wpmath0000003_6

Gaussian_orbital.html

  1. Φ ( 𝐫 ) = R l ( r ) Y l m ( θ , ϕ ) \ \Phi(\mathbf{r})=R_{l}(r)Y_{lm}(\theta,\phi)
  2. Y l m ( θ , ϕ ) Y_{lm}(\theta,\phi)
  3. l l
  4. m m
  5. z z
  6. r , θ , ϕ r,\theta,\phi
  7. R l ( r ) = A ( l , α ) r l e - α r , \ R_{l}(r)=A(l,\alpha)r^{l}e^{-\alpha r},
  8. A ( l , α ) A(l,\alpha)
  9. R l ( r ) = B ( l , α ) r l e - α r 2 , \ R_{l}(r)=B(l,\alpha)r^{l}e^{-\alpha r^{2}},
  10. B ( l , α ) B(l,\alpha)
  11. R l ( r ) = r l p = 1 , P c p A ( l , α p ) exp ( - α p r 2 ) \ R_{l}(r)=r^{l}\sum_{p=1,P}c_{p}A(l,\alpha_{p})\exp(-\alpha_{p}r^{2})
  12. c p c_{p}
  13. α p \alpha_{p}

Gaussian_period.html

  1. 2 cos ( 2 π 17 ) = ζ + ζ 16 2\cos\left(\frac{2\pi}{17}\right)=\zeta+\zeta^{16}\,
  2. ζ = exp ( 2 π i 17 ) . \zeta=\exp\left(\frac{2\pi i}{17}\right).
  3. G = ( / n ) × G=(\mathbb{Z}/n\mathbb{Z})^{\times}
  4. ζ = exp ( 2 π i n ) . \zeta=\exp\left(\frac{2\pi i}{n}\right).
  5. ζ a \zeta^{a}
  6. a a
  7. P = Tr ( ζ ) / L ( ζ j ) P=\operatorname{Tr}_{\mathbb{Q}(\zeta)/L}(\zeta^{j})
  8. P = ζ + ζ 4 + ζ 9 + P=\zeta+\zeta^{4}+\zeta^{9}+\cdots
  9. P + P * = - 1 P+P^{*}=-1
  10. P = { - 1 + p 2 , if p = 4 m + 1 , - 1 + i p 2 , if p = 4 m + 3. P=\begin{cases}\frac{-1+\sqrt{p}}{2},&\,\text{if }p=4m+1,\\ \frac{-1+i\sqrt{p}}{2},&\,\text{if }p=4m+3.\end{cases}
  11. χ ( a ) ζ a \sum\chi(a)\zeta^{a}
  12. χ ( a ) \chi(a)
  13. G ( k , χ ) = m = 1 n χ ( m ) exp ( 2 π i m k n ) . G(k,\chi)=\sum_{m=1}^{n}\chi(m)\exp\left(\frac{2\pi imk}{n}\right).
  14. χ = χ 1 \chi=\chi_{1}
  15. G ( k , χ 1 ) = c n ( k ) = m = 1 ; ( m , n ) = 1 n exp ( 2 π i m k n ) = d | ( n , k ) d μ ( n d ) G(k,\chi_{1})=c_{n}(k)=\sum_{m=1;(m,n)=1}^{n}\exp\left(\frac{2\pi imk}{n}% \right)=\sum_{d|(n,k)}d\mu\left(\frac{n}{d}\right)
  16. G ( k , χ ) G(k,\chi)
  17. G ( 1 , χ ) G(1,\chi)
  18. G ( 1 , χ ) G(1,\chi)

Gaussian_rational.html

  1. p / q p/q
  2. 1 / q q ¯ 1/q\bar{q}
  3. q ¯ \bar{q}
  4. q q
  5. P / Q P/Q
  6. p / q p/q
  7. | P q - p Q | = 1 |Pq-pQ|=1

Gay-Lussac's_law.html

  1. P T {P}\propto{T}
  2. P T = k \frac{P}{T}=k
  3. P 1 T 1 = P 2 T 2 or P 1 T 2 = P 2 T 1 . \frac{P_{1}}{T_{1}}=\frac{P_{2}}{T_{2}}\qquad\mathrm{or}\qquad{P_{1}}{T_{2}}={% P_{2}}{T_{1}}.

Geiger–Marsden_experiment.html

  1. Δ p = F Δ t = k Q α Q n r 2 2 r v α \Delta p=F\Delta t=k\cdot\frac{Q_{\alpha}Q_{n}}{r^{2}}\cdot\frac{2r}{v_{\alpha}}
  2. θ Δ p p < k 2 Q α Q n m α r v α 2 = 8.998 10 9 × 2 × 3.204 10 - 19 × 1.266 10 - 17 6.645 10 - 27 × 1.44 10 - 10 × ( 1.53 10 7 ) 2 \theta\approx\frac{\Delta p}{p}<k\cdot\frac{2Q_{\alpha}Q_{n}}{m_{\alpha}rv_{% \alpha}^{2}}=8.998\cdot 10^{9}\times\frac{2\times 3.204\cdot 10^{-19}\times 1.% 266\cdot 10^{-17}}{6.645\cdot 10^{-27}\times 1.44\cdot 10^{-10}\times(1.53% \cdot 10^{7})^{2}}
  3. θ < 0.000326 rad ( or 0.0186 ) \theta<0.000326~{}\mathrm{rad}~{}(\mathrm{or}~{}0.0186^{\circ})
  4. s = X n t csc 4 ϕ 2 16 r 2 ( 2 Q n Q α m v 2 ) 2 s=\frac{Xnt\csc^{4}\tfrac{\phi}{2}}{16r^{2}}\cdot(\frac{2Q_{n}Q_{\alpha}}{mv^{% 2}})^{2}

Gel_permeation_chromatography.html

  1. V t = V g + V i + V o Vt=Vg+Vi+Vo

Gelfand_representation.html

  1. 1 ( 𝐙 ) \ell^{1}({\mathbf{Z}})
  2. a ^ : Φ A \widehat{a}:\Phi_{A}\to{\mathbb{C}}
  3. a ^ ( ϕ ) = ϕ ( a ) \widehat{a}(\phi)=\phi(a)
  4. a ^ \widehat{a}
  5. a a ^ a\mapsto\widehat{a}
  6. a ^ \widehat{a}
  7. f ~ \tilde{f}
  8. f {\mathcal{L}}f
  9. φ x A * \varphi_{x}\in A^{*}
  10. φ x ( f ) = f ( x ) \varphi_{x}(f)=f(x)
  11. φ x \varphi_{x}
  12. C 0 ( X ) C 0 ( Φ A ) . C_{0}(X)\to C_{0}(\Phi_{A}).
  13. γ : A C 0 ( X ) \gamma:A\to C_{0}(X)

Gelfond–Schneider_theorem.html

  1. ( 2 2 ) 2 = 2 2 2 = 2 2 = 2. {\left(\sqrt{2}^{\sqrt{2}}\right)}^{\sqrt{2}}=\sqrt{2}^{\sqrt{2}\cdot\sqrt{2}}% =\sqrt{2}^{2}=2.
  2. 2 2 2^{\sqrt{2}}
  3. 2 2 . \sqrt{2}^{\sqrt{2}}.
  4. e π = ( e i π ) - i = ( - 1 ) - i = 23.14069263 e^{\pi}=\left(e^{i\pi}\right)^{-i}=(-1)^{-i}=23.14069263\ldots
  5. i i = ( e i π / 2 ) i = e - π / 2 = 0.207879576 . i^{i}=\left(e^{i\pi/2}\right)^{i}=e^{-\pi/2}=0.207879576\ldots.

Gell-Mann_matrices.html

  1. g i g_{i}
  2. [ g i , g j ] = i f i j k g k [g_{i},g_{j}]=if^{ijk}g_{k}\,
  3. f i j k f^{ijk}
  4. f 123 = 1 , f 147 = f 165 = f 246 = f 257 = f 345 = f 376 = 1 2 , f 458 = f 678 = 3 2 . f^{123}=1\ ,\quad f^{147}=f^{165}=f^{246}=f^{257}=f^{345}=f^{376}=\frac{1}{2}% \ ,\quad f^{458}=f^{678}=\frac{\sqrt{3}}{2}\ .
  5. exp ( i θ j g j ) \mathrm{exp}(i\theta_{j}g_{j})
  6. θ j \theta_{j}
  7. λ 1 = ( 0 1 0 1 0 0 0 0 0 ) \lambda_{1}=\begin{pmatrix}0&1&0\\ 1&0&0\\ 0&0&0\end{pmatrix}
  8. λ 2 = ( 0 - i 0 i 0 0 0 0 0 ) \lambda_{2}=\begin{pmatrix}0&-i&0\\ i&0&0\\ 0&0&0\end{pmatrix}
  9. λ 3 = ( 1 0 0 0 - 1 0 0 0 0 ) \lambda_{3}=\begin{pmatrix}1&0&0\\ 0&-1&0\\ 0&0&0\end{pmatrix}
  10. λ 4 = ( 0 0 1 0 0 0 1 0 0 ) \lambda_{4}=\begin{pmatrix}0&0&1\\ 0&0&0\\ 1&0&0\end{pmatrix}
  11. λ 5 = ( 0 0 - i 0 0 0 i 0 0 ) \lambda_{5}=\begin{pmatrix}0&0&-i\\ 0&0&0\\ i&0&0\end{pmatrix}
  12. λ 6 = ( 0 0 0 0 0 1 0 1 0 ) \lambda_{6}=\begin{pmatrix}0&0&0\\ 0&0&1\\ 0&1&0\end{pmatrix}
  13. λ 7 = ( 0 0 0 0 0 - i 0 i 0 ) \lambda_{7}=\begin{pmatrix}0&0&0\\ 0&0&-i\\ 0&i&0\end{pmatrix}
  14. λ 8 = 1 3 ( 1 0 0 0 1 0 0 0 - 2 ) \lambda_{8}=\frac{1}{\sqrt{3}}\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&0&-2\end{pmatrix}
  15. g i = λ i / 2 g_{i}=\lambda_{i}/2
  16. tr ( λ i λ j ) = 2 δ i j \mathrm{tr}(\lambda_{i}\lambda_{j})=2\delta_{ij}
  17. λ 3 \lambda_{3}
  18. λ 8 \lambda_{8}
  19. { λ 1 , λ 2 , λ 3 } \{\lambda_{1},\lambda_{2},\lambda_{3}\}
  20. { λ 4 , λ 5 , x } \{\lambda_{4},\lambda_{5},x\}
  21. { λ 6 , λ 7 , y } \{\lambda_{6},\lambda_{7},y\}
  22. λ 3 \lambda_{3}
  23. λ 8 \lambda_{8}
  24. C = i = 1 8 λ i λ i = 16 / 3 C=\sum_{i=1}^{8}\lambda_{i}\lambda_{i}=16/3

Gene_regulatory_network.html

  1. N N
  2. S 1 ( t ) , S 2 ( t ) , , S N ( t ) S_{1}(t),S_{2}(t),\ldots,S_{N}(t)
  3. N N
  4. t t
  5. d S j d t = f j ( S 1 , S 2 , , S N ) \frac{dS_{j}}{dt}=f_{j}\left(S_{1},S_{2},\ldots,S_{N}\right)
  6. f j f_{j}
  7. S j S_{j}
  8. f j f_{j}
  9. f j f_{j}
  10. d S j d t = 0 \frac{dS_{j}}{dt}=0
  11. j j
  12. RNAP + Pro i k i , b a s Pro i ( τ i 1 ) + RBS i ( τ i 1 ) + RNAP ( τ i 2 ) \,\text{RNAP}+\,\text{Pro}_{i}\overset{k_{i,bas}}{\longrightarrow}\,\text{Pro}% _{i}(\tau_{i}^{1})+\,\text{RBS}_{i}(\tau_{i}^{1})+\,\text{RNAP}(\tau_{i}^{2})

General_position.html

  1. d + 1 d+1
  2. d d
  3. d d
  4. d d
  5. k k
  6. k k
  7. ( k - 2 ) (k-2)
  8. k k
  9. d + 1 d+1
  10. d + 1 d+1
  11. n - 1 n-1
  12. 3 × 3 = 9 3\times 3=9
  13. g 2 g\geq 2

Generalized_coordinates.html

  1. f j ( q 1 , , q n , t ) = 0 , j = 1 , , k , f_{j}(q_{1},...,q_{n},t)=0,j=1,...,k,
  2. g j ( q 1 , , q n , q ˙ 1 , , q ˙ n , t ) = 0 , j = 1 , . , k . g_{j}(q_{1},...,q_{n},\dot{q}_{1},...,\dot{q}_{n},t)=0,j=1,....,k.
  3. f ( x , y ) = x 2 + y 2 - L 2 = 0 , f(x,y)=x^{2}+y^{2}-L^{2}=0,
  4. f ˙ ( x , y ) = 2 x x ˙ + 2 y y ˙ = 0. \dot{f}(x,y)=2x\dot{x}+2y\dot{y}=0.
  5. 𝐫 = ( x , y ) = ( L sin θ , - L cos θ ) . \mathbf{r}=(x,y)=(L\sin\theta,-L\cos\theta).
  6. 𝐅 = ( 0 , - m g ) , \mathbf{F}=(0,-mg),
  7. δ W = 𝐅 δ 𝐫 . \delta W=\mathbf{F}\cdot\delta\mathbf{r}.
  8. δ 𝐫 = ( δ x , δ y ) = ( L cos θ , L sin θ ) δ θ . \delta\mathbf{r}=(\delta x,\delta y)=(L\cos\theta,L\sin\theta)\delta\theta.
  9. δ W = - m g δ y = - m g L sin θ δ θ . \delta W=-mg\delta y=-mgL\sin\theta\delta\theta.
  10. F θ = - m g L sin θ . F_{\theta}=-mgL\sin\theta.
  11. 𝐯 = ( x ˙ , y ˙ ) = ( L cos θ , L sin θ ) θ ˙ , \mathbf{v}=(\dot{x},\dot{y})=(L\cos\theta,L\sin\theta)\dot{\theta},
  12. T = 1 2 m 𝐯 𝐯 = 1 2 m ( x ˙ 2 + y ˙ 2 ) = 1 2 m L 2 θ ˙ 2 . T=\frac{1}{2}m\mathbf{v}\cdot\mathbf{v}=\frac{1}{2}m(\dot{x}^{2}+\dot{y}^{2})=% \frac{1}{2}mL^{2}\dot{\theta}^{2}.
  13. d d t T x ˙ - T x = F x + λ f x , d d t T y ˙ - T y = F y + λ f y . \frac{d}{dt}\frac{\partial T}{\partial\dot{x}}-\frac{\partial T}{\partial x}=F% _{x}+\lambda\frac{\partial f}{\partial x},\quad\frac{d}{dt}\frac{\partial T}{% \partial\dot{y}}-\frac{\partial T}{\partial y}=F_{y}+\lambda\frac{\partial f}{% \partial y}.
  14. m x ¨ = λ ( 2 x ) , m y ¨ = - m g + λ ( 2 y ) , x 2 + y 2 - L 2 = 0 , m\ddot{x}=\lambda(2x),\quad m\ddot{y}=-mg+\lambda(2y),\quad x^{2}+y^{2}-L^{2}=0,
  15. d d t T θ ˙ - T θ = F θ , \frac{d}{dt}\frac{\partial T}{\partial\dot{\theta}}-\frac{\partial T}{\partial% \theta}=F_{\theta},
  16. m L 2 θ ¨ = - m g L sin θ , mL^{2}\ddot{\theta}=-mgL\sin\theta,
  17. θ ¨ + g L sin θ = 0. \ddot{\theta}+\frac{g}{L}\sin\theta=0.
  18. f 1 ( x 1 , y 1 , x 2 , y 2 ) = 𝐫 1 𝐫 1 - L 1 2 = 0 , f 2 ( x 1 , y 1 , x 2 , y 2 ) = ( 𝐫 2 - 𝐫 1 ) ( 𝐫 2 - 𝐫 1 ) - L 2 2 = 0. f_{1}(x_{1},y_{1},x_{2},y_{2})=\mathbf{r}_{1}\cdot\mathbf{r}_{1}-L_{1}^{2}=0,% \quad f_{2}(x_{1},y_{1},x_{2},y_{2})=(\mathbf{r}_{2}-\mathbf{r}_{1})\cdot(% \mathbf{r}_{2}-\mathbf{r}_{1})-L_{2}^{2}=0.
  19. 𝐫 1 = ( L 1 sin θ 1 , - L 1 cos θ 1 ) , 𝐫 2 = ( L 1 sin θ 1 , - L 1 cos θ 1 ) + ( L 2 sin θ 2 , - L 2 cos θ 2 ) . \mathbf{r}_{1}=(L_{1}\sin\theta_{1},-L_{1}\cos\theta_{1}),\quad\mathbf{r}_{2}=% (L_{1}\sin\theta_{1},-L_{1}\cos\theta_{1})+(L_{2}\sin\theta_{2},-L_{2}\cos% \theta_{2}).
  20. 𝐅 1 = ( 0 , - m 1 g ) , 𝐅 2 = ( 0 , - m 2 g ) \mathbf{F}_{1}=(0,-m_{1}g),\quad\mathbf{F}_{2}=(0,-m_{2}g)
  21. δ W = 𝐅 1 δ 𝐫 1 + 𝐅 2 δ 𝐫 2 . \delta W=\mathbf{F}_{1}\cdot\delta\mathbf{r}_{1}+\mathbf{F}_{2}\cdot\delta% \mathbf{r}_{2}.
  22. δ 𝐫 1 = ( L 1 cos θ 1 , L 1 sin θ 1 ) δ θ 1 , δ 𝐫 2 = ( L 1 cos θ 1 , L 1 sin θ 1 ) δ θ 1 + ( L 2 cos θ 2 , L 2 sin θ 2 ) δ θ 2 \delta\mathbf{r}_{1}=(L_{1}\cos\theta_{1},L_{1}\sin\theta_{1})\delta\theta_{1}% ,\quad\delta\mathbf{r}_{2}=(L_{1}\cos\theta_{1},L_{1}\sin\theta_{1})\delta% \theta_{1}+(L_{2}\cos\theta_{2},L_{2}\sin\theta_{2})\delta\theta_{2}
  23. δ W = - ( m 1 + m 2 ) g L 1 sin θ 1 δ θ 1 - m 2 g L 2 sin θ 2 δ θ 2 , \delta W=-(m_{1}+m_{2})gL_{1}\sin\theta_{1}\delta\theta_{1}-m_{2}gL_{2}\sin% \theta_{2}\delta\theta_{2},
  24. F θ 1 = - ( m 1 + m 2 ) g L 1 sin θ 1 , F θ 2 = - m 2 g L 2 sin θ 2 . F_{\theta_{1}}=-(m_{1}+m_{2})gL_{1}\sin\theta_{1},\quad F_{\theta_{2}}=-m_{2}% gL_{2}\sin\theta_{2}.
  25. T = 1 2 m 1 𝐯 1 𝐯 1 + 1 2 m 2 𝐯 2 𝐯 2 = 1 2 ( m 1 + m 2 ) L 1 2 θ ˙ 1 2 + 1 2 m 2 L 2 2 θ ˙ 2 2 + m 2 L 1 L 2 cos ( θ 2 - θ 1 ) θ ˙ 1 θ ˙ 2 . T=\frac{1}{2}m_{1}\mathbf{v}_{1}\cdot\mathbf{v}_{1}+\frac{1}{2}m_{2}\mathbf{v}% _{2}\cdot\mathbf{v}_{2}=\frac{1}{2}(m_{1}+m_{2})L_{1}^{2}\dot{\theta}_{1}^{2}+% \frac{1}{2}m_{2}L_{2}^{2}\dot{\theta}_{2}^{2}+m_{2}L_{1}L_{2}\cos(\theta_{2}-% \theta_{1})\dot{\theta}_{1}\dot{\theta}_{2}.
  26. ( m 1 + m 2 ) L 1 2 θ ¨ 1 + m 2 L 1 L 2 θ ¨ 2 cos ( θ 2 - θ 1 ) + m 2 L 1 L 2 θ 2 ¨ 2 sin ( θ 1 - θ 2 ) = - ( m 1 + m 2 ) g L 1 sin θ 1 , (m_{1}+m_{2})L_{1}^{2}\ddot{\theta}_{1}+m_{2}L_{1}L_{2}\ddot{\theta}_{2}\cos(% \theta_{2}-\theta_{1})+m_{2}L_{1}L_{2}\ddot{\theta_{2}}^{2}\sin(\theta_{1}-% \theta_{2})=-(m_{1}+m_{2})gL_{1}\sin\theta_{1},
  27. m 2 L 2 2 θ ¨ 2 + m 2 L 1 L 2 θ ¨ 1 cos ( θ 2 - θ 1 ) + m 2 L 1 L 2 θ 1 ¨ 2 sin ( θ 2 - θ 1 ) = - m 2 g L 2 sin θ 2 . m_{2}L_{2}^{2}\ddot{\theta}_{2}+m_{2}L_{1}L_{2}\ddot{\theta}_{1}\cos(\theta_{2% }-\theta_{1})+m_{2}L_{1}L_{2}\ddot{\theta_{1}}^{2}\sin(\theta_{2}-\theta_{1})=% -m_{2}gL_{2}\sin\theta_{2}.
  28. δ W = j = 1 m 𝐅 j δ 𝐫 j . \delta W=\sum_{j=1}^{m}\mathbf{F}_{j}\cdot\delta\mathbf{r}_{j}.
  29. δ 𝐫 j = 𝐫 j q 1 δ q 1 + + 𝐫 j q n δ q n , \delta\mathbf{r}_{j}=\frac{\partial\mathbf{r}_{j}}{\partial q_{1}}\delta{q}_{1% }+\ldots+\frac{\partial\mathbf{r}_{j}}{\partial q_{n}}\delta{q}_{n},
  30. δ W = ( j = 1 m 𝐅 j 𝐫 j q 1 ) δ q 1 + + ( j = 1 m 𝐅 j 𝐫 j q n ) δ q n . \delta W=\left(\sum_{j=1}^{m}\mathbf{F}_{j}\cdot\frac{\partial\mathbf{r}_{j}}{% \partial q_{1}}\right)\delta{q}_{1}+\ldots+\left(\sum_{j=1}^{m}\mathbf{F}_{j}% \cdot\frac{\partial\mathbf{r}_{j}}{\partial q_{n}}\right)\delta{q}_{n}.
  31. F i = j = 1 m 𝐅 j 𝐫 j q i , i = 1 , , n , F_{i}=\sum_{j=1}^{m}\mathbf{F}_{j}\cdot\frac{\partial\mathbf{r}_{j}}{\partial q% _{i}},\quad i=1,\ldots,n,
  32. F i = j = 1 m 𝐅 j 𝐯 j q ˙ i , i = 1 , , n , F_{i}=\sum_{j=1}^{m}\mathbf{F}_{j}\cdot\frac{\partial\mathbf{v}_{j}}{\partial% \dot{q}_{i}},\quad i=1,\ldots,n,
  33. δ W = 0 F i = 0 , i = 1 , , n . \delta W=0\quad\Rightarrow\quad F_{i}=0,i=1,\ldots,n.

Generalized_flag_variety.html

  1. { 0 } = V 0 \sub V 1 \sub V 2 \sub \sub V k = V . \{0\}=V_{0}\sub V_{1}\sub V_{2}\sub\cdots\sub V_{k}=V.
  2. 0 = d 0 < d 1 < d 2 < < d k = n , 0=d_{0}<d_{1}<d_{2}<\cdots<d_{k}=n,
  3. U ( n ) / T n U(n)/T^{n}
  4. F ( d 1 , d 2 , d k , 𝔽 ) F(d_{1},d_{2},\ldots d_{k},\mathbb{F})
  5. U ( n ) / U ( n 1 ) × × U ( n k ) U(n)/U(n_{1})\times\cdots\times U(n_{k})
  6. O ( n ) / O ( n 1 ) × × O ( n k ) O(n)/O(n_{1})\times\cdots\times O(n_{k})
  7. E r + 1 0 , r E r + 1 r + 1 , 0 E_{r+1}^{0,r}\to E_{r+1}^{r+1,0}
  8. H * ( G / H ) H * ( B T ) W ( H ) / ( H ~ * ( B T ) W ( G ) ) , H^{*}(G/H)\cong H^{*}(BT)^{W(H)}/\big(\widetilde{H}^{*}(BT)^{W(G)}\big),
  9. H ~ * \widetilde{H}^{*}
  10. H * ( U ( n ) / T n ) [ t 1 , , t n ] / ( σ 1 , , σ n ) , H^{*}\big(U(n)/T^{n}\big)\cong\mathbb{Q}[t_{1},\ldots,t_{n}]/(\sigma_{1},% \ldots,\sigma_{n}),
  11. H * ( U ( 2 ) / T 2 ) [ t 1 , t 2 ] / ( t 1 + t 2 , t 1 t 2 ) [ t 1 ] / ( t 1 2 ) , H^{*}\big(U(2)/T^{2}\big)\cong\mathbb{Q}[t_{1},t_{2}]/(t_{1}+t_{2},t_{1}t_{2})% \cong\mathbb{Q}[t_{1}]/(t_{1}^{2}),

Generalized_Gauss–Bonnet_theorem.html

  1. Ω \Omega
  2. Ω \Omega
  3. 𝔰 𝔬 ( 2 n ) \mathfrak{s}\mathfrak{o}(2n)
  4. Ω \Omega
  5. even T * M \wedge^{\hbox{even}}\,T^{*}M
  6. Pf ( Ω ) \mbox{Pf}~{}(\Omega)
  7. M Pf ( Ω ) = ( 2 π ) n χ ( M ) \int_{M}\mbox{Pf}~{}(\Omega)=(2\pi)^{n}\chi(M)
  8. χ ( M ) \chi(M)
  9. n = 4 n=4
  10. χ ( M ) = 1 32 π 2 M ( | R m | 2 - 4 | R c | 2 + R 2 ) d μ \chi(M)=\frac{1}{32\pi^{2}}\int_{M}\left(|Rm|^{2}-4|Rc|^{2}+R^{2}\right)d\mu
  11. R m Rm
  12. R c Rc
  13. R R
  14. D D
  15. D * D^{*}

Generalized_hypergeometric_function.html

  1. β 0 + β 1 z + β 2 z 2 + = n 0 β n z n \beta_{0}+\beta_{1}z+\beta_{2}z^{2}+\dots=\sum_{n\geqslant 0}\beta_{n}z^{n}
  2. β n + 1 β n = A ( n ) B ( n ) \frac{\beta_{n+1}}{\beta_{n}}=\frac{A(n)}{B(n)}
  3. 1 + z 1 ! + z 2 2 ! + z 3 3 ! + c , 1+\frac{z}{1!}+\frac{z^{2}}{2!}+\frac{z^{3}}{3!}+c\dots,
  4. β n = 1 n ! , β n + 1 β n = 1 n + 1 . \beta_{n}=\frac{1}{n!},\qquad\frac{\beta_{n+1}}{\beta_{n}}=\frac{1}{n+1}.
  5. A ( n ) = 1 A(n)=1
  6. B ( n ) = n + 1 B(n)=n+1
  7. c ( a 1 + n ) ( a p + n ) d ( b 1 + n ) ( b q + n ) ( 1 + n ) \frac{c(a_{1}+n)\dots(a_{p}+n)}{d(b_{1}+n)\dots(b_{q}+n)(1+n)}
  8. 1 + a 1 a p b 1 b q .1 c z d + a 1 a p b 1 b q .1 ( a 1 + 1 ) ( a p + 1 ) ( b 1 + 1 ) ( b q + 1 ) .2 ( c z d ) 2 + 1+\frac{a_{1}\dots a_{p}}{b_{1}\dots b_{q}.1}\frac{cz}{d}+\frac{a_{1}\dots a_{% p}}{b_{1}\dots b_{q}.1}\frac{(a_{1}+1)\dots(a_{p}+1)}{(b_{1}+1)\dots(b_{q}+1).% 2}\left(\frac{cz}{d}\right)^{2}+\dots
  9. 1 + a 1 a p b 1 b q z 1 ! + a 1 ( a 1 + 1 ) a p ( a p + 1 ) b 1 ( b 1 + 1 ) b q ( b q + 1 ) z 2 2 ! + 1+\frac{a_{1}\dots a_{p}}{b_{1}\dots b_{q}}\frac{z}{1!}+\frac{a_{1}(a_{1}+1)% \dots a_{p}(a_{p}+1)}{b_{1}(b_{1}+1)\dots b_{q}(b_{q}+1)}\frac{z^{2}}{2!}+\dots
  10. F q p ( a 1 , , a p ; b 1 , , b q ; z ) {}_{p}F_{q}(a_{1},\ldots,a_{p};b_{1},\ldots,b_{q};z)
  11. F q p [ a 1 a 2 a p b 1 b 2 b q ; z ] \,{}_{p}F_{q}\left[\begin{matrix}a_{1}&a_{2}&\ldots&a_{p}\\ b_{1}&b_{2}&\ldots&b_{q}\end{matrix};z\right]
  12. ( a ) 0 \displaystyle(a)_{0}
  13. F q p ( a 1 , , a p ; b 1 , , b q ; z ) = n = 0 ( a 1 ) n ( a p ) n ( b 1 ) n ( b q ) n z n n ! \,{}_{p}F_{q}(a_{1},\ldots,a_{p};b_{1},\ldots,b_{q};z)=\sum_{n=0}^{\infty}% \frac{(a_{1})_{n}\dots(a_{p})_{n}}{(b_{1})_{n}\dots(b_{q})_{n}}\,\frac{z^{n}}{% n!}
  14. Li 2 ( x ) = n > 0 x n n - 2 = x F 2 3 ( 1 , 1 , 1 ; 2 , 2 ; x ) \operatorname{Li}_{2}(x)=\sum_{n>0}\,{x^{n}}{n^{-2}}=x\;{}_{3}F_{2}(1,1,1;2,2;x)
  15. Q n ( x ; a , b , N ) = F 2 3 ( - n , - x , n + a + b + 1 ; a + 1 , - N + 1 ; 1 ) . Q_{n}(x;a,b,N)={}_{3}F_{2}(-n,-x,n+a+b+1;a+1,-N+1;1).
  16. p n ( t 2 ) = ( a + b ) n ( a + c ) n ( a + d ) n F 3 4 ( - n a + b + c + d + n - 1 a - t a + t a + b a + c a + d ; 1 ) . p_{n}(t^{2})=(a+b)_{n}(a+c)_{n}(a+d)_{n}\;{}_{4}F_{3}\left(\begin{matrix}-n&a+% b+c+d+n-1&a-t&a+t\\ a+b&a+c&a+d\end{matrix};1\right).
  17. Γ ( a , z ) z a - 1 e - z ( 1 + a - 1 z + ( a - 1 ) ( a - 2 ) z 2 ) \Gamma(a,z)\sim z^{a-1}e^{-z}\left(1+\frac{a-1}{z}+\frac{(a-1)(a-2)}{z^{2}}% \dots\right)
  18. ( b k - a j ) > 0 \Re\left(\sum b_{k}-\sum a_{j}\right)>0
  19. i = 1 p a i j = 1 q b j \sum_{i=1}^{p}a_{i}\geq\sum_{j=1}^{q}b_{j}
  20. lim z 1 ( 1 - z ) d log ( p F q ( a 1 , , a p ; b 1 , , b q ; z p ) ) d z = i = 1 p a i - j = 1 q b j \lim_{z\rightarrow 1}(1-z)\frac{d\log(_{p}F_{q}(a_{1},\ldots,a_{p};b_{1},% \ldots,b_{q};z^{p}))}{dz}=\sum_{i=1}^{p}a_{i}-\sum_{j=1}^{q}b_{j}
  21. F 1 2 ( 3 , 1 ; 1 ; z ) = F 1 2 ( 1 , 3 ; 1 ; z ) = F 0 1 ( 3 ; ; z ) \,{}_{2}F_{1}(3,1;1;z)=\,{}_{2}F_{1}(1,3;1;z)=\,{}_{1}F_{0}(3;;z)
  22. F B + 1 A + 1 [ a 1 , , a A , c b 1 , , b B , d ; z ] = Γ ( d ) Γ ( c ) Γ ( d - c ) 0 1 t c - 1 ( 1 - t ) < m t p l > F B d - c - 1 A [ a 1 , , a A b 1 , , b B ; t z ] d t {}_{A+1}F_{B+1}\left[\begin{array}[]{c}a_{1},\ldots,a_{A},c\\ b_{1},\ldots,b_{B},d\end{array};z\right]=\frac{\Gamma(d)}{\Gamma(c)\Gamma(d-c)% }\int_{0}^{1}t^{c-1}(1-t)_{<}mtpl>{}^{d-c-1}\ {}_{A}F_{B}\left[\begin{array}[]% {c}a_{1},\ldots,a_{A}\\ b_{1},\ldots,b_{B}\end{array};tz\right]dt
  23. ( z d d z + a j ) F q p [ a 1 , , a j , , a p b 1 , , b q ; z ] = a j F q p [ a 1 , , a j + 1 , , a p b 1 , , b q ; z ] ( z d d z + b k - 1 ) F q p [ a 1 , , a p b 1 , , b k , , b q ; z ] = ( b k - 1 ) F q p [ a 1 , , a p b 1 , , b k - 1 , , b q ; z ] d d z F q p [ a 1 , , a p b 1 , , b q ; z ] = i = 1 p a i j = 1 q b j F q p [ a 1 + 1 , , a p + 1 b 1 + 1 , , b q + 1 ; z ] \begin{aligned}\displaystyle\left(z\frac{{\rm{d}}}{{\rm{d}}z}+a_{j}\right){}_{% p}F_{q}\left[\begin{array}[]{c}a_{1},\dots,a_{j},\dots,a_{p}\\ b_{1},\dots,b_{q}\end{array};z\right]&\displaystyle=a_{j}\;{}_{p}F_{q}\left[% \begin{array}[]{c}a_{1},\dots,a_{j}+1,\dots,a_{p}\\ b_{1},\dots,b_{q}\end{array};z\right]\\ \displaystyle\left(z\frac{{\rm{d}}}{{\rm{d}}z}+b_{k}-1\right){}_{p}F_{q}\left[% \begin{array}[]{c}a_{1},\dots,a_{p}\\ b_{1},\dots,b_{k},\dots,b_{q}\end{array};z\right]&\displaystyle=(b_{k}-1)\;{}_% {p}F_{q}\left[\begin{array}[]{c}a_{1},\dots,a_{p}\\ b_{1},\dots,b_{k}-1,\dots,b_{q}\end{array};z\right]\\ \displaystyle\frac{{\rm{d}}}{{\rm{d}}z}\;{}_{p}F_{q}\left[\begin{array}[]{c}a_% {1},\dots,a_{p}\\ b_{1},\dots,b_{q}\end{array};z\right]&\displaystyle=\frac{\prod_{i=1}^{p}a_{i}% }{\prod_{j=1}^{q}b_{j}}\;{}_{p}F_{q}\left[\begin{array}[]{c}a_{1}+1,\dots,a_{p% }+1\\ b_{1}+1,\dots,b_{q}+1\end{array};z\right]\end{aligned}
  24. z n = 1 p ( z d d z + a n ) w = z d d z n = 1 q ( z d d z + b n - 1 ) w z\prod_{n=1}^{p}\left(z\frac{{\rm{d}}}{{\rm{d}}z}+a_{n}\right)w=z\frac{{\rm{d}% }}{{\rm{d}}z}\prod_{n=1}^{q}\left(z\frac{{\rm{d}}}{{\rm{d}}z}+b_{n}-1\right)w
  25. ϑ = z d d z . \vartheta=z\frac{{\rm{d}}}{{\rm{d}}z}.
  26. F q p ( a 1 , , a p ; b 1 , , b q ; z ) , ϑ F q p ( a 1 , , a p ; b 1 , , b q ; z ) {}_{p}F_{q}(a_{1},\dots,a_{p};b_{1},\dots,b_{q};z),\vartheta\;{}_{p}F_{q}(a_{1% },\dots,a_{p};b_{1},\dots,b_{q};z)
  27. F q p ( a 1 , , a j + 1 , , a p ; b 1 , , b q ; z ) , {}_{p}F_{q}(a_{1},\dots,a_{j}+1,\dots,a_{p};b_{1},\dots,b_{q};z),
  28. F q p ( a 1 , , a p ; b 1 , , b k - 1 , , b q ; z ) , {}_{p}F_{q}(a_{1},\dots,a_{p};b_{1},\dots,b_{k}-1,\dots,b_{q};z),
  29. z F q p ( a 1 + 1 , , a p + 1 ; b 1 + 1 , , b q + 1 ; z ) , z\;{}_{p}F_{q}(a_{1}+1,\dots,a_{p}+1;b_{1}+1,\dots,b_{q}+1;z),
  30. F q p ( a 1 , , a p ; b 1 , , b q ; z ) . {}_{p}F_{q}(a_{1},\dots,a_{p};b_{1},\dots,b_{q};z).
  31. F q p {}_{p}F_{q}
  32. F 1 0 ( ; a ; z ) = ( 1 ) F 1 0 ( ; a ; z ) \;{}_{0}F_{1}(;a;z)=(1)\;{}_{0}F_{1}(;a;z)
  33. F 1 0 ( ; a - 1 ; z ) = ( ϑ a - 1 + 1 ) F 1 0 ( ; a ; z ) \;{}_{0}F_{1}(;a-1;z)=(\frac{\vartheta}{a-1}+1)\;{}_{0}F_{1}(;a;z)
  34. z F 1 0 ( ; a + 1 ; z ) = ( a ϑ ) F 1 0 ( ; a ; z ) z\;{}_{0}F_{1}(;a+1;z)=(a\vartheta)\;{}_{0}F_{1}(;a;z)
  35. F 1 0 ( ; a - 1 ; z ) - F 1 0 ( ; a ; z ) = z a ( a - 1 ) F 1 0 ( ; a + 1 ; z ) \;{}_{0}F_{1}(;a-1;z)-\;{}_{0}F_{1}(;a;z)=\frac{z}{a(a-1)}\;{}_{0}F_{1}(;a+1;z)
  36. F 1 1 ( a + 1 ; b ; z ) - F 1 1 ( a ; b ; z ) = z b F 1 1 ( a + 1 ; b + 1 ; z ) \;{}_{1}F_{1}(a+1;b;z)-\,{}_{1}F_{1}(a;b;z)=\frac{z}{b}\;{}_{1}F_{1}(a+1;b+1;z)
  37. F 1 1 ( a ; b - 1 ; z ) - F 1 1 ( a ; b ; z ) = a z b ( b - 1 ) F 1 1 ( a + 1 ; b + 1 ; z ) \;{}_{1}F_{1}(a;b-1;z)-\,{}_{1}F_{1}(a;b;z)=\frac{az}{b(b-1)}\;{}_{1}F_{1}(a+1% ;b+1;z)
  38. F 1 1 ( a ; b - 1 ; z ) - F 1 1 ( a + 1 ; b ; z ) = ( a - b + 1 ) z b ( b - 1 ) F 1 1 ( a + 1 ; b + 1 ; z ) \;{}_{1}F_{1}(a;b-1;z)-\,{}_{1}F_{1}(a+1;b;z)=\frac{(a-b+1)z}{b(b-1)}\;{}_{1}F% _{1}(a+1;b+1;z)
  39. F 1 2 ( a + 1 , b ; c ; z ) - F 1 2 ( a , b ; c ; z ) = b z c F 1 2 ( a + 1 , b + 1 ; c + 1 ; z ) \;{}_{2}F_{1}(a+1,b;c;z)-\,{}_{2}F_{1}(a,b;c;z)=\frac{bz}{c}\;{}_{2}F_{1}(a+1,% b+1;c+1;z)
  40. F 1 2 ( a + 1 , b ; c ; z ) - F 1 2 ( a , b + 1 ; c ; z ) = ( b - a ) z c F 1 2 ( a + 1 , b + 1 ; c + 1 ; z ) \;{}_{2}F_{1}(a+1,b;c;z)-\,{}_{2}F_{1}(a,b+1;c;z)=\frac{(b-a)z}{c}\;{}_{2}F_{1% }(a+1,b+1;c+1;z)
  41. F 1 2 ( a , b ; c - 1 ; z ) - F 1 2 ( a + 1 , b ; c ; z ) = ( a - c + 1 ) b z c ( c - 1 ) F 1 2 ( a + 1 , b + 1 ; c + 1 ; z ) \;{}_{2}F_{1}(a,b;c-1;z)-\,{}_{2}F_{1}(a+1,b;c;z)=\frac{(a-c+1)bz}{c(c-1)}\;{}% _{2}F_{1}(a+1,b+1;c+1;z)
  42. ( p + q + 3 2 ) {\left({{p+q+3}\atop{2}}\right)}
  43. { 1 , ϑ , ϑ 2 } F q p ( a 1 , , a p ; b 1 , , b q ; z ) , \{1,\vartheta,\vartheta^{2}\}\;{}_{p}F_{q}(a_{1},\dots,a_{p};b_{1},\dots,b_{q}% ;z),
  44. F q p ( a 1 , , a p ; b 1 , , b q ; z ) {}_{p}F_{q}(a_{1},\dots,a_{p};b_{1},\dots,b_{q};z)
  45. F q p ( a 1 , , a p ; b 1 , , b q ; z ) . {}_{p}F_{q}(a_{1},\dots,a_{p};b_{1},\dots,b_{q};z).
  46. F 1 0 ( ; a ; z ) {}_{0}F_{1}(;a;z)
  47. F 1 1 ( a ; b ; z ) {}_{1}F_{1}(a;b;z)
  48. F 1 2 ( a , b ; c ; z ) {}_{2}F_{1}(a,b;c;z)
  49. F 2 3 ( a , b , - n ; c , 1 + a + b - c - n ; 1 ) = ( c - a ) n ( c - b ) n ( c ) n ( c - a - b ) n . {}_{3}F_{2}(a,b,-n;c,1+a+b-c-n;1)=\frac{(c-a)_{n}(c-b)_{n}}{(c)_{n}(c-a-b)_{n}}.
  50. F 2 3 ( a , b , c ; 1 + a - b , 1 + a - c ; 1 ) = Γ ( 1 + a 2 ) Γ ( 1 + a 2 - b - c ) Γ ( 1 + a - b ) Γ ( 1 + a - c ) Γ ( 1 + a ) Γ ( 1 + a - b - c ) Γ ( 1 + a 2 - b ) Γ ( 1 + a 2 - c ) . {}_{3}F_{2}(a,b,c;1+a-b,1+a-c;1)=\frac{\Gamma(1+\frac{a}{2})\Gamma(1+\frac{a}{% 2}-b-c)\Gamma(1+a-b)\Gamma(1+a-c)}{\Gamma(1+a)\Gamma(1+a-b-c)\Gamma(1+\frac{a}% {2}-b)\Gamma(1+\frac{a}{2}-c)}.
  51. F 6 7 \displaystyle{}_{7}F_{6}
  52. 1 + 2 a = b + c + d + e - m . 1+2a=b+c+d+e-m.
  53. e - x F 2 2 ( a , 1 + d ; c , d ; x ) = F 2 2 ( c - a - 1 , f + 1 ; c , f ; - x ) e^{-x}\;{}_{2}F_{2}(a,1+d;c,d;x)={}_{2}F_{2}(c-a-1,f+1;c,f;-x)
  54. f = d ( a - c + 1 ) a - d f=\frac{d(a-c+1)}{a-d}
  55. e - x 2 F 2 2 ( a , 1 + b ; 2 a + 1 , b ; x ) = F 1 0 ( ; a + 1 2 ; x 2 16 ) - x ( 1 - 2 a b ) 2 ( 2 a + 1 ) F 1 0 ( ; a + 3 2 ; x 2 16 ) , e^{-\frac{x}{2}}\,{}_{2}F_{2}\left(a,1+b;2a+1,b;x\right)={}_{0}F_{1}\left(;a+% \tfrac{1}{2};\tfrac{x^{2}}{16}\right)-\frac{x\left(1-\tfrac{2a}{b}\right)}{2(2% a+1)}\;{}_{0}F_{1}\left(;a+\tfrac{3}{2};\tfrac{x^{2}}{16}\right),
  56. e - x 2 F 1 1 ( a , 2 a , x ) = F 1 0 ( ; a + 1 2 ; x 2 16 ) e^{-\frac{x}{2}}\,{}_{1}F_{1}(a,2a,x)={}_{0}F_{1}\left(;a+\tfrac{1}{2};\tfrac{% x^{2}}{16}\right)
  57. F 2 2 ( a , b ; c , d ; x ) = i = 0 ( b - d i ) ( a + i - 1 i ) ( c + i - 1 i ) ( d + i - 1 i ) F 1 1 ( a + i ; c + i ; x ) x i i ! = e x i = 0 ( b - d i ) ( a + i - 1 i ) ( c + i - 1 i ) ( d + i - 1 i ) F 1 1 ( c - a ; c + i ; - x ) x i i ! , \begin{aligned}\displaystyle{}_{2}F_{2}(a,b;c,d;x)=&\displaystyle\sum_{i=0}% \frac{{b-d\choose i}{a+i-1\choose i}}{{c+i-1\choose i}{d+i-1\choose i}}\;{}_{1% }F_{1}(a+i;c+i;x)\frac{x^{i}}{i!}\\ \displaystyle=&\displaystyle e^{x}\sum_{i=0}\frac{{b-d\choose i}{a+i-1\choose i% }}{{c+i-1\choose i}{d+i-1\choose i}}\;{}_{1}F_{1}(c-a;c+i;-x)\frac{x^{i}}{i!},% \end{aligned}
  58. F 1 2 ( 2 a , 2 b ; a + b + 1 2 ; x ) = F 1 2 ( a , b ; a + b + 1 2 ; 4 x ( 1 - x ) ) . {}_{2}F_{1}\left(2a,2b;a+b+\tfrac{1}{2};x\right)={}_{2}F_{1}\left(a,b;a+b+% \tfrac{1}{2};4x(1-x)\right).
  59. F 2 3 ( 2 c - 2 s - 1 , 2 s , c - 1 2 ; 2 c - 1 , c ; x ) = F 1 2 ( c - s - 1 2 , s ; c ; x ) 2 {}_{3}F_{2}(2c-2s-1,2s,c-\tfrac{1}{2};2c-1,c;x)=\,{}_{2}F_{1}(c-s-\tfrac{1}{2}% ,s;c;x)^{2}
  60. F 0 0 ( ; ; z ) = e z {}_{0}F_{0}(;;z)=e^{z}
  61. d d z w = w \frac{d}{dz}w=w
  62. w = k e z w=ke^{z}
  63. F 0 1 ( a ; ; z ) = ( 1 - z ) - a . {}_{1}F_{0}(a;;z)=(1-z)^{-a}.
  64. d d z w = ( z d d z + a ) w , \frac{d}{dz}w=\left(z\frac{d}{dz}+a\right)w,
  65. ( 1 - z ) d w d z = a w , (1-z)\frac{dw}{dz}=aw,
  66. w = k ( 1 - z ) - a w=k(1-z)^{-a}
  67. F 0 1 ( 1 ; ; z ) = ( 1 - z ) - 1 {}_{1}F_{0}(1;;z)=(1-z)^{-1}
  68. F 1 0 ( ; a ; z ) {}_{0}F_{1}(;a;z)
  69. J α ( x ) = ( x 2 ) α Γ ( α + 1 ) F 1 0 ( ; α + 1 ; - 1 4 x 2 ) . J_{\alpha}(x)=\frac{(\tfrac{x}{2})^{\alpha}}{\Gamma(\alpha+1)}{}_{0}F_{1}\left% (;\alpha+1;-\tfrac{1}{4}x^{2}\right).
  70. w = ( z d d z + a ) d w d z w=\left(z\frac{d}{dz}+a\right)\frac{dw}{dz}
  71. z d 2 w d z 2 + a d w d z - w = 0. z\frac{d^{2}w}{dz^{2}}+a\frac{dw}{dz}-w=0.
  72. w = z 1 - a u , w=z^{1-a}u,
  73. z 1 - a F 1 0 ( ; 2 - a ; z ) , z^{1-a}\;{}_{0}F_{1}(;2-a;z),
  74. k F 1 0 ( ; a ; z ) + l z 1 - a F 1 0 ( ; 2 - a ; z ) k\;{}_{0}F_{1}(;a;z)+lz^{1-a}\;{}_{0}F_{1}(;2-a;z)
  75. F 1 1 ( a ; b ; z ) {}_{1}F_{1}(a;b;z)
  76. M ( a ; b ; z ) M(a;b;z)
  77. γ ( a , z ) \gamma(a,z)
  78. ( z d d z + a ) w = ( z d d z + b ) d w d z \left(z\frac{d}{dz}+a\right)w=\left(z\frac{d}{dz}+b\right)\frac{dw}{dz}
  79. z d 2 w d z 2 + ( b - z ) d w d z - a w = 0. z\frac{d^{2}w}{dz^{2}}+(b-z)\frac{dw}{dz}-aw=0.
  80. w = z 1 - b u , w=z^{1-b}u,
  81. z 1 - b F 1 1 ( 1 + a - b ; 2 - b ; z ) , z^{1-b}\;{}_{1}F_{1}(1+a-b;2-b;z),
  82. k F 1 1 ( a ; b ; z ) + l z 1 - b F 1 1 ( 1 + a - b ; 2 - b ; z ) k\;{}_{1}F_{1}(a;b;z)+lz^{1-b}\;{}_{1}F_{1}(1+a-b;2-b;z)
  83. F 1 1 ( - n ; b ; z ) {}_{1}F_{1}(-n;b;z)
  84. F 1 2 ( a , b ; c ; z ) {}_{2}F_{1}(a,b;c;z)
  85. ( z d d z + a ) ( z d d z + b ) w = ( z d d z + c ) d w d z \left(z\frac{d}{dz}+a\right)\left(z\frac{d}{dz}+b\right)w=\left(z\frac{d}{dz}+% c\right)\frac{dw}{dz}
  86. z ( 1 - z ) d 2 w d z 2 + ( c - ( a + b + 1 ) z ) d w d z - a b w = 0. z(1-z)\frac{d^{2}w}{dz^{2}}+(c-(a+b+1)z)\frac{dw}{dz}-abw=0.
  87. w = z 1 - c u w=z^{1-c}u
  88. z 1 - c F 1 2 ( 1 + a - c , 1 + b - c ; 2 - c ; z ) , z^{1-c}\;{}_{2}F_{1}(1+a-c,1+b-c;2-c;z),
  89. F 1 2 ( - n , b ; c ; z ) {}_{2}F_{1}(-n,b;c;z)
  90. 0 x 1 + y α d y = x 2 + α { α F 1 2 ( 1 α , 1 2 ; 1 + 1 α ; - x α ) + 2 x α + 1 } , α 0. \int_{0}^{x}\sqrt{1+y^{\alpha}}\,\mathrm{d}y=\frac{x}{2+\alpha}\left\{\alpha\;% {}_{2}F_{1}\left(\tfrac{1}{\alpha},\tfrac{1}{2};1+\tfrac{1}{\alpha};-x^{\alpha% }\right)+2\sqrt{x^{\alpha}+1}\right\},\qquad\alpha\neq 0.

Geodesic_curvature.html

  1. k g k_{g}
  2. γ \gamma
  3. M ¯ \bar{M}
  4. γ \gamma
  5. γ \gamma
  6. M M
  7. M ¯ \bar{M}
  8. γ \gamma
  9. M M
  10. γ \gamma
  11. M ¯ \bar{M}
  12. k k
  13. γ \gamma
  14. M M
  15. γ \gamma
  16. k n k_{n}
  17. γ \gamma
  18. M M
  19. k g k_{g}
  20. k = k g 2 + k n 2 k=\sqrt{k_{g}^{2}+k_{n}^{2}}
  21. M M
  22. k = k n k=k_{n}
  23. γ \gamma
  24. M ¯ \bar{M}
  25. T = d γ / d s T=d\gamma/ds
  26. T T
  27. k = D T / d s k=\|DT/ds\|
  28. γ \gamma
  29. M M
  30. D T / d s DT/ds
  31. D T / d s DT/ds
  32. n \mathbb{R}^{n}
  33. D T / d s DT/ds
  34. d T / d s dT/ds
  35. M M
  36. S 2 S^{2}
  37. S 2 S^{2}
  38. k = 1 k=1
  39. r r
  40. 1 / r 1/r
  41. k g = 1 - r 2 / r k_{g}=\sqrt{1-r^{2}}/r
  42. M M
  43. M M
  44. M ¯ \bar{M}
  45. M M
  46. D T / d s DT/ds
  47. M M
  48. k n k_{n}
  49. T T
  50. D T / d s DT/ds
  51. ¯ \bar{\nabla}
  52. D T / d s = ¯ T T DT/ds=\bar{\nabla}_{T}T
  53. ¯ T T = T T + ( ¯ T T ) \bar{\nabla}_{T}T=\nabla_{T}T+(\bar{\nabla}_{T}T)^{\perp}
  54. T T \nabla_{T}T
  55. M M
  56. I I ( T , T ) \mathrm{I\!I}(T,T)
  57. I I \mathrm{I\!I}

Geographic_coordinate_conversion.html

  1. decimal degrees = degrees + minutes / 60 + seconds / 3600 \rm{decimal\ degrees}=\rm{degrees}+\rm{minutes}/60+\rm{seconds}/3600
  2. degrees \displaystyle\rm{degrees}
  3. x \lfloor x\rfloor
  4. x x
  5. N ( ϕ ) \,N(\phi)
  6. e 2 N ( ϕ ) \,e^{2}N(\phi)
  7. ( X , Y , Z ) \,(X,Y,Z)
  8. ϕ \ \phi
  9. λ \ \lambda
  10. h h
  11. X \displaystyle X
  12. N ( ϕ ) = a 1 - e 2 sin 2 ϕ , N(\phi)=\frac{a}{\sqrt{1-e^{2}\sin^{2}\phi}},
  13. a a
  14. e e
  15. N ( ϕ ) \,N(\phi)
  16. p cos ϕ - Z sin ϕ - e 2 N ( ϕ ) = 0 , \frac{p}{\cos\phi}-\frac{Z}{\sin\phi}-e^{2}N(\phi)=0,
  17. p = X 2 + Y 2 p=\sqrt{X^{2}+Y^{2}}
  18. ( d X d Y d Z ) \displaystyle\begin{pmatrix}dX\\ dY\\ dZ\\ \end{pmatrix}
  19. M ( ϕ ) = a ( 1 - e 2 ) ( 1 - e 2 sin 2 ϕ ) 3 / 2 M(\phi)=\frac{a(1-e^{2})}{\left(1-e^{2}\sin^{2}\phi\right)^{3/2}}
  20. λ \,\lambda
  21. κ - 1 - e 2 a κ p 2 + ( 1 - e 2 ) z 2 κ 2 = 0 , \kappa-1-\frac{e^{2}a\kappa}{\sqrt{p^{2}+(1-e^{2})z^{2}\kappa^{2}}}=0,
  22. κ = p z tan ϕ \kappa=\frac{p}{z}\tan\phi
  23. p = x 2 + y 2 . p=\sqrt{x^{2}+y^{2}}.
  24. h = e - 2 ( κ - 1 - κ 0 - 1 ) p 2 + z 2 κ 2 , h=e^{-2}(\kappa^{-1}-{\kappa_{0}}^{-1})\sqrt{p^{2}+z^{2}\kappa^{2}},
  25. κ 0 = ( 1 - e 2 ) - 1 . \kappa_{0}=\left(1-e^{2}\right)^{-1}.
  26. κ i + 1 = c i + ( 1 - e 2 ) z 2 κ i 3 c i - p 2 = 1 + p 2 + ( 1 - e 2 ) z 2 κ i 3 c i - p 2 , \kappa_{i+1}=\frac{c_{i}+\left(1-e^{2}\right)z^{2}\kappa_{i}^{3}}{c_{i}-p^{2}}% =1+\frac{p^{2}+\left(1-e^{2}\right)z^{2}\kappa_{i}^{3}}{c_{i}-p^{2}},
  27. c i = ( p 2 + ( 1 - e 2 ) z 2 κ i 2 ) 3 / 2 a e 2 . c_{i}=\frac{\left(p^{2}+\left(1-e^{2}\right)z^{2}\kappa_{i}^{2}\right)^{3/2}}{% ae^{2}}.
  28. κ 0 \,\kappa_{0}
  29. h 0 h\approx 0
  30. ζ \displaystyle\zeta
  31. { a , b , e , e } \{a,b,e,e^{\prime}\}
  32. r = X 2 + Y 2 E 2 = a 2 - b 2 F = 54 b 2 Z 2 G = r 2 + ( 1 - e 2 ) Z 2 - e 2 E 2 C = e 4 F r 2 G 3 S = 1 + C + C 2 + 2 C 3 P = F 3 ( S + 1 S + 1 ) 2 G 2 Q = 1 + 2 e 4 P r 0 = - ( P e 2 r ) 1 + Q + 1 2 a 2 ( 1 + 1 / Q ) - P ( 1 - e 2 ) Z 2 Q ( 1 + Q ) - 1 2 P r 2 U = ( r - e 2 r 0 ) 2 + Z 2 V = ( r - e 2 r 0 ) 2 + ( 1 - e 2 ) Z 2 Z 0 = b 2 Z a V h = U ( 1 - b 2 a V ) ϕ = arctan [ Z + e 2 Z 0 r ] λ = arctan 2 [ Y , X ] \begin{matrix}r&=&\sqrt{X^{2}+Y^{2}}\\ E^{2}&=&a^{2}-b^{2}\\ F&=&54b^{2}Z^{2}\\ G&=&r^{2}+(1-e^{2})Z^{2}-e^{2}E^{2}\\ C&=&\frac{e^{4}Fr^{2}}{G^{3}}\\ S&=&\sqrt[3]{1+C+\sqrt{C^{2}+2C}}\\ P&=&\frac{F}{3\left(S+\frac{1}{S}+1\right)^{2}G^{2}}\\ Q&=&\sqrt{1+2e^{4}P}\\ r_{0}&=&\frac{-(Pe^{2}r)}{1+Q}+\sqrt{\frac{1}{2}a^{2}\left(1+1/Q\right)-\frac{% P(1-e^{2})Z^{2}}{Q(1+Q)}-\frac{1}{2}Pr^{2}}\\ U&=&\sqrt{(r-e^{2}r_{0})^{2}+Z^{2}}\\ V&=&\sqrt{(r-e^{2}r_{0})^{2}+(1-e^{2})Z^{2}}\\ Z_{0}&=&\frac{b^{2}Z}{aV}\\ h&=&U\left(1-\frac{b^{2}}{aV}\right)\\ \phi&=&\arctan\left[\frac{Z+e^{\prime 2}Z_{0}}{r}\right]\\ \lambda&=&\arctan 2[Y,X]\end{matrix}
  33. { X r , Y r , Z r } \{X_{r},Y_{r},Z_{r}\}
  34. { X p , Y p , Z p } \{X_{p},Y_{p},Z_{p}\}
  35. [ x y z ] = [ - sin λ r cos λ r 0 - sin ϕ r cos λ r - sin ϕ r sin λ r cos ϕ r cos ϕ r cos λ r cos ϕ r sin λ r sin ϕ r ] [ X p - X r Y p - Y r Z p - Z r ] \begin{bmatrix}x\\ y\\ z\\ \end{bmatrix}=\begin{bmatrix}-\sin\lambda_{r}&\cos\lambda_{r}&0\\ -\sin\phi_{r}\cos\lambda_{r}&-\sin\phi_{r}\sin\lambda_{r}&\cos\phi_{r}\\ \cos\phi_{r}\cos\lambda_{r}&\cos\phi_{r}\sin\lambda_{r}&\sin\phi_{r}\end{% bmatrix}\begin{bmatrix}X_{p}-X_{r}\\ Y_{p}-Y_{r}\\ Z_{p}-Z_{r}\end{bmatrix}
  36. ϕ \ \phi
  37. ϕ \ \phi^{\prime}
  38. tan ϕ = Z r X r 2 + Y r 2 = N ( ϕ ) ( 1 - f ) 2 + h N ( ϕ ) + h tan ϕ \tan\phi^{\prime}=\frac{Z_{r}}{\sqrt{X_{r}^{2}+Y_{r}^{2}}}=\frac{N(\phi)(1-f)^% {2}+h}{N(\phi)+h}\tan\phi
  39. tan λ = Y r X r \tan\lambda=\frac{Y_{r}}{X_{r}}
  40. ϕ \ \phi
  41. λ \ \lambda
  42. [ X Y Z ] = [ - sin λ - sin ϕ cos λ cos ϕ cos λ cos λ - sin ϕ sin λ cos ϕ sin λ 0 cos ϕ sin ϕ ] [ x y z ] + [ X r Y r Z r ] \begin{bmatrix}X\\ Y\\ Z\\ \end{bmatrix}=\begin{bmatrix}-\sin\lambda&-\sin\phi\cos\lambda&\cos\phi\cos% \lambda\\ \cos\lambda&-\sin\phi\sin\lambda&\cos\phi\sin\lambda\\ 0&\cos\phi&\sin\phi\end{bmatrix}\begin{bmatrix}x\\ y\\ z\end{bmatrix}+\begin{bmatrix}X_{r}\\ Y_{r}\\ Z_{r}\end{bmatrix}
  43. A A
  44. B B
  45. ϕ B , λ B , h B \phi_{B},\lambda_{B},h_{B}
  46. B B
  47. ϕ A , λ A , h A \phi_{A},\lambda_{A},h_{A}
  48. A A
  49. ϕ B \phi_{B}
  50. Δ ϕ = a 0 + a 1 U + a 2 V + a 3 U 2 + a 4 U V + a 5 V 2 + \Delta\phi=a_{0}+a_{1}U+a_{2}V+a_{3}U^{2}+a_{4}UV+a_{5}V^{2}+\cdots
  51. a i = parameters fitted by multiple regression U = K ( ϕ A - ϕ m ) V = K ( λ A - λ m ) K = scale factor ϕ m , λ m = origin of the datum A \begin{aligned}\displaystyle a_{i}&\displaystyle=\rm{parameters\ fitted\ by\ % multiple\ regression}\\ \displaystyle U&\displaystyle=K(\phi_{A}-\phi_{m})\\ \displaystyle V&\displaystyle=K(\lambda_{A}-\lambda_{m})\\ \displaystyle K&\displaystyle=\rm{scale\ factor}\\ \displaystyle\phi_{m},\lambda_{m}&\displaystyle=\rm{origin\ of\ the\ datum\ }A% \\ \end{aligned}
  52. Δ λ \Delta\lambda
  53. Δ h \Delta h
  54. ( A , B ) (A,B)
  55. A A
  56. B B
  57. A A
  58. A B A\to B
  59. A A
  60. B B
  61. B B
  62. [ X B Y B Z B ] = [ c x c y c z ] + ( 1 + s × 10 - 6 ) [ 1 - r z r y r z 1 - r x - r y r x 1 ] [ X A Y A Z A ] . \begin{bmatrix}X_{B}\\ Y_{B}\\ Z_{B}\end{bmatrix}=\begin{bmatrix}c_{x}\\ c_{y}\\ c_{z}\end{bmatrix}+(1+s\times 10^{-6})\cdot\begin{bmatrix}1&-r_{z}&r_{y}\\ r_{z}&1&-r_{x}\\ -r_{y}&r_{x}&1\end{bmatrix}\cdot\begin{bmatrix}X_{A}\\ Y_{A}\\ Z_{A}\end{bmatrix}.
  63. c x , c y , c z c_{x},c_{y},c_{z}
  64. r x , r y , r z r_{x},r_{y},r_{z}
  65. s s
  66. A A
  67. A B A\to B
  68. A A
  69. B B
  70. B B
  71. [ X B Y B Z B ] = [ X A Y A Z A ] + [ Δ X A Δ Y A Δ Z A ] + [ 1 - r z r y r z 1 - r x - r y r x 1 ] [ X A - X A 0 Y A - Y A 0 Z A - Z A 0 ] + Δ S [ X A - X A 0 Y A - Y A 0 Z A - Z A 0 ] . \begin{bmatrix}X_{B}\\ Y_{B}\\ Z_{B}\end{bmatrix}=\begin{bmatrix}X_{A}\\ Y_{A}\\ Z_{A}\end{bmatrix}+\begin{bmatrix}\Delta X_{A}\\ \Delta Y_{A}\\ \Delta Z_{A}\end{bmatrix}+\begin{bmatrix}1&-r_{z}&r_{y}\\ r_{z}&1&-r_{x}\\ -r_{y}&r_{x}&1\end{bmatrix}\cdot\begin{bmatrix}X_{A}-X^{0}_{A}\\ Y_{A}-Y^{0}_{A}\\ Z_{A}-Z^{0}_{A}\end{bmatrix}+\Delta S\begin{bmatrix}X_{A}-X^{0}_{A}\\ Y_{A}-Y^{0}_{A}\\ Z_{A}-Z^{0}_{A}\end{bmatrix}.
  72. ( X A 0 , Y A 0 , Z A 0 ) (X^{0}_{A},Y^{0}_{A},Z^{0}_{A})
  73. Δ S \Delta S

Geometric_dimensioning_and_tolerancing.html

  1. 𝖠 - - | {\displaystyle\Box}\!\!\!\!{\scriptstyle\mathsf{A}}\!-\!\!\!-\!\!\!% \blacktriangleleft\!\!\!|

Geometric_hashing.html

  1. ( 12 , 17 ) ; (12,17);
  2. ( 45 , 13 ) ; (45,13);
  3. ( 40 , 46 ) ; (40,46);
  4. ( 20 , 35 ) ; (20,35);
  5. ( 35 , 25 ) (35,25)
  6. x x^{\prime}
  7. y y^{\prime}
  8. x x^{\prime}
  9. ( - 0.75 , - 1.25 ) ; (-0.75,-1.25);
  10. ( 1.00 , 0.00 ) ; (1.00,0.00);
  11. ( - 0.50 , 1.25 ) ; (-0.50,1.25);
  12. ( - 1.00 , 0.00 ) ; (-1.00,0.00);
  13. ( 0.00 , 0.25 ) (0.00,0.25)
  14. x x^{\prime}
  15. y y^{\prime}
  16. ( - 0.75 , - 1.25 ) ; (-0.75,-1.25);
  17. ( 1.00 , 0.00 ) ; (1.00,0.00);
  18. ( - 0.50 , 1.25 ) ; (-0.50,1.25);
  19. ( - 1.00 , 0.00 ) ; (-1.00,0.00);
  20. ( 0.00 , 0.25 ) (0.00,0.25)
  21. ( 1.00 , 0.00 ) ; (1.00,0.00);
  22. ( 0.00 , 1.25 ) ; (0.00,1.25);
  23. ( - 1.00 , 0.00 ) ; (-1.00,0.00);
  24. ( 0.00 , - 0.25 ) ; (0.00,-0.25);
  25. ( 0.00 , 0.50 ) (0.00,0.50)

Geometric_phase.html

  1. γ \gamma
  2. γ [ C ] = i C n , t | ( R | n , t ) d R \gamma[C]=i\oint_{C}\!\langle n,t|\left(\nabla_{R}|n,t\rangle\right)\,dR\,
  3. R R
  4. C C
  5. C C
  6. C C
  7. M × M M\times M
  8. τ i j μ = ψ i | μ ψ j \tau_{ij}^{\mu}=\left\langle\psi_{i}|\partial^{\mu}\psi_{j}\right\rangle
  9. ψ i \psi_{i}
  10. R μ R_{\mu}
  11. Γ \Gamma
  12. R μ ( t ) R_{\mu}\left(t\right)
  13. t [ 0 , 1 ] t\in\left[0,1\right]
  14. R μ ( t + 1 ) = R μ ( t ) R_{\mu}\left(t+1\right)=R_{\mu}\left(t\right)
  15. D [ Γ ] = P ^ e Γ τ μ d R μ D\left[\Gamma\right]=\hat{P}e^{\oint_{\Gamma}{\tau^{\mu}dR_{\mu}}}
  16. P ^ {\hat{P}}
  17. M M
  18. e i β j e^{i\beta_{j}}
  19. β j \beta_{j}
  20. j j
  21. e i β j = ( - 1 ) N j e^{i\beta_{j}}=\left(-1\right)^{N_{j}}
  22. N j N_{j}
  23. ψ j \psi_{j}
  24. Γ \Gamma
  25. N + 1 N+1
  26. ( n = 0 , , N ) \left(n=0,...,N\right)
  27. R ( t n ) R\left(t_{n}\right)
  28. t 0 = 0 t_{0}=0
  29. t N = 1 t_{N}=1
  30. ψ j [ R ( t n ) ] \psi_{j}\left[R\left(t_{n}\right)\right]
  31. I j ( Γ , N ) = n = 0 N - 1 ψ j [ R ( t n ) ] | ψ j [ R ( t n + 1 ) ] I_{j}\left(\Gamma,N\right)=\prod\limits_{n=0}^{N-1}{\left\langle\psi_{j}\left[% R\left(t_{n}\right)\right]|\psi_{j}\left[R\left(t_{n+1}\right)\right]\right\rangle}
  32. N N\to\infty
  33. I j ( Γ , N ) e i β j I_{j}\left(\Gamma,N\right)\to e^{i\beta_{j}}
  34. B B
  35. R c R_{c}
  36. R c R_{c}
  37. R c R_{c}
  38. E = ( n + α ) ω c , α = 1 / 2 E=(n+\alpha)\hbar\omega_{c},\alpha=1/2
  39. E = v 2 ( n + α ) e B , α = 0 E=v\sqrt{2(n+\alpha)eB\hbar},\alpha=0
  40. n = 0 , 1 , 2 , n=0,1,2,\ldots
  41. d 𝐫 𝐤 - e d 𝐫 𝐀 + γ = 2 π ( n + 1 / 2 ) \hbar\oint d\mathbf{r}\cdot\mathbf{k}-e\oint d\mathbf{r}\cdot\mathbf{A}+\hbar% \gamma=2\pi\hbar(n+1/2)
  42. γ \gamma
  43. γ = 0 \gamma=0
  44. γ = π \gamma=\pi
  45. α = 1 / 2 \alpha=1/2
  46. α = 0 \alpha=0
  47. ω c = e B / m \omega_{c}=eB/m
  48. v v

Geometric_standard_deviation.html

  1. σ g = exp ( i = 1 n ( ln A i μ g ) 2 n ) . ( 1 ) \sigma_{g}=\exp\left(\sqrt{\sum_{i=1}^{n}(\ln{A_{i}\over\mu_{g}})^{2}\over n}% \right).\qquad\qquad(1)
  2. μ g = A 1 A 2 A n n . \mu_{g}=\sqrt[n]{A_{1}A_{2}\cdots A_{n}}.\,
  3. ln μ g = 1 n ln ( A 1 A 2 A n ) . \ln\mu_{g}={1\over n}\ln(A_{1}A_{2}\cdots A_{n}).
  4. A i A_{i}
  5. i i
  6. ln μ g = 1 n [ ln A 1 + ln A 2 + + ln A n ] . \ln\mu_{g}={1\over n}[\ln A_{1}+\ln A_{2}+\cdots+\ln A_{n}].\,
  7. ln μ g \ln\,\mu_{g}
  8. { ln A 1 , ln A 2 , , ln A n } \{\ln A_{1},\ln A_{2},\dots,\ln A_{n}\}
  9. ln σ g = i = 1 n ( ln A i - ln μ g ) 2 n . \ln\sigma_{g}=\sqrt{\sum_{i=1}^{n}(\ln A_{i}-\ln\mu_{g})^{2}\over n}.
  10. σ g = exp i = 1 n ( ln A i μ g ) 2 n . \sigma_{g}=\exp{\sqrt{\sum_{i=1}^{n}(\ln{A_{i}\over\mu_{g}})^{2}\over n}}.
  11. z = ln ( x ) - ln ( μ g ) ln σ g = log σ g ( x / μ g ) . z={{\ln(x)-\ln(\mu_{g})}\over\ln\sigma_{g}}={\log_{\sigma_{g}}(x/\mu_{g})}.\,
  12. x = μ g σ g z . x=\mu_{g}{\sigma_{g}}^{z}.
  13. σ g = exp ( stdev ( ln ( A ) ) ) \sigma_{g}=\exp(\operatorname{stdev}(\ln(A)))

Geometric_topology.html

  1. = M - 1 M 0 M 1 M 2 M m - 1 M m = M \emptyset=M_{-1}\subset M_{0}\subset M_{1}\subset M_{2}\subset\dots\subset M_{% m-1}\subset M_{m}=M
  2. M i M_{i}
  3. M i - 1 M_{i-1}
  4. i i
  5. x N , x\in N,
  6. U M U\subset M
  7. ( U , U N ) (U,U\cap N)
  8. ( n , d ) (\mathbb{R}^{n},\mathbb{R}^{d})
  9. d \mathbb{R}^{d}
  10. n \mathbb{R}^{n}
  11. U R n U\to R^{n}
  12. U N U\cap N
  13. d \mathbb{R}^{d}

Geometrized_unit_system.html

  1. c = 1 c=1
  2. G = 1 G=1
  3. ϵ 0 \epsilon_{0}
  4. k B = 1 k_{\mathrm{B}}=1
  5. ϵ 0 = 1 \epsilon_{0}=1

Geostrophic_wind.html

  1. D s y m b o l U D t = - 2 s y m b o l Ω × s y m b o l U - 1 ρ p + s y m b o l g + s y m b o l F r {Dsymbol{U}\over Dt}=-2symbol{\Omega}\times symbol{U}-{1\over\rho}\nabla p+% symbol{g}+symbol{F}_{r}
  2. s y m b o l U symbol{U}
  3. s y m b o l Ω symbol{\Omega}
  4. ρ \rho
  5. p p
  6. s y m b o l F r symbol{F}_{r}
  7. s y m b o l g symbol{g}
  8. D D t {D\;\over Dt}
  9. D u D t = - 1 ρ P x + f v {Du\over Dt}=-{1\over\rho}{\partial P\over\partial x}+f\cdot v
  10. D v D t = - 1 ρ P y - f u {Dv\over Dt}=-{1\over\rho}{\partial P\over\partial y}-f\cdot u
  11. 0 = - g - 1 ρ P z 0=-g-{1\over\rho}{\partial P\over\partial z}
  12. f = 2 Ω sin ϕ f=2\Omega\sin{\phi}
  13. f v = 1 ρ P x f\cdot v={1\over\rho}{\partial P\over\partial x}
  14. f u = - 1 ρ P y f\cdot u=-{1\over\rho}{\partial P\over\partial y}
  15. f v = g P / x P / z = g Z x f\cdot v=g\frac{\partial P/\partial x}{\partial P/\partial z}=g{\partial Z% \over\partial x}
  16. f u = - g P / y P / z = - g Z y f\cdot u=-g\frac{\partial P/\partial y}{\partial P/\partial z}=-g{\partial Z% \over\partial y}
  17. P x d x + P y d y + P z d Z = 0 {\partial P\over\partial x}dx+{\partial P\over\partial y}dy+{\partial P\over% \partial z}dZ=0
  18. ( u g , v g ) (u_{g},v_{g})
  19. u g = - g f Z y u_{g}=-{g\over f}{\partial Z\over\partial y}
  20. v g = g f Z x v_{g}={g\over f}{\partial Z\over\partial x}
  21. V g = k ^ f × p Φ \overrightarrow{V_{g}}={\hat{k}\over f}\times\nabla_{p}\Phi

Germ_(mathematics).html

  1. S U = T U . S\,\cap\,U=T\,\cap\,U.
  2. f x g or S x T . f\sim_{x}g\quad\,\text{or}\quad S\sim_{x}T.
  3. [ f ] x = { g : X Y g x f } . [f]_{x}=\{g:X\to Y\mid g\sim_{x}f\}.
  4. f : ( X , x ) ( Y , y ) . f:(X,x)\to(Y,y).
  5. S x T 1 S x 𝟏 T . S\sim_{x}T\;\Longleftrightarrow\;\mathbf{1}_{S}\sim_{x}\mathbf{1}_{T}.
  6. S U = T U \scriptstyle S\,\cap\,U\;=\;T\,\cap\,U
  7. f | S V = g | T V \scriptstyle f|_{S\cap V}=g|_{T\cap V}
  8. x V U \scriptstyle x\in V\subset U
  9. U V \scriptstyle U\cap V
  10. J x k ( X , Y ) \scriptstyle J_{x}^{k}(X,Y)
  11. \scriptstyle\mathcal{F}
  12. ( U ) \scriptstyle\mathcal{F}(U)
  13. V U \scriptstyle V\subset U
  14. res V U : ( U ) ( V ) \scriptstyle\mathrm{res}_{VU}:\mathcal{F}(U)\to\mathcal{F}(V)
  15. f ( U ) \scriptstyle f\in\mathcal{F}(U)
  16. g ( V ) \scriptstyle g\in\mathcal{F}(V)
  17. W U V \scriptstyle W\subset U\cap V
  18. ( W ) \scriptstyle\mathcal{F}(W)
  19. x \scriptstyle\mathcal{F}_{x}
  20. \scriptstyle\mathcal{F}
  21. X X
  22. Y Y
  23. \scriptstyle\mathcal{F}
  24. X , Y X,Y
  25. C 0 ( X , Y ) Hom ( X , Y ) C^{0}(X,Y)\subset\mbox{Hom}~{}(X,Y)\,
  26. X X
  27. Y Y
  28. C k ( X , Y ) Hom ( X , Y ) C^{k}(X,Y)\subset\mbox{Hom}~{}(X,Y)\,
  29. k k
  30. C ( X , Y ) = k C k ( X , Y ) Hom ( X , Y ) C^{\infty}(X,Y)=\bigcap_{k}C^{k}(X,Y)\subset\mbox{Hom}~{}(X,Y)\,
  31. C ω ( X , Y ) Hom ( X , Y ) C^{\omega}(X,Y)\subset\mbox{Hom}~{}(X,Y)
  32. ω \omega
  33. C k C^{k}
  34. C C
  35. X , Y X,Y
  36. X , Y X,Y
  37. \scriptstyle\mathcal{F}
  38. X X
  39. x x
  40. X X
  41. x \scriptstyle\mathcal{F}_{x}
  42. 𝒞 x 0 \mathcal{C}_{x}^{0}\,
  43. x x
  44. 𝒞 x k \mathcal{C}_{x}^{k}\,
  45. k k
  46. k k
  47. x x
  48. 𝒞 x \mathcal{C}_{x}^{\infty}\,
  49. x x
  50. 𝒞 x ω \mathcal{C}_{x}^{\omega}\,
  51. x x
  52. 𝒪 x \mathcal{O}_{x}\,
  53. x x
  54. 𝔙 x \mathfrak{V}_{x}\,
  55. x x
  56. x x
  57. X X
  58. x = 0 x=0
  59. X = n X=n
  60. 𝒞 0 n , 𝒞 k n , 𝒞 n , 𝒞 ω n , 𝒪 n , 𝔙 n {{}_{n}\mathcal{C}^{0}},{{}_{n}\mathcal{C}^{k}},{{}_{n}\mathcal{C}^{\infty}},{% {}_{n}\mathcal{C}^{\omega}},{{}_{n}\mathcal{O}},{{}_{n}\mathfrak{V}}\,
  61. X X
  62. n n
  63. x = 0 x=0
  64. P P
  65. V n V_{n}
  66. P P
  67. V n V_{n}

GG.html

  1. 10 9 10^{9}

Gibbs_phenomenon.html

  1. π / 4 \pi/4
  2. sin ( x ) + 1 3 sin ( 3 x ) + 1 5 sin ( 5 x ) + . \sin(x)+\frac{1}{3}\sin(3x)+\frac{1}{5}\sin(5x)+\cdots.
  3. π / 4 \pi/4
  4. 2 n π 2n\pi
  5. ( 2 n + 1 ) π (2n+1)\pi
  6. - π / 4 -\pi/4
  7. ( 2 n + 1 ) π (2n+1)\pi
  8. ( 2 n + 2 ) π (2n+2)\pi
  9. π / 2 \pi/2
  10. π \pi
  11. π / 4 \pi/4
  12. 1 2 0 π sin t t d t - π 4 = π 2 ( 0.089490 ) \frac{1}{2}\int_{0}^{\pi}\frac{\sin t}{t}\,dt-\frac{\pi}{4}=\frac{\pi}{2}\cdot% (0.089490\dots)
  13. a ( 0.089392 ) a\cdot(0.089392\dots)
  14. 0 π sin t t d t = ( 1.851937052 ) = π 2 + π ( 0.089392 ) \int_{0}^{\pi}\frac{\sin t}{t}\ dt=(1.851937052\dots)=\frac{\pi}{2}+\pi\cdot(0% .089392\dots)
  15. f : f:{\mathbb{R}}\to{\mathbb{R}}
  16. L > 0 L>0
  17. x 0 x_{0}
  18. f ( x 0 - ) f(x_{0}^{-})
  19. f ( x 0 + ) f(x_{0}^{+})
  20. f f
  21. a a
  22. f ( x 0 + ) - f ( x 0 - ) = a 0. f(x_{0}^{+})-f(x_{0}^{-})=a\neq 0.
  23. S N f ( x ) := - N n N f ^ ( n ) e 2 i π n x L = 1 2 a 0 + n = 1 N ( a n cos ( 2 π n x L ) + b n sin ( 2 π n x L ) ) , S_{N}f(x):=\sum_{-N\leq n\leq N}\hat{f}(n)e^{\frac{2i\pi nx}{L}}=\frac{1}{2}a_% {0}+\sum_{n=1}^{N}\left(a_{n}\cos\left(\frac{2\pi nx}{L}\right)+b_{n}\sin\left% (\frac{2\pi nx}{L}\right)\right),
  24. f ^ ( n ) , a n , b n \hat{f}(n),a_{n},b_{n}
  25. f ^ ( n ) := 1 L 0 L f ( x ) e - 2 i π n x / L d x \hat{f}(n):=\frac{1}{L}\int_{0}^{L}f(x)e^{-2i\pi nx/L}\,dx
  26. a n := 2 L 0 L f ( x ) cos ( 2 π n x L ) d x a_{n}:=\frac{2}{L}\int_{0}^{L}f(x)\cos\left(\frac{2\pi nx}{L}\right)\,dx
  27. b n := 2 L 0 L f ( x ) sin ( 2 π n x L ) d x . b_{n}:=\frac{2}{L}\int_{0}^{L}f(x)\sin\left(\frac{2\pi nx}{L}\right)\,dx.
  28. lim N S N f ( x 0 + L 2 N ) = f ( x 0 + ) + a ( 0.089392 ) \lim_{N\to\infty}S_{N}f\left(x_{0}+\frac{L}{2N}\right)=f(x_{0}^{+})+a\cdot(0.0% 89392\dots)
  29. lim N S N f ( x 0 - L 2 N ) = f ( x 0 - ) - a ( 0.089392 ) \lim_{N\to\infty}S_{N}f\left(x_{0}-\frac{L}{2N}\right)=f(x_{0}^{-})-a\cdot(0.0% 89392\dots)
  30. lim N S N f ( x 0 ) = f ( x 0 - ) + f ( x 0 + ) 2 . \lim_{N\to\infty}S_{N}f(x_{0})=\frac{f(x_{0}^{-})+f(x_{0}^{+})}{2}.
  31. x N x_{N}
  32. x 0 x_{0}
  33. N N\to\infty
  34. lim sup N S N f ( x N ) f ( x 0 + ) + a ( 0.089392 ) \limsup_{N\to\infty}S_{N}f(x_{N})\leq f(x_{0}^{+})+a\cdot(0.089392\dots)
  35. lim inf N S N f ( x N ) f ( x 0 - ) - a ( 0.089392 ) . \liminf_{N\to\infty}S_{N}f(x_{N})\geq f(x_{0}^{-})-a\cdot(0.089392\dots).
  36. - - 1 sin ( π x ) π x d x . \int_{-\infty}^{-1}\frac{\sin(\pi x)}{\pi x}\,dx.
  37. 2 π 2\pi
  38. x 0 x_{0}
  39. π / 2 \pi/2
  40. S N f ( x ) = sin ( x ) + 1 3 sin ( 3 x ) + + 1 N - 1 sin ( ( N - 1 ) x ) . S_{N}f(x)=\sin(x)+\frac{1}{3}\sin(3x)+\cdots+\frac{1}{N-1}\sin((N-1)x).
  41. x = 0 x=0
  42. S N f ( 0 ) = 0 = - π 4 + π 4 2 = f ( 0 - ) + f ( 0 + ) 2 S_{N}f(0)=0=\frac{-\frac{\pi}{4}+\frac{\pi}{4}}{2}=\frac{f(0^{-})+f(0^{+})}{2}
  43. S N f ( 2 π 2 N ) = sin ( π N ) + 1 3 sin ( 3 π N ) + + 1 N - 1 sin ( ( N - 1 ) π N ) . S_{N}f\left(\frac{2\pi}{2N}\right)=\sin\left(\frac{\pi}{N}\right)+\frac{1}{3}% \sin\left(\frac{3\pi}{N}\right)+\cdots+\frac{1}{N-1}\sin\left(\frac{(N-1)\pi}{% N}\right).
  44. sinc ( x ) \operatorname{sinc}(x)\,
  45. S N f ( 2 π 2 N ) = π 2 [ 2 N sinc ( 1 N ) + 2 N sinc ( 3 N ) + + 2 N sinc ( ( N - 1 ) N ) ] . S_{N}f\left(\frac{2\pi}{2N}\right)=\frac{\pi}{2}\left[\frac{2}{N}\operatorname% {sinc}\left(\frac{1}{N}\right)+\frac{2}{N}\operatorname{sinc}\left(\frac{3}{N}% \right)+\cdots+\frac{2}{N}\operatorname{sinc}\left(\frac{(N-1)}{N}\right)% \right].
  46. 0 1 sinc ( x ) d x \int_{0}^{1}\operatorname{sinc}(x)\ dx
  47. 2 / N 2/N
  48. N N\to\infty
  49. lim N S N f ( 2 π 2 N ) \displaystyle\lim_{N\to\infty}S_{N}f\left(\frac{2\pi}{2N}\right)
  50. lim N S N f ( - 2 π 2 N ) = - π 2 0 1 sinc ( x ) d x = - π 4 - π 2 ( 0.089490 ) . \lim_{N\to\infty}S_{N}f\left(-\frac{2\pi}{2N}\right)=-\frac{\pi}{2}\int_{0}^{1% }\operatorname{sinc}(x)\ dx=-\frac{\pi}{4}-\frac{\pi}{2}\cdot(0.089490\dots).

Gibbs_sampling.html

  1. k \left.k\right.
  2. 𝐗 = ( x 1 , , x n ) \mathbf{X}=(x_{1},\dots,x_{n})
  3. p ( x 1 , , x n ) \left.p(x_{1},\dots,x_{n})\right.
  4. i i
  5. 𝐗 ( i ) = ( x 1 ( i ) , , x n ( i ) ) \mathbf{X}^{(i)}=(x_{1}^{(i)},\dots,x_{n}^{(i)})
  6. 𝐗 ( 0 ) \mathbf{X}^{(0)}
  7. i + 1 i+1
  8. x j ( i + 1 ) x_{j}^{(i+1)}
  9. j j
  10. p ( x j | x 1 ( i + 1 ) , , x j - 1 ( i + 1 ) , x j + 1 ( i ) , , x n ( i ) ) p(x_{j}|x_{1}^{(i+1)},\dots,x_{j-1}^{(i+1)},x_{j+1}^{(i)},\dots,x_{n}^{(i)})
  11. j + 1 j+1
  12. i i
  13. i + 1 i+1
  14. k k
  15. n n
  16. n n
  17. p ( x j | x 1 , , x j - 1 , x j + 1 , , x n ) = p ( x 1 , , x n ) p ( x 1 , , x j - 1 , x j + 1 , , x n ) p ( x 1 , , x n ) p(x_{j}|x_{1},\dots,x_{j-1},x_{j+1},\dots,x_{n})=\frac{p(x_{1},\dots,x_{n})}{p% (x_{1},\dots,x_{j-1},x_{j+1},\dots,x_{n})}\propto p(x_{1},\dots,x_{n})
  18. x j x_{j}
  19. x j x_{j}
  20. x j x_{j}
  21. x j x_{j}
  22. x j x_{j}
  23. x j x_{j}
  24. X \left.X\right.
  25. θ Θ \theta\in\Theta\,\!
  26. d \left.d\right.
  27. g ( θ 1 , , θ d ) g(\theta_{1},\ldots,\theta_{d})
  28. d \left.d\right.
  29. θ i \left.\theta_{i}\right.
  30. Θ \left.\Theta\right.
  31. 1 j d 1\leq j\leq d
  32. θ j \left.\theta_{j}\right.
  33. g ( θ 1 , , θ j - 1 , , θ j + 1 , , θ d ) g(\theta_{1},\ldots,\theta_{j-1},\,\cdot\,,\theta_{j+1},\ldots,\theta_{d})
  34. g \left.g\right.
  35. x j y x\sim_{j}y
  36. x i = y i \left.x_{i}=y_{i}\right.
  37. i j i\neq j
  38. p x y \left.p_{xy}\right.
  39. x Θ x\in\Theta
  40. y Θ y\in\Theta
  41. p x y = { 1 d g ( y ) z Θ : z j x g ( z ) x j y 0 otherwise p_{xy}=\begin{cases}\frac{1}{d}\frac{g(y)}{\sum_{z\in\Theta:z\sim_{j}x}g(z)}&x% \sim_{j}y\\ 0&\,\text{otherwise}\end{cases}
  42. g ( x ) p x y = 1 d g ( x ) g ( y ) z Θ : z j x g ( z ) = 1 d g ( y ) g ( x ) z Θ : z j y g ( z ) = g ( y ) p y x g(x)p_{xy}=\frac{1}{d}\frac{g(x)g(y)}{\sum_{z\in\Theta:z\sim_{j}x}g(z)}=\frac{% 1}{d}\frac{g(y)g(x)}{\sum_{z\in\Theta:z\sim_{j}y}g(z)}=g(y)p_{yx}
  43. x j y x\sim_{j}y
  44. g \left.g\right.
  45. j \left.j\right.
  46. x j ( i ) x_{j}^{(i)}
  47. x j ( i - 1 ) x_{j}^{(i-1)}
  48. x j ( i - 1 ) x_{j}^{(i-1)}
  49. x j ( i ) x_{j}^{(i)}
  50. 1 2 ( 2 100 - 1 ) \frac{1}{2(2^{100}-1)}
  51. 2 100 2^{100}
  52. 2 99 2^{99}
  53. 2 99 2^{99}
  54. 2 99 2^{99}

Gift_wrapping_algorithm.html

  1. O ( n h ) O(nh)
  2. O ( n log n ) O(n\log n)
  3. O ( n log h ) O(n\log h)

Gimbal_lock.html

  1. x x
  2. y y
  3. z z
  4. X X
  5. Y Y
  6. Z Z
  7. α \alpha
  8. β \beta
  9. γ \gamma
  10. R \displaystyle R
  11. β = π 2 \beta=\tfrac{\pi}{2}
  12. cos π 2 = 0 \cos\tfrac{\pi}{2}=0
  13. sin π 2 = 1 \sin\tfrac{\pi}{2}=1
  14. R \displaystyle R
  15. R \displaystyle R
  16. R \displaystyle R
  17. α \alpha
  18. γ \gamma
  19. α + γ \alpha+\gamma
  20. Z Z
  21. α \alpha
  22. γ \gamma
  23. β \beta
  24. α \alpha
  25. π 2 \tfrac{\pi}{2}
  26. γ \gamma

Gimel_function.html

  1. : κ κ cf ( κ ) \gimel\colon\kappa\mapsto\kappa^{\mathrm{cf}(\kappa)}
  2. \gimel
  3. ( κ ) > κ \gimel(\kappa)>\kappa
  4. κ \kappa
  5. ( κ ) = 2 κ \gimel(\kappa)=2^{\kappa}
  6. κ \kappa
  7. ( κ ) \gimel(\kappa)
  8. 2 κ = ( κ ) 2^{\kappa}=\gimel(\kappa)
  9. 2 κ = 2 < κ × ( κ ) 2^{\kappa}=2^{<\kappa}\times\gimel(\kappa)
  10. 2 κ = ( 2 < κ ) 2^{\kappa}=\gimel(2^{<\kappa})

Ginzburg–Landau_theory.html

  1. F = F n + α | ψ | 2 + β 2 | ψ | 4 + 1 2 m | ( - i - 2 e 𝐀 ) ψ | 2 + | 𝐁 | 2 2 μ 0 F=F_{n}+\alpha|\psi|^{2}+\frac{\beta}{2}|\psi|^{4}+\frac{1}{2m}\left|\left(-i% \hbar\nabla-2e\mathbf{A}\right)\psi\right|^{2}+\frac{|\mathbf{B}|^{2}}{2\mu_{0}}
  2. 𝐁 = × 𝐀 \mathbf{B}=\nabla\times\mathbf{A}
  3. α ψ + β | ψ | 2 ψ + 1 2 m ( - i - 2 e 𝐀 ) 2 ψ = 0 \alpha\psi+\beta|\psi|^{2}\psi+\frac{1}{2m}\left(-i\hbar\nabla-2e\mathbf{A}% \right)^{2}\psi=0
  4. × 𝐁 = μ 0 𝐣 ; 𝐣 = 2 e m Re { ψ * ( - i - 2 e 𝐀 ) ψ } \nabla\times\mathbf{B}=\mu_{0}\mathbf{j}\;\;;\;\;\mathbf{j}=\frac{2e}{m}% \mathrm{Re}\left\{\psi^{*}\left(-i\hbar\nabla-2e\mathbf{A}\right)\psi\right\}
  5. α ψ + β | ψ | 2 ψ = 0. \alpha\psi+\beta|\psi|^{2}\psi=0.\,
  6. | ψ | 2 = - α β . |\psi|^{2}=-\frac{\alpha}{\beta}.
  7. | ψ | 2 = - α 0 ( T - T c ) β , |\psi|^{2}=-\frac{\alpha_{0}(T-T_{c})}{\beta},
  8. ξ = 2 2 m | α | . \xi=\sqrt{\frac{\hbar^{2}}{2m|\alpha|}}.
  9. ξ = 2 4 m | α | . \xi=\sqrt{\frac{\hbar^{2}}{4m|\alpha|}}.
  10. λ = m 4 μ 0 e 2 ψ 0 2 , \lambda=\sqrt{\frac{m}{4\mu_{0}e^{2}\psi_{0}^{2}}},

Girsanov_theorem.html

  1. { W t } \{W_{t}\}
  2. { Ω , , P } \{\Omega,\mathcal{F},P\}
  3. X t X_{t}
  4. { t W } \{\mathcal{F}^{W}_{t}\}
  5. X t X_{t}
  6. X 0 = 0 X_{0}=0
  7. Z t = ( X ) t , Z_{t}=\mathcal{E}(X)_{t},\,
  8. ( X ) \mathcal{E}(X)
  9. ( X ) t = exp ( X t - 1 2 [ X ] t ) , \mathcal{E}(X)_{t}=\exp\left(X_{t}-\frac{1}{2}[X]_{t}\right),
  10. [ X ] t [X]_{t}
  11. X t X_{t}
  12. Z t Z_{t}
  13. { Ω , } \{\Omega,\mathcal{F}\}
  14. d Q d P | t = Z t = ( X ) t \frac{dQ}{dP}|_{\mathcal{F}_{t}}=Z_{t}=\mathcal{E}(X)_{t}
  15. t W \mathcal{F}^{W}_{t}
  16. t W . \mathcal{F}^{W}_{t}.\,
  17. Y ~ t = Y t - [ Y , X ] t \tilde{Y}_{t}=Y_{t}-\left[Y,X\right]_{t}
  18. { Ω , F , Q , { F t W } } \{\Omega,F,Q,\{F^{W}_{t}\}\}
  19. W ~ t = W t - [ W , X ] t \tilde{W}_{t}=W_{t}-\left[W,X\right]_{t}
  20. W ~ t \tilde{W}_{t}
  21. [ W ~ ] t = [ W t , W t ] - 2 [ W t , [ W , X ] t ] + [ [ W , X ] t , [ W , X ] t ] = [ W ] t = t \left[\tilde{W}\right]_{t}=\left[W_{t},W_{t}\right]-2\left[W_{t},[W,X]_{t}% \right]+\left[[W,X]_{t},[W,X]_{t}\right]=\left[W\right]_{t}=t
  22. X t = 0 t Y s d W s . X_{t}=\int_{0}^{t}Y_{s}\,dW_{s}.
  23. ( X ) \mathcal{E}(X)
  24. E P [ exp ( 1 2 0 T Y s 2 d s ) ] < . E_{P}\left[\exp\left(\frac{1}{2}\int_{0}^{T}Y_{s}^{2}\,ds\right)\right]<\infty.
  25. ( X ) \mathcal{E}(X)
  26. Z t = 1 + 0 t Z s d X s . Z_{t}=1+\int_{0}^{t}Z_{s}\,dX_{s}.\,
  27. \mathcal{F}_{\infty}
  28. λ 0 \lambda\neq 0
  29. d Q d P = ( 0 r - μ σ d W s ) \frac{dQ}{dP}=\mathcal{E}\left(\int_{0}^{\cdot}\frac{r-\mu}{\sigma}\,dW_{s}\right)
  30. r r
  31. μ \mu
  32. σ \sigma

Global_optimization.html

  1. f f
  2. x P \vec{x}\in P
  3. x D \vec{x}\in D
  4. D D
  5. g ( x ) g(x)
  6. f ( x ) := ( - 1 ) g ( x ) f(x):=(-1)\cdot g(x)
  7. f f

Global_value_numbering.html

  1. [ w 1 , x 1 , y 2 , z 2 ] [{w}\mapsto 1,{x}\mapsto 1,{y}\mapsto 2,{z}\mapsto 2]

Glossary_of_differential_geometry_and_topology.html

  1. f f
  2. f : M N f:M\to N
  3. f - 1 : N M f^{-1}:N\to M
  4. T p ( M ) T_{p}(M)
  5. T p ( N ) T_{p}(N)
  6. B B × B B\to B\times B

Glossary_of_graph_theory.html

  1. u u
  2. v v
  3. { u , v } \{u,v\}
  4. G G
  5. { u , v } \{u,v\}
  6. G G
  7. ( u , v ) (u,v)
  8. ( v , u ) (v,u)
  9. v v
  10. G ¯ \bar{G}
  11. G ¯ \bar{G}
  12. n n
  13. n n
  14. \mathcal{F}
  15. \mathcal{F}
  16. \mathcal{F}
  17. Q n Q_{n}
  18. G 2 G_{2}
  19. G 1 G_{1}
  20. G 2 G_{2}
  21. G 1 G_{1}
  22. G 2 = ( V 2 , E 2 ) G_{2}=(V_{2},E_{2})
  23. G 1 = ( V 1 , E 1 ) G_{1}=(V_{1},E_{1})
  24. V 2 V_{2}
  25. V 1 V_{1}
  26. E 2 E_{2}
  27. G 1 G_{1}
  28. G 2 = ( V 2 , E 2 ) G_{2}=(V_{2},E_{2})
  29. G 1 = ( V 1 , E 1 ) G_{1}=(V_{1},E_{1})
  30. V 2 V_{2}
  31. V 1 V_{1}
  32. E 2 E_{2}
  33. G 1 G_{1}
  34. α ( G ) \alpha^{\prime}(G)
  35. e = ( u , v ) e=(u,v)
  36. ( u , v ) (u,v)
  37. ( v , u ) (v,u)

Glossary_of_order_theory.html

  1. \bigvee
  2. \bigvee
  3. \bigwedge
  4. \bigvee
  5. X X
  6. ν : X [ 0 , 1 ] \nu:X\to[0,1]
  7. ν ( ) = 0 \nu(\emptyset)=0
  8. ν ( U ) + ν ( V ) = ν ( U V ) + ν ( U V ) \nu(U)+\nu(V)=\nu(U\cup V)+\nu(U\cap V)

Glossary_of_Riemannian_and_metric_geometry.html

  1. | x y | X |xy|_{X}
  2. f : X Y f:X\to Y
  3. c | x y | X | f ( x ) f ( y ) | Y C | x y | X c|xy|_{X}\leq|f(x)f(y)|_{Y}\leq C|xy|_{X}
  4. B γ ( p ) = lim t ( | γ ( t ) - p | - t ) B_{\gamma}(p)=\lim_{t\to\infty}(|\gamma(t)-p|-t)
  5. p 1 , p 2 , , p k p_{1},p_{2},\dots,p_{k}
  6. f ( x ) = i | p i x | 2 f(x)=\sum_{i}|p_{i}x|^{2}
  7. | p i p j | |p_{i}p_{j}|
  8. γ \gamma
  9. γ \gamma
  10. γ \gamma
  11. f γ f\circ\gamma
  12. λ \lambda
  13. γ \gamma
  14. t t
  15. f γ ( t ) - λ t 2 f\circ\gamma(t)-\lambda t^{2}
  16. ( γ ( t ) , γ ( t ) ) (\gamma(t),\gamma^{\prime}(t))
  17. γ \gamma
  18. N F N\rtimes F
  19. N F N\rtimes F
  20. γ τ \gamma_{\tau}
  21. γ 0 = γ \gamma_{0}=\gamma
  22. J ( t ) = γ τ ( t ) / τ | τ = 0 . J(t)=\partial\gamma_{\tau}(t)/\partial\tau|_{\tau=0}.\,
  23. ϵ \epsilon
  24. ϵ \leq\epsilon
  25. S 1 S^{1}
  26. N {\mathbb{R}}^{N}
  27. N {\mathbb{R}}^{N}
  28. T p M T_{p}M
  29. f : 𝐑 Y f:\,\textbf{R}\to Y
  30. K > 0 K>0
  31. C 0 C\geq 0
  32. 1 K d ( x , y ) - C d ( f ( x ) , f ( y ) ) K d ( x , y ) + C . {1\over K}d(x,y)-C\leq d(f(x),f(y))\leq Kd(x,y)+C.
  33. f : X Y f:X\to Y
  34. K 1 K\geq 1
  35. C 0 C\geq 0
  36. 1 K d ( x , y ) - C d ( f ( x ) , f ( y ) ) K d ( x , y ) + C . {1\over K}d(x,y)-C\leq d(f(x),f(y))\leq Kd(x,y)+C.
  37. II ( v , w ) = S ( v ) , w \,\text{II}(v,w)=\langle S(v),w\rangle
  38. S ( v ) = ± v n S(v)=\pm\nabla_{v}n
  39. s y s t k ( M ) syst_{k}(M)

Glossary_of_tensor_theory.html

  1. v w v\otimes w\,
  2. V W . V\otimes W.\,
  3. \otimes

Goldstone_boson.html

  1. φ φ
  2. λ ( ϕ * ϕ - v 2 ) 2 , \lambda(\phi^{*}\phi-v^{2})^{2}~{},
  3. λ λ→∞
  4. δ φ = i ε φ δφ=iεφ
  5. θ θ
  6. ϕ = v e i θ \phi=ve^{i\theta}\,
  7. θ θ
  8. v θ
  9. θ θ
  10. δ θ = ϵ , \delta\theta=\epsilon~{},
  11. | 0 |0〉
  12. = - 1 2 ( μ ϕ * ) μ ϕ + m 2 ϕ * ϕ = - 1 2 ( - i v e - i θ μ θ ) ( i v e i θ μ θ ) + m 2 v 2 , {\mathcal{L}}=-\frac{1}{2}(\partial^{\mu}\phi^{*})\partial_{\mu}\phi+m^{2}\phi% ^{*}\phi=-\frac{1}{2}(-ive^{-i\theta}\partial^{\mu}\theta)(ive^{i\theta}% \partial_{\mu}\theta)+m^{2}v^{2},
  13. = - v 2 2 ( μ θ ) ( μ θ ) + m 2 v 2 . =-\frac{v^{2}}{2}(\partial^{\mu}\theta)(\partial_{\mu}\theta)+m^{2}v^{2}~{}.
  14. m ² v ² m²v²
  15. J μ = - v 2 μ θ . J_{\mu}=-v^{2}\partial_{\mu}\theta~{}.
  16. θ θ
  17. θ = 0 〈θ〉=0
  18. θ = ε 〈θ〉=−ε
  19. d d t Q = d d t x J 0 ( x ) = 0 . {d\over dt}Q={d\over dt}\int_{x}J^{0}(x)=0~{}.
  20. d d t Q A = d d t x e - x 2 2 A 2 J 0 ( x ) = - x e - x 2 2 A 2 J = x ( e - x 2 2 A 2 ) J , {d\over dt}Q_{A}={d\over dt}\int_{x}e^{-x^{2}\over 2A^{2}}J^{0}(x)=-\int_{x}e^% {-x^{2}\over 2A^{2}}\nabla\cdot J=\int_{x}\nabla(e^{-x^{2}\over 2A^{2}})\cdot J% ~{},
  21. d d t Q A | 0 1 A Q A | 0 . \|{d\over dt}Q_{A}|0\rangle\|\approx{1\over A}\|Q_{A}|0\rangle\|.
  22. d d t | θ = H | θ m 0 | θ . \|{d\over dt}|\theta\rangle\|=\|H|\theta\rangle\|\geq m_{0}\|\;|\theta\rangle% \|~{}.

Googlewhack.html

  1. β \beta
  2. β = 0.52 \beta=0.52

Gorenstein_ring.html

  1. ( R , m , k ) (R,m,k)
  2. n n
  3. R R
  4. R R
  5. R R
  6. n n
  7. R R
  8. Ext R i ( k , R ) = 0 \operatorname{Ext}^{i}_{R}(k,R)=0
  9. i n i\neq n
  10. Ext R n ( k , R ) \operatorname{Ext}^{n}_{R}(k,R)
  11. k k
  12. Ext R i ( k , R ) = 0 \operatorname{Ext}^{i}_{R}(k,R)=0
  13. i > n i>n
  14. Ext R i ( k , R ) = 0 \operatorname{Ext}^{i}_{R}(k,R)=0
  15. i < n i<n
  16. Ext R n ( k , R ) \operatorname{Ext}^{n}_{R}(k,R)
  17. k k
  18. R R
  19. n n

GOST_(block_cipher).html

  1. 2 256 2^{256}
  2. 2 228 2^{228}
  3. 2 178 2^{178}
  4. 2 70 2^{70}
  5. 2 64 2^{64}
  6. 2 101 2^{101}
  7. 2 100 2^{100}

Graham_scan.html

  1. [ 0 , π ) [0,\pi)
  2. P 1 = ( x 1 , y 1 ) P_{1}=(x_{1},y_{1})
  3. P 2 = ( x 2 , y 2 ) P_{2}=(x_{2},y_{2})
  4. P 3 = ( x 3 , y 3 ) P_{3}=(x_{3},y_{3})
  5. P 1 P 2 \overrightarrow{P_{1}P_{2}}
  6. P 1 P 3 \overrightarrow{P_{1}P_{3}}
  7. ( x 2 - x 1 ) ( y 3 - y 1 ) - ( y 2 - y 1 ) ( x 3 - x 1 ) (x_{2}-x_{1})(y_{3}-y_{1})-(y_{2}-y_{1})(x_{3}-x_{1})
  8. ( x 2 , y 2 ) (x_{2},y_{2})
  9. ( x 3 , y 3 ) (x_{3},y_{3})
  10. ( x 2 , y 2 ) (x_{2},y_{2})
  11. ( x 2 , y 2 ) (x_{2},y_{2})

Graph_(abstract_data_type).html

  1. \infty
  2. O ( | V | + | E | ) O(|V|+|E|)
  3. O ( | V | 2 ) O(|V|^{2})
  4. O ( | V | | E | ) O(|V|\cdot|E|)
  5. O ( 1 ) O(1)
  6. O ( | V | 2 ) O(|V|^{2})
  7. O ( | V | | E | ) O(|V|\cdot|E|)
  8. O ( 1 ) O(1)
  9. O ( 1 ) O(1)
  10. O ( | V | | E | ) O(|V|\cdot|E|)
  11. O ( | E | ) O(|E|)
  12. O ( | V | 2 ) O(|V|^{2})
  13. O ( | V | | E | ) O(|V|\cdot|E|)
  14. O ( | E | ) O(|E|)
  15. O ( 1 ) O(1)
  16. O ( | V | | E | ) O(|V|\cdot|E|)
  17. O ( | V | ) O(|V|)
  18. O ( 1 ) O(1)
  19. O ( | E | ) O(|E|)

Graph_(mathematics).html

  1. | V | |V|
  2. | E | |E|
  3. 𝐒𝐞𝐭 D \mathbf{Set}\downarrow D
  4. D : 𝐒𝐞𝐭 𝐒𝐞𝐭 D:\mathbf{Set}\rightarrow\mathbf{Set}
  5. s s
  6. s × s s\times s

Graph_coloring.html

  1. χ ( G ) = min { k : P ( G , k ) > 0 } . \chi(G)=\min\{k\,\colon\,P(G,k)>0\}.
  2. t ( t - 1 ) ( t - 2 ) t(t-1)(t-2)
  3. t ( t - 1 ) ( t - 2 ) ( t - ( n - 1 ) ) t(t-1)(t-2)\cdots(t-(n-1))
  4. t ( t - 1 ) n - 1 t(t-1)^{n-1}
  5. ( t - 1 ) n + ( - 1 ) n ( t - 1 ) (t-1)^{n}+(-1)^{n}(t-1)
  6. t ( t - 1 ) ( t - 2 ) ( t 7 - 12 t 6 + 67 t 5 - 230 t 4 + 529 t 3 - 814 t 2 + 775 t - 352 ) t(t-1)(t-2)(t^{7}-12t^{6}+67t^{5}-230t^{4}+529t^{3}-814t^{2}+775t-352)
  7. d d
  8. d \mathbb{Z}^{d}
  9. 1 χ ( G ) n . 1\leq\chi(G)\leq n.\,
  10. K n K_{n}
  11. χ ( K n ) = n \chi(K_{n})=n
  12. χ ( G ) ( χ ( G ) - 1 ) 2 m . \chi(G)(\chi(G)-1)\leq 2m.\,
  13. χ ( G ) ω ( G ) . \chi(G)\geq\omega(G).\,
  14. χ ( G ) Δ ( G ) + 1. \chi(G)\leq\Delta(G)+1.\,
  15. χ ( G ) = n \chi(G)=n
  16. Δ ( G ) = n - 1 \Delta(G)=n-1
  17. χ ( G ) = 3 \chi(G)=3
  18. Δ ( G ) = 2 \Delta(G)=2
  19. χ ( G ) Δ ( G ) \chi(G)\leq\Delta(G)
  20. W W
  21. W i , j = 0 W_{i,j}=0
  22. ( i , j ) (i,j)
  23. G G
  24. χ W ( G ) = 1 - λ m a x ( W ) / λ m i n ( W ) \chi_{W}(G)=1-\lambda_{max}(W)/\lambda_{min}(W)
  25. λ m a x ( W ) , λ m i n ( W ) \lambda_{max}(W),\lambda_{min}(W)
  26. W W
  27. χ H ( G ) = max W χ W ( G ) \chi_{H}(G)=\max_{W}\chi_{W}(G)
  28. W W
  29. χ H ( G ) χ ( G ) \chi_{H}(G)\leq\chi(G)
  30. W W
  31. W i , j - 1 / ( k - 1 ) W_{i,j}\leq-1/(k-1)
  32. ( i , j ) (i,j)
  33. G G
  34. χ V ( G ) \chi_{V}(G)
  35. W W
  36. χ V ( G ) χ ( G ) \chi_{V}(G)\leq\chi(G)
  37. ϑ ( G ¯ ) χ ( G ) \vartheta(\bar{G})\leq\chi(G)
  38. χ f ( G ) χ ( G ) \chi_{f}(G)\leq\chi(G)
  39. χ H ( G ) χ V ( G ) ϑ ( G ¯ ) χ f ( G ) χ ( G ) \chi_{H}(G)\leq\chi_{V}(G)\leq\vartheta(\bar{G})\leq\chi_{f}(G)\leq\chi(G)
  40. L ( G ) L(G)
  41. χ ( G ) = χ ( L ( G ) ) . \chi^{\prime}(G)=\chi(L(G)).\,
  42. Δ ( G ) \Delta(G)
  43. χ ( G ) Δ ( G ) . \chi^{\prime}(G)\geq\Delta(G).\,
  44. χ ( G ) = Δ ( G ) \chi^{\prime}(G)=\Delta(G)
  45. Δ \Delta
  46. Δ \Delta
  47. Δ + 1 \Delta+1
  48. P ( G , t ) P(G,t)
  49. [ 4 , ) [4,\infty)
  50. [ 5 , ) [5,\infty)
  51. P ( G , 4 ) 0 P(G,4)\neq 0
  52. k n k^{n}
  53. k = 1 , , n - 1 k=1,\ldots,n-1
  54. O ( 2.445 n ) O(2.445^{n})
  55. O ( 2 n n ) O(2^{n}n)
  56. O ( 1.3289 n ) O(1.3289^{n})
  57. O ( 1.7272 n ) O(1.7272^{n})
  58. G / u v G/uv
  59. χ ( G ) = min { χ ( G + u v ) , χ ( G / u v ) } \chi(G)=\,\text{min}\{\chi(G+uv),\chi(G/uv)\}
  60. G + u v G+uv
  61. u v uv
  62. P ( G - u v , k ) = P ( G / u v , k ) + P ( G , k ) P(G-uv,k)=P(G/uv,k)+P(G,k)
  63. G - u v G-uv
  64. u v uv
  65. P ( G - u v , k ) P(G-uv,k)
  66. ( ( 1 + 5 ) / 2 ) n + m = O ( 1.6180 n + m ) ((1+\sqrt{5})/2)^{n+m}=O(1.6180^{n+m})
  67. t ( G ) t(G)
  68. v 1 v_{1}
  69. v n v_{n}
  70. v i v_{i}
  71. v i v_{i}
  72. v 1 v_{1}
  73. v i - 1 v_{i-1}
  74. χ ( G ) \chi(G)
  75. n / 2 n/2
  76. max i min { d ( x i ) + 1 , i } \,\text{max}_{i}\,\text{ min}\{d(x_{i})+1,i\}
  77. 2 O ( log n ) 2^{O\left(\sqrt{\log n}\right)}
  78. χ ( G , k ) \chi(G,k)
  79. K 6 K_{6}

Graph_minor.html

  1. O ( n h log h ) \scriptstyle O(nh\sqrt{\log h})
  2. O ( h log h ) \scriptstyle O(h\sqrt{\log h})
  3. O ( h log h ) \scriptstyle O(h\sqrt{\log h})
  4. O ( n ) \scriptstyle O(\sqrt{n})
  5. O ( n ) \scriptstyle O(\sqrt{n})

Graph_reduction.html

  1. ( ( 2 + 2 ) + ( 2 + 2 ) ) + ( 3 + 3 ) \displaystyle{}\qquad((2+2)+(2+2))+(3+3)
  2. ( ( 2 + 2 ) + ( 2 + 2 ) ) + ( 3 + 3 ) \displaystyle{}\qquad((2+2)+(2+2))+(3+3)

Graphical_model.html

  1. X 1 , , X n X_{1},\ldots,X_{n}
  2. P [ X 1 , , X n ] = i = 1 n P [ X i | p a i ] P[X_{1},\ldots,X_{n}]=\prod_{i=1}^{n}P[X_{i}|pa_{i}]
  3. p a i pa_{i}
  4. X i X_{i}
  5. A , B , C , D A,B,C,D
  6. P [ A , B , C , D ] = P [ A ] P [ B ] P [ C | B , D ] P [ D | A , B , C ] . P[A,B,C,D]=P[A]\cdot P[B]\cdot P[C|B,D]\cdot P[D|A,B,C].

Grassmannian.html

  1. 𝐆𝐫 ( r , V ) \mathbf{Gr}(r,V)
  2. V V
  3. r r
  4. 𝐆𝐫 ( 1 , V ) \mathbf{Gr}(1,V)
  5. V V
  6. V V
  7. V V
  8. 𝐆𝐫 ( V , r ) \mathbf{Gr}(V,r)
  9. 𝐆𝐫 ( r , V ) \mathbf{Gr}(r,V)
  10. 𝐆𝐫 ( r , n ) \mathbf{Gr}(r,n)
  11. 𝐆𝐫 ( n , r ) \mathbf{Gr}(n,r)
  12. r r
  13. n n
  14. M M
  15. r r
  16. x x
  17. M M
  18. M M
  19. x x
  20. M M
  21. 𝐆𝐫 ( r , n ) \mathbf{Gr}(r,n)
  22. M M
  23. x x
  24. r r
  25. M M
  26. M M
  27. r = 1 r=1
  28. 𝐆𝐫 ( 1 , 3 ) \mathbf{Gr}(1,3)
  29. r = 2 r=2
  30. 𝐆𝐫 ( 2 , 4 ) \mathbf{Gr}(2,4)
  31. V V
  32. k k
  33. 𝐆𝐫 ( r , V ) \mathbf{Gr}(r,V)
  34. r r
  35. V V
  36. V V
  37. n n
  38. 𝐆𝐫 ( r , n ) \mathbf{Gr}(r,n)
  39. V V
  40. 𝐏 ( V ) \mathbf{P}(V)
  41. 𝐏 ( V ) \mathbf{P}(V)
  42. 𝐆𝐫 ( r 1 , 𝐏 ( V ) ) \mathbf{Gr}(r−1,\mathbf{P}(V))
  43. 𝐆𝐫 ( r 1 , n 1 ) \mathbf{Gr}(r−1,n−1)
  44. G L ( V ) GL(V)
  45. r r
  46. V V
  47. H H
  48. 𝐆𝐫 ( r , V ) = G L ( V ) / H \mathbf{Gr}(r,V)=GL(V)/H
  49. 𝐑 \mathbf{R}
  50. 𝐂 \mathbf{C}
  51. G L ( V ) GL(V)
  52. V V
  53. 𝐑 \mathbf{R}
  54. G L ( V ) GL(V)
  55. O ( V ) O(V)
  56. 𝐆𝐫 ( r , n ) = O ( n ) / ( O ( r ) × O ( n r ) ) \mathbf{Gr}(r,n)=O(n)/(O(r)×O(n–r))
  57. r ( n r ) r(n–r)
  58. 𝐂 \mathbf{C}
  59. G L ( V ) GL(V)
  60. U ( V ) U(V)
  61. W W
  62. V V
  63. V V
  64. W W
  65. d ( W , W ) = P W - P W , d(W,W^{\prime})=\lVert P_{W}-P_{W^{\prime}}\rVert,
  66. [ u ! ! ] [ u ! ! ] [u^{\prime}!!^{\prime}]⋅[u^{\prime}!!^{\prime}]
  67. 𝐆𝐫 ( r , V ) \mathbf{Gr}(r,V)
  68. V V
  69. k k
  70. G L ( V ) GL(V)
  71. H H
  72. G L ( V ) GL(V)
  73. \mathcal{E}
  74. S S
  75. r r
  76. S S
  77. T T
  78. T := O S O T \mathcal{E}_{T}:=\mathcal{E}\otimes_{O_{S}}O_{T}
  79. r r
  80. T T
  81. 𝐆𝐫 ( r , T ) \mathbf{Gr}(r,\mathcal{E}_{T})
  82. S S
  83. 𝐆𝐫 ( r , ) \mathbf{Gr}(r,\mathcal{E})
  84. \mathcal{E}
  85. S S
  86. k k
  87. \mathcal{E}
  88. V V
  89. V V
  90. S S
  91. S S′
  92. 𝐆𝐫 ( r , ) × S S 𝐆𝐫 ( r , S ) \mathbf{Gr}(r,\mathcal{E})\times_{S}S^{\prime}\simeq\mathbf{Gr}(r,\mathcal{E}_% {S^{\prime}})
  93. s s
  94. S S
  95. 𝐆𝐫 ( r , ) s \mathbf{Gr}(r,\mathcal{E})_{s}
  96. G r ( r , O S k ( s ) ) {Gr}(r,\mathcal{E}\otimes_{O_{S}}k(s))
  97. k ( s ) k(s)
  98. 𝒢 \mathcal{G}
  99. 𝐆𝐫 ( r , 𝐆𝐫 ( r , ) ) , \mathbf{Gr}\left(r,\mathcal{E}_{\mathbf{Gr}(r,\mathcal{E})}\right),
  100. 𝒢 \mathcal{G}
  101. 𝐆𝐫 ( r , ) \mathcal{E}_{\mathbf{Gr}(r,\mathcal{E})}
  102. r r
  103. 𝐆𝐫 ( r , ) \mathbf{Gr}(r,\mathcal{E})
  104. 𝐏 ( 𝒢 ) \mathbf{P}(\mathcal{G})
  105. 𝐏 ( 𝒢 ) 𝐏 ( 𝐆𝐫 ( r , ) ) = 𝐏 ( ) × S 𝐆𝐫 ( r , ) . \mathbf{P}(\mathcal{G})\to\mathbf{P}\left(\mathcal{E}_{\mathbf{Gr}(r,\mathcal{% E})}\right)=\mathbf{P}({\mathcal{E}})\times_{S}\mathbf{Gr}(r,\mathcal{E}).
  106. S S
  107. T 𝐆𝐫 ( r , ) , T\to\mathbf{Gr}(r,\mathcal{E}),
  108. 𝐏 ( 𝒢 T ) 𝐏 ( ) × S T . \mathbf{P}(\mathcal{G}_{T})\to\mathbf{P}(\mathcal{E})\times_{S}T.
  109. T \mathcal{E}_{T}
  110. r r
  111. 𝐆𝐫 ( r , ) ( T ) \mathbf{Gr}(r,\mathcal{E})(T)
  112. r r
  113. 𝐏 ( ) × S T . \mathbf{P}(\mathcal{E})\times_{S}T.
  114. T = S T=S
  115. k k
  116. \mathcal{E}
  117. V V
  118. 𝐆𝐫 ( r , ) ( k ) \mathbf{Gr}(r,\mathcal{E})(k)
  119. r 1 r−1
  120. 𝐏 ( V ) \mathbf{P}(V)
  121. 𝐏 ( 𝒢 ) ( k ) \mathbf{P}(\mathcal{G})(k)
  122. 𝐏 ( V ) × k 𝐆𝐫 ( r , ) \mathbf{P}(V)\times_{k}\mathbf{Gr}(r,\mathcal{E})
  123. { ( x , v ) 𝐏 ( V ) ( k ) × 𝐆𝐫 ( r , ) ( k ) x v } . \{(x,v)\in\mathbf{P}(V)(k)\times\mathbf{Gr}(r,\mathcal{E})(k)\mid x\in v\}.
  124. ψ : 𝐆𝐫 ( r , V ) 𝐏 ( r V ) . \psi:\mathbf{Gr}(r,V)\to\mathbf{P}\left(\wedge^{r}V\right).
  125. W W
  126. r r
  127. V V
  128. ψ ( W ) ψ(W)
  129. W W
  130. ψ ( W ) ψ(W)
  131. ψ ( W ) = w 1 w r . \psi(W)=w_{1}\wedge\cdots\wedge w_{r}.
  132. W W
  133. ψ ψ
  134. ψ ψ
  135. W W
  136. ψ ( W ) ψ(W)
  137. w w
  138. w ψ ( W ) = 0 w∧ψ(W)=0
  139. r r
  140. W W
  141. Z Z
  142. V V
  143. k 0 k≥0
  144. ψ ( W ) ψ ( Z ) - i 1 < < i k ( v 1 v i 1 - 1 w 1 v i 1 + 1 v i k - 1 w k v i k + 1 v r ) ( v i 1 v i k w k + 1 w r ) = 0. \psi(W)\cdot\psi(Z)-\sum_{i_{1}<\cdots<i_{k}}(v_{1}\wedge\cdots\wedge v_{i_{1}% -1}\wedge w_{1}\wedge v_{i_{1}+1}\wedge\cdots\wedge v_{i_{k}-1}\wedge w_{k}% \wedge v_{i_{k}+1}\wedge\cdots\wedge v_{r})\cdot(v_{i_{1}}\wedge\cdots\wedge v% _{i_{k}}\wedge w_{k+1}\cdots\wedge w_{r})=0.
  145. d i m ( V ) = 4 dim(V)=4
  146. r = 2 r=2
  147. 𝐆𝐫 ( 2 , V ) \mathbf{Gr}(2,V)
  148. r r
  149. M ( n , 𝐑 ) M(n,\mathbf{R})
  150. n × n n×n
  151. A ( r , n ) M ( n , 𝐑 ) A(r,n)⊂M(n,\mathbf{R})
  152. X A ( r , n ) X∈A(r,n)
  153. X X
  154. X X
  155. X X
  156. r r
  157. t r ( X ) = r tr(X)=r
  158. A ( r , n ) A(r,n)
  159. X A ( r , n ) X∈A(r,n)
  160. X X
  161. r r
  162. W W
  163. V V
  164. ( n r ) (n–r)
  165. V / W V/W
  166. V V
  167. 0 W V V / W 0 0→W→V→V/W→0
  168. r r
  169. V V
  170. ( n r ) (n–r)
  171. V V
  172. 𝐆𝐫 ( r , V ) \mathbf{Gr}(r,V)
  173. 𝐆𝐫 ( n r , V ) \mathbf{Gr}(n−r,V)
  174. V V
  175. r r
  176. ( n r ) (n–r)
  177. 𝐆𝐫 ( r , n ) \mathbf{Gr}(r,n)
  178. 𝐆𝐫 ( r , n ) \mathbf{Gr}(r,n)
  179. W W
  180. i i
  181. i = 1 , , r i=1,...,r
  182. r r
  183. 1 1
  184. 𝐆𝐫 ( r , n ) \mathbf{Gr}(r,n)
  185. r r
  186. 𝐑 \mathbf{R}
  187. 𝐆𝐫 ( r 1 , n 1 ) \mathbf{Gr}(r−1,n−1)
  188. r r
  189. 𝐆𝐫 ( r , n 1 ) \mathbf{Gr}(r,n−1)
  190. χ r , n = χ r - 1 , n - 1 + ( - 1 ) r χ r , n - 1 , χ 0 , n = χ n , n = 1. \chi_{r,n}=\chi_{r-1,n-1}+(-1)^{r}\chi_{r,n-1},\qquad\chi_{0,n}=\chi_{n,n}=1.
  191. n n
  192. r r
  193. χ r , n = ( n 2 r 2 ) . \chi_{r,n}={\lfloor\frac{n}{2}\rfloor\choose\lfloor\frac{r}{2}\rfloor}.
  194. 𝐆𝐫 ( r , n ) \mathbf{Gr}(r,n)
  195. r r
  196. n n
  197. E E
  198. ( n r ) (n−r)
  199. F F
  200. E E
  201. E E
  202. F F
  203. E E
  204. F F
  205. c ( E ) c ( F ) = 1. c(E)c(F)=1.
  206. c k ( E ) c n - k ( F ) = ( - 1 ) n - r c_{k}(E)c_{n-k}(F)=(-1)^{n-r}
  207. 2 n 2n
  208. 2 n 2n
  209. V V
  210. n n
  211. 𝐆𝐫 ( r , n ) \mathbf{Gr}(r,n)
  212. O ( n ) O(n)
  213. V V
  214. 𝐆𝐫 ( r , n ) \mathbf{Gr}(r,n)
  215. A 𝐆𝐫 ( r , n ) A⊆\mathbf{Gr}(r,n)
  216. γ r , n ( A ) = θ n { g O ( n ) : g V A } . \gamma_{r,n}(A)=\theta_{n}\{g\in O(n):gV\in A\}.
  217. O ( n ) O(n)
  218. g g
  219. O ( n ) O(n)
  220. r r
  221. 𝐆𝐫 ( r , n ) \mathbf{Gr}(r,n)
  222. 𝐆𝐫 ~ ( r , n ) . \tilde{\mathbf{Gr}}(r,n).
  223. S O ( n ) / ( S O ( r ) × S O ( n - r ) ) . SO(n)/(SO(r)\times SO(n-r)).

Gravitational_potential.html

  1. U = m V , U=mV,
  2. Δ U = m g Δ h . \Delta U=mg\Delta h.
  3. V ( x ) = W m = 1 m x F d x = 1 m x G m M x 2 d x = - G M x , V(x)=\frac{W}{m}=\frac{1}{m}\int\limits_{\infty}^{x}F\ dx=\frac{1}{m}\int% \limits_{\infty}^{x}\frac{GmM}{x^{2}}dx=-\frac{GM}{x},
  4. 𝐚 = - G M x 3 𝐱 = - G M x 2 𝐱 ^ , \mathbf{a}=-\frac{GM}{x^{3}}\mathbf{x}=-\frac{GM}{x^{2}}\hat{\mathbf{x}},
  5. 𝐱 ^ \hat{\mathbf{x}}
  6. | 𝐚 | = G M x 2 . |\mathbf{a}|=\frac{GM}{x^{2}}.
  7. V ( 𝐱 ) = i = 1 n - G m i | 𝐱 - 𝐱 𝐢 | . V(\mathbf{x})=\sum_{i=1}^{n}-\frac{Gm_{i}}{|\mathbf{x}-\mathbf{x_{i}}|}.
  8. V ( 𝐱 ) = - 𝐑 3 G | 𝐱 - 𝐫 | d m ( 𝐫 ) , V(\mathbf{x})=-\int_{\mathbf{R}^{3}}\frac{G}{|\mathbf{x}-\mathbf{r}|}\,dm(% \mathbf{r}),
  9. V ( 𝐱 ) = - 𝐑 3 G | 𝐱 - 𝐫 | ρ ( 𝐫 ) d v ( 𝐫 ) . V(\mathbf{x})=-\int_{\mathbf{R}^{3}}\frac{G}{|\mathbf{x}-\mathbf{r}|}\,\rho(% \mathbf{r})dv(\mathbf{r}).
  10. ρ ( 𝐱 ) = 1 4 π G Δ V ( 𝐱 ) . \rho(\mathbf{x})=\frac{1}{4\pi G}\Delta V(\mathbf{x}).
  11. V ( r ) = 2 3 π G ρ ( r 2 - 3 R 2 ) , r R , V(r)=\frac{2}{3}\pi G\rho(r^{2}-3R^{2}),\qquad r\leq R,
  12. V ( 𝐱 ) = - 3 G | 𝐱 - 𝐫 | d m ( 𝐫 ) . V(\mathbf{x})=-\int_{\mathbb{R}^{3}}\frac{G}{|\mathbf{x}-\mathbf{r}|}\ dm(% \mathbf{r}).
  13. V ( 𝐱 ) = - 3 G | 𝐱 | 2 - 2 𝐱 𝐫 + | 𝐫 | 2 d m ( 𝐫 ) = - 1 | 𝐱 | 3 G / 1 - 2 r | 𝐱 | cos θ + ( r | 𝐱 | ) 2 d m ( 𝐫 ) \begin{aligned}\displaystyle V(\mathbf{x})&\displaystyle=-\int_{\mathbb{R}^{3}% }\frac{G}{\sqrt{|\mathbf{x}|^{2}-2\mathbf{x}\cdot\mathbf{r}+|\mathbf{r}|^{2}}}% \,dm(\mathbf{r})\\ &\displaystyle=-\frac{1}{|\mathbf{x}|}\int_{\mathbb{R}^{3}}G\,\left/\,\sqrt{1-% 2\frac{r}{|\mathbf{x}|}\cos\theta+\left(\frac{r}{|\mathbf{x}|}\right)^{2}}% \right.\,dm(\mathbf{r})\end{aligned}
  14. ( 1 - 2 X Z + Z 2 ) - 1 2 = n = 0 Z n P n ( X ) \left(1-2XZ+Z^{2}\right)^{-\frac{1}{2}}\ =\sum_{n=0}^{\infty}Z^{n}P_{n}(X)
  15. r cos θ d m \int r\cos\theta dm
  16. V ( 𝐱 ) = - G M | 𝐱 | - G | 𝐱 | ( r | 𝐱 | ) 2 3 cos 2 θ - 1 2 d m ( 𝐫 ) + V(\mathbf{x})=-\frac{GM}{|\mathbf{x}|}-\frac{G}{|\mathbf{x}|}\int\left(\frac{r% }{|\mathbf{x}|}\right)^{2}\frac{3\cos^{2}\theta-1}{2}dm(\mathbf{r})+\cdots

Gravity_current.html

  1. Fr = u f / g h \mathrm{Fr}=u_{f}/\sqrt{g^{\prime}h}
  2. h l = Q hl=Q
  3. F r Fr
  4. u f u_{f}
  5. g g^{\prime}
  6. h h
  7. l l
  8. Q Q
  9. h h
  10. F r Fr
  11. u f = d l / d t u_{f}=dl/dt
  12. l 0 l_{0}
  13. Q Q
  14. F r Fr
  15. l 3 / 2 = l 0 3 / 2 + 3 F r 2 g Q t . l^{3/2}=l_{0}^{3/2}+\frac{3Fr}{2}\sqrt{g^{\prime}Q}t.

Grayscale.html

  1. Y Y
  2. C linear = { C srgb 12.92 , C srgb 0.04045 ( C srgb + 0.055 1.055 ) 2.4 , C srgb > 0.04045 C_{\mathrm{linear}}=\begin{cases}\frac{C_{\mathrm{srgb}}}{12.92},&C_{\mathrm{% srgb}}\leq 0.04045\\ \left(\frac{C_{\mathrm{srgb}}+0.055}{1.055}\right)^{2.4},&C_{\mathrm{srgb}}>0.% 04045\end{cases}
  3. R R
  4. G G
  5. B B
  6. Y Y
  7. Y = 0.2126 R + 0.7152 G + 0.0722 B Y=0.2126R+0.7152G+0.0722B
  8. Y Y
  9. Y srgb = { 12.92 Y , Y 0.0031308 1.055 Y 1 / 2.4 - 0.055 , Y > 0.0031308. Y_{\mathrm{srgb}}=\begin{cases}12.92\ Y,&Y\leq 0.0031308\\ 1.055\ Y^{1/2.4}-0.055,&Y>0.0031308.\end{cases}
  10. ) \mathbf{)}
  11. ) \mathbf{)}
  12. Y = 0.299 R + 0.587 G + 0.114 B Y^{\prime}=0.299R^{\prime}+0.587G^{\prime}+0.114B^{\prime}
  13. Y = 0.2126 R + 0.7152 G + 0.0722 B Y^{\prime}=0.2126R^{\prime}+0.7152G^{\prime}+0.0722B^{\prime}
  14. $\mathbf{ }$
  15. Y Y
  16. Y < s u b > s r g b Y<sub>srgb

Great-circle_distance.html

  1. π r \pi r
  2. ϕ 1 , λ 1 \phi_{1},\lambda_{1}
  3. ϕ 2 , λ 2 \phi_{2},\lambda_{2}
  4. Δ ϕ , Δ λ \Delta\phi,\Delta\lambda
  5. Δ σ \Delta\sigma
  6. Δ σ = arccos ( sin ϕ 1 sin ϕ 2 + cos ϕ 1 cos ϕ 2 cos Δ λ ) . \Delta\sigma=\arccos\bigl(\sin\phi_{1}\sin\phi_{2}+\cos\phi_{1}\cos\phi_{2}% \cos\Delta\lambda\bigr).
  7. Δ σ \Delta\sigma
  8. d = r Δ σ . d=r\,\Delta\sigma.
  9. Δ σ = 2 arcsin ( sin 2 ( Δ ϕ 2 ) + cos ϕ 1 cos ϕ 2 sin 2 ( Δ λ 2 ) ) . \Delta\sigma=2\arcsin\left(\sqrt{\sin^{2}\left(\frac{\Delta\phi}{2}\right)+% \cos{\phi_{1}}\cos{\phi_{2}}\sin^{2}\left(\frac{\Delta\lambda}{2}\right)}% \right).\;\!
  10. Δ σ = arctan ( ( cos ϕ 2 sin Δ λ ) 2 + ( cos ϕ 1 sin ϕ 2 - sin ϕ 1 cos ϕ 2 cos Δ λ ) 2 sin ϕ 1 sin ϕ 2 + cos ϕ 1 cos ϕ 2 cos Δ λ ) . \Delta\sigma=\arctan\left(\frac{\sqrt{\left(\cos\phi_{2}\sin\Delta\lambda% \right)^{2}+\left(\cos\phi_{1}\sin\phi_{2}-\sin\phi_{1}\cos\phi_{2}\cos\Delta% \lambda\right)^{2}}}{\sin\phi_{1}\sin\phi_{2}+\cos\phi_{1}\cos\phi_{2}\cos% \Delta\lambda}\right).
  11. Δ σ \Delta\sigma
  12. Δ σ = arccos ( 𝐧 1 𝐧 2 ) Δ σ = arcsin ( | 𝐧 1 × 𝐧 2 | ) Δ σ = arctan ( | 𝐧 1 × 𝐧 2 | 𝐧 1 𝐧 2 ) \begin{aligned}\displaystyle\Delta\sigma&\displaystyle=\arccos(\mathbf{n}_{1}% \cdot\mathbf{n}_{2})\\ \displaystyle\Delta\sigma&\displaystyle=\arcsin\left(\left|\mathbf{n}_{1}% \times\mathbf{n}_{2}\right|\right)\\ \displaystyle\Delta\sigma&\displaystyle=\arctan\left(\frac{\left|\mathbf{n}_{1% }\times\mathbf{n}_{2}\right|}{\mathbf{n}_{1}\cdot\mathbf{n}_{2}}\right)\\ \end{aligned}\,\!
  13. 𝐧 1 \mathbf{n}_{1}
  14. 𝐧 2 \mathbf{n}_{2}
  15. C h C_{h}\,\!
  16. Δ X \displaystyle\Delta{X}
  17. Δ σ = 2 arcsin ( C 2 ) . \Delta\sigma=2\arcsin\left(\frac{C}{2}\right).
  18. d = r Δ σ . d=r\Delta\sigma.
  19. a a
  20. b b
  21. b 2 / a b^{2}/a
  22. a 2 / b a^{2}/b
  23. R 1 = 1 3 ( 2 a + b ) 6371 km R_{1}=\frac{1}{3}(2a+b)\approx 6371\,\mathrm{km}

Grothendieck's_Galois_theory.html

  1. \Z ^ \hat{\Z}
  2. \Z ^ \hat{\Z}

Group_algebra.html

  1. [ f * g ] ( t ) = G f ( s ) g ( s - 1 t ) d μ ( s ) . [f*g](t)=\int_{G}f(s)g\left(s^{-1}t\right)\,d\mu(s).
  2. Support ( f * g ) Support ( f ) Support ( g ) \operatorname{Support}(f*g)\subseteq\operatorname{Support}(f)\cdot% \operatorname{Support}(g)
  3. f * ( s ) = f ( s - 1 ) ¯ Δ ( s - 1 ) f^{*}(s)=\overline{f(s^{-1})}\Delta(s^{-1})
  4. f 1 := G | f ( s ) | d μ ( s ) , \|f\|_{1}:=\int_{G}|f(s)|d\mu(s),
  5. V f V ( g ) d μ ( g ) = 1. \int_{V}f_{V}(g)\,d\mu(g)=1.
  6. π U ( f ) = G f ( g ) U ( g ) d μ ( g ) \pi_{U}(f)=\int_{G}f(g)U(g)\,d\mu(g)
  7. U π U U\mapsto\pi_{U}
  8. { π ( f ) ξ : f C c ( G ) , ξ H π } \left\{\pi(f)\xi:f\in\operatorname{C}_{c}(G),\xi\in H_{\pi}\right\}
  9. f C * := sup π π ( f ) , \|f\|_{C^{*}}:=\sup_{\pi}\|\pi(f)\|,
  10. π ( f ) f 1 , \|\pi(f)\|\leq\|f\|_{1},
  11. 𝐂 [ G ] C max * ( G ) . \mathbf{C}[G]\hookrightarrow C^{*}_{\,\text{max}}(G).
  12. f C r * := sup { f * g 2 : g 2 = 1 } , \|f\|_{C^{*}_{r}}:=\sup\left\{\|f*g\|_{2}:\|g\|_{2}=1\right\},
  13. f 2 = G | f | 2 d μ \|f\|_{2}=\sqrt{\int_{G}|f|^{2}d\mu}

Group_extension.html

  1. 1 N G Q 1. 1\rightarrow N\rightarrow G\rightarrow Q\rightarrow 1.\,\!
  2. Ext 1 ( Q , N ) ; \operatorname{Ext}^{1}_{\mathbb{Z}}(Q,N);
  3. 1 K i G π H 1 1\rightarrow K\stackrel{i}{\rightarrow}G\stackrel{\pi}{\rightarrow}H\rightarrow 1
  4. 1 K i G π H 1 1\to K\stackrel{i^{\prime}}{\rightarrow}G^{\prime}\stackrel{\pi^{\prime}}{% \rightarrow}H\rightarrow 1
  5. T : G G T:G\to G^{\prime}
  6. 1 K G H 1 1\rightarrow K\rightarrow G\rightarrow H\rightarrow 1
  7. 1 K K × H H 1 1\rightarrow K\rightarrow K\times H\rightarrow H\rightarrow 1
  8. K × H K\times H
  9. 1 K G H 1 1\rightarrow K\rightarrow G\rightarrow H\rightarrow 1
  10. s : H G s\colon H\rightarrow G
  11. π s = id H \pi\circ s=\mathrm{id}_{H}
  12. H Aut ( K ) H\to\operatorname{Aut}(K)
  13. Ext ( Q , N ) \operatorname{Ext}(Q,N)
  14. 1 A E G 1 1\rightarrow A\rightarrow E\rightarrow G\rightarrow 1
  15. 0 𝔞 𝔢 𝔤 0 0\rightarrow\mathfrak{a}\rightarrow\mathfrak{e}\rightarrow\mathfrak{g}\rightarrow 0
  16. 𝔞 \mathfrak{a}
  17. 𝔢 \mathfrak{e}
  18. G Out ( A ) G\to\operatorname{Out}(A)
  19. G G
  20. G G
  21. π : G * G \pi:G^{*}\to G
  22. G < s u p > G<sup>∗

Group_ring.html

  1. x α f ( x ) x\mapsto\alpha\cdot f(x)
  2. x f ( x ) + g ( x ) x\mapsto f(x)+g(x)
  3. x u v = x f ( u ) g ( v ) = u G f ( u ) g ( u - 1 x ) . x\mapsto\sum_{uv=x}f(u)g(v)=\sum_{u\in G}f(u)g(u^{-1}x).
  4. g G f ( g ) g , \sum_{g\in G}f(g)g,
  5. g G f g g , \sum_{g\in G}f_{g}g,
  6. r = z 0 1 G + z 1 a + z 2 a 2 r=z_{0}1_{G}+z_{1}a+z_{2}a^{2}\,
  7. s = w 0 1 G + w 1 a + w 2 a 2 s=w_{0}1_{G}+w_{1}a+w_{2}a^{2}\,
  8. r + s = ( z 0 + w 0 ) 1 G + ( z 1 + w 1 ) a + ( z 2 + w 2 ) a 2 r+s=(z_{0}+w_{0})1_{G}+(z_{1}+w_{1})a+(z_{2}+w_{2})a^{2}\,
  9. r s = ( z 0 w 0 + z 1 w 2 + z 2 w 1 ) 1 G + ( z 0 w 1 + z 1 w 0 + z 2 w 2 ) a + ( z 0 w 2 + z 2 w 0 + z 1 w 1 ) a 2 . rs=(z_{0}w_{0}+z_{1}w_{2}+z_{2}w_{1})1_{G}+(z_{0}w_{1}+z_{1}w_{0}+z_{2}w_{2})a% +(z_{0}w_{2}+z_{2}w_{0}+z_{1}w_{1})a^{2}.
  10. f ( g ) = 1 1 G + g 1 G 0 g = 𝟏 { 1 G } ( g ) = { 1 g = 1 G 0 g 1 G , f(g)=1\cdot 1_{G}+\sum_{g\not=1_{G}}0\cdot g=\mathbf{1}_{\{1_{G}\}}(g)=\begin{% cases}1&g=1_{G}\\ 0&g\neq 1_{G}\end{cases},
  11. f ( g ) = 1 s + g s 0 g = 𝟏 { s } ( g ) = { 1 g = s 0 g s f(g)=1\cdot s+\sum_{g\not=s}0\cdot g=\mathbf{1}_{\{s\}}(g)=\begin{cases}1&g=s% \\ 0&g\neq s\end{cases}
  12. x = g G a g g . x=\sum_{g\in G}a_{g}g.
  13. g h = g h , g\cdot h=gh,
  14. e g e h = e g h . e_{g}\cdot e_{h}=e_{gh}.
  15. x = g G a g g x=\sum_{g\in G}a_{g}g
  16. ( x , f ) = g G a g f ( g ) , (x,f)=\sum_{g\in G}a_{g}f(g),
  17. \mapsto
  18. ρ ( g ) e h = e g h \rho(g)\cdot e_{h}=e_{gh}
  19. ρ ( g ) r = h G k h ρ ( g ) e h = h G k h e g h . \rho(g)\cdot r=\sum_{h\in G}k_{h}\rho(g)\cdot e_{h}=\sum_{h\in G}k_{h}e_{gh}.
  20. ρ ~ : K [ G ] End ( V ) . \tilde{\rho}:K[G]\rightarrow\mbox{End}~{}(V).
  21. ρ ~ ( r ) End ( V ) . \tilde{\rho}(r)\in\mbox{End}~{}(V).
  22. ρ ~ ( r ) \tilde{\rho}(r)
  23. ρ ~ ( r ) ( v 1 + v 2 ) = ρ ~ ( r ) v 1 + ρ ~ ( r ) v 2 \tilde{\rho}(r)\cdot(v_{1}+v_{2})=\tilde{\rho}(r)\cdot v_{1}+\tilde{\rho}(r)% \cdot v_{2}
  24. ρ ~ \tilde{\rho}
  25. ρ ~ ( r + s ) v = ρ ~ ( r ) v + ρ ~ ( s ) v \tilde{\rho}(r+s)\cdot v=\tilde{\rho}(r)\cdot v+\tilde{\rho}(s)\cdot v
  26. ρ ~ ( r s ) v = ρ ~ ( r ) ρ ~ ( s ) v . \tilde{\rho}(rs)\cdot v=\tilde{\rho}(r)\cdot\tilde{\rho}(s)\cdot v.
  27. ρ ~ ( 1 ) v = v \tilde{\rho}(1)\cdot v=v
  28. 1 = e e 1=e_{e}
  29. ρ ~ \tilde{\rho}
  30. ρ ~ \tilde{\rho}
  31. ρ ~ ( a r ) v 1 + ρ ~ ( b r ) v 2 = a ρ ~ ( r ) v 1 + b ρ ~ ( r ) v 2 = ρ ~ ( r ) ( a v 1 + b v 2 ) \tilde{\rho}(ar)\cdot v_{1}+\tilde{\rho}(br)\cdot v_{2}=a\tilde{\rho}(r)\cdot v% _{1}+b\tilde{\rho}(r)\cdot v_{2}=\tilde{\rho}(r)\cdot(av_{1}+bv_{2})
  32. ρ : G Aut ( V ) , \rho:G\rightarrow\mbox{Aut}~{}(V),
  33. ρ ~ : K [ G ] End ( V ) , \tilde{\rho}:K[G]\rightarrow\mbox{End}~{}(V),
  34. ρ ~ ( e g ) = ρ ( g ) \tilde{\rho}(e_{g})=\rho(g)
  35. Z ( K [ G ] ) := { z K [ G ] : r K [ G ] , z r = r z } . Z(K[G]):=\left\{z\in K[G]:\forall r\in K[G],zr=rz\right\}.
  36. Z ( K [ G ] ) = { g G a g g : g , h G , a g = a h - 1 g h } . Z(K[G])=\left\{\sum_{g\in G}a_{g}g:\forall g,h\in G,a_{g}=a_{h^{-1}gh}\right\}.
  37. g G a g g , g G b g g = 1 | G | g G a ¯ g b g . \left\langle\sum_{g\in G}a_{g}g,\sum_{g\in G}b_{g}g\right\rangle=\frac{1}{|G|}% \sum_{g\in G}\bar{a}_{g}b_{g}.
  38. GrpRng : Grp R - Alg \operatorname{GrpRng}\colon\mathbf{\operatorname{Grp}}\to R\mathbf{% \operatorname{-Alg}}
  39. GrpUnits : R - Alg Grp \operatorname{GrpUnits}\colon R\mathbf{\operatorname{-Alg}}\to\mathbf{% \operatorname{Grp}}
  40. a n = 1 a^{n}=1
  41. a \langle a\rangle
  42. x = ( a - 1 ) b ( 1 + a + a 2 + + a n - 1 ) x=(a-1)b\left(1+a+a^{2}+...+a^{n-1}\right)
  43. ( 1 + x ) ( 1 - x ) = 1 (1+x)(1-x)=1

Group_scheme.html

  1. 𝐆 m ^ \widehat{\mathbf{G}_{m}}
  2. / n \mathbb{Z}/n
  3. μ n \mu_{n}

Growth_rate_(group_theory).html

  1. x T x\in T
  2. x - 1 T x^{-1}\in T
  3. x G x\in G
  4. x = a 1 a 2 a k where a i T . x=a_{1}\cdot a_{2}\cdots a_{k}\mbox{ where }~{}a_{i}\in T.
  5. B n ( G , T ) = { x G x = a 1 a 2 a k where a i T and k n } . B_{n}(G,T)=\{x\in G\mid x=a_{1}\cdot a_{2}\cdots a_{k}\mbox{ where }~{}a_{i}% \in T\mbox{ and }~{}k\leq n\}.
  6. B n ( G , T ) = { x G d ( x , e ) n } . B_{n}(G,T)=\{x\in G\mid d(x,e)\leq n\}.
  7. B n ( G , T ) B_{n}(G,T)
  8. a b a\sim b
  9. a ( n / C ) b ( n ) a ( C n ) , a(n/C)\leq b(n)\leq a(Cn),\,
  10. p n q n p^{n}\sim q^{n}
  11. p , q > 1 p,q>1
  12. # ( n ) = | B n ( G , T ) | , \#(n)=|B_{n}(G,T)|,
  13. | B n ( G , T ) | |B_{n}(G,T)|
  14. B n ( G , T ) B_{n}(G,T)
  15. # ( n ) \#(n)
  16. B n ( G , T ) B_{n}(G,T)
  17. 1 C d F ( x , y ) d E ( x , y ) C d F ( x , y ) . {1\over C}\ d_{F}(x,y)\leq d_{E}(x,y)\leq C\ d_{F}(x,y).
  18. # ( n ) C ( n k + 1 ) \#(n)\leq C(n^{k}+1)
  19. C , k < C,k<\infty
  20. k 0 k_{0}
  21. k 0 k_{0}
  22. # ( n ) n k 0 \#(n)\sim n^{k_{0}}
  23. # ( n ) a n \#(n)\geq a^{n}
  24. a > 1 a>1
  25. b > 1 b>1
  26. # ( n ) b n \#(n)\leq b^{n}
  27. # ( n ) \#(n)
  28. π 1 ( M ) \pi_{1}(M)
  29. π 1 ( M ) \pi_{1}(M)

Gröbner_basis.html

  1. R = K [ x 1 , , x n ] R=K[x_{1},\ldots,x_{n}]
  2. K K
  3. K K
  4. c 1 M 1 + + c m M m c_{1}M_{1}+\cdots+c_{m}M_{m}
  5. c i c_{i}
  6. K K
  7. M i M_{i}
  8. M M
  9. M = x 1 a 1 x n a n , M=x_{1}^{a_{1}}\cdots x_{n}^{a_{n}},
  10. a i a_{i}
  11. A = [ a 1 , , a n ] A=[a_{1},\ldots,a_{n}]
  12. M M
  13. x 1 a 1 x n a n = X A x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}=X^{A}
  14. F = { f 1 , , f k } F=\{f_{1},\ldots,f_{k}\}
  15. R R
  16. F F
  17. F F
  18. R R
  19. f 1 , , f k = { i = 1 k g i f i | g 1 , , g k K [ x 1 , , x n ] } . \langle f_{1},\ldots,f_{k}\rangle=\left\{\sum_{i=1}^{k}g_{i}f_{i}\;|\;g_{1},% \ldots,g_{k}\in K[x_{1},\ldots,x_{n}]\right\}.
  20. M M
  21. N N
  22. P P
  23. M < N M P < N P M<N\Longleftrightarrow MP<NP
  24. M < M P M<MP
  25. p p
  26. p p
  27. p p
  28. p p
  29. red 1 ( f , g ) = f - c lc ( g ) q g . \operatorname{red}_{1}(f,g)=f-\frac{c}{\operatorname{lc}(g)}\,q\,g.
  30. red 1 ( f , G ) = red 1 ( f , g ) \operatorname{red}_{1}(f,G)=\operatorname{red}_{1}(f,g)
  31. red 1 \operatorname{red}_{1}
  32. red ( f , G ) \operatorname{red}(f,G)
  33. f = h + g G q g g , f=h+\sum_{g\in G}q_{g}\,g,
  34. q g q_{g}
  35. R [ e 1 , , e l ] / { e i e j | 1 i j l } , R[e_{1},\ldots,e_{l}]/\left\langle\{e_{i}e_{j}|1\leq i\leq j\leq l\}\right\rangle,
  36. e 1 , , e l e_{1},\ldots,e_{l}
  37. g 1 , , g k g_{1},\ldots,g_{k}
  38. R [ e 1 , , e l ] R[e_{1},\ldots,e_{l}]
  39. g 1 , , g k g_{1},\ldots,g_{k}
  40. e i e j e_{i}e_{j}
  41. 1 i j l 1\leq i\leq j\leq l
  42. i = 0 d i t i \sum_{i=0}^{\infty}d_{i}t^{i}
  43. d i d_{i}
  44. i = 0 d i t i = P ( t ) ( 1 - t ) d , \sum_{i=0}^{\infty}d_{i}t^{i}=\frac{P(t)}{(1-t)^{d}},
  45. P ( t ) P(t)
  46. P ( 1 ) P(1)
  47. P ( t ) P(t)
  48. K [ x 1 , , x n , y 1 , , y m ] = K [ X , Y ] , K[x_{1},\ldots,x_{n},y_{1},\ldots,y_{m}]=K[X,Y],
  49. G K [ Y ] G\cap K[Y]
  50. I K [ Y ] I\cap K[Y]
  51. G K [ Y ] G\cap K[Y]
  52. G K [ Y ] G\cap K[Y]
  53. G K [ Y ] G\cap K[Y]
  54. I K [ Y ] . I\cap K[Y].
  55. x 1 > > x n x_{1}>\cdots>x_{n}
  56. { x 1 , , x k } , { x k + 1 , , x n } . \{x_{1},\ldots,x_{k}\},\{x_{k+1},\ldots,x_{n}\}.
  57. K = t 2 , t f 1 , , t f m , ( 1 - t ) g 1 , , ( 1 - t ) g k . K=\langle t^{2},tf_{1},\ldots,tf_{m},(1-t)g_{1},\ldots,(1-t)g_{k}\rangle.
  58. ( a - b ) t + b (a-b)t+b
  59. a I a\in I
  60. b J b\in J
  61. b I J . b\in I\cap J.
  62. x 1 = f 1 ( t ) g 1 ( t ) x n = f n ( t ) g n ( t ) , \begin{aligned}\displaystyle x_{1}&\displaystyle=\frac{f_{1}(t)}{g_{1}(t)}\\ \displaystyle\vdots\\ \displaystyle x_{n}&\displaystyle=\frac{f_{n}(t)}{g_{n}(t)},\end{aligned}
  63. f i ( t ) f_{i}(t)
  64. g i ( t ) g_{i}(t)
  65. f i ( t ) f_{i}(t)
  66. g i ( t ) g_{i}(t)
  67. Res t ( g 1 x 1 - f 1 , g 2 x 2 - f 2 ) . \,\text{Res}_{t}(g_{1}x_{1}-f_{1},g_{2}x_{2}-f_{2}).
  68. g 1 x 1 - f 1 , , g n x 2 - f n . \langle g_{1}x_{1}-f_{1},\ldots,g_{n}x_{2}-f_{n}\rangle.
  69. t ( x 1 , x 2 ) t\mapsto(x_{1},x_{2})
  70. R f = R [ t ] / ( 1 - f t ) , R_{f}=R[t]/(1-ft),
  71. R f I R_{f}I
  72. R f . R_{f}.
  73. R f R_{f}
  74. R f I R_{f}I
  75. R f . R_{f}.
  76. I : f = { g R | ( k ) f k g I } I:f^{\infty}=\{g\in R|(\exists k\in\mathbb{N})f^{k}g\in I\}
  77. I : f = J R . I:f^{\infty}=J\cap R.
  78. I : f I:f^{\infty}
  79. F { 1 - t f } , F\cup\{1-tf\},
  80. F { 1 - t f } F\cup\{1-tf\}
  81. f 1 , , f k f_{1},\ldots,f_{k}
  82. f = f 1 f k , f=f_{1}\ldots f_{k},
  83. f = f 1 f k f=f_{1}\ldots f_{k}
  84. f 1 , f_{1},
  85. f 2 , f_{2},
  86. 1 - t 1 f 1 , , 1 - t k f k , 1-t_{1}f_{1},\ldots,1-t_{k}f_{k},
  87. t i t_{i}
  88. x 1 = p 1 p 0 x n = p n p 0 , \begin{aligned}\displaystyle x_{1}&\displaystyle=\frac{p_{1}}{p_{0}}\\ \displaystyle\vdots\\ \displaystyle x_{n}&\displaystyle=\frac{p_{n}}{p_{0}},\end{aligned}
  89. p 0 , , p n p_{0},\ldots,p_{n}
  90. t 1 , , t k . t_{1},\ldots,t_{k}.
  91. t 1 , , t k t_{1},\ldots,t_{k}
  92. x 1 , , x n x_{1},\ldots,x_{n}
  93. I = p 0 x 1 - p 1 , , p 0 x n - p n . I=\left\langle p_{0}x_{1}-p_{1},\ldots,p_{0}x_{n}-p_{n}\right\rangle.
  94. p i p_{i}
  95. p i p_{i}
  96. t i t_{i}
  97. I I
  98. p 0 p_{0}
  99. G G
  100. t i t_{i}
  101. G G

Grünwald–Letnikov_derivative.html

  1. 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}
  2. f ′′ ( x ) = lim h 0 f ( x + h ) - f ( x ) h f^{\prime\prime}(x)=\lim_{h\to 0}\frac{f^{\prime}(x+h)-f^{\prime}(x)}{h}
  3. = lim h 1 0 lim h 2 0 f ( x + h 1 + h 2 ) - f ( x + h 1 ) h 2 - lim h 2 0 f ( x + h 2 ) - f ( x ) h 2 h 1 =\lim_{h_{1}\to 0}\frac{\lim_{h_{2}\to 0}\frac{f(x+h_{1}+h_{2})-f(x+h_{1})}{h_% {2}}-\lim_{h_{2}\to 0}\frac{f(x+h_{2})-f(x)}{h_{2}}}{h_{1}}
  4. = lim h 0 f ( x + 2 h ) - 2 f ( x + h ) + f ( x ) h 2 , =\lim_{h\to 0}\frac{f(x+2h)-2f(x+h)+f(x)}{h^{2}},
  5. f ( n ) ( x ) = lim h 0 0 m n ( - 1 ) m ( n m ) f ( x + ( n - m ) h ) h n . f^{(n)}(x)=\lim_{h\to 0}\frac{\sum_{0\leq m\leq n}(-1)^{m}{n\choose m}f(x+(n-m% )h)}{h^{n}}.
  6. 𝔻 q f ( x ) = lim h 0 1 h q 0 m < ( - 1 ) m ( q m ) f ( x + ( q - m ) h ) . \mathbb{D}^{q}f(x)=\lim_{h\to 0}\frac{1}{h^{q}}\sum_{0\leq m<\infty}(-1)^{m}{q% \choose m}f(x+(q-m)h).
  7. Δ h q f ( x ) = 0 m < ( - 1 ) m ( q m ) f ( x + ( q - m ) h ) . \Delta^{q}_{h}f(x)=\sum_{0\leq m<\infty}(-1)^{m}{q\choose m}f(x+(q-m)h).
  8. 𝔻 q f ( x ) = lim h 0 Δ h q f ( x ) h q . \mathbb{D}^{q}f(x)=\lim_{h\to 0}\frac{\Delta^{q}_{h}f(x)}{h^{q}}.
  9. f ( n ) ( x ) = lim h 0 ( - 1 ) n h n 0 m n ( - 1 ) m ( n m ) f ( x + m h ) . f^{(n)}(x)=\lim_{h\to 0}\frac{(-1)^{n}}{h^{n}}\sum_{0\leq m\leq n}(-1)^{m}{n% \choose m}f(x+mh).
  10. 𝔻 q f ( x ) = lim h 0 ( - 1 ) q h q 0 m < ( - 1 ) m ( q m ) f ( x + m h ) . \mathbb{D}^{q}f(x)=\lim_{h\to 0}\frac{(-1)^{q}}{h^{q}}\sum_{0\leq m<\infty}(-1% )^{m}{q\choose m}f(x+mh).
  11. 𝔻 q f ( x ) = lim h 0 1 h q 0 m < ( - 1 ) m ( q m ) f ( x - m h ) . \mathbb{D}^{q}f(x)=\lim_{h\to 0}\frac{1}{h^{q}}\sum_{0\leq m<\infty}(-1)^{m}{q% \choose m}f(x-mh).

Gummel_plot.html

  1. I c I_{c}
  2. I b I_{b}
  3. V b e V_{be}
  4. V b c V_{bc}
  5. β \beta
  6. α \alpha
  7. n n
  8. β \beta

Gyrator.html

  1. v 2 = R i 1 v_{2}=Ri_{1}
  2. v 1 = - R i 2 v_{1}=-Ri_{2}
  3. R \scriptstyle{R}
  4. P = v 1 i 1 + v 2 i 2 = ( - R i 2 ) i 1 + ( R i 1 ) i 2 0 P=v_{1}i_{1}+v_{2}i_{2}=(-Ri_{2})i_{1}+(Ri_{1})i_{2}\equiv 0
  5. Z = [ 0 - R R 0 ] , Y = [ 0 G - G 0 ] , G = 1 R Z=\begin{bmatrix}0&-R\\ R&0\end{bmatrix},\quad Y=\begin{bmatrix}0&G\\ -G&0\end{bmatrix},\quad G=\frac{1}{R}
  6. S = [ 0 - 1 1 0 ] S=\begin{bmatrix}0&-1\\ 1&0\end{bmatrix}
  7. Z in = R 2 Z load \ Z_{\mathrm{in}}=\frac{R^{2}}{Z_{\mathrm{load}}}
  8. Y = [ 0 G 1 - G 2 0 ] Y=\begin{bmatrix}0&G_{1}\\ -G_{2}&0\end{bmatrix}
  9. R 1 \scriptstyle{R_{1}}
  10. R 2 \scriptstyle{R_{2}}
  11. R 1 : R 2 \scriptstyle{R_{1}:R_{2}}
  12. R 1 R 3 R 2 \scriptstyle{\frac{R_{1}R_{3}}{R_{2}}}
  13. Z = R L + j ω L Z=R_{\mathrm{L}}+j\omega L\,\!
  14. Z in = ( R L + j ω R L R C ) ( R + 1 j ω C ) Z_{\mathrm{in}}=\left(R_{\mathrm{L}}+j\omega R_{\mathrm{L}}RC\right)\|\left(R+% {1\over{j\omega C}}\right)
  15. Z in = R L + j ω R L R C Z_{\mathrm{in}}=R_{\mathrm{L}}+j\omega R_{\mathrm{L}}RC\,\!

H-alpha.html

  1. 3 2 \scriptstyle 3\rightarrow 2
  2. Z = 1 Z=1

H-theorem.html

  1. H ( t ) = 0 f ( E , t ) [ log ( f ( E , t ) E ) - 1 ] d E H(t)=\int_{0}^{\infty}f(E,t)\left[\log\left(\frac{f(E,t)}{\sqrt{E}}\right)-1% \right]\,dE
  2. δ q 1 δ p r \delta q_{1}...\delta p_{r}
  3. δ n = f ( q 1 p r , t ) δ q 1 δ p r . \delta n=f(q_{1}...p_{r},t)\delta q_{1}...\delta p_{r}.\,
  4. H = i f i ln f i δ q 1 δ p r H=\sum_{i}f_{i}\ln f_{i}\,\delta q_{1}\cdots\delta p_{r}
  5. H = f ln f d q 1 d p r H=\int\cdots\int f\ln f\,dq_{1}\cdots dp_{r}
  6. H \displaystyle H
  7. H = - ln P + constant H=-\ln P+\,\text{constant}\,
  8. H = - ln G + constant H=-\ln G+\,\text{constant}\,
  9. H = def P ( ln P ) d 3 v = ln P \displaystyle H\ \stackrel{\mathrm{def}}{=}\ \int{P({\ln P})\,d^{3}v}=\left% \langle\ln P\right\rangle
  10. S = def - N k H S\ \stackrel{\mathrm{def}}{=}\ -NkH
  11. H = i p i ln p i , H=\sum_{i}p_{i}\ln p_{i},\,
  12. S = - k i p i ln p i S=-k\sum_{i}p_{i}\ln p_{i}\;
  13. d S d t = - k i ( d p i d t ln p i + d p i d t ) = - k i d p i d t ln p i \begin{aligned}\displaystyle\frac{dS}{dt}&\displaystyle=-k\sum_{i}\left(\frac{% dp_{i}}{dt}\ln p_{i}+\frac{dp_{i}}{dt}\right)\\ &\displaystyle=-k\sum_{i}\frac{dp_{i}}{dt}\ln p_{i}\\ \end{aligned}
  14. d p α d t = β ν α β ( p β - p α ) d p β d t = α ν α β ( p α - p β ) \begin{aligned}\displaystyle\frac{dp_{\alpha}}{dt}&\displaystyle=\sum_{\beta}% \nu_{\alpha\beta}(p_{\beta}-p_{\alpha})\\ \displaystyle\frac{dp_{\beta}}{dt}&\displaystyle=\sum_{\alpha}\nu_{\alpha\beta% }(p_{\alpha}-p_{\beta})\\ \end{aligned}
  15. d S d t = 1 2 k α , β ν α β ( ln p β - ln p α ) ( p β - p α ) . \frac{dS}{dt}=\frac{1}{2}k\sum_{\alpha,\beta}\nu_{\alpha\beta}(\ln p_{\beta}-% \ln p_{\alpha})(p_{\beta}-p_{\alpha}).
  16. Δ S 0 \Delta S\geq 0\,

Hadamard's_inequality.html

  1. | det ( N ) | i = 1 n v i , |\det(N)|\leq\prod_{i=1}^{n}\|v_{i}\|,
  2. | det ( N ) | B n n n / 2 . |\det(N)|\leq B^{n}n^{n/2}.
  3. | det ( N ) | n n / 2 . |\det(N)|\leq n^{n/2}.
  4. det ( P ) = det ( N ) 2 i = 1 n v i 2 = i = 1 n p i i . \det(P)=\det(N)^{2}\leq\prod_{i=1}^{n}\|v_{i}\|^{2}=\prod_{i=1}^{n}p_{ii}.
  5. | det M | 1 , |\det M|\leq 1,
  6. | det N | = ( i = 1 n v i ) | det M | i = 1 n v i . |\det N|=\bigg(\prod_{i=1}^{n}\|v_{i}\|\bigg)|\det M|\leq\prod_{i=1}^{n}\|v_{i% }\|.
  7. det P = i = 1 n λ i ( 1 n i = 1 n λ i ) n = ( 1 n tr P ) n = 1 n = 1 , \det P=\prod_{i=1}^{n}\lambda_{i}\leq\bigg({1\over n}\sum_{i=1}^{n}\lambda_{i}% \bigg)^{n}=\left({1\over n}\mathrm{tr}P\right)^{n}=1^{n}=1,
  8. | det M | = det P 1. |\det M|=\sqrt{\det P}\leq 1.

Hadamard_matrix.html

  1. H H T = n I n HH^{\mathrm{T}}=nI_{n}
  2. n \sqrt{n}
  3. det ( H ) = ± n n 2 , \operatorname{det}(H)=\pm n^{\frac{n}{2}},
  4. | det ( M ) | n n / 2 . |\operatorname{det}(M)|\leq n^{n/2}.
  5. [ H H H - H ] \begin{bmatrix}H&H\\ H&-H\end{bmatrix}
  6. H 1 = [ 1 ] , H_{1}=\begin{bmatrix}1\end{bmatrix},
  7. H 2 = [ 1 1 1 - 1 ] , H_{2}=\begin{bmatrix}1&1\\ 1&-1\end{bmatrix},
  8. H 2 k = [ H 2 k - 1 H 2 k - 1 H 2 k - 1 - H 2 k - 1 ] = H 2 H 2 k - 1 , H_{2^{k}}=\begin{bmatrix}H_{2^{k-1}}&H_{2^{k-1}}\\ H_{2^{k-1}}&-H_{2^{k-1}}\end{bmatrix}=H_{2}\otimes H_{2^{k-1}},
  9. 2 k N 2\leq k\in N
  10. \left.\otimes\right.
  11. { 1 , - 1 , × } { 0 , 1 , } \{1,-1,\times\}\mapsto\{0,1,\oplus\}
  12. F n F_{n}
  13. n × 2 n n\times 2^{n}
  14. F n F_{n}
  15. F 1 = [ 0 1 ] F_{1}=\begin{bmatrix}0&1\end{bmatrix}
  16. F n = [ 0 1 × 2 n - 1 1 1 × 2 n - 1 F n - 1 F n - 1 ] . F_{n}=\begin{bmatrix}0_{1\times 2^{n-1}}&1_{1\times 2^{n-1}}\\ F_{n-1}&F_{n-1}\end{bmatrix}.
  17. H 2 n = F n T F n . H_{2^{n}}=F_{n}^{\rm T}F_{n}.
  18. H 2 n H_{2^{n}}
  19. 2 n 2^{n}
  20. 2 n - 1 2^{n-1}
  21. F n . F_{n}.
  22. H 2 n H_{2^{n}}
  23. H n H_{n}
  24. H m H_{m}
  25. H n H m H_{n}\otimes H_{m}
  26. H T + H = 2 I . H^{\rm T}+H=2I.\,
  27. W W T = w I WW^{T}=wI

Hadwiger's_theorem.html

  1. v ( S ) + v ( T ) = v ( S T ) + v ( S T ) . v(S)+v(T)=v(S\cap T)+v(S\cup T)~{}.
  2. Vol n ( K + t B ) = j = 0 n ( n j ) W j ( K ) t j , \mathrm{Vol}_{n}(K+tB)=\sum_{j=0}^{n}{\left({{n}\atop{j}}\right)}W_{j}(K)t^{j}% ~{},
  3. W j ( t K ) = t n - j W j ( K ) , t 0 . W_{j}(tK)=t^{n-j}W_{j}(K)~{},\quad t\geq 0~{}.
  4. v ( S ) = j = 0 n c j W j ( S ) . v(S)=\sum_{j=0}^{n}c_{j}W_{j}(S)~{}.

Halbach_array.html

  1. π 2 \scriptstyle\frac{\pi}{2}
  2. π 2 \scriptstyle\frac{\pi}{2}
  3. sin ( x ) cos ( y ) \scriptstyle\sin(x)\cos(y)
  4. F ( x , y ) = F 0 e i k x e - k y F(x,y)=F_{0}e^{ikx}e^{-ky}
  5. F ( x , y ) \scriptstyle F(x,y)
  6. F x + i F y \scriptstyle F_{x}+iF_{y}
  7. F 0 \scriptstyle F_{0}
  8. k \scriptstyle k
  9. 2 π λ \scriptstyle\frac{2\pi}{\lambda}
  10. H y cos ( k x ) H_{y}\approx\cos(kx)
  11. M = M r [ cos ( k ϕ ) ρ ^ + sin ( k ϕ ) ϕ ^ ] M=M_{r}\left[\cos(k\phi)\hat{\rho}+\sin(k\phi)\hat{\phi}\right]
  12. H = M r ln ( R o R i ) y ^ H=M_{r}\ln\left(\frac{R_{o}}{R_{i}}\right)\hat{y}
  13. B = 4 3 B 0 ln ( R o R i ) B=\frac{4}{3}B_{0}\ln\left(\frac{R_{o}}{R_{i}}\right)

Ham_(disambiguation).html

  1. Ham ( M , ω ) \mathop{\rm Ham}(M,\omega)
  2. ( M , ω ) (M,\omega)

Hand_axe.html

  1. L a < 2.75 {\frac{L}{a}}<2.75
  2. 2.75 < L a < 3.75 2.75<{\frac{L}{a}}<3.75
  3. 3.75 < L a 3.75<{\frac{L}{a}}
  4. L m < 1.3 {\frac{L}{m}}<1.3
  5. 1 , 3 < L m < 1.6 1,3<{\frac{L}{m}}<1.6
  6. 1.6 < L m 1.6<{\frac{L}{m}}
  7. m e < 2.35 {\frac{m}{e}}<2.35
  8. m e > 2.35 {\frac{m}{e}}>2.35

Handicap_principle.html

  1. C L C_{L}
  2. C H C_{H}
  3. S L * S^{*}_{L}
  4. S H * S^{*}_{H}
  5. B L B_{L}
  6. B H B_{H}
  7. S L * S^{*}_{L}
  8. S H * S^{*}_{H}

Hankel_matrix.html

  1. [ a b c d e b c d e f c d e f g d e f g h e f g h i ] . \begin{bmatrix}a&b&c&d&e\\ b&c&d&e&f\\ c&d&e&f&g\\ d&e&f&g&h\\ e&f&g&h&i\\ \end{bmatrix}.
  2. A i , j = A i - 1 , j + 1 . A_{i,j}=A_{i-1,j+1}.
  3. ( A i , j ) i , j 1 (A_{i,j})_{i,j\geq 1}
  4. A i , j A_{i,j}
  5. i + j i+j
  6. { h n } n 0 \{h_{n}\}_{n\geq 0}
  7. { b n } n 0 \{b_{n}\}_{n\geq 0}
  8. h n = det ( b i + j - 2 ) 1 i , j n + 1 . h_{n}=\det(b_{i+j-2})_{1\leq i,j\leq n+1}.
  9. a i , j = b i + j - 2 a_{i,j}=b_{i+j-2}
  10. { b n } \{b_{n}\}
  11. c n = k = 0 n ( n k ) b k c_{n}=\sum_{k=0}^{n}{n\choose k}b_{k}
  12. { b n } \{b_{n}\}
  13. det ( b i + j - 2 ) 1 i , j n + 1 = det ( c i + j - 2 ) 1 i , j n + 1 . \det(b_{i+j-2})_{1\leq i,j\leq n+1}=\det(c_{i+j-2})_{1\leq i,j\leq n+1}.
  14. J n J_{n}
  15. n n
  16. 5 5
  17. J 5 = [ 1 1 1 1 1 ] . J_{5}=\begin{bmatrix}&&&&1\\ &&&1&\\ &&1&&\\ &1&&&\\ 1&&&&\\ \end{bmatrix}.
  18. H ( m , n ) H(m,n)
  19. m × n m\times n
  20. H ( m , n ) = T ( m , n ) J n H(m,n)=T(m,n)\,J_{n}
  21. T ( m , n ) T(m,n)
  22. m × n m\times n

Haplotype.html

  1. H = N N - 1 ( 1 - i x i 2 ) H=\frac{N}{N-1}(1-\sum_{i}x_{i}^{2})
  2. x i x_{i}
  3. N N

Happened-before.html

  1. \to\;
  2. a a\;
  3. b b\;
  4. a b a\to b\;
  5. a a\;
  6. b b\;
  7. a a\;
  8. b b\;
  9. a a\;
  10. a b a\to b\;
  11. a , b , c \forall a,b,c
  12. a b a\to b\;
  13. b c b\to c\;
  14. a c a\to c\;
  15. a , a a \forall a,a\nrightarrow a
  16. a , b , \forall a,b,
  17. a b b a a\to b\land b\to a
  18. a = b a=b
  19. a b a\to b\;
  20. b a b\nrightarrow a

Happy_number.html

  1. n = n 0 n=n_{0}
  2. n 1 n_{1}
  3. n 2 n_{2}
  4. n i + 1 n_{i+1}
  5. n i n_{i}
  6. n i = 1 n_{i}=1
  7. 9 2 m 9^{2}m
  8. 81 m 81m
  9. m = 4 m=4
  10. n 10 m - 1 > 81 m n\geq 10^{m-1}>81m
  11. [ 1 , 10 122 ] [1,10^{122}]
  12. × 10 7 5000 \times 10^{7}5000
  13. 1 2 + 7 2 + 4 2 + 2 2 + 6 2 + 2 2 + 4 2 + 7 2 + 1 2 = 176 1^{2}+7^{2}+4^{2}+2^{2}+6^{2}+2^{2}+4^{2}+7^{2}+1^{2}=176
  14. 2 42643801 - 1 2^{42643801}-1
  15. 100 2 100_{2}
  16. 123 5 = 1 5 2 + 2 5 + 3 = 38. 123_{5}=1\cdot 5^{2}+2\cdot 5+3=38.
  17. 1 b , 10 b , 100 b , 1000 b , 1_{b},10_{b},100_{b},1000_{b},...
  18. 1000 b 1000_{b}
  19. n < 1000 b n<1000_{b}
  20. 3 ( b - 1 ) 2 3(b-1)^{2}
  21. b 3 b^{3}
  22. b 5 b\geq 5
  23. 1000 b 1000_{b}
  24. 1000 b 1000_{b}
  25. 1000 2 1000_{2}
  26. 111 2 11 2 10 2 1 111_{2}\rightarrow 11_{2}\rightarrow 10_{2}\rightarrow 1
  27. 110 2 10 2 1 110_{2}\rightarrow 10_{2}\rightarrow 1
  28. 101 2 10 2 1 101_{2}\rightarrow 10_{2}\rightarrow 1
  29. 100 2 1. 100_{2}\rightarrow 1.
  30. 153 , 370 , 371 , 153,370,371,
  31. 407 407
  32. 133 55 250 133 55 250... 133\rightarrow 55\rightarrow 250\rightarrow 133\rightarrow 55\rightarrow 250...
  33. 217 352 160 217 352 160... 217\rightarrow 352\rightarrow 160\rightarrow 217\rightarrow 352\rightarrow 160% ...
  34. 1459 1459
  35. 919 919
  36. 136 136
  37. 244 244
  38. 1634 , 8208 1634,8208
  39. 9474 9474

Harmonic_divisor_number.html

  1. 4 1 1 + 1 2 + 1 3 + 1 6 = 2. \frac{4}{\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{6}}=2.
  2. 12 1 1 + 1 2 + 1 4 + 1 5 + 1 7 + 1 10 + 1 14 + 1 20 + 1 28 + 1 35 + 1 70 + 1 140 = 5 \frac{12}{\frac{1}{1}+\frac{1}{2}+\frac{1}{4}+\frac{1}{5}+\frac{1}{7}+\frac{1}% {10}+\frac{1}{14}+\frac{1}{20}+\frac{1}{28}+\frac{1}{35}+\frac{1}{70}+\frac{1}% {140}}=5

Harmonograph.html

  1. x ( t ) = A sin ( t f + p ) e - d t , x(t)=A\sin(tf+p)e^{-dt},\,\!
  2. f f
  3. p p
  4. A A
  5. d d
  6. t t
  7. x ( t ) = A 1 sin ( t f 1 + p 1 ) e - d 1 t + A 2 sin ( t f 2 + p 2 ) e - d 2 t . x(t)=A_{1}\sin(tf_{1}+p_{1})e^{-d_{1}t}+A_{2}\sin(tf_{2}+p_{2})e^{-d_{2}t}.\,\!
  8. x ( t ) = A 1 sin ( t f 1 + p 1 ) e - d 1 t + A 2 sin ( t f 2 + p 2 ) e - d 2 t , x(t)=A_{1}\sin(tf_{1}+p_{1})e^{-d_{1}t}+A_{2}\sin(tf_{2}+p_{2})e^{-d_{2}t},\,\!
  9. y ( t ) = A 3 sin ( t f 3 + p 3 ) e - d 3 t + A 4 sin ( t f 4 + p 4 ) e - d 4 t . y(t)=A_{3}\sin(tf_{3}+p_{3})e^{-d_{3}t}+A_{4}\sin(tf_{4}+p_{4})e^{-d_{4}t}.\,\!

Harris–Todaro_model.html

  1. w r < l e l u s w u \ w_{r}<\frac{l_{e}}{l_{us}}w_{u}
  2. w r > l e l u s w u \ w_{r}>\frac{l_{e}}{l_{us}}w_{u}
  3. w r = l e l u s w u \ w_{r}=\frac{l_{e}}{l_{us}}w_{u}

Hartley_oscillator.html

  1. f = 1 2 π L C f={1\over 2\pi\sqrt{LC}}\,
  2. L = L 1 + L 2 L=L_{1}+L_{2}\,
  3. L = L 1 + L 2 + k L 1 L 2 L=L_{1}+L_{2}+k\sqrt{L_{1}L_{2}}\,

Hartley_transform.html

  1. H ( ω ) = { f } ( ω ) = 1 2 π - f ( t ) cas ( ω t ) d t , H(\omega)=\left\{\mathcal{H}f\right\}(\omega)=\frac{1}{\sqrt{2\pi}}\int_{-% \infty}^{\infty}f(t)\,\mbox{cas}~{}(\omega t)\mathrm{d}t,
  2. ω \omega
  3. cas ( t ) = cos ( t ) + sin ( t ) = 2 sin ( t + π / 4 ) = 2 cos ( t - π / 4 ) \mbox{cas}~{}(t)=\cos(t)+\sin(t)=\sqrt{2}\sin(t+\pi/4)=\sqrt{2}\cos(t-\pi/4)\,
  4. f = { { f } } . f=\{\mathcal{H}\{\mathcal{H}f\}\}.
  5. 1 / 2 π {1}/{\sqrt{2\pi}}
  6. 1 / 2 π {1}/{2\pi}
  7. 1 / 2 π {1}/{2\pi}
  8. 2 π ν t 2\pi\nu t
  9. ω t \omega t
  10. 1 / 2 π {1}/{\sqrt{2\pi}}
  11. F ( ω ) = { f ( t ) } ( ω ) F(\omega)=\mathcal{F}\{f(t)\}(\omega)
  12. exp ( - i ω t ) = cos ( ω t ) - i sin ( ω t ) , \exp\left({-i\omega t}\right)=\cos(\omega t)-i\sin(\omega t),
  13. 1 / 2 π 1/\sqrt{2\pi}
  14. F ( ω ) = H ( ω ) + H ( - ω ) 2 - i H ( ω ) - H ( - ω ) 2 . F(\omega)=\frac{H(\omega)+H(-\omega)}{2}-i\frac{H(\omega)-H(-\omega)}{2}.
  15. { f } = { f } - { f } = { f ( 1 + i ) } \{\mathcal{H}f\}=\Re\{\mathcal{F}f\}-\Im\{\mathcal{F}f\}=\Re\{\mathcal{F}f% \cdot(1+i)\}
  16. \Re
  17. \Im
  18. x ( t ) x(t)
  19. y ( t ) y(t)
  20. X ( ω ) X(\omega)
  21. Y ( ω ) Y(\omega)
  22. z ( t ) = x * y z(t)=x*y
  23. Z ( ω ) = { ( x * y ) } = 2 π ( X ( ω ) [ Y ( ω ) + Y ( - ω ) ] + X ( - ω ) [ Y ( ω ) - Y ( - ω ) ] ) / 2. Z(\omega)=\{\mathcal{H}(x*y)\}=\sqrt{2\pi}\left(X(\omega)\left[Y(\omega)+Y(-% \omega)\right]+X(-\omega)\left[Y(\omega)-Y(-\omega)\right]\right)/2.
  24. cas ( t ) = 2 sin ( t + π / 4 ) \mbox{cas}~{}(t)=\sqrt{2}\sin(t+\pi/4)
  25. 2 cas ( a + b ) = cas ( a ) cas ( b ) + cas ( - a ) cas ( b ) + cas ( a ) cas ( - b ) - cas ( - a ) cas ( - b ) . 2\mbox{cas}~{}(a+b)=\mbox{cas}~{}(a)\mbox{cas}~{}(b)+\mbox{cas}~{}(-a)\mbox{% cas}~{}(b)+\mbox{cas}~{}(a)\mbox{cas}~{}(-b)-\mbox{cas}~{}(-a)\mbox{cas}~{}(-b% ).\,
  26. cas ( a + b ) = cos ( a ) cas ( b ) + sin ( a ) cas ( - b ) = cos ( b ) cas ( a ) + sin ( b ) cas ( - a ) \mbox{cas}~{}(a+b)=\cos(a)\mbox{cas}~{}(b)+\sin(a)\mbox{cas}~{}(-b)=\cos(b)% \mbox{cas}~{}(a)+\sin(b)\mbox{cas}~{}(-a)\,
  27. cas ( a ) = d d a cas ( a ) = cos ( a ) - sin ( a ) = cas ( - a ) . \mbox{cas}~{}^{\prime}(a)=\frac{\mbox{d}~{}}{\mbox{d}~{}a}\mbox{cas}~{}(a)=% \cos(a)-\sin(a)=\mbox{cas}~{}(-a).

Hartree–Fock_method.html

  1. E = - 1 / n 2 E=-1/n^{2}
  2. E = - 1 / ( n + d ) 2 E=-1/(n+d)^{2}
  3. F ^ [ { ϕ j } ] ( 1 ) = H ^ core ( 1 ) + j = 1 N / 2 [ 2 J ^ j ( 1 ) - K ^ j ( 1 ) ] \hat{F}[\{\phi_{j}\}](1)=\hat{H}^{\,\text{core}}(1)+\sum_{j=1}^{N/2}[2\hat{J}_% {j}(1)-\hat{K}_{j}(1)]
  4. F ^ [ { ϕ j } ] ( 1 ) \hat{F}[\{\phi_{j}\}](1)
  5. ϕ j \phi_{j}
  6. H ^ core ( 1 ) = - 1 2 1 2 - α Z α r 1 α \hat{H}^{\,\text{core}}(1)=-\frac{1}{2}\nabla^{2}_{1}-\sum_{\alpha}\frac{Z_{% \alpha}}{r_{1\alpha}}
  7. J ^ j ( 1 ) \hat{J}_{j}(1)
  8. K ^ j ( 1 ) \hat{K}_{j}(1)
  9. F ^ ( 1 ) ϕ i ( 1 ) = ϵ i ϕ i ( 1 ) \hat{F}(1)\phi_{i}(1)=\epsilon_{i}\phi_{i}(1)
  10. ϕ i ( 1 ) \phi_{i}\;(1)

Hasse_diagram.html

  1. \subseteq

Hasse_principle.html

  1. 𝔭 \mathfrak{p}
  2. 𝔭 \mathfrak{p}
  3. H 1 ( k , G ) s H 1 ( k s , G ) H^{1}(k,G)\rightarrow\prod_{s}H^{1}(k_{s},G)

Hausdorff_distance.html

  1. d H ( X , Y ) = max { sup x X inf y Y d ( x , y ) , sup y Y inf x X d ( x , y ) } , d_{\mathrm{H}}(X,Y)=\max\{\,\sup_{x\in X}\inf_{y\in Y}d(x,y),\,\sup_{y\in Y}% \inf_{x\in X}d(x,y)\,\}\mbox{,}~{}\!
  2. d H ( X , Y ) = inf { ϵ 0 ; X Y ϵ and Y X ϵ } d_{H}(X,Y)=\inf\{\epsilon\geq 0\,;\ X\subseteq Y_{\epsilon}\ \mbox{and}~{}\ Y% \subseteq X_{\epsilon}\}
  3. X ϵ := x X { z M ; d ( z , x ) ϵ } X_{\epsilon}:=\bigcup_{x\in X}\{z\in M\,;\ d(z,x)\leq\epsilon\}
  4. ϵ \epsilon
  5. X X
  6. ϵ \epsilon
  7. X X
  8. ϵ \epsilon
  9. X X
  10. d H ( X , Y ) = ϵ d_{H}(X,Y)=\epsilon
  11. X Y ϵ and Y X ϵ X\subseteq Y_{\epsilon}\ \mbox{and}~{}\ Y\subseteq X_{\epsilon}
  12. \mathbb{R}
  13. d d
  14. d ( x , y ) := | y - x | , x , y d(x,y):=|y-x|,\quad x,y\in\mathbb{R}
  15. X := ( 0 , 1 ] and Y := [ - 1 , 0 ) X:=(0,1]\quad\mbox{and}~{}\quad Y:=[-1,0)
  16. d H ( X , Y ) = 1 d_{H}(X,Y)=1
  17. X Y 1 X\nsubseteq Y_{1}
  18. Y 1 = [ - 2 , 1 ) Y_{1}=[-2,1)
  19. 1 X 1\in X
  20. d ( x , Y ) = inf { d ( x , y ) | y Y } d(x,Y)=\inf\{d(x,y)|y\in Y\}
  21. d ( X , Y ) = sup { d ( x , Y ) | x X } d(X,Y)=\sup\{d(x,Y)|x\in X\}
  22. X Y X\subseteq Y
  23. d H ( X , Y ) = max { d ( X , Y ) , d ( Y , X ) } . d_{\mathrm{H}}(X,Y)=\max\{d(X,Y),d(Y,X)\}\,.

Hausdorff_measure.html

  1. ( X , ρ ) (X,\rho)
  2. U X \scriptstyle U\subset X
  3. diam U \mathrm{diam}\;U
  4. diam U := sup { ρ ( x , y ) | x , y U } , diam := 0 \mathrm{diam}\;U:=\sup\{\rho(x,y)|x,y\in U\},\quad\mathrm{diam}\;\emptyset:=0
  5. S S
  6. X X
  7. δ > 0 \delta>0
  8. H δ d ( S ) = inf { i = 1 ( diam U i ) d : i = 1 U i S , diam U i < δ } . H^{d}_{\delta}(S)=\inf\Bigl\{\sum_{i=1}^{\infty}(\operatorname{diam}\;U_{i})^{% d}:\bigcup_{i=1}^{\infty}U_{i}\supseteq S,\,\operatorname{diam}\;U_{i}<\delta% \Bigr\}.
  9. S S
  10. U i X \scriptstyle U_{i}\subset X
  11. diam U i < δ \scriptstyle\operatorname{diam}\;U_{i}<\delta
  12. H δ d ( S ) \scriptstyle H^{d}_{\delta}(S)
  13. δ \delta
  14. δ \delta
  15. lim δ 0 H δ d ( S ) \scriptstyle\lim_{\delta\to 0}H^{d}_{\delta}(S)
  16. H d ( S ) := sup δ > 0 H δ d ( S ) = lim δ 0 H δ d ( S ) . H^{d}(S):=\sup_{\delta>0}H^{d}_{\delta}(S)=\lim_{\delta\to 0}H^{d}_{\delta}(S).
  17. H d ( S ) H^{d}(S)
  18. d d
  19. S S
  20. X X
  21. H d H^{d}
  22. H δ d ( S ) \scriptstyle H^{d}_{\delta}(S)
  23. X X
  24. λ d \lambda_{d}
  25. λ d ( E ) = 2 - d α d H d ( E ) \lambda_{d}(E)=2^{-d}\alpha_{d}H^{d}(E)\,
  26. α d = Γ ( 1 2 ) d Γ ( d 2 + 1 ) = π d / 2 Γ ( d 2 + 1 ) . \alpha_{d}=\frac{\Gamma(\frac{1}{2})^{d}}{\Gamma(\frac{d}{2}+1)}=\frac{\pi^{d/% 2}}{\Gamma(\frac{d}{2}+1)}.
  27. dim Haus ( S ) = inf { d 0 : H d ( S ) = 0 } = sup ( { d 0 : H d ( S ) = } { 0 } ) , \operatorname{dim}_{\mathrm{Haus}}(S)=\inf\{d\geq 0:H^{d}(S)=0\}=\sup\bigl(\{d% \geq 0:H^{d}(S)=\infty\}\cup\{0\}\bigr),
  28. inf = . \inf\emptyset=\infty.\,
  29. n \scriptstyle\mathbb{R}^{n}
  30. m m
  31. n \scriptstyle\mathbb{R}^{n}
  32. m < n m<n
  33. m m
  34. m m
  35. n \scriptstyle\mathbb{R}^{n}
  36. 2 - m α m 2^{-m}\alpha_{m}
  37. m m
  38. d d
  39. d d
  40. | U i | d |U_{i}|^{d}
  41. ϕ ( U i ) \phi(U_{i})
  42. ϕ \phi
  43. ϕ ( ) = 0 \phi(\emptyset)=0
  44. S S
  45. ϕ \phi
  46. ϕ \phi
  47. d d
  48. S S
  49. H d ( S ) = 0 H^{d}(S)=0
  50. H ϕ ( S ) ( 0 , ) \scriptstyle H^{\phi}(S)\in(0,\infty)
  51. ϕ . \phi.
  52. ϕ ( t ) = t 2 log log 1 t \scriptstyle\phi(t)=t^{2}\,\log\log\frac{1}{t}
  53. ϕ ( t ) = t 2 log 1 t log log log 1 t \scriptstyle\phi(t)=t^{2}\log\frac{1}{t}\log\log\log\frac{1}{t}
  54. σ \sigma
  55. n \scriptstyle\mathbb{R}^{n}
  56. n > 2 n>2
  57. n = 2 n=2

Haversine_formula.html

  1. haversin ( d r ) = haversin ( ϕ 2 - ϕ 1 ) + cos ( ϕ 1 ) cos ( ϕ 2 ) haversin ( λ 2 - λ 1 ) \operatorname{haversin}\left(\frac{d}{r}\right)=\operatorname{haversin}(\phi_{% 2}-\phi_{1})+\cos(\phi_{1})\cos(\phi_{2})\operatorname{haversin}(\lambda_{2}-% \lambda_{1})
  2. haversin ( θ ) = sin 2 ( θ 2 ) = 1 - cos ( θ ) 2 \operatorname{haversin}(\theta)=\sin^{2}\left(\frac{\theta}{2}\right)=\frac{1-% \cos(\theta)}{2}
  3. ϕ 1 , ϕ 2 \phi_{1},\phi_{2}
  4. λ 1 , λ 2 \lambda_{1},\lambda_{2}
  5. d = r haversin - 1 ( h ) = 2 r arcsin ( h ) d=r\operatorname{haversin}^{-1}(h)=2r\arcsin\left(\sqrt{h}\right)
  6. d = 2 r arcsin ( haversin ( ϕ 2 - ϕ 1 ) + cos ( ϕ 1 ) cos ( ϕ 2 ) haversin ( λ 2 - λ 1 ) ) d=2r\arcsin\left(\sqrt{\operatorname{haversin}(\phi_{2}-\phi_{1})+\cos(\phi_{1% })\cos(\phi_{2})\operatorname{haversin}(\lambda_{2}-\lambda_{1})}\right)
  7. = 2 r arcsin ( sin 2 ( ϕ 2 - ϕ 1 2 ) + cos ( ϕ 1 ) cos ( ϕ 2 ) sin 2 ( λ 2 - λ 1 2 ) ) =2r\arcsin\left(\sqrt{\sin^{2}\left(\frac{\phi_{2}-\phi_{1}}{2}\right)+\cos(% \phi_{1})\cos(\phi_{2})\sin^{2}\left(\frac{\lambda_{2}-\lambda_{1}}{2}\right)}\right)
  8. haversin ( c ) = haversin ( a - b ) + sin ( a ) sin ( b ) haversin ( C ) . \operatorname{haversin}(c)=\operatorname{haversin}(a-b)+\sin(a)\sin(b)\,% \operatorname{haversin}(C).
  9. cos ( c ) = cos ( a ) cos ( b ) + sin ( a ) sin ( b ) cos ( C ) . \cos(c)=\cos(a)\cos(b)+\sin(a)\sin(b)\cos(C).\,

Headstone.html

  1. α ω \alpha\omega
  2. χ ρ \chi\rho

Heaps'_law.html

  1. V R ( n ) = K n β V_{R}(n)=Kn^{\beta}

Heat_death_of_the_universe.html

  1. 10 10 56 10^{10^{56}}

Heat_sink.html

  1. q k q_{k}
  2. q k = - k A d T d x q_{k}=-kA\frac{dT}{dx}
  3. Q ˙ = m ˙ c p , i n ( T a i r , o u t - T a i r , i n ) \dot{Q}=\dot{m}c_{p,in}(T_{air,out}-T_{air,in})
  4. Q ˙ = T h s - T a i r , a v R h s \dot{Q}=\frac{T_{hs}-T_{air,av}}{R_{hs}}
  5. T a i r , a v = T a i r , i n + T a i r , o u t 2 T_{air,av}=\frac{T_{air,in}+T_{air,out}}{2}
  6. m ˙ \dot{m}
  7. η f = tanh ( m L c ) m L c \eta_{f}=\frac{\tanh(mL_{c})}{mL_{c}}
  8. m L c = 2 h f k t f L f mL_{c}=\sqrt{\frac{2h_{f}}{kt_{f}}}L_{f}
  9. R b R_{b}
  10. R f R_{f}
  11. R b R_{b}
  12. R b = t b k A b R_{b}=\frac{t_{b}}{kA_{b}}
  13. t b t_{b}
  14. k k
  15. A b A_{b}
  16. R f R_{f}
  17. R f = 1 n h f W f ( t f + 2 η f L f ) R_{f}=\frac{1}{nh_{f}W_{f}\left(t_{f}+2\eta_{f}L_{f}\right)}
  18. η f = tanh m L c m L c \eta_{f}=\frac{\tanh{mL_{c}}}{mL_{c}}
  19. m L c = 2 h f k t f L f mL_{c}=\sqrt{\frac{2h_{f}}{kt_{f}}}L_{f}
  20. D h = 4 A c h P c h D_{h}=\frac{4A_{ch}}{P_{ch}}
  21. R e = 4 G ˙ ρ n π D h μ Re=\frac{4\dot{G}\rho}{n\pi D_{h}\mu}
  22. f = ( 0.79 ln R e - 1.64 ) - 2 f=(0.79\ln Re-1.64)^{-2}
  23. N u = ( f / 8 ) ( R e - 1000 ) P r 1 + 12.7 ( f / 8 ) 0.5 ( P r 2 3 - 1 ) Nu=\frac{(f/8)(Re-1000)Pr}{1+12.7(f/8)^{0.5}(Pr^{\frac{2}{3}}-1)}
  24. h f = N u k a i r D h h_{f}=\frac{Nuk_{air}}{D_{h}}
  25. ρ = P a t m R a T i n \rho=\frac{P_{atm}}{R_{a}T_{in}}
  26. R h s R_{hs}
  27. R h s = R b + R f R_{hs}=R_{b}+R_{f}

Hebbian_theory.html

  1. w i j = x i x j \,w_{ij}=x_{i}x_{j}
  2. w i j w_{ij}
  3. j j
  4. i i
  5. x i x_{i}
  6. i i
  7. w i j w_{ij}
  8. i = j i=j
  9. w i j = 1 p k = 1 p x i k x j k w_{ij}=\frac{1}{p}\sum_{k=1}^{p}x_{i}^{k}x_{j}^{k}\,
  10. w i j w_{ij}
  11. j j
  12. i i
  13. p p
  14. x i k x_{i}^{k}
  15. k k
  16. i i
  17. w i j w_{ij}
  18. i = j i=j
  19. Δ w i = η x i y , \,\Delta w_{i}=\eta x_{i}y,
  20. i i
  21. w i w_{i}
  22. η \eta
  23. i i
  24. x i x_{i}
  25. y y
  26. y = j w j x j , \,y=\sum_{j}w_{j}x_{j},

Hecke_operator.html

  1. Δ ( q ) = q ( n = 1 ( 1 - q n ) ) 24 = n = 1 τ ( n ) q n , q = e 2 π i τ , \Delta(q)=q\left(\prod_{n=1}^{\infty}(1-q^{n})\right)^{24}=\sum_{n=1}^{\infty}% \tau(n)q^{n},\quad q=e^{2\pi i\tau},
  2. τ ( m n ) = τ ( m ) τ ( n ) for ( m , n ) = 1. \tau(mn)=\tau(m)\tau(n)\quad\,\text{ for }(m,n)=1.
  3. f ( Λ ) \sum f(\Lambda^{\prime})
  4. T m f ( z ) = m k - 1 ( a b c d ) Γ \ M m ( c z + d ) - k f ( a z + b c z + d ) , T_{m}f(z)=m^{k-1}\sum_{\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)\in\Gamma\backslash M_{m}}(cz+d)^{-k}f\left(\frac{% az+b}{cz+d}\right),
  5. T m f ( z ) = m k - 1 a , d > 0 , a d = m 1 d k b ( mod d ) f ( a z + b d ) , T_{m}f(z)=m^{k-1}\sum_{a,d>0,ad=m}\frac{1}{d^{k}}\sum_{b\;\;(\mathop{{\rm mod}% }d)}f\left(\frac{az+b}{d}\right),
  6. b n = r > 0 , r | ( m , n ) r k - 1 a m n / r 2 . b_{n}=\sum_{r>0,r|(m,n)}r^{k-1}a_{mn/r^{2}}.
  7. T m f = a m f , a m a n = r > 0 , r | ( m , n ) r k - 1 a m n / r 2 , m , n 1. T_{m}f=a_{m}f,\quad a_{m}a_{n}=\sum_{r>0,r|(m,n)}r^{k-1}a_{mn/r^{2}},\ m,n\geq 1.

Heegner_number.html

  1. d \sqrt{−}{d}
  2. n 2 - n + 41 , n^{2}-n+41,\,
  3. n n
  4. n 2 + n + 41 , n^{2}+n+41,\,
  5. n n
  6. n 2 + n + p n^{2}+n+p\,
  7. n = 0 , , p - 2 n=0,\dots,p-2
  8. 1 - 4 p 1-4p
  9. p - 1 p-1
  10. p 2 p^{2}
  11. p - 2 p-2
  12. 7 , 11 , 19 , 43 , 67 , 163 7,11,19,43,67,163
  13. 2 , 3 , 5 , 11 , 17 , 41 2,3,5,11,17,41
  14. e π 163 e^{\pi\sqrt{163}}
  15. e π 163 = 262 537 412 640 768 743.999 999 999 999 25... e^{\pi\sqrt{163}}=262\,537\,412\,640\,768\,743.999\,999\,999\,999\,25...
  16. 640 320 3 + 744. \approx 640\,320^{3}+744.
  17. j ( ( 1 + - d ) / 2 ) j((1+\sqrt{-d})/2)
  18. e π d - j ( ( 1 + - d ) / 2 ) + 744 e^{\pi\sqrt{d}}\approx-j((1+\sqrt{-d})/2)+744
  19. τ \tau
  20. | Cl ( 𝐐 ( τ ) ) | |\mbox{Cl}~{}(\mathbf{Q}(\tau))|
  21. 𝐐 ( τ ) \mathbf{Q}(\tau)
  22. 𝐐 ( τ ) \mathbf{Q}(\tau)
  23. q = exp ( 2 π i τ ) q=\exp(2\pi i\tau)
  24. j ( q ) = 1 q + 744 + 196 884 q + . j(q)=\frac{1}{q}+744+196\,884q+\cdots.
  25. c n c_{n}
  26. ln ( c n ) 4 π n + O ( ln ( n ) ) \ln(c_{n})\sim 4\pi\sqrt{n}+O(\ln(n))
  27. 200 000 n 200\,000^{n}
  28. q 1 / 200 000 q\ll 1/200\,000
  29. τ = ( 1 + - 163 ) / 2 \tau=(1+\sqrt{-163})/2
  30. q = - exp ( - π 163 ) q=-\exp(-\pi\sqrt{163})
  31. 1 q = - exp ( π 163 ) \frac{1}{q}=-\exp(\pi\sqrt{163})
  32. j ( ( 1 + - 163 ) / 2 ) = ( - 640 320 ) 3 j((1+\sqrt{-163})/2)=(-640\,320)^{3}
  33. ( - 640 320 ) 3 = - e π 163 + 744 + O ( e - π 163 ) . (-640\,320)^{3}=-e^{\pi\sqrt{163}}+744+O\left(e^{-\pi\sqrt{163}}\right).
  34. e π 163 = 640 320 3 + 744 + O ( e - π 163 ) e^{\pi\sqrt{163}}=640\,320^{3}+744+O\left(e^{-\pi\sqrt{163}}\right)
  35. - 196 884 / e π 163 196 884 / ( 640 320 3 + 744 ) - 0.000 000 000 000 75 -196\,884/e^{\pi\sqrt{163}}\approx 196\,884/(640\,320^{3}+744)\approx-0.000\,0% 00\,000\,000\,75
  36. e π 163 e^{\pi\sqrt{163}}
  37. 1 π = 12 640 320 3 / 2 k = 0 ( 6 k ) ! ( 163 3 344 418 k + 13 591 409 ) ( 3 k ) ! ( k ! ) 3 ( - 640 320 ) 3 k \frac{1}{\pi}=\frac{12}{640\,320^{3/2}}\sum_{k=0}^{\infty}\frac{(6k)!(163\cdot 3% \,344\,418k+13\,591\,409)}{(3k)!(k!)^{3}(-640\,320)^{3k}}
  38. j ( 1 + - 163 2 ) = - 640 320 3 j\big(\tfrac{1+\sqrt{-163}}{2}\big)=-640\,320^{3}
  39. e π 19 96 3 + 744 - 0.22 e π 43 960 3 + 744 - 0.000 22 e π 67 5 280 3 + 744 - 0.000 0013 e π 163 640 320 3 + 744 - 0.000 000 000 000 75 \begin{aligned}\displaystyle e^{\pi\sqrt{19}}&\displaystyle\approx 96^{3}+744-% 0.22\\ \displaystyle e^{\pi\sqrt{43}}&\displaystyle\approx 960^{3}+744-0.000\,22\\ \displaystyle e^{\pi\sqrt{67}}&\displaystyle\approx 5\,280^{3}+744-0.000\,0013% \\ \displaystyle e^{\pi\sqrt{163}}&\displaystyle\approx 640\,320^{3}+744-0.000\,0% 00\,000\,000\,75\end{aligned}
  40. e π 19 12 3 ( 3 2 - 1 ) 3 + 744 - 0.22 e π 43 12 3 ( 9 2 - 1 ) 3 + 744 - 0.000 22 e π 67 12 3 ( 21 2 - 1 ) 3 + 744 - 0.000 0013 e π 163 12 3 ( 231 2 - 1 ) 3 + 744 - 0.000 000 000 000 75 \begin{aligned}\displaystyle e^{\pi\sqrt{19}}&\displaystyle\approx 12^{3}(3^{2% }-1)^{3}+744-0.22\\ \displaystyle e^{\pi\sqrt{43}}&\displaystyle\approx 12^{3}(9^{2}-1)^{3}+744-0.% 000\,22\\ \displaystyle e^{\pi\sqrt{67}}&\displaystyle\approx 12^{3}(21^{2}-1)^{3}+744-0% .000\,0013\\ \displaystyle e^{\pi\sqrt{163}}&\displaystyle\approx 12^{3}(231^{2}-1)^{3}+744% -0.000\,000\,000\,000\,75\end{aligned}
  41. d < 19 d<19
  42. d = 19 d=19
  43. 12 3 ( n 2 - 1 ) 3 = ( 2 2 3 ( n - 1 ) ( n + 1 ) ) 3 12^{3}(n^{2}-1)^{3}=(2^{2}\cdot 3\cdot(n-1)\cdot(n+1))^{3}
  44. j ( ( 1 + - 19 ) / 2 ) = 96 3 = ( 2 5 3 ) 3 j ( ( 1 + - 43 ) / 2 ) = 960 3 = ( 2 6 3 5 ) 3 j ( ( 1 + - 67 ) / 2 ) = 5 280 3 = ( 2 5 3 5 11 ) 3 j ( ( 1 + - 163 ) / 2 ) = 640 320 3 = ( 2 6 3 5 23 29 ) 3 . \begin{aligned}\displaystyle j((1+\sqrt{-19})/2)&\displaystyle=96^{3}=(2^{5}% \cdot 3)^{3}\\ \displaystyle j((1+\sqrt{-43})/2)&\displaystyle=960^{3}=(2^{6}\cdot 3\cdot 5)^% {3}\\ \displaystyle j((1+\sqrt{-67})/2)&\displaystyle=5\,280^{3}=(2^{5}\cdot 3\cdot 5% \cdot 11)^{3}\\ \displaystyle j((1+\sqrt{-163})/2)&\displaystyle=640\,320^{3}=(2^{6}\cdot 3% \cdot 5\cdot 23\cdot 29)^{3}.\end{aligned}
  45. e π 19 x 24 - 24 ; x 3 - 2 x - 2 = 0 e π 43 x 24 - 24 ; x 3 - 2 x 2 - 2 = 0 e π 67 x 24 - 24 ; x 3 - 2 x 2 - 2 x - 2 = 0 e π 163 x 24 - 24 ; x 3 - 6 x 2 + 4 x - 2 = 0 \begin{aligned}\displaystyle e^{\pi\sqrt{19}}&\displaystyle\approx x^{24}-24;x% ^{3}-2x-2=0\\ \displaystyle e^{\pi\sqrt{43}}&\displaystyle\approx x^{24}-24;x^{3}-2x^{2}-2=0% \\ \displaystyle e^{\pi\sqrt{67}}&\displaystyle\approx x^{24}-24;x^{3}-2x^{2}-2x-% 2=0\\ \displaystyle e^{\pi\sqrt{163}}&\displaystyle\approx x^{24}-24;x^{3}-6x^{2}+4x% -2=0\end{aligned}
  46. e π 19 3 5 ( 3 - 2 ( - 3 + 1 3 19 ) ) - 2 - 12.000 06 e π 43 3 5 ( 9 - 2 ( - 39 + 7 3 43 ) ) - 2 - 12.000 000 061 e π 67 3 5 ( 21 - 2 ( - 219 + 31 3 67 ) ) - 2 - 12.000 000 000 36 e π 163 3 5 ( 231 - 2 ( - 26 679 + 2 413 3 163 ) ) - 2 - 12.000 000 000 000 000 21 \begin{aligned}\displaystyle e^{\pi\sqrt{19}}&\displaystyle\approx 3^{5}\left(% 3-\sqrt{2(-3+1\sqrt{3\cdot 19})}\right)^{-2}-12.000\,06\dots\\ \displaystyle e^{\pi\sqrt{43}}&\displaystyle\approx 3^{5}\left(9-\sqrt{2(-39+7% \sqrt{3\cdot 43})}\right)^{-2}-12.000\,000\,061\dots\\ \displaystyle e^{\pi\sqrt{67}}&\displaystyle\approx 3^{5}\left(21-\sqrt{2(-219% +31\sqrt{3\cdot 67})}\right)^{-2}-12.000\,000\,000\,36\dots\\ \displaystyle e^{\pi\sqrt{163}}&\displaystyle\approx 3^{5}\left(231-\sqrt{2(-2% 6\,679+2\,413\sqrt{3\cdot 163})}\right)^{-2}-12.000\,000\,000\,000\,000\,21% \dots\end{aligned}
  47. n = 3 , 9 , 21 , 231 n=3,9,21,231
  48. 2 6 3 ( - 3 2 + 3 19 1 2 ) = 96 2 2 6 3 ( - 39 2 + 3 43 7 2 ) = 960 2 2 6 3 ( - 219 2 + 3 67 31 2 ) = 5 280 2 2 6 3 ( - 26679 2 + 3 163 2413 2 ) = 640 320 2 \begin{aligned}&\displaystyle 2^{6}\cdot 3(-3^{2}+3\cdot 19\cdot 1^{2})=96^{2}% \\ &\displaystyle 2^{6}\cdot 3(-39^{2}+3\cdot 43\cdot 7^{2})=960^{2}\\ &\displaystyle 2^{6}\cdot 3(-219^{2}+3\cdot 67\cdot 31^{2})=5\,280^{2}\\ &\displaystyle 2^{6}\cdot 3(-26679^{2}+3\cdot 163\cdot 2413^{2})=640\,320^{2}% \end{aligned}
  49. e π 19 ( 5 x ) 3 - 6.000 010 e π 43 ( 5 x ) 3 - 6.000 000 010 e π 67 ( 5 x ) 3 - 6.000 000 000 061 e π 163 ( 5 x ) 3 - 6.000 000 000 000 000 034 \begin{aligned}\displaystyle e^{\pi\sqrt{19}}&\displaystyle\approx(5x)^{3}-6.0% 00\,010\dots\\ \displaystyle e^{\pi\sqrt{43}}&\displaystyle\approx(5x)^{3}-6.000\,000\,010% \dots\\ \displaystyle e^{\pi\sqrt{67}}&\displaystyle\approx(5x)^{3}-6.000\,000\,000\,0% 61\dots\\ \displaystyle e^{\pi\sqrt{163}}&\displaystyle\approx(5x)^{3}-6.000\,000\,000\,% 000\,000\,034\dots\end{aligned}
  50. 5 x 6 - 96 x 5 - 10 x 3 + 1 = 0 5 x 6 - 960 x 5 - 10 x 3 + 1 = 0 5 x 6 - 5 280 x 5 - 10 x 3 + 1 = 0 5 x 6 - 640 320 x 5 - 10 x 3 + 1 = 0 \begin{aligned}&\displaystyle 5x^{6}-96x^{5}-10x^{3}+1=0\\ &\displaystyle 5x^{6}-960x^{5}-10x^{3}+1=0\\ &\displaystyle 5x^{6}-5\,280x^{5}-10x^{3}+1=0\\ &\displaystyle 5x^{6}-640\,320x^{5}-10x^{3}+1=0\end{aligned}
  51. 5 \mathbb{Q}\sqrt{5}
  52. τ = ( 1 + - 163 ) / 2 \tau=(1+\sqrt{-163})/2
  53. e π 163 = ( e π i / 24 η ( τ ) η ( 2 τ ) ) 24 - 24.000 000 000 000 001 05 e π 163 = ( e π i / 12 η ( τ ) η ( 3 τ ) ) 12 - 12.000 000 000 000 000 21 e π 163 = ( e π i / 6 η ( τ ) η ( 5 τ ) ) 6 - 6.000 000 000 000 000 034 \begin{aligned}\displaystyle e^{\pi\sqrt{163}}&\displaystyle=\left(\frac{e^{% \pi i/24}\eta(\tau)}{\eta(2\tau)}\right)^{24}-24.000\,000\,000\,000\,001\,05% \dots\\ \displaystyle e^{\pi\sqrt{163}}&\displaystyle=\left(\frac{e^{\pi i/12}\eta(% \tau)}{\eta(3\tau)}\right)^{12}-12.000\,000\,000\,000\,000\,21\dots\\ \displaystyle e^{\pi\sqrt{163}}&\displaystyle=\left(\frac{e^{\pi i/6}\eta(\tau% )}{\eta(5\tau)}\right)^{6}-6.000\,000\,000\,000\,000\,034\dots\end{aligned}
  54. k 2 ( mod p ) k^{2}\;\;(\mathop{{\rm mod}}p)
  55. k = 0 , 1 , , ( p - 1 ) / 2 k=0,1,\dots,(p-1)/2
  56. ( p - k ) 2 k 2 ( mod p ) (p-k)^{2}\equiv k^{2}\;\;(\mathop{{\rm mod}}p)
  57. e π d , d Z * e^{\pi\sqrt{d}},d\in Z^{*}
  58. e π d - 744 3 \sqrt[3]{e^{\pi\sqrt{d}}-744}
  59. 196 884 / e π d 196\,884/e^{\pi\sqrt{d}}
  60. [ 0 , 1 s i z e = 120 % ] [0,1size=120\%]
  61. [ 0 , 0.5 s i z e = 120 % ] [0, 0.5size=120\%]

Heinrich_Lenz.html

  1. L L

Heinz_Hopf.html

  1. S 3 S 2 S^{3}\to S^{2}

Heisenberg_group.html

  1. ( 1 a c 0 1 b 0 0 1 ) \begin{pmatrix}1&a&c\\ 0&1&b\\ 0&0&1\\ \end{pmatrix}
  2. ( 1 a c 0 1 b 0 0 1 ) ( 1 a c 0 1 b 0 0 1 ) = ( 1 a + a c + c + a b 0 1 b + b 0 0 1 ) . \begin{pmatrix}1&a&c\\ 0&1&b\\ 0&0&1\\ \end{pmatrix}\begin{pmatrix}1&a^{\prime}&c^{\prime}\\ 0&1&b^{\prime}\\ 0&0&1\\ \end{pmatrix}=\begin{pmatrix}1&a+a^{\prime}&c+c^{\prime}+ab^{\prime}\\ 0&1&b+b^{\prime}\\ 0&0&1\\ \end{pmatrix}\,.
  3. ( 1 a c 0 1 b 0 0 1 ) - 1 = ( 1 - a a b - c 0 1 - b 0 0 1 ) . \begin{pmatrix}1&a&c\\ 0&1&b\\ 0&0&1\\ \end{pmatrix}^{-1}=\begin{pmatrix}1&-a&ab-c\\ 0&1&-b\\ 0&0&1\\ \end{pmatrix}\,.
  4. a , b , c a,b,c
  5. a , b , c a,b,c
  6. x = ( 1 1 0 0 1 0 0 0 1 ) , y = ( 1 0 0 0 1 1 0 0 1 ) x=\begin{pmatrix}1&1&0\\ 0&1&0\\ 0&0&1\\ \end{pmatrix},\ \ y=\begin{pmatrix}1&0&0\\ 0&1&1\\ 0&0&1\\ \end{pmatrix}
  7. z = x y x - 1 y - 1 , x z = z x , y z = z y z=xyx^{-1}y^{-1},\ xz=zx,\ yz=zy
  8. z = ( 1 0 1 0 1 0 0 0 1 ) z=\begin{pmatrix}1&0&1\\ 0&1&0\\ 0&0&1\\ \end{pmatrix}
  9. ( 1 a c 0 1 b 0 0 1 ) = y b z c x a . \begin{pmatrix}1&a&c\\ 0&1&b\\ 0&0&1\\ \end{pmatrix}=y^{b}z^{c}x^{a}\,.
  10. z = x y x - 1 y - 1 , x p = y p = z p = 1 , x z = z x , y z = z y . z=xyx^{-1}y^{-1},\ x^{p}=y^{p}=z^{p}=1,\ xz=zx,\ yz=zy.
  11. x = ( 1 1 0 0 1 0 0 0 1 ) , y = ( 1 0 0 0 1 1 0 0 1 ) x=\begin{pmatrix}1&1&0\\ 0&1&0\\ 0&0&1\\ \end{pmatrix},\ \ y=\begin{pmatrix}1&0&0\\ 0&1&1\\ 0&0&1\\ \end{pmatrix}
  12. x y = ( 1 1 1 0 1 1 0 0 1 ) , xy=\begin{pmatrix}1&1&1\\ 0&1&1\\ 0&0&1\\ \end{pmatrix},
  13. y x = ( 1 1 0 0 1 1 0 0 1 ) . yx=\begin{pmatrix}1&1&0\\ 0&1&1\\ 0&0&1\\ \end{pmatrix}.
  14. [ 1 𝐚 c 0 I n 𝐛 0 0 1 ] \begin{bmatrix}1&\mathbf{a}&c\\ 0&I_{n}&\mathbf{b}\\ 0&0&1\end{bmatrix}
  15. [ 1 𝐚 c 0 I n 𝐛 0 0 1 ] [ 1 𝐚 c 0 I n 𝐛 0 0 1 ] = [ 1 𝐚 + 𝐚 c + c + 𝐚 𝐛 0 I n 𝐛 + 𝐛 0 0 1 ] \begin{bmatrix}1&\mathbf{a}&c\\ 0&I_{n}&\mathbf{b}\\ 0&0&1\end{bmatrix}\cdot\begin{bmatrix}1&\mathbf{a}^{\prime}&c^{\prime}\\ 0&I_{n}&\mathbf{b}^{\prime}\\ 0&0&1\end{bmatrix}=\begin{bmatrix}1&\mathbf{a}+\mathbf{a}^{\prime}&c+c^{\prime% }+\mathbf{a}\cdot\mathbf{b}^{\prime}\\ 0&I_{n}&\mathbf{b}+\mathbf{b}^{\prime}\\ 0&0&1\end{bmatrix}
  16. [ 1 𝐚 c 0 I n 𝐛 0 0 1 ] [ 1 - 𝐚 - c + 𝐚 𝐛 0 I n - 𝐛 0 0 1 ] = [ 1 0 0 0 I n 0 0 0 1 ] . \begin{bmatrix}1&\mathbf{a}&c\\ 0&I_{n}&\mathbf{b}\\ 0&0&1\end{bmatrix}\cdot\begin{bmatrix}1&-\mathbf{a}&-c+\mathbf{a}\cdot\mathbf{% b}\\ 0&I_{n}&-\mathbf{b}\\ 0&0&1\end{bmatrix}=\begin{bmatrix}1&0&0\\ 0&I_{n}&0\\ 0&0&1\end{bmatrix}.
  17. [ 0 𝐚 c 0 0 n 𝐛 0 0 0 ] , \begin{bmatrix}0&\mathbf{a}&c\\ 0&0_{n}&\mathbf{b}\\ 0&0&0\end{bmatrix},
  18. exp [ 0 𝐚 c 0 0 n 𝐛 0 0 0 ] = k = 0 1 k ! [ 0 𝐚 c 0 0 n 𝐛 0 0 0 ] k = [ 1 𝐚 c + 1 2 𝐚 𝐛 0 I n 𝐛 0 0 1 ] . \exp\begin{bmatrix}0&\mathbf{a}&c\\ 0&0_{n}&\mathbf{b}\\ 0&0&0\end{bmatrix}=\sum_{k=0}^{\infty}\frac{1}{k!}\begin{bmatrix}0&\mathbf{a}&% c\\ 0&0_{n}&\mathbf{b}\\ 0&0&0\end{bmatrix}^{k}=\begin{bmatrix}1&\mathbf{a}&c+{1\over 2}\mathbf{a}\cdot% \mathbf{b}\\ 0&I_{n}&\mathbf{b}\\ 0&0&1\end{bmatrix}.
  19. p i = [ 0 e i T 0 0 0 n 0 0 0 0 ] , p_{i}=\begin{bmatrix}0&\operatorname{e}_{i}^{\mathrm{T}}&0\\ 0&0_{n}&0\\ 0&0&0\end{bmatrix},
  20. q j = [ 0 0 0 0 0 n e j 0 0 0 ] , q_{j}=\begin{bmatrix}0&0&0\\ 0&0_{n}&\operatorname{e}_{j}\\ 0&0&0\end{bmatrix},
  21. z = [ 0 0 1 0 0 n 0 0 0 0 ] , z=\begin{bmatrix}0&0&1\\ 0&0_{n}&0\\ 0&0&0\end{bmatrix},
  22. ( v , t ) ( v , t ) = ( v + v , t + t + 1 2 ω ( v , v ) ) . (v,t)\cdot(v^{\prime},t^{\prime})=\left(v+v^{\prime},t+t^{\prime}+\tfrac{1}{2}% \omega(v,v^{\prime})\right).
  23. 0 𝐑 H ( V ) V 0. 0\to\mathbf{R}\to H(V)\to V\to 0.
  24. v = q a 𝐞 a + p a 𝐟 a . v=q^{a}\mathbf{e}_{a}+p_{a}\mathbf{f}^{a}.
  25. v = q a 𝐞 a + p a 𝐟 a + t E v=q^{a}\mathbf{e}_{a}+p_{a}\mathbf{f}^{a}+tE
  26. ( p , q , t ) ( p , q , t ) = ( p + p , q + q , t + t + 1 2 ( p q - p q ) ) . (p,q,t)\cdot(p^{\prime},q^{\prime},t^{\prime})=\left(p+p^{\prime},q+q^{\prime}% ,t+t^{\prime}+\frac{1}{2}(pq^{\prime}-p^{\prime}q)\right).
  27. [ ( v 1 , t 1 ) , ( v 2 , t 2 ) ] = ω ( v 1 , v 2 ) \begin{bmatrix}(v_{1},t_{1}),(v_{2},t_{2})\end{bmatrix}=\omega(v_{1},v_{2})
  28. [ 𝐞 a , 𝐟 b ] = δ a b [\mathbf{e}_{a},\mathbf{f}^{b}]=\delta_{a}^{b}
  29. v = q a 𝐞 a + p a 𝐟 a + u E v=q^{a}\mathbf{e}_{a}+p_{a}\mathbf{f}^{a}+uE
  30. ( p , q , u ) ( p , q , u ) = ( p + p , q + q , u + u + p q ) . (p,q,u)\cdot(p^{\prime},q^{\prime},u^{\prime})=(p+p^{\prime},q+q^{\prime},u+u^% {\prime}+pq^{\prime}).
  31. v = q a 𝐞 a + p a 𝐟 a + u E v=q^{a}\mathbf{e}_{a}+p_{a}\mathbf{f}^{a}+uE
  32. [ 1 p u 0 I n q 0 0 1 ] \begin{bmatrix}1&p&u\\ 0&I_{n}&q\\ 0&0&1\end{bmatrix}
  33. u = t + 1 2 p q u=t+\tfrac{1}{2}pq
  34. u + u + p q - 1 2 ( p + p ) ( q + q ) u+u^{\prime}+pq^{\prime}-\tfrac{1}{2}(p+p^{\prime})(q+q^{\prime})
  35. = t + 1 2 p q + t + 1 2 p q + p q - 1 2 ( p + p ) ( q + q ) =t+\tfrac{1}{2}pq+t^{\prime}+\tfrac{1}{2}p^{\prime}q^{\prime}+pq^{\prime}-% \tfrac{1}{2}(p+p^{\prime})(q+q^{\prime})
  36. = t + t + 1 2 ( p q - p q ) =t+t^{\prime}+\tfrac{1}{2}(pq^{\prime}-p^{\prime}q)
  37. 𝔥 n \mathfrak{h}_{n}
  38. U ( 𝔥 n ) U(\mathfrak{h}_{n})
  39. 𝔥 n \mathfrak{h}_{n}
  40. z j p 1 k 1 p 2 k 2 p n k n q 1 1 q 2 2 q n n z^{j}p_{1}^{k_{1}}p_{2}^{k_{2}}\cdots p_{n}^{k_{n}}q_{1}^{\ell_{1}}q_{2}^{\ell% _{2}}\cdots q_{n}^{\ell_{n}}
  41. U ( 𝔥 n ) U(\mathfrak{h}_{n})
  42. j , k , c j k z j p 1 k 1 p 2 k 2 p n k n q 1 1 q 2 2 q n n \sum_{j,\vec{k},\vec{\ell}}c_{j\vec{k}\vec{\ell}}\,\,z^{j}p_{1}^{k_{1}}p_{2}^{% k_{2}}\cdots p_{n}^{k_{n}}q_{1}^{\ell_{1}}q_{2}^{\ell_{2}}\cdots q_{n}^{\ell_{% n}}
  43. p k p = p p k , q k q = q q k , p k q - q p k = δ k z , z p k - p k z = 0 , z q k - q k z = 0. p_{k}p_{\ell}=p_{\ell}p_{k},\quad q_{k}q_{\ell}=q_{\ell}q_{k},\quad p_{k}q_{% \ell}-q_{\ell}p_{k}=\delta_{k\ell}z,\quad zp_{k}-p_{k}z=0,\quad zq_{k}-q_{k}z=0.
  44. U ( 𝔥 n ) U(\mathfrak{h}_{n})
  45. P = k , c k x 1 k 1 x 2 k 2 x n k n x 1 1 x 2 2 x n n P=\sum_{\vec{k},\vec{\ell}}c_{\vec{k}\vec{\ell}}\,\,\partial_{x_{1}}^{k_{1}}% \partial_{x_{2}}^{k_{2}}\cdots\partial_{x_{n}}^{k_{n}}x_{1}^{\ell_{1}}x_{2}^{% \ell_{2}}\cdots x_{n}^{\ell_{n}}
  46. U ( 𝔥 n ) U(\mathfrak{h}_{n})
  47. z j p 1 k 1 p 2 k 2 p n k n q 1 1 q 2 2 q n n x 1 k 1 x 2 k 2 x n k n x 1 1 x 2 2 x n n . z^{j}p_{1}^{k_{1}}p_{2}^{k_{2}}\cdots p_{n}^{k_{n}}q_{1}^{\ell_{1}}q_{2}^{\ell% _{2}}\cdots q_{n}^{\ell_{n}}\,\mapsto\,\partial_{x_{1}}^{k_{1}}\partial_{x_{2}% }^{k_{2}}\cdots\partial_{x_{n}}^{k_{n}}x_{1}^{\ell_{1}}x_{2}^{\ell_{2}}\cdots x% _{n}^{\ell_{n}}.
  48. Θ p = d z - 1 2 ( x d y - y d x ) . \Theta_{p}=dz-\frac{1}{2}\left(xdy-ydx\right).
  49. Θ p : T p 𝐑 3 𝐑 \Theta_{p}:T_{p}\mathbf{R}^{3}\to\mathbf{R}
  50. H p = { v T p 𝐑 3 Θ p ( v ) = 0 } . H_{p}=\{v\in T_{p}\mathbf{R}^{3}\mid\Theta_{p}(v)=0\}.
  51. v = ( v 1 , v 2 , v 3 ) v=(v_{1},v_{2},v_{3})
  52. w = ( w 1 , w 2 , w 3 ) w=(w_{1},w_{2},w_{3})
  53. v , w = v 1 w 1 + v 2 w 2 . \langle v,w\rangle=v_{1}w_{1}+v_{2}w_{2}.
  54. X = x - 1 2 y z , X=\frac{\partial}{\partial x}-\frac{1}{2}y\frac{\partial}{\partial z},
  55. Y = y + 1 2 x z , Y=\frac{\partial}{\partial y}+\frac{1}{2}x\frac{\partial}{\partial z},
  56. Z = z , Z=\frac{\partial}{\partial z},
  57. γ ( t ) = ( x ( t ) , y ( t ) , z ( t ) ) \gamma(t)=(x(t),y(t),z(t))
  58. c ( t ) = ( x ( t ) , y ( t ) ) c(t)=(x(t),y(t))
  59. z ( t ) = 1 2 c x d y - y d x z(t)=\frac{1}{2}\int_{c}xdy-ydx

Helical_antenna.html

  1. R 140 ( C λ ) R\simeq 140\left(\frac{C}{\lambda}\right)
  2. D o 15 N C 2 S λ 3 D_{o}\simeq 15N\frac{C^{2}S}{\lambda^{3}}
  3. HPBW 52 λ 3 / 2 C N S degrees \,\text{HPBW}\simeq\frac{52\lambda^{3/2}}{C\sqrt{NS}}\,\,\text{degrees}
  4. FNBW 115 λ 3 / 2 C N S degrees \,\text{FNBW}\simeq\frac{115\lambda^{3/2}}{C\sqrt{NS}}\,\,\text{degrees}

Helioseismology.html

  1. f f
  2. f f
  3. f f
  4. l l
  5. m m
  6. n n
  7. n n
  8. l l
  9. m m
  10. ω n l m \omega_{nlm}
  11. ω n l \omega_{nl}
  12. m m
  13. δ ω n l m = ω n l m - ω n l \delta\omega_{nlm}=\omega_{nlm}-\omega_{nl}
  14. l l
  15. m m

Hellinger–Toeplitz_theorem.html

  1. | \langle\cdot|\cdot\rangle
  2. A x | y = x | A y \langle Ax|y\rangle=\langle x|Ay\rangle
  3. [ H f ] ( x ) = - 1 2 d 2 d x 2 f ( x ) + 1 2 x 2 f ( x ) . [Hf](x)=-\frac{1}{2}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}f(x)+\frac{1}{2}x^{2% }f(x).

Helmholtz's_theorems.html

  1. Γ = A ω n d A = c u d s \Gamma=\int_{A}\vec{\omega}\cdot\vec{n}dA=\oint_{c}\vec{u}\cdot d\vec{s}
  2. Γ \Gamma
  3. ω \vec{\omega}
  4. n \vec{n}
  5. u \vec{u}
  6. D Γ D t = 0 \frac{D\Gamma}{Dt}=0

Hemocytometer.html

  1. concentration of cells in original mixture = ( number of cells counted ( proportion of chamber counted ) ( volume of squares counted ) ) ( volume of diluted sample volume of original mixture in sample ) \mbox{concentration of cells in original mixture}~{}=\left(\frac{\mbox{number % of cells counted}~{}}{(\mbox{proportion of chamber counted}~{})(\mbox{volume % of squares counted}~{})}\right)\left(\frac{\mbox{volume of diluted sample}~{}}% {\mbox{volume of original mixture in sample}~{}}\right)

Hemorheology.html

  1. μ = ( 3 4 ) 10 - 3 P a s \mu=(3\sim 4)\cdot 10^{-3}\,Pa\cdot s
  2. ν = μ ρ = ( 3 4 ) 10 - 3 1.06 10 3 = ( 2.8 3.8 ) 10 - 6 m 2 s \nu=\frac{\mu}{\rho}=\frac{(3\sim 4)\cdot 10^{-3}}{1.06\cdot 10^{3}}=(2.8\sim 3% .8)\cdot 10^{-6}\,\frac{m^{2}}{s}
  3. τ = F A \tau=\frac{F}{A}
  4. γ = D H \gamma=\frac{D}{H}
  5. γ ˙ = V H \dot{\gamma}=\frac{V}{H}
  6. τ \tau
  7. γ \gamma
  8. ϕ \phi
  9. ϕ = 0 \phi=0
  10. ϕ \phi
  11. e i ω t e^{i\omega t}
  12. ω = 2 π f \omega=2\pi f
  13. f f
  14. τ * = τ e - i ϕ \tau^{*}=\tau e^{-i\phi}
  15. γ * = γ e - i π 2 \gamma^{*}=\gamma e^{-i\frac{\pi}{2}}
  16. γ ˙ * = γ ˙ e - i 0 \dot{\gamma}^{*}=\dot{\gamma}e^{-i0}
  17. τ * = τ - i τ ′′ \tau^{*}=\tau^{\prime}-i\tau^{\prime\prime}
  18. τ \tau^{\prime}
  19. τ ′′ \tau^{\prime\prime}
  20. η * \eta^{*}
  21. η * = τ * γ ˙ * = ( τ γ ˙ + i τ ′′ γ ˙ ) = η + i η ′′ \eta^{*}=\frac{\tau^{*}}{\dot{\gamma}^{*}}=(\frac{\tau^{\prime}}{\dot{\gamma}}% +i\frac{\tau^{\prime\prime}}{\dot{\gamma}})=\eta^{\prime}+i\eta^{\prime\prime}
  22. G = τ * γ * = ( τ ′′ γ + i τ γ ) G=\frac{\tau^{*}}{\gamma^{*}}=(\frac{\tau^{\prime\prime}}{\gamma}+i\frac{\tau^% {\prime}}{\gamma})
  23. G = G + i G ′′ G=G^{\prime}+iG^{\prime\prime}
  24. η * = η d a s h 1 + i ω ( η d a s h E s p r i n g ) = η - i η ′′ \eta^{*}=\frac{\eta_{dash}}{1+i\omega(\frac{\eta_{dash}}{E_{spring}})}=\eta^{% \prime}-i\eta^{\prime\prime}
  25. μ ( h , d ) \mu(h,d)
  26. S + γ [ D S D t - Δ V S - S ( Δ V ) T ] = μ ( h , d ) [ B + γ ( D B D t - Δ V B - B ( Δ V ) T ) ] - g A + C 1 ( g A - C 2 I μ ( h , d ) 2 ) S+\gamma\left[\frac{DS}{Dt}-\Delta V\cdot S-S\cdot{(\Delta V)}^{T}\right]=\mu(% h,d)\left[B+\gamma\left(\frac{DB}{Dt}-\Delta V\cdot B-B\cdot{(\Delta V)}^{T}% \right)\right]-gA+C_{1}\left(gA-\frac{C_{2}I}{\mu(h,d)^{2}}\right)
  27. γ \gamma
  28. S = μ B + g A S=\mu B+gA
  29. B = Δ V + ( Δ V ) T B=\Delta V+(\Delta V)^{T}
  30. T s ( t ) T_{s}(t)
  31. T s ( t ) = c H cos θ T_{s}(t)=cH\cos\theta
  32. θ {\theta}
  33. G = G + i G ′′ G=G^{\prime}+iG^{\prime\prime}
  34. G G^{\prime}
  35. G ′′ G^{\prime\prime}
  36. G = σ 0 ε 0 cos ϕ G^{\prime}=\frac{\sigma_{0}}{\varepsilon_{0}}\cos\phi
  37. G ′′ = σ 0 ε 0 sin ϕ G^{\prime\prime}=\frac{\sigma_{0}}{\varepsilon_{0}}\sin\phi
  38. σ 0 \sigma_{0}
  39. ε 0 \varepsilon_{0}
  40. ϕ \phi
  41. T s ( t ) T_{s}(t)
  42. T s ( t ) T_{s}(t)
  43. ϕ \phi
  44. ϕ = sin - 1 4 A π Δ T s Δ d \phi=\sin^{-1}\frac{4A}{\pi\Delta T_{s}\Delta d}
  45. G = Δ T s Δ d cos ϕ G^{\prime}=\frac{\Delta T_{s}}{\Delta d}\cos\phi
  46. G ′′ = Δ T s Δ d sin ϕ = 4 A π ω Δ d 2 G^{\prime\prime}=\frac{\Delta T_{s}}{\Delta d}\sin\phi=\frac{4A}{\pi\omega% \Delta d^{2}}
  47. γ ˙ > 700 s - 1 \dot{\gamma}>700\,s^{-1}
  48. μ a = μ 0 × ( 1 + k H ) \mu_{a}={{\mu_{0}}\times{(1+kH)}}
  49. τ 0.5 = a | γ | 0.5 + b 0.5 {\tau}^{0.5}={{a}{|\gamma|}^{0.5}+b^{0.5}}
  50. μ a = μ 0 ( 1 - 0.5 k H ) - 2 \mu_{a}={{\mu_{0}}{{(1-0.5kH)}^{-2}}}
  51. k = k 0 + k inf γ 0.5 r 1 + γ 0.5 r k={{k_{0}+k_{\inf}{\gamma^{0.5}}_{r}}\over{1+{\gamma^{0.5}}_{r}}}
  52. γ r = γ γ c \gamma_{r}={{\gamma}\over{\gamma_{c}}}

Hendecagon.html

  1. A = 11 4 a 2 cot π 11 9.36564 a 2 . A=\frac{11}{4}a^{2}\cot\frac{\pi}{11}\simeq 9.36564\,a^{2}.

Henderson–Hasselbalch_equation.html

  1. pH = p K a + log 10 ( [ A - ] [ HA ] ) \mathrm{pH}=\mathrm{p}K_{\mathrm{a}}+\log_{10}\left(\frac{[\mathrm{A}^{-}]}{[% \mathrm{HA}]}\right)
  2. H A A HAA
  3. A A⁻⁻
  4. p K a = - log 10 ( K a ) = - log 10 ( [ H 3 O + ] [ A - ] [ HA ] ) \mathrm{p}K_{\mathrm{a}}=-\log_{10}(K_{\mathrm{a}})=-\log_{10}\left(\frac{[% \mathrm{H}_{3}\mathrm{O}^{+}][\mathrm{A}^{-}]}{[\mathrm{HA}]}\right)
  5. HA + H 2 O A - + H 3 O + \mathrm{HA}+\mathrm{H}_{2}\mathrm{O}\rightleftharpoons\mathrm{A}^{-}+\mathrm{H% }_{3}\mathrm{O}^{+}
  6. A A⁻
  7. B + H + BH + \mathrm{B}+\mathrm{H}^{+}\rightleftharpoons\mathrm{BH}^{+}
  8. K b K_{\mathrm{b}}
  9. K b K_{\mathrm{b}}
  10. p K b = - log 10 ( K b ) = - log 10 ( [ OH - ] [ HA ] [ A - ] ) \mathrm{p}K_{\mathrm{b}}=-\log_{10}(K_{\mathrm{b}})=-\log_{10}\left(\frac{[% \mathrm{O}\mathrm{H}^{-}][\mathrm{HA}]}{[\mathrm{A}^{-}]}\right)
  11. pOH = p K b + log 10 ( [ BH + ] [ B ] ) \mathrm{pOH}=\mathrm{p}K_{\mathrm{b}}+\log_{10}\left(\frac{[\mathrm{BH}^{+}]}{% [\mathrm{B}]}\right)
  12. pH = p K a + log 10 ( [ B ] [ BH + ] ) \mathrm{pH}=\mathrm{p}K_{\mathrm{a}}+\log_{10}\left(\frac{[\mathrm{B}]}{[% \mathrm{BH}^{+}]}\right)
  13. K a = [ H + ] [ A - ] [ HA ] K_{\mathrm{a}}=\frac{[\mathrm{H}^{+}][\mathrm{A}^{-}]}{[\mathrm{HA}]}
  14. log 10 K a = log 10 ( [ H + ] [ A - ] [ HA ] ) \log_{10}K_{\mathrm{a}}=\log_{10}\left(\frac{[\mathrm{H}^{+}][\mathrm{A}^{-}]}% {[\mathrm{HA}]}\right)
  15. log 10 K a = log 10 [ H + ] + log 10 ( [ A - ] [ HA ] ) \log_{10}K_{\mathrm{a}}=\log_{10}[\mathrm{H}^{+}]+\log_{10}\left(\frac{[% \mathrm{A}^{-}]}{[\mathrm{HA}]}\right)
  16. log 10 [ H + ] \log_{10}[\mathrm{H}^{+}]
  17. - p K a = - pH + log 10 ( [ A - ] [ HA ] ) -\mathrm{p}K_{\mathrm{a}}=-\mathrm{pH}+\log_{10}\left(\frac{[\mathrm{A}^{-}]}{% [\mathrm{HA}]}\right)
  18. pH = p K a + log 10 ( [ A - ] [ HA ] ) \mathrm{pH}=\mathrm{p}K_{\mathrm{a}}+\log_{10}\left(\frac{[\mathrm{A}^{-}]}{[% \mathrm{HA}]}\right)
  19. [ A - ] / [ HA ] [\mathrm{A}^{-}]/[\mathrm{HA}]
  20. n A - / n H A n_{A^{-}}/n_{HA}
  21. α A - / α H A \alpha_{A^{-}}/\alpha_{HA}
  22. α A - + α H A = 1 \alpha_{A^{-}}+\alpha_{HA}=1
  23. pH = p K a H 2 CO 3 + log 10 ( [ HCO 3 - ] [ H 2 CO 3 ] ) , \mathrm{pH}=\mathrm{p}K_{\mathrm{a}~{}\mathrm{H}_{2}\mathrm{CO}_{3}}+\log_{10}% \left(\frac{[\mathrm{HCO}_{3}^{-}]}{[\mathrm{H}_{2}\mathrm{CO}_{3}]}\right),
  24. [ H 2 CO 3 ] = k H CO 2 × p CO 2 , [\mathrm{H}_{2}\mathrm{CO}_{3}]=k_{\rm H~{}CO_{2}}\,\times p_{\mathrm{CO}_{2}},
  25. pH = 6.1 + log 10 ( [ HCO 3 - ] 0.0307 × p CO 2 ) , \mathrm{pH}=6.1+\log_{10}\left(\frac{[\mathrm{HCO}_{3}^{-}]}{0.0307\times p_{% \mathrm{CO}_{2}}}\right),
  26. p H pH
  27. [ - F o r m u l a E r r o r - ] [-FormulaError-]

Henryk_Grossman.html

  1. k + a c c 100 + a v v 100 k+{a_{c}\cdot c\over 100}+{a_{v}\cdot v\over 100}
  2. c 0 c_{0}
  3. v 0 v_{0}
  4. r = 1 + a c 100 r=1+{a_{c}\over 100}
  5. w = 1 + a v 100 w=1+{a_{v}\over 100}
  6. c j = c o r j c_{j}=c_{o}\cdot r^{j}
  7. v j = v o w j v_{j}=v_{o}\cdot w^{j}
  8. S = k + c o r j a c 100 + v o w j a v 100 = s v o w j 100 S=k+{c_{o}\cdot r^{j}\cdot a_{c}\over 100}+{v_{o}\cdot w^{j}\cdot a_{v}\over 1% 00}={s\cdot v_{o}\cdot w^{j}\over 100}
  9. k = v o w j ( s - a v ) 100 - c o r j a c 100 k={v_{o}\cdot w^{j}(s-a_{v})\over 100}-{c_{o}\cdot r^{j}\cdot a_{c}\over 100}
  10. v o w j ( s - a v ) 100 > c o r j a c 100 {v_{o}\cdot w^{j}(s-a_{v})\over 100}>{c_{o}\cdot r^{j}\cdot a_{c}\over 100}
  11. v o w n ( s - a v ) 100 = c o r n a c 100 {v_{o}\cdot w^{n}(s-a_{v})\over 100}={c_{o}\cdot r^{n}\cdot a_{c}\over 100}
  12. ( r w ) n = s - a v Ω a c ({r\over w})^{n}={s-a_{v}\over\Omega\cdot a_{c}}
  13. l o g ( s - a v Ω a c ) l o g ( 100 + a c 100 + a v ) {log\left(\frac{s-a_{v}}{\Omega\cdot a_{c}}\right)}\over{log\left(\frac{100+a_% {c}}{100+a_{v}}\right)}

Henstock–Kurzweil_integral.html

  1. f ( x ) = 1 x sin ( 1 x 3 ) . f(x)=\frac{1}{x}\sin\left(\frac{1}{x^{3}}\right).
  2. a = u 0 < u 1 < < u n = b , t i [ u i - 1 , u i ] a=u_{0}<u_{1}<\cdots<u_{n}=b,\ \ t_{i}\in[u_{i-1},u_{i}]
  3. δ : [ a , b ] ( 0 , ) , \delta\colon[a,b]\to(0,\infty),\,
  4. δ \delta
  5. i t i - δ ( t i ) < u i - 1 t i u i < t i + δ ( t i ) . \forall i\ \ t_{i}-\delta(t_{i})<u_{i-1}\leq t_{i}\leq u_{i}<t_{i}+\delta(t_{i% }).
  6. f : [ a , b ] f\colon[a,b]\to\mathbb{R}
  7. P f = i = 1 n ( u i - u i - 1 ) f ( t i ) . \sum_{P}f=\sum_{i=1}^{n}(u_{i}-u_{i-1})f(t_{i}).
  8. f : [ a , b ] , f\colon[a,b]\to\mathbb{R},
  9. δ \delta
  10. δ \delta
  11. | P f - I | < ε . {\Big|}\sum_{P}f-I{\Big|}<\varepsilon.
  12. δ \delta
  13. δ \delta
  14. a b f ( x ) d x = a c f ( x ) d x + c b f ( x ) d x . \int_{a}^{b}f(x)\,dx=\int_{a}^{c}f(x)\,dx+\int_{c}^{b}f(x)\,dx.
  15. a b α f ( x ) + β g ( x ) d x = α a b f ( x ) d x + β a b g ( x ) d x . \int_{a}^{b}\alpha f(x)+\beta g(x)\,dx=\alpha\int_{a}^{b}f(x)\,dx+\beta\int_{a% }^{b}g(x)\,dx.
  16. a b f ( x ) d x = lim c b - a c f ( x ) d x \int_{a}^{b}f(x)\,dx=\lim_{c\to b^{-}}\int_{a}^{c}f(x)\,dx
  17. 0 1 sin ( 1 / x ) x d x \int_{0}^{1}\frac{\sin(1/x)}{x}\,dx
  18. a f ( x ) d x := lim b a b f ( x ) d x . \int_{a}^{\infty}f(x)\,dx:=\lim_{b\to\infty}\int_{a}^{b}f(x)\,dx.
  19. F ( x ) - F ( a ) = a x F ( t ) d t . F(x)-F(a)=\int_{a}^{x}F^{\prime}(t)\,dt.
  20. F ( x ) = a x f ( t ) d t , F(x)=\int_{a}^{x}f(t)\,dt,
  21. i u i - u i - 1 < δ ( t i ) \forall i\ \ u_{i}-u_{i-1}<\delta(t_{i})
  22. i [ u i - 1 , u i ] U δ ( t i ) ( t i ) \forall i\ \ [u_{i-1},u_{i}]\subset U_{\delta(t_{i})}(t_{i})
  23. U ε ( a ) U_{\varepsilon}(a)
  24. ε \varepsilon
  25. δ \delta
  26. t i [ u i - 1 , u i ] t_{i}\in[u_{i-1},u_{i}]

Heptagon.html

  1. A = 7 4 a 2 cot π 7 3.633912444 a 2 . A=\frac{7}{4}a^{2}\cot\frac{\pi}{7}\simeq 3.633912444a^{2}.
  2. A = 1 4 7 3 ( 35 + 2 196 ( 13 - 3 i 3 ) 3 + 2 196 ( 13 + 3 i 3 ) 3 ) a 2 . A=\frac{1}{4}\sqrt{\frac{7}{3}\left(35+2\sqrt[3]{196(13-3i\sqrt{3})}+2\sqrt[3]% {196(13+3i\sqrt{3})}\right)}a^{2}.
  3. i i
  4. 2 cos 2 π 7 1.247 \scriptstyle{2\cos{\tfrac{2\pi}{7}}\approx 1.247}
  5. 2 π / 7 {2π}/{7}
  6. \scriptstyle\angle{}
  7. \scriptstyle\angle{}
  8. B D = 1 2 B C \scriptstyle{BD={1\over 2}BC}
  9. S = 3 2 11 \scriptstyle{S=3\tfrac{2}{11}}
  10. R = 3 2 3 \scriptstyle{R=3\tfrac{2}{3}}
  11. S R = 2 sin π 7 1 - ( 4 11 ) 2 \scriptstyle{\tfrac{S}{R}=\ 2\sin{\tfrac{\pi}{7}}\approx 1-(\tfrac{4}{11})^{2}}

Heptagonal_number.html

  1. 5 n 2 - 3 n 2 \frac{5n^{2}-3n}{2}
  2. T n + T n 2 , T_{n}+T_{\lfloor\frac{n}{2}\rfloor},
  3. n = 1 2 n ( 5 n - 3 ) = 1 15 π 25 - 10 5 + 2 3 ln ( 5 ) + 1 + 5 3 ln ( 1 2 10 - 2 5 ) + 1 - 5 3 ln ( 1 2 10 + 2 5 ) \sum_{n=1}^{\infty}\frac{2}{n(5n-3)}=\frac{1}{15}{\pi}{\sqrt{25-10\sqrt{5}}}+% \frac{2}{3}\ln(5)+\frac{{1}+\sqrt{5}}{3}\ln\left(\frac{1}{2}\sqrt{10-2\sqrt{5}% }\right)+\frac{{1}-\sqrt{5}}{3}\ln\left(\frac{1}{2}\sqrt{10+2\sqrt{5}}\right)
  4. n = 40 x + 9 + 3 10 . n=\frac{\sqrt{40x+9}+3}{10}.
  5. x = 5 n 2 - 3 n 2 x=\frac{5n^{2}-3n}{2}
  6. 2 x = 5 n 2 - 3 n 2x=5n^{2}-3n
  7. 5 n 2 - 3 n - 2 x = 0 5n^{2}-3n-2x=0
  8. n = - ( - 3 ) ± ( - 3 ) 2 - ( 4 × 5 × - 2 x ) 2 × 5 n=\frac{-(-3)\pm\sqrt{(-3)^{2}-(4\times 5\times-2x)}}{2\times 5}
  9. n = 3 ± 9 - ( - 40 x ) 10 n=\frac{3\pm\sqrt{9-(-40x)}}{10}
  10. n = 3 ± 9 + 40 x 10 n=\frac{3\pm\sqrt{9+40x}}{10}
  11. n = ± 40 x + 9 + 3 10 , n=\frac{\pm\sqrt{40x+9}+3}{10},
  12. 2 x 2 = y 2 ( 5 y - 3 ) 2 ± 2 2x^{2}=y^{2}(5y-3)^{2}\pm 2

Heptagonal_pyramidal_number.html

  1. n ( n + 1 ) ( 5 n - 2 ) 6 . \frac{n(n+1)(5n-2)}{6}.

Hereditarily_finite_set.html

  1. k = 0 V k = V ω . \bigcup_{k=0}^{\infty}V_{k}=V_{\omega}.
  2. H 0 H_{\aleph_{0}}
  3. 0 \aleph_{0}
  4. f ( 2 a + 2 b + ) = { f ( a ) , f ( b ) , } f(2^{a}+2^{b}+\cdots)=\{f(a),f(b),\ldots\}

Hessenberg_matrix.html

  1. [ 1 4 2 3 3 4 1 7 0 2 3 4 0 0 1 3 ] \begin{bmatrix}1&4&2&3\\ 3&4&1&7\\ 0&2&3&4\\ 0&0&1&3\\ \end{bmatrix}
  2. [ 1 2 0 0 5 2 3 0 3 4 3 7 5 6 1 1 ] \begin{bmatrix}1&2&0&0\\ 5&2&3&0\\ 3&4&3&7\\ 5&6&1&1\\ \end{bmatrix}

Hessian_matrix.html

  1. f ( 𝐱 ) f(\mathbf{x})∈ℝ
  2. f f
  3. 𝐇 \mathbf{H}
  4. f f
  5. n × n n×n
  6. H = [ 2 f x 1 2 2 f x 1 x 2 2 f x 1 x n 2 f x 2 x 1 2 f x 2 2 2 f x 2 x n 2 f x n x 1 2 f x n x 2 2 f x n 2 ] . H=\begin{bmatrix}\dfrac{\partial^{2}f}{\partial x_{1}^{2}}&\dfrac{\partial^{2}% f}{\partial x_{1}\,\partial x_{2}}&\cdots&\dfrac{\partial^{2}f}{\partial x_{1}% \,\partial x_{n}}\\ \dfrac{\partial^{2}f}{\partial x_{2}\,\partial x_{1}}&\dfrac{\partial^{2}f}{% \partial x_{2}^{2}}&\cdots&\dfrac{\partial^{2}f}{\partial x_{2}\,\partial x_{n% }}\\ \vdots&\vdots&\ddots&\vdots\\ \dfrac{\partial^{2}f}{\partial x_{n}\,\partial x_{1}}&\dfrac{\partial^{2}f}{% \partial x_{n}\,\partial x_{2}}&\cdots&\dfrac{\partial^{2}f}{\partial x_{n}^{2% }}\end{bmatrix}.
  7. H i , j = 2 f x i x j . H_{i,j}=\frac{\partial^{2}f}{\partial x_{i}\partial x_{j}}.
  8. 𝐇 ( f ) ( 𝐱 ) = 𝐉 ( f ) ( 𝐱 ) \mathbf{H}(f)(\mathbf{x})=\mathbf{J}(∇f)(\mathbf{x})
  9. x i ( f x j ) = x j ( f x i ) . \frac{\partial}{\partial x_{i}}\left(\frac{\partial f}{\partial x_{j}}\right)=% \frac{\partial}{\partial x_{j}}\left(\frac{\partial f}{\partial x_{i}}\right).
  10. f f
  11. D D
  12. f f
  13. D D
  14. f f
  15. 𝐱 \mathbf{x}
  16. f f
  17. 𝐱 \mathbf{x}
  18. 𝐱 \mathbf{x}
  19. 𝐱 \mathbf{x}
  20. f f
  21. f f
  22. f f
  23. x x
  24. x x
  25. f f
  26. x x
  27. x x
  28. f f
  29. x x
  30. x x
  31. f f
  32. f f
  33. x x
  34. x x
  35. f f
  36. g g
  37. g ( 𝐱 ) = c g(\mathbf{x})=c
  38. H ( f , g ) = [ 0 g x 1 g x 2 g x n g x 1 2 f x 1 2 2 f x 1 x 2 2 f x 1 x n g x 2 2 f x 2 x 1 2 f x 2 2 2 f x 2 x n g x n 2 f x n x 1 2 f x n x 2 2 f x n 2 ] H(f,g)=\begin{bmatrix}0&\dfrac{\partial g}{\partial x_{1}}&\dfrac{\partial g}{% \partial x_{2}}&\cdots&\dfrac{\partial g}{\partial x_{n}}\\ \dfrac{\partial g}{\partial x_{1}}&\dfrac{\partial^{2}f}{\partial x_{1}^{2}}&% \dfrac{\partial^{2}f}{\partial x_{1}\,\partial x_{2}}&\cdots&\dfrac{\partial^{% 2}f}{\partial x_{1}\,\partial x_{n}}\\ \dfrac{\partial g}{\partial x_{2}}&\dfrac{\partial^{2}f}{\partial x_{2}\,% \partial x_{1}}&\dfrac{\partial^{2}f}{\partial x_{2}^{2}}&\cdots&\dfrac{% \partial^{2}f}{\partial x_{2}\,\partial x_{n}}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ \dfrac{\partial g}{\partial x_{n}}&\dfrac{\partial^{2}f}{\partial x_{n}\,% \partial x_{1}}&\dfrac{\partial^{2}f}{\partial x_{n}\,\partial x_{2}}&\cdots&% \dfrac{\partial^{2}f}{\partial x_{n}^{2}}\end{bmatrix}
  39. 𝐳 \mathbf{z}
  40. f f
  41. f ( x ) = ( f 1 ( x ) , f 2 ( x ) , , f m ( x ) ) , f(x)=(f_{1}(x),f_{2}(x),\dots,f_{m}(x)),
  42. n × n n×n
  43. m m
  44. 𝐟 \mathbf{f}
  45. H ( f ) = ( H ( f 1 ) , H ( f 2 ) , , H ( f m ) ) H(f)=(H(f_{1}),H(f_{2}),\dots,H(f_{m}))
  46. m m
  47. ( M , g ) (M,g)
  48. \nabla
  49. f : M f:M\to\mathbb{R}
  50. Hess ( f ) Γ ( T * M T * M ) \displaystyle\mbox{Hess}~{}(f)\in\Gamma(T^{*}M\otimes T^{*}M)
  51. Hess ( f ) := f = d f \mbox{Hess}~{}(f):=\nabla\nabla f=\nabla df
  52. { x i } \{x^{i}\}
  53. Hess ( f ) = i j f d x i d x j = ( 2 f x i x j - Γ i j k f x k ) d x i d x j \mbox{Hess}~{}(f)=\nabla_{i}\,\partial_{j}f\ dx^{i}\!\otimes\!dx^{j}=\left(% \frac{\partial^{2}f}{\partial x^{i}\partial x^{j}}-\Gamma_{ij}^{k}\frac{% \partial f}{\partial x^{k}}\right)dx^{i}\otimes dx^{j}
  54. Γ i j k \Gamma^{k}_{ij}
  55. Hess ( f ) ( X , Y ) = X grad f , Y \mbox{Hess}~{}(f)(X,Y)=\langle\nabla_{X}\mbox{grad}~{}f,Y\rangle
  56. Hess ( f ) ( X , Y ) = X ( Y f ) - d f ( X Y ) \mbox{Hess}~{}(f)(X,Y)=X(Yf)-df(\nabla_{X}Y)
  57. y = f ( 𝐱 + Δ 𝐱 ) f ( 𝐱 ) + f ( 𝐱 ) Δ 𝐱 + 1 2 Δ 𝐱 T 𝐇 ( 𝐱 ) Δ 𝐱 y=f(\mathbf{x}+\Delta\mathbf{x})\approx f(\mathbf{x})+\nabla f(\mathbf{x})% \Delta\mathbf{x}+\frac{1}{2}\Delta\mathbf{x}^{\mathrm{T}}\mathbf{H}(\mathbf{x}% )\Delta\mathbf{x}
  58. f ∇f
  59. 𝐇 ( 𝐯 ) \mathbf{H}(\mathbf{v})
  60. f ( 𝐱 + Δ 𝐱 ) = f ( 𝐱 ) + 𝐇 ( Δ 𝐱 ) + O ( Δ 𝐱 2 ) \nabla f(\mathbf{x}+\Delta\mathbf{x})=\nabla f(\mathbf{x})+\mathbf{H}(\Delta% \mathbf{x})+O(\|\Delta\mathbf{x}\|^{2})
  61. Δ 𝐱 = r 𝐯 Δ\mathbf{x}=r\mathbf{v}
  62. r r
  63. 𝐇 ( Δ 𝐱 ) = 𝐇 ( r 𝐯 ) = r 𝐇 ( 𝐯 ) = f ( 𝐱 + r 𝐯 ) - f ( 𝐱 ) + O ( r 2 ) , \mathbf{H}(\Delta\mathbf{x})=\mathbf{H}(r\mathbf{v})=r\mathbf{H}(\mathbf{v})=% \nabla f(\mathbf{x}+r\mathbf{v})-\nabla f(\mathbf{x})+O(r^{2}),
  64. 𝐇 ( 𝐯 ) = 1 r [ f ( 𝐱 + r 𝐯 ) - f ( 𝐱 ) ] + O ( r ) \mathbf{H}(\mathbf{v})=\frac{1}{r}\Bigl[\nabla f(\mathbf{x}+r\mathbf{v})-% \nabla f(\mathbf{x})\Bigr]+O(r)
  65. r r
  66. O O

Hexagonal_number.html

  1. h n = 2 n 2 - n = n ( 2 n - 1 ) = 2 n × ( 2 n - 1 ) 2 . h_{n}=2n^{2}-n=n(2n-1)={{2n}\times{(2n-1)}\over 2}.\,\!
  2. M p 2 p - 1 = M p ( M p + 1 ) / 2 = h ( M p + 1 ) / 2 = h 2 p - 1 M_{p}2^{p-1}=M_{p}(M_{p}+1)/2=h_{(M_{p}+1)/2}=h_{2^{p-1}}
  3. n = 8 x + 1 + 1 4 . n=\frac{\sqrt{8x+1}+1}{4}.
  4. h n = i = 0 n - 1 ( 4 i + 1 ) h_{n}=\sum_{i=0}^{n-1}{(4i+1)}

Hénon_map.html

  1. { x n + 1 = 1 - a x n 2 + y n y n + 1 = b x n . \begin{cases}x_{n+1}=1-ax_{n}^{2}+y_{n}\\ y_{n+1}=bx_{n}.\end{cases}
  2. x = 609 - 7 28 0.631354477 , x=\frac{\sqrt{609}-7}{28}\approx 0.631354477,
  3. y = 3 ( 609 - 7 ) 280 0.189406343. y=\frac{3\left(\sqrt{609}-7\right)}{280}\approx 0.189406343.
  4. ( x 1 , y 1 ) = ( x , 1 - a x 2 + y ) (x_{1},y_{1})=(x,1-ax^{2}+y)\,
  5. ( x 2 , y 2 ) = ( b x 1 , y 1 ) (x_{2},y_{2})=(bx_{1},y_{1})\,
  6. ( x 3 , y 3 ) = ( y 2 , x 2 ) (x_{3},y_{3})=(y_{2},x_{2})\,

Hierarchical_clustering.html

  1. O ( n 3 ) O(n^{3})
  2. O ( 2 n ) O(2^{n})
  3. O ( n 2 ) O(n^{2})
  4. 2 \scriptstyle\sqrt{2}
  5. a - b 2 = i ( a i - b i ) 2 \|a-b\|_{2}=\sqrt{\sum_{i}(a_{i}-b_{i})^{2}}
  6. a - b 2 2 = i ( a i - b i ) 2 \|a-b\|_{2}^{2}=\sum_{i}(a_{i}-b_{i})^{2}
  7. a - b 1 = i | a i - b i | \|a-b\|_{1}=\sum_{i}|a_{i}-b_{i}|
  8. a - b = max i | a i - b i | \|a-b\|_{\infty}=\max_{i}|a_{i}-b_{i}|
  9. ( a - b ) S - 1 ( a - b ) \sqrt{(a-b)^{\top}S^{-1}(a-b)}
  10. max { d ( a , b ) : a A , b B } . \max\,\{\,d(a,b):a\in A,\,b\in B\,\}.
  11. min { d ( a , b ) : a A , b B } . \min\,\{\,d(a,b):a\in A,\,b\in B\,\}.
  12. 1 | A | | B | a A b B d ( a , b ) . \frac{1}{|A||B|}\sum_{a\in A}\sum_{b\in B}d(a,b).
  13. || c s - c t || ||c_{s}-c_{t}||
  14. c s c_{s}
  15. c t c_{t}
  16. 2 n m i , j = 1 n , m a i - b j 2 - 1 n 2 i , j = 1 n a i - a j 2 - 1 m 2 i , j = 1 m b i - b j 2 \frac{2}{nm}\sum_{i,j=1}^{n,m}\|a_{i}-b_{j}\|_{2}-\frac{1}{n^{2}}\sum_{i,j=1}^% {n}\|a_{i}-a_{j}\|_{2}-\frac{1}{m^{2}}\sum_{i,j=1}^{m}\|b_{i}-b_{j}\|_{2}
  17. 𝒜 \mathcal{A}
  18. \mathcal{B}
  19. max { d ( x , y ) : x 𝒜 , y } . \max\{\,d(x,y):x\in\mathcal{A},\,y\in\mathcal{B}\,\}.
  20. min { d ( x , y ) : x 𝒜 , y } . \min\{\,d(x,y):x\in\mathcal{A},\,y\in\mathcal{B}\,\}.
  21. 1 | 𝒜 | | | x 𝒜 y d ( x , y ) . {1\over{|\mathcal{A}|\cdot|\mathcal{B}|}}\sum_{x\in\mathcal{A}}\sum_{y\in% \mathcal{B}}d(x,y).

Higman–Sims_group.html

  1. × 10 7 \times 10^{7}
  2. T 10 A ( τ ) T_{10A}(\tau)
  3. j 10 A ( τ ) = T 10 A ( τ ) + 4 = ( ( η ( τ ) η ( 5 τ ) η ( 2 τ ) η ( 10 τ ) ) 2 + 2 2 ( η ( 2 τ ) η ( 10 τ ) η ( τ ) η ( 5 τ ) ) 2 ) 2 = ( ( η ( τ ) η ( 2 τ ) η ( 5 τ ) η ( 10 τ ) ) + 5 ( η ( 5 τ ) η ( 10 τ ) η ( τ ) η ( 2 τ ) ) ) 2 - 4 = 1 q + 4 + 22 q + 56 q 2 + 177 q 3 + 352 q 4 + 870 q 5 + 1584 q 6 + \begin{aligned}\displaystyle j_{10A}(\tau)&\displaystyle=T_{10A}(\tau)+4\\ &\displaystyle=\Big(\big(\tfrac{\eta(\tau)\,\eta(5\tau)}{\eta(2\tau)\,\eta(10% \tau)}\big)^{2}+2^{2}\big(\tfrac{\eta(2\tau)\,\eta(10\tau)}{\eta(\tau)\,\eta(5% \tau)}\big)^{2}\Big)^{2}\\ &\displaystyle=\Big(\big(\tfrac{\eta(\tau)\,\eta(2\tau)}{\eta(5\tau)\,\eta(10% \tau)}\big)+5\big(\tfrac{\eta(5\tau)\,\eta(10\tau)}{\eta(\tau)\,\eta(2\tau)}% \big)\Big)^{2}-4\\ &\displaystyle=\frac{1}{q}+4+22q+56q^{2}+177q^{3}+352q^{4}+870q^{5}+1584q^{6}+% \dots\end{aligned}

Hilbert's_seventh_problem.html

  1. a b a^{b}
  2. a { 0 , 1 } a\not\in\{0,1\}
  3. b b
  4. a b a^{b}
  5. b ln α + ln β = 0 b\ln{\alpha}+\ln{\beta}=0

Hilbert's_sixteenth_problem.html

  1. n 2 - 3 n + 4 2 {n^{2}-3n+4\over 2}
  2. d x d t = P ( x , y ) , d y d t = Q ( x , y ) {dx\over dt}=P(x,y),\qquad{dy\over dt}=Q(x,y)

Hilbert_transform.html

  1. H ( u ) ( t ) = p.v. - u ( τ ) h ( t - τ ) d τ = 1 π p.v. - u ( τ ) t - τ d τ H(u)(t)=\,\text{p.v.}\int_{-\infty}^{\infty}u(\tau)h(t-\tau)\,d\tau=\frac{1}{% \pi}\ \,\text{p.v.}\int_{-\infty}^{\infty}\frac{u(\tau)}{t-\tau}\,d\tau
  2. H ( u ) ( t ) = - 1 π lim ε 0 ε u ( t + τ ) - u ( t - τ ) τ d τ . H(u)(t)=-\frac{1}{\pi}\lim_{\varepsilon\rightarrow 0}\int_{\varepsilon}^{% \infty}\frac{u(t+\tau)-u(t-\tau)}{\tau}\,d\tau.
  3. H ( H ( u ) ) ( t ) = - u ( t ) H(H(u))(t)=-u(t)
  4. u ^ ( t ) \widehat{u}(t)\,
  5. u ~ ( t ) \tilde{u}(t)
  6. ( H ( u ) ) ( ω ) = ( - i sgn ( ω ) ) ( u ) ( ω ) \mathcal{F}(H(u))(\omega)=(-i\,\operatorname{sgn}(\omega))\cdot\mathcal{F}(u)(\omega)
  7. \mathcal{F}
  8. \mathcal{F}
  9. σ H ( ω ) = { i = e + i π 2 , for ω < 0 0 , for ω = 0 - i = e - i π 2 , for ω > 0 \sigma_{H}(\omega)=\begin{cases}i=e^{+\frac{i\pi}{2}},&\mbox{for }~{}\omega<0% \\ 0,&\mbox{for }~{}\omega=0\\ -i=e^{-\frac{i\pi}{2}},&\mbox{for }~{}\omega>0\end{cases}
  10. ( σ H ( ω ) ) 2 = e ± i π = - 1 for ω 0 \big(\sigma_{H}(\omega)\big)^{2}=e^{\pm i\pi}=-1\qquad\,\text{for }\omega\neq 0
  11. u ( t ) u(t)\,
  12. H ( u ) ( t ) H(u)(t)
  13. sin ( t ) \sin(t)
  14. - cos ( t ) -\cos(t)
  15. cos ( t ) \cos(t)
  16. sin ( t ) \sin(t)\,
  17. exp ( i t ) \exp\left(it\right)
  18. - i exp ( i t ) -i\exp\left(it\right)
  19. exp ( - i t ) \exp\left(-it\right)
  20. i exp ( - i t ) i\exp\left(-it\right)
  21. 1 t 2 + 1 1\over t^{2}+1
  22. t t 2 + 1 t\over t^{2}+1
  23. e - t 2 e^{-t^{2}}
  24. 2 π - 1 / 2 F ( t ) 2\pi^{-1/2}F(t)
  25. sin ( t ) t \sin(t)\over t
  26. 1 - cos ( t ) t 1-\cos(t)\over t
  27. ( t ) \sqcap(t)
  28. 1 π log | t + 1 2 t - 1 2 | {1\over\pi}\log\left|{t+{1\over 2}\over t-{1\over 2}}\right|
  29. δ ( t ) \delta(t)\,
  30. 1 π t {1\over\pi t}
  31. χ [ a , b ] ( t ) \chi_{[a,b]}(t)\,
  32. 1 π ln | t - a t - b | {\frac{1}{\pi}\,\text{ln}\left|\frac{t-a}{t-b}\right|}
  33. - 1 π ϵ u ( t + τ ) - u ( t - τ ) τ d τ H ( u ) ( t ) -\frac{1}{\pi}\int_{\epsilon}^{\infty}\frac{u(t+\tau)-u(t-\tau)}{\tau}\,d\tau% \to H(u)(t)
  34. H u p C p u p \|Hu\|_{p}\leq C_{p}\|u\|_{p}
  35. C p = { tan π 2 p for 1 < p 2 cot π 2 p for 2 < p < C_{p}=\begin{cases}\tan\frac{\pi}{2p}&\,\text{for }1<p\leq 2\\ \cot\frac{\pi}{2p}&\,\text{for }2<p<\infty\end{cases}
  36. S R f = - R R f ^ ( ξ ) e 2 π i x ξ d ξ S_{R}f=\int_{-R}^{R}\hat{f}({\xi})e^{2\pi ix\xi}\,d\xi
  37. H ( H ( u ) ) = - u H(H(u))=-u
  38. H - 1 = - H H^{-1}=-H
  39. H ( d u d t ) = d d t H ( u ) H\left(\frac{du}{dt}\right)=\frac{d}{dt}H(u)
  40. H ( d k u d t k ) = d k d t k H ( u ) H\left(\frac{d^{k}u}{dt^{k}}\right)=\frac{d^{k}}{dt^{k}}H(u)
  41. h ( t ) = p.v. 1 π t h(t)=\,\text{p.v. }\frac{1}{\pi t}
  42. H ( u ) = h * u H(u)=h*u
  43. H ( u ) ( t ) = d d t ( 1 π ( u * log | | ) ( t ) ) H(u)(t)=\frac{d}{dt}\left(\frac{1}{\pi}(u*\log|\cdot|)(t)\right)
  44. H ( u * v ) = H ( u ) * v = u * H ( v ) H(u*v)=H(u)*v=u*H(v)
  45. h * ( u * v ) = ( h * u ) * v = u * ( h * v ) h*(u*v)=(h*u)*v=u*(h*v)
  46. 1 < 1 p + 1 r 1<\frac{1}{p}+\frac{1}{r}
  47. U g - 1 f ( x ) = ( c x + d ) - 1 f ( a x + b c x + d ) , g = ( a b c d ) \displaystyle{U_{g}^{-1}f(x)=(cx+d)^{-1}f\left({ax+b\over cx+d}\right),\,\,\,g% =\begin{pmatrix}a&b\\ c&d\end{pmatrix}}
  48. 𝒟 L p = lim n W n , p ( ) \mathcal{D}_{L^{p}}=\underset{n\to\infty}{\underset{\longleftarrow}{\lim}}W^{n% ,p}(\mathbb{R})
  49. 𝒟 L p \mathcal{D}_{L^{p}}
  50. 𝒟 L p \mathcal{D}_{L^{p}}^{\prime}
  51. u 𝒟 L p u\in\mathcal{D}^{\prime}_{L^{p}}
  52. H ( u ) 𝒟 L p H(u)\in\mathcal{D}^{\prime}_{L^{p}}
  53. H u , v = def u , - H v \langle Hu,v\rangle\overset{\mathrm{def}}{=}\langle u,-Hv\rangle
  54. v 𝒟 L p v\in\mathcal{D}_{L^{p}}
  55. H ( u ) ( t ) = p.v. - u ( τ ) { h ( t - τ ) - h 0 ( - τ ) } d τ H(u)(t)=\,\text{p.v.}\int_{-\infty}^{\infty}u(\tau)\left\{h(t-\tau)-h_{0}(-% \tau)\right\}\,d\tau
  56. h 0 ( x ) = { 0 if | x | < 1 1 π x otherwise h_{0}(x)=\begin{cases}0&\mathrm{if\ }|x|<1\\ \frac{1}{\pi x}&\mathrm{otherwise}\end{cases}
  57. F ( x ) = f ( x ) + i g ( x ) F(x)=f(x)+ig(x)
  58. u ( x + i y ) = u ( x , y ) = 1 π - f ( s ) y ( x - s ) 2 + y 2 d s u(x+iy)=u(x,y)=\frac{1}{\pi}\int_{-\infty}^{\infty}f(s)\frac{y}{(x-s)^{2}+y^{2% }}\,ds
  59. P ( x , y ) = 1 π y x 2 + y 2 P(x,y)=\frac{1}{\pi}\frac{y}{x^{2}+y^{2}}
  60. lim y v ( x + i y ) = 0 \lim_{y\to\infty}v(x+iy)=0
  61. Q ( x , y ) = 1 π x x 2 + y 2 Q(x,y)=\frac{1}{\pi}\frac{x}{x^{2}+y^{2}}
  62. v ( x , y ) = 1 π - f ( s ) x - s ( x - s ) 2 + y 2 d s v(x,y)=\frac{1}{\pi}\int_{-\infty}^{\infty}f(s)\frac{x-s}{(x-s)^{2}+y^{2}}\,ds
  63. i π z = P ( x , y ) + i Q ( x , y ) \frac{i}{\pi z}=P(x,y)+iQ(x,y)
  64. H ( f ) = lim y 0 Q ( - , y ) f H(f)=\lim_{y\to 0}Q(-,y)\star f
  65. - | F ( x + i y ) | 2 d x < K \int_{-\infty}^{\infty}|F(x+iy)|^{2}\,dx<K
  66. ( F ) ( x ) \mathcal{F}(F)(x)
  67. - | F ( x + i y ) | p d x < K \int_{-\infty}^{\infty}|F(x+iy)|^{p}\,dx<K
  68. F ( x ) = f ( x ) - i g ( x ) F(x)=f(x)-ig(x)
  69. - | g ( x ) | p 1 + x 2 d x < \int_{-\infty}^{\infty}\frac{|g(x)|^{p}}{1+x^{2}}\,dx<\infty
  70. F + ( x ) - F - ( x ) = f ( x ) F_{+}(x)-F_{-}(x)=f(x)
  71. f ( x ) = F + ( x ) - F - ( x ) f(x)=F_{+}(x)-F_{-}(x)
  72. H ( f ) ( x ) = 1 i ( F + ( x ) + F - ( x ) ) H(f)(x)=\frac{1}{i}(F_{+}(x)+F_{-}(x))
  73. f ~ ( x ) = 1 2 π p.v. 0 2 π f ( t ) cot ( x - t 2 ) d t \tilde{f}(x)=\frac{1}{2\pi}\,\text{ p.v.}\int_{0}^{2\pi}f(t)\cot\left(\frac{x-% t}{2}\right)\,dt
  74. cot ( x - t 2 ) \scriptstyle\cot\left(\frac{x-t}{2}\right)
  75. 1 2 cot ( x 2 ) = 1 x + n = 1 ( 1 x + 2 n π + 1 x - 2 n π ) \frac{1}{2}\cot\left(\frac{x}{2}\right)=\frac{1}{x}+\sum_{n=1}^{\infty}\left(% \frac{1}{x+2n\pi}+\frac{1}{x-2n\pi}\right)
  76. U f ( x ) = π - 1 2 ( x + i ) - 1 f ( C ( x ) ) \displaystyle{Uf(x)=\pi^{-\frac{1}{2}}(x+i)^{-1}f(C(x))}
  77. H ( f L P ( t ) f H P ( t ) ) = f L P ( t ) H ( f H P ( t ) ) H(f_{LP}(t)f_{HP}(t))=f_{LP}(t)H(f_{HP}(t))
  78. u ( t ) = u m ( t ) cos ( ω t + ϕ ) u(t)=u_{m}(t)\cdot\cos(\omega t+\phi)
  79. ω 2 π Hz, \frac{\omega}{2\pi}\,\text{ Hz,}
  80. H ( u ) ( t ) = u m ( t ) sin ( ω t + ϕ ) H(u)(t)=u_{m}(t)\cdot\sin(\omega t+\phi)
  81. u a ( t ) = u ( t ) + i H ( u ) ( t ) u_{a}(t)=u(t)+i\cdot H(u)(t)
  82. u a ( t ) u_{a}(t)
  83. = u m ( t ) cos ( ω t + ϕ ) + i u m ( t ) sin ( ω t + ϕ ) =u_{m}(t)\cdot\cos(\omega t+\phi)+i\cdot u_{m}(t)\cdot\sin(\omega t+\phi)
  84. = u m ( t ) [ cos ( ω t + ϕ ) + i sin ( ω t + ϕ ) ] =u_{m}(t)\cdot\left[\cos(\omega t+\phi)+i\cdot\sin(\omega t+\phi)\right]
  85. u ( t ) = A cos ( ω t + ϕ m ( t ) ) u(t)=A\cdot\cos(\omega t+\phi_{m}(t))
  86. ω + ϕ m ( t ) . \omega+\phi_{m}^{\prime}(t).
  87. ϕ m \phi_{m}^{\prime}
  88. H ( u ) ( t ) A sin ( ω t + ϕ m ( t ) ) H(u)(t)\approx A\cdot\sin(\omega t+\phi_{m}(t))
  89. u a ( t ) A e i ( ω t + ϕ m ( t ) ) u_{a}(t)\approx A\cdot e^{i(\omega t+\phi_{m}(t))}
  90. u m ( t ) = m ( t ) + i m ^ ( t ) u_{m}(t)=m(t)+i\cdot\widehat{m}(t)
  91. u a ( t ) = ( m ( t ) + i m ^ ( t ) ) e i ( ω t + ϕ ) u_{a}(t)=(m(t)+i\cdot\widehat{m}(t))\cdot e^{i(\omega t+\phi)}
  92. u ( t ) \displaystyle u(t)
  93. u [ n ] , u[n],
  94. U ( ω ) , U(\omega),
  95. u ^ [ n ] , \hat{u}[n],
  96. u ^ [ n ] \hat{u}[n]
  97. u ^ [ n ] = D T F T - 1 ( U ( ω ) ) * D T F T - 1 ( - i sgn ( ω ) ) = u [ n ] * 1 2 π - π π ( - i sgn ( ω ) ) e i ω n d ω = u [ n ] * 1 2 π [ - π 0 i e i ω n d ω - 0 π i e i ω n d ω ] h [ n ] , \begin{aligned}\displaystyle\hat{u}[n]&\displaystyle=\scriptstyle{DTFT}^{-1}% \displaystyle(U(\omega))\ *\ \scriptstyle{DTFT}^{-1}\displaystyle(-i\cdot% \operatorname{sgn}(\omega))\\ &\displaystyle=u[n]\ *\ \frac{1}{2\pi}\int_{-\pi}^{\pi}(-i\cdot\operatorname{% sgn}(\omega))\cdot e^{i\omega n}d\omega\\ &\displaystyle=u[n]\ *\ \underbrace{\frac{1}{2\pi}\left[\int_{-\pi}^{0}i\cdot e% ^{i\omega n}d\omega-\int_{0}^{\pi}i\cdot e^{i\omega n}d\omega\right]}_{h[n]},% \end{aligned}
  98. h [ n ] = def { 0 , for n even 2 π n for n odd , h[n]\ \stackrel{\mathrm{def}}{=}\ \begin{cases}0,&\mbox{for }~{}n\mbox{ even}% \\ \frac{2}{\pi n}&\mbox{for }~{}n\mbox{ odd}~{},\end{cases}
  99. u ^ [ n ] \hat{u}[n]
  100. u [ n ] , u[n],
  101. h ~ [ n ] , \tilde{h}[n],
  102. u ^ . \hat{u}.
  103. h ~ N [ n ] = def m = - h ~ [ n - m N ] , \tilde{h}_{N}[n]\ \stackrel{\,\text{def}}{=}\ \sum_{m=-\infty}^{\infty}\tilde{% h}[n-mN],
  104. ( h ~ ) . \left(\tilde{h}\right).

Hill_climbing.html

  1. f ( 𝐱 ) f(\mathbf{x})
  2. 𝐱 \mathbf{x}
  3. 𝐱 \mathbf{x}
  4. f ( 𝐱 ) f(\mathbf{x})
  5. 𝐱 \mathbf{x}
  6. f ( 𝐱 ) f(\mathbf{x})
  7. f ( 𝐱 ) f(\mathbf{x})
  8. 𝐱 \mathbf{x}
  9. 𝐱 \mathbf{x}
  10. f ( 𝐱 ) f(\mathbf{x})
  11. x m x_{m}
  12. x 0 x_{0}
  13. x m x_{m}
  14. x m x_{m}

History_of_Scania.html

  1. * *
  2. * *

Hodge_theory.html

  1. 0 Ω 0 ( M ) d 0 Ω 1 ( M ) d 1 d n - 1 Ω n ( M ) d n 0 0\rightarrow\Omega^{0}(M)\xrightarrow{d_{0}}\Omega^{1}(M)\xrightarrow{d_{1}}% \cdots\xrightarrow{d_{n-1}}\Omega^{n}(M)\xrightarrow{d_{n}}0
  2. H k ( M ) = ker d k im d k - 1 . H^{k}(M)=\frac{\ker d_{k}}{\mathrm{im}\,d_{k-1}}.
  3. M d α , β k + 1 d V = M α , δ β k d V \int_{M}\langle d\alpha,\beta\rangle_{k+1}\,dV=\int_{M}\langle\alpha,\delta% \beta\rangle_{k}\,dV
  4. , k \langle\ ,\ \rangle_{k}
  5. Δ k ( M ) = { α Ω k ( M ) Δ α = 0 } . \mathcal{H}_{\Delta}^{k}(M)=\{\alpha\in\Omega^{k}(M)\mid\Delta\alpha=0\}.
  6. d Δ k ( M ) = 0 d\mathcal{H}_{\Delta}^{k}(M)=0
  7. φ : Δ k ( M ) H k ( M ) \varphi:\mathcal{H}_{\Delta}^{k}(M)\rightarrow H^{k}(M)
  8. φ \varphi
  9. E 0 , E 1 , , E N E_{0},E_{1},\dots,E_{N}
  10. L i : Γ ( E i ) Γ ( E i + 1 ) L_{i}:\Gamma(E_{i})\rightarrow\Gamma(E_{i+1})
  11. Γ ( E 0 ) Γ ( E 1 ) Γ ( E N ) \Gamma(E_{0})\rightarrow\Gamma(E_{1})\rightarrow\cdots\rightarrow\Gamma(E_{N})
  12. = i Γ ( E i ) \mathcal{E}^{\bullet}=\bigoplus_{i}\Gamma(E_{i})
  13. L = L i : L=\bigoplus L_{i}:\mathcal{E}^{\bullet}\rightarrow\mathcal{E}^{\bullet}
  14. = { e Δ e = 0 } . \mathcal{H}=\{e\in\mathcal{E}^{\bullet}\mid\Delta e=0\}.
  15. H : H:\mathcal{E}^{\bullet}\rightarrow\mathcal{H}
  16. H ( E j ) ( E j ) H(E_{j})\cong\mathcal{H}(E_{j})
  17. H k ( V ) H^{k}(V)
  18. b k = dim H k ( V ) = p + q = k h p , q , b_{k}=\dim H^{k}(V)=\sum_{p+q=k}h^{p,q},\,
  19. h p , q = dim H p , q . h^{p,q}=\dim H^{p,q}.\,
  20. H q ( V , Ω p ) H^{q}(V,\Omega^{p})