forall xP(x)\to\exists x\lnot P(x))
  15. ( ¬ [ x P ( x ) ] ) x [ ¬ P ( x ) ] . (\lnot[\forall xP(x)])\to\exists x[\lnot P(x)].
  16. \to
  17. \rightarrow
  18. \wedge
  19. x y ( P ( f ( x ) ) ¬ ( P ( x ) Q ( f ( y ) , x , z ) ) ) \forall x\forall y(P(f(x))\rightarrow\neg(P(x)\rightarrow Q(f(y),x,z)))
  20. y P ( x , y ) \forall y\,P(x,y)
  21. ¬ \neg
  22. ¬ \neg
  23. \rightarrow
  24. \rightarrow
  25. \rightarrow
  26. \forall
  27. \forall
  28. \exists
  29. \forall
  30. \forall
  31. \forall
  32. \rightarrow
  33. P ( x ) x Q ( x ) P(x)\rightarrow\forall x\,Q(x)
  34. P ( x ) x Q ( x ) P(x)\rightarrow\forall x\,Q(x)
  35. P ( x ) P(x)
  36. x x
  37. x Q ( x ) \forall x\,Q(x)
  38. x Phil ( x ) \exists x\,\,\text{Phil}(x)
  39. ( x y ( + ( x , y ) , z ) x y + ( x , y ) = 0 ) (\forall x\forall y\,\mathop{\leq}(\mathop{+}(x,y),z)\to\forall x\,\forall y\,% \mathop{+}(x,y)=0)
  40. x y ( x + y z ) x y ( x + y = 0 ) . \forall x\forall y(x+y\leq z)\to\forall x\forall y(x+y=0).
  41. x P ( x ) \exists xP(x)
  42. D D
  43. I ( c ) = 10 I(c)=10
  44. c c
  45. D n D^{n}
  46. D D
  47. D n D^{n}
  48. D n D^{n}
  49. { t r u e , f a l s e } \{true,false\}
  50. y = x y=x
  51. t 1 , , t n t_{1},\ldots,t_{n}
  52. d 1 , , d n d_{1},\ldots,d_{n}
  53. f ( t 1 , , t n ) f(t_{1},\ldots,t_{n})
  54. ( I ( f ) ) ( d 1 , , d n ) (I(f))(d_{1},\ldots,d_{n})
  55. P ( t 1 , , t n ) P(t_{1},\ldots,t_{n})
  56. v 1 , , v n I ( P ) \langle v_{1},\ldots,v_{n}\rangle\in I(P)
  57. v 1 , , v n v_{1},\ldots,v_{n}
  58. t 1 , , t n t_{1},\ldots,t_{n}
  59. I ( P ) I(P)
  60. P P
  61. D n D^{n}
  62. t 1 = t 2 t_{1}=t_{2}
  63. t 1 t_{1}
  64. t 2 t_{2}
  65. ¬ ϕ \neg\phi
  66. ϕ ψ \phi\rightarrow\psi
  67. x ϕ ( x ) \exists x\phi(x)
  68. μ \mu
  69. μ \mu^{\prime}
  70. μ \mu
  71. μ \mu^{\prime}
  72. x ϕ ( x ) \exists x\phi(x)
  73. x ϕ ( x ) \forall x\phi(x)
  74. μ \mu
  75. μ \mu^{\prime}
  76. μ \mu
  77. x ϕ ( x ) \forall x\phi(x)
  78. μ \mu
  79. μ \mu^{\prime}
  80. x ϕ ( x ) \exists x\phi(x)
  81. ϕ ( c d ) \phi(c_{d})
  82. ϕ ( c d ) \phi(c_{d})
  83. x ϕ ( x ) \forall x\phi(x)
  84. ϕ ( c d ) \phi(c_{d})
  85. M φ M\vDash\varphi
  86. ϕ x ψ \phi\lor\exists x\psi
  87. x ( ϕ ψ ) \exists x(\phi\lor\psi)
  88. x ( x = y ) \exists x(x=y)
  89. x ( x = x + 1 ) \exists x(x=x+1)
  90. z ( z = x + 1 ) \exists z(z=x+1)
  91. A 1 , , A n B 1 , , B k , A_{1},\ldots,A_{n}\vdash B_{1},\ldots,B_{k},
  92. \vdash
  93. ( A 1 A n ) (A_{1}\land\cdots\land A_{n})
  94. ( B 1 B k ) (B_{1}\lor\cdots\lor B_{k})
  95. ¬ A \lnot A
  96. C D C\lor D
  97. C D C\lor D
  98. A 1 A k C A_{1}\lor\cdots\lor A_{k}\lor C
  99. B 1 B l ¬ C B_{1}\lor\cdots\lor B_{l}\lor\lnot C
  100. A 1 A k B 1 B l A_{1}\lor\cdots\lor A_{k}\lor B_{1}\lor\cdots\lor B_{l}
  101. ¬ x P ( x ) x ¬ P ( x ) \lnot\forall x\,P(x)\Leftrightarrow\exists x\,\lnot P(x)
  102. ¬ x P ( x ) x ¬ P ( x ) \lnot\exists x\,P(x)\Leftrightarrow\forall x\,\lnot P(x)
  103. x y P ( x , y ) y x P ( x , y ) \forall x\,\forall y\,P(x,y)\Leftrightarrow\forall y\,\forall x\,P(x,y)
  104. x y P ( x , y ) y x P ( x , y ) \exists x\,\exists y\,P(x,y)\Leftrightarrow\exists y\,\exists x\,P(x,y)
  105. x P ( x ) x Q ( x ) x ( P ( x ) Q ( x ) ) \forall x\,P(x)\land\forall x\,Q(x)\Leftrightarrow\forall x\,(P(x)\land Q(x))
  106. x P ( x ) x Q ( x ) x ( P ( x ) Q ( x ) ) \exists x\,P(x)\lor\exists x\,Q(x)\Leftrightarrow\exists x\,(P(x)\lor Q(x))
  107. P x Q ( x ) x ( P Q ( x ) ) P\land\exists x\,Q(x)\Leftrightarrow\exists x\,(P\land Q(x))
  108. x x
  109. P P
  110. P x Q ( x ) x ( P Q ( x ) ) P\lor\forall x\,Q(x)\Leftrightarrow\forall x\,(P\lor Q(x))
  111. x x
  112. P P
  113. \wedge
  114. \in
  115. \forall
  116. \in
  117. \leftrightarrow
  118. \in
  119. \wedge
  120. \forall
  121. \in
  122. \leftrightarrow
  123. \in
  124. x y [ z ( z x z y ) x = y ] \forall x\forall y[\forall z(z\in x\Leftrightarrow z\in y)\Rightarrow x=y]
  125. x y [ z ( z x z y ) z ( x z y z ) ] \forall x\forall y[\forall z(z\in x\Leftrightarrow z\in y)\Rightarrow\forall z% (x\in z\Leftrightarrow y\in z)]
  126. Σ 1 1 \Sigma_{1}^{1}
  127. n \exists^{\geq n}
  128. n \exists^{\leq n}
  129. x ϕ ( x ) \exists x\phi(x)
  130. ¬ x ¬ ϕ ( x ) \neg\forall x\neg\phi(x)
  131. x ϕ ( x ) \forall x\phi(x)
  132. ¬ x ¬ ϕ ( x ) \neg\exists x\neg\phi(x)
  133. \exists
  134. \forall
  135. ϕ ψ \phi\lor\psi
  136. ¬ ( ¬ ϕ ¬ ψ ) \lnot(\lnot\phi\land\lnot\psi)
  137. ϕ ψ \phi\land\psi
  138. ¬ ( ¬ ϕ ¬ ψ ) \lnot(\lnot\phi\lor\lnot\psi)
  139. \vee
  140. \wedge
  141. ¬ \neg
  142. \vee
  143. ¬ \neg
  144. \wedge
  145. ¬ \neg
  146. \rightarrow
  147. 0 \;0
  148. 0 ( x ) \;0(x)
  149. x = 0 \;x=0
  150. P ( 0 , y ) \;P(0,y)
  151. x ( 0 ( x ) P ( x , y ) ) \forall x\;(0(x)\rightarrow P(x,y))
  152. f ( x 1 , x 2 , , x n ) f(x_{1},x_{2},...,x_{n})
  153. F ( x 1 , x 2 , , x n , y ) F(x_{1},x_{2},...,x_{n},y)
  154. y = f ( x 1 , x 2 , , x n ) y=f(x_{1},x_{2},...,x_{n})
  155. P 1 ( x ) P_{1}(x)
  156. P 2 ( x ) P_{2}(x)
  157. x ( P 1 ( x ) P 2 ( x ) ) ¬ x ( P 1 ( x ) P 2 ( x ) ) \forall x(P_{1}(x)\lor P_{2}(x))\land\lnot\exists x(P_{1}(x)\land P_{2}(x))
  158. P 1 P_{1}
  159. P 2 P_{2}
  160. x ( P 1 ( x ) ϕ ( x ) ) \exists x(P_{1}(x)\land\phi(x))
  161. ! \exists!
  162. \exists
  163. \wedge\forall
  164. \rightarrow
  165. a ( Phil ( a ) ) \exists a(\,\text{Phil}(a))
  166. Phil ( Phil ( a ) ) \exists\,\text{Phil}(\,\text{Phil}(a))