wpmath0000009_5

Harmonic_wavelet_transform.html

  1. w ( 2 j t - k ) w(2^{j}t-k)\!
  2. w ( t ) = e i 4 π t - e i 2 π t i 2 π t . w(t)=\frac{e^{i4\pi t}-e^{i2\pi t}}{i2\pi t}.
  3. - w * ( 2 j t - k ) w ( 2 j t - k ) d t = 1 2 j δ j , j δ k , k \int_{-\infty}^{\infty}w^{*}(2^{j}t-k)\cdot w(2^{j^{\prime}}t-k^{\prime})\,dt=% \frac{1}{2^{j}}\delta_{j,j^{\prime}}\delta_{k,k^{\prime}}
  4. - w ( 2 j t - k ) w ( 2 j t - k ) d t = 0 \int_{-\infty}^{\infty}w(2^{j}t-k)\cdot w(2^{j^{\prime}}t-k^{\prime})\,dt=0
  5. δ \delta
  6. φ ( t - k ) \varphi(t-k)
  7. φ ( t ) = e i 2 π t - 1 i 2 π t . \varphi(t)=\frac{e^{i2\pi t}-1}{i2\pi t}.
  8. - φ * ( t - k ) φ ( t - k ) d t = δ k , k \int_{-\infty}^{\infty}\varphi^{*}(t-k)\cdot\varphi(t-k^{\prime})\,dt=\delta_{% k,k^{\prime}}
  9. - w * ( 2 j t - k ) φ ( t - k ) d t = 0 for j 0 \int_{-\infty}^{\infty}w^{*}(2^{j}t-k)\cdot\varphi(t-k^{\prime})\,dt=0\,\text{% for }j\geq 0
  10. - φ ( t - k ) φ ( t - k ) d t = 0 \int_{-\infty}^{\infty}\varphi(t-k)\cdot\varphi(t-k^{\prime})\,dt=0
  11. - w ( 2 j t - k ) φ ( t - k ) d t = 0 for j 0. \int_{-\infty}^{\infty}w(2^{j}t-k)\cdot\varphi(t-k^{\prime})\,dt=0\,\text{ for% }j\geq 0.
  12. f ( t ) f(t)
  13. f ( t ) = j = - k = - [ a j , k w ( 2 j t - k ) + a ~ j , k w * ( 2 j t - k ) ] , f(t)=\sum_{j=-\infty}^{\infty}\sum_{k=-\infty}^{\infty}\left[a_{j,k}w(2^{j}t-k% )+\tilde{a}_{j,k}w^{*}(2^{j}t-k)\right],
  14. f ( t ) = k = - [ a k φ ( t - k ) + a ~ k φ * ( t - k ) ] + j = 0 k = - [ a j , k w ( 2 j t - k ) + a ~ j , k w * ( 2 j t - k ) ] . f(t)=\sum_{k=-\infty}^{\infty}\left[a_{k}\varphi(t-k)+\tilde{a}_{k}\varphi^{*}% (t-k)\right]+\sum_{j=0}^{\infty}\sum_{k=-\infty}^{\infty}\left[a_{j,k}w(2^{j}t% -k)+\tilde{a}_{j,k}w^{*}(2^{j}t-k)\right].
  15. a j , k \displaystyle a_{j,k}
  16. a ~ j , k = a j , k * \tilde{a}_{j,k}=a_{j,k}^{*}
  17. a ~ k = a k * \tilde{a}_{k}=a_{k}^{*}
  18. j = - k = - 2 - j ( | a j , k | 2 + | a ~ j , k | 2 ) \displaystyle\sum_{j=-\infty}^{\infty}\sum_{k=-\infty}^{\infty}2^{-j}\left(|a_% {j,k}|^{2}+|\tilde{a}_{j,k}|^{2}\right)

Harmony_search.html

  1. 𝐱 \mathbf{x}
  2. 𝐱 1 , , 𝐱 h m s \mathbf{x}^{1},\ldots,\mathbf{x}^{hms}
  3. h m s hms
  4. 𝐇𝐌 = [ x 1 1 x n 1 | f ( 𝐱 1 ) | x 1 h m s x n h m s | f ( 𝐱 h m s ) ] . \mathbf{HM}=\begin{bmatrix}x^{1}_{1}&\cdots&x^{1}_{n}&|&f(\mathbf{x}^{1})\\ \vdots&\ddots&\vdots&|&\vdots\\ x^{hms}_{1}&\cdots&x^{hms}_{n}&|&f(\mathbf{x}^{hms})\\ \end{bmatrix}.
  5. 𝐱 \mathbf{x}^{\prime}
  6. x i x^{{}^{\prime}}_{i}
  7. h m c r hmcr
  8. h m c r hmcr
  9. x i x i i n t ( u ( 0 , 1 ) * h m s ) + 1 x^{{}^{\prime}}_{i}\leftarrow x^{int(u(0,1)*hms)+1}_{i}
  10. 1 - h m c r 1-hmcr
  11. p a r par
  12. p a r par
  13. x i x^{{}^{\prime}}_{i}
  14. x i x i + δ x^{{}^{\prime}}_{i}\leftarrow x^{{}^{\prime}}_{i}+\delta
  15. x i x i - δ x^{{}^{\prime}}_{i}\leftarrow x^{{}^{\prime}}_{i}-\delta
  16. x i x i + f w u ( - 1 , 1 ) x^{{}^{\prime}}_{i}\leftarrow x^{{}^{\prime}}_{i}+fw\cdot u(-1,1)
  17. 1 - p a r 1-par
  18. 𝐱 \mathbf{x}^{{}^{\prime}}
  19. 𝐱 W o r s t \mathbf{x}^{Worst}
  20. 𝐱 W o r s t \mathbf{x}^{Worst}
  21. 𝐱 \mathbf{x}^{\prime}
  22. h m s hms
  23. h m c r hmcr
  24. p a r par
  25. δ \delta
  26. f w fw

Harrod–Johnson_diagram.html

  1. Y 1 = F 1 ( K , L ) Y_{1}=F_{1}(K,L)\,
  2. Y s = F s ( K , L ) Y_{s}=F_{s}(K,L)\,
  3. p 1 D K [ F 1 ( K , L ) ] = r = p 2 D K [ F 2 ( K , L ) ] p_{1}D_{K}[F_{1}(K,L)]=r=p_{2}D_{K}[F_{2}(K,L)]\,
  4. p 1 D L [ F 1 ( K , L ) ] = w = p 2 D L [ F 2 ( K , L ) ] p_{1}D_{L}[F_{1}(K,L)]=w=p_{2}D_{L}[F_{2}(K,L)]\,
  5. ω \omega
  6. ω = w / r = p i D L [ F i ( K , L ) ] , p i D K [ F i ( K , L ) ] \omega=w/r=\frac{p_{i}D_{L}[F_{i}(K,L)],p_{i}D_{K}[F_{i}(K,L)]}{\,}
  7. i = { 1 , 2 } . i=\{1,2\}.
  8. k i = K i / L i k_{i}=K_{i}/L_{i}
  9. k i , k_{i},
  10. p 1 D K [ F 1 ( K , L ) ] = p 2 D K [ F 2 ( K , L ) ] p_{1}D_{K}[F_{1}(K,L)]=p_{2}D_{K}[F_{2}(K,L)]
  11. p 1 D L [ F 1 ( K , L ) ] = p 2 D L [ F 2 ( K , L ) ] p_{1}D_{L}[F_{1}(K,L)]=p_{2}D_{L}[F_{2}(K,L)]\,
  12. p 1 / P 2 , p_{1}/P_{2},

Hartman–Grobman_theorem.html

  1. u ( t ) n u(t)\in\mathbb{R}^{n}
  2. d u / d t = f ( u ) du/dt=f(u)
  3. f : n n f:\mathbb{R}^{n}\to\mathbb{R}^{n}
  4. u * n u^{*}\in\mathbb{R}^{n}
  5. f ( u * ) = 0 f(u^{*})=0
  6. A = [ f i / x j ] A=[\partial f_{i}/\partial x_{j}]
  7. f f
  8. u * u^{*}
  9. N N
  10. u * u^{*}
  11. h : N n h:N\to\mathbb{R}^{n}
  12. h ( u * ) = 0 h(u^{*})=0
  13. N N
  14. d u / d t = f ( u ) du/dt=f(u)
  15. U = h ( u ) U=h(u)
  16. d U / d t = A U dU/dt=AU
  17. f f
  18. h h
  19. A A
  20. d u / d t = f ( u , t ) du/dt=f(u,t)
  21. u = ( y , z ) u=(y,z)
  22. d y / d t = - 3 y + y z and d z / d t = z + y 2 . dy/dt=-3y+yz\quad\,\text{and}\quad dz/dt=z+y^{2}.
  23. u * = 0 u^{*}=0
  24. u = h - 1 ( U ) u=h^{-1}(U)
  25. U = ( Y , Z ) U=(Y,Z)
  26. y Y + Y Z + 1 42 Y 3 + 1 2 Y Z 2 y\approx Y+YZ+\dfrac{1}{42}Y^{3}+\dfrac{1}{2}YZ^{2}
  27. z Z - 1 7 Y 2 - 1 3 Y 2 Z z\approx Z-\dfrac{1}{7}Y^{2}-\dfrac{1}{3}Y^{2}Z
  28. u = ( y , z ) u=(y,z)
  29. U = ( Y , Z ) U=(Y,Z)
  30. d Y / d t = - 3 Y and d Z / d t = Z . dY/dt=-3Y\quad\,\text{and}\quad dZ/dt=Z.
  31. α \alpha

Hasegawa–Mima_equation.html

  1. 1 ω c i t 1 \frac{1}{\omega_{ci}}\frac{\partial}{\partial t}\ll 1
  2. ω c i \omega_{ci}
  3. n e Z n i n_{e}\approx Zn_{i}\,
  4. n = n 0 e e ϕ / T e . n=n_{0}e^{e\phi/T_{e}}.\,
  5. t ( 2 ϕ - ϕ ) - [ ( ϕ × 𝐳 ^ ) ] [ 2 ϕ - ln ( n 0 ω c i ) ] = 0. \frac{\partial}{\partial t}\left(\nabla^{2}\phi-\phi\right)-\left[\left(\nabla% \phi\times\mathbf{\hat{z}}\right)\cdot\nabla\right]\left[\nabla^{2}\phi-\ln% \left(\frac{n_{0}}{\omega_{ci}}\right)\right]=0.
  6. n t + n 𝐯 = 0. \frac{\partial n}{\partial t}+\nabla\cdot n\mathbf{v}=0.
  7. 𝐯 𝐄 = 𝐄 × 𝐁 c B 2 = - ϕ × 𝐳 ^ c B . \mathbf{v_{E}}=\frac{\mathbf{E}\times\mathbf{B}}{cB^{2}}=\frac{-\nabla\phi% \times\mathbf{\hat{z}}}{cB}.
  8. t ( 2 ψ ) - [ ( ψ × 𝐳 ^ ) ] 2 ψ = 0 \frac{\partial}{\partial t}\left(\nabla^{2}\psi\right)-\left[\left(\nabla\psi% \times\mathbf{\hat{z}}\right)\cdot\nabla\right]\nabla^{2}\psi=0
  9. 𝐯 = - ψ × 𝐳 ^ . \mathbf{v}=-\nabla\psi\times\mathbf{\hat{z}}.
  10. ϕ \phi^{\prime}
  11. t = ω c i t , ω c i = e Z B m i c . t^{\prime}=\omega_{ci}t,\ \ \ \ \ \ \ \ \ \ \ \ \omega_{ci}=\frac{eZB}{m_{i}c}.
  12. x = x ρ s , ρ s 2 T e m i ω c i 2 . x^{\prime}=\frac{x}{\rho_{s}},\ \ \ \ \ \ \ \ \ \ \ \ \rho_{s}^{2}\equiv\frac{% T_{e}}{m_{i}\omega_{ci}^{2}}.
  13. ϕ = e ϕ T e . \phi^{\prime}=\frac{e\phi}{T_{e}}.
  14. 1 ω c i t ( ρ s 2 2 e ϕ T e - e ϕ T e ) - [ ( ρ s e ϕ T e × 𝐳 ^ ) ρ s ] [ ρ s 2 2 e ϕ T e - ln ( n 0 ω c i ) ] = 0. \frac{1}{\omega_{ci}}\frac{\partial}{\partial t}\left(\rho_{s}^{2}\nabla^{2}% \frac{e\phi}{T_{e}}-\frac{e\phi}{T_{e}}\right)-\left[\left(\rho_{s}\nabla\frac% {e\phi}{T_{e}}\times\mathbf{\hat{z}}\right)\cdot\rho_{s}\nabla\right]\left[% \rho_{s}^{2}\nabla^{2}\frac{e\phi}{T_{e}}-\ln\left(\frac{n_{0}}{\omega_{ci}}% \right)\right]=0.
  15. [ A , B ] A x B y - A y B x . \left[A,B\right]\equiv\frac{\partial A}{\partial x}\frac{\partial B}{\partial y% }-\frac{\partial A}{\partial y}\frac{\partial B}{\partial x}.
  16. t ( 2 ϕ - ϕ ) + [ ϕ , 2 ϕ ] - [ ϕ , ln ( n 0 ω c i ) ] = 0. \frac{\partial}{\partial t}\left(\nabla^{2}\phi-\phi\right)+\left[\phi,\nabla^% {2}\phi\right]-\left[\phi,\ln\left(\frac{n_{0}}{\omega_{ci}}\right)\right]=0.
  17. ( ψ ) 2 d V = v x 2 + v y 2 d V . \int\left(\nabla\psi\right)^{2}dV=\int v_{x}^{2}+v_{y}^{2}\,dV.
  18. ( 2 ψ ) 2 d V = ( × 𝐯 ) 2 d V . \int\left(\nabla^{2}\psi\right)^{2}\,dV=\int\left(\nabla\times\mathbf{v}\right% )^{2}\,dV.
  19. [ ϕ 2 + ( ϕ ) 2 ] d V . \int\left[\phi^{2}+\left(\nabla\phi\right)^{2}\right]\,dV.
  20. [ ( ϕ ) 2 + ( 2 ϕ ) 2 ] d V . \int\left[\left(\nabla\phi\right)^{2}+\left(\nabla^{2}\phi\right)^{2}\right]\,dV.

Hat_operator.html

  1. 𝐲 ^ = H 𝐲 . \hat{\mathbf{y}}=H\mathbf{y}.
  2. 𝐚 × 𝐛 = 𝐚 ^ 𝐛 \mathbf{a}\times\mathbf{b}=\mathbf{\hat{a}}\mathbf{b}
  3. 𝐚 × 𝐛 = [ a x a y a z ] × [ b x b y b z ] = [ 0 - a z a y a z 0 - a x - a y a x 0 ] [ b x b y b z ] = 𝐚 ^ 𝐛 \mathbf{a}\times\mathbf{b}=\begin{bmatrix}a_{x}\\ a_{y}\\ a_{z}\end{bmatrix}\times\begin{bmatrix}b_{x}\\ b_{y}\\ b_{z}\end{bmatrix}=\begin{bmatrix}0&-a_{z}&a_{y}\\ a_{z}&0&-a_{x}\\ -a_{y}&a_{x}&0\end{bmatrix}\begin{bmatrix}b_{x}\\ b_{y}\\ b_{z}\end{bmatrix}=\mathbf{\hat{a}}\mathbf{b}

Hausdorff_density.html

  1. μ \mu
  2. a n a\in\mathbb{R}^{n}
  3. Θ * s ( μ , a ) = lim sup r 0 μ ( B r ( a ) ) r s \Theta^{*s}(\mu,a)=\limsup_{r\rightarrow 0}\frac{\mu(B_{r}(a))}{r^{s}}
  4. Θ * s ( μ , a ) = lim inf r 0 μ ( B r ( a ) ) r s \Theta_{*}^{s}(\mu,a)=\liminf_{r\rightarrow 0}\frac{\mu(B_{r}(a))}{r^{s}}
  5. B r ( a ) B_{r}(a)
  6. Θ * s ( μ , a ) Θ * s ( μ , a ) \Theta_{*}^{s}(\mu,a)\leq\Theta^{*s}(\mu,a)
  7. a n a\in\mathbb{R}^{n}
  8. μ \mu
  9. Θ s ( μ , a ) \Theta^{s}(\mu,a)
  10. μ \mu
  11. d \mathbb{R}^{d}
  12. Θ s ( μ , a ) \Theta^{s}(\mu,a)
  13. μ \mu
  14. μ \mu
  15. d \mathbb{R}^{d}
  16. 1 \geq 1
  17. Θ m ( μ , a ) \Theta^{m}(\mu,a)
  18. μ \mu
  19. μ \mu
  20. μ \mu
  21. μ H m \mu\ll H^{m}
  22. μ \mu
  23. H m H^{m}
  24. μ \mu

Hausner_ratio.html

  1. H = ρ T ρ B H=\frac{\rho_{T}}{\rho_{B}}
  2. ρ B \rho_{B}
  3. ρ T \rho_{T}
  4. H = 100 / ( 100 - C ) H=100/(100-C)

Head_(vessel).html

  1. r 1 = D o r_{1}=Do
  2. r 2 = 0.1 × D o r_{2}=0.1\times Do
  3. h 3.5 × t h\geq 3.5\times t
  4. h 2 h_{2}
  5. h 2 = 0.1935 × D o - 0.455 × t h_{2}=0.1935\times Do-0.455\times t
  6. r 1 = 0.8 × D o r_{1}=0.8\times Do
  7. r 2 = 0.154 × D o r_{2}=0.154\times Do
  8. h 3 × t h\geq 3\times t
  9. h 2 h_{2}
  10. h 2 = 0.255 × D o - 0.635 × t h_{2}=0.255\times Do-0.635\times t

Heat_capacity_rate.html

  1. C = c p d m d t C=c_{p}\frac{dm}{dt}

Heat_loss_due_to_linear_thermal_bridging.html

  1. H T B H_{T}B
  2. H T B H_{T}B
  3. H T B = y A e x p H_{T}B=y\sum A_{exp}
  4. y = 0.08 y=0.08
  5. y = 0.15 y=0.15
  6. A e x p \sum A_{exp}

Heavy_fermion.html

  1. T c o h T_{coh}
  2. T c o h T_{coh}
  3. σ ( ω ) = n e 2 m * τ * 1 + ω 2 τ * 2 \sigma(\omega)=\frac{ne^{2}}{m^{*}}\frac{\tau^{*}}{1+\omega^{2}\tau^{*2}}
  4. m * m^{*}
  5. 1 τ * = m m * 1 τ \frac{1}{\tau^{*}}=\frac{m}{m^{*}}\frac{1}{\tau}
  6. C P = C P , e l + C P , p h = γ T + β T 3 C_{P}=C_{P,el}+C_{P,ph}=\gamma T+\beta T^{3}
  7. C P , e l = γ T = π 2 2 k B ϵ F n k B T C_{P,el}=\gamma T=\frac{\pi^{2}}{2}\frac{k_{B}}{\epsilon_{F}}nk_{B}T
  8. ϵ F = 2 k F 2 2 m \epsilon_{F}=\frac{\hbar^{2}k_{F}^{2}}{2m}

Heavy_Rydberg_system.html

  1. H + / H - H^{+}/H^{-}
  2. H - H^{-}
  3. H + / H - H^{+}/H^{-}
  4. H - H^{-}
  5. O 2 O_{2}
  6. H 2 S H_{2}S
  7. H F HF
  8. μ = m 1 m 2 m 1 + m 2 \mu={m_{1}m_{2}\over m_{1}+m_{2}}
  9. τ 2 = 4 π 2 μ k Z e 2 a 3 \tau^{2}={4\pi^{2}\mu\over kZe^{2}}a^{3}
  10. τ \tau
  11. μ \mu
  12. a a
  13. k = 1 / ( 4 π ϵ 0 ) k=1/(4\pi\epsilon_{0})

Height_of_a_polynomial.html

  1. P = a 0 + a 1 x + a 2 x 2 + + a n x n , P=a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{n}x^{n},
  2. H ( P ) = max 𝑖 | a i | H(P)=\underset{i}{\max}\,|a_{i}|\,
  3. L ( P ) = i = 0 n | a i | . L(P)=\sum_{i=0}^{n}|a_{i}|.\,
  4. ( n n / 2 ) - 1 H ( P ) M ( P ) H ( P ) n + 1 ; {\left({{n}\atop{\lfloor n/2\rfloor}}\right)}^{-1}H(P)\leq M(P)\leq H(P)\sqrt{% n+1};
  5. L ( p ) 2 n M ( p ) 2 n L ( p ) ; L(p)\leq 2^{n}M(p)\leq 2^{n}L(p);
  6. H ( p ) L ( p ) n H ( p ) H(p)\leq L(p)\leq nH(p)
  7. ( n n / 2 ) \scriptstyle{\left({{n}\atop{\lfloor n/2\rfloor}}\right)}

Helix–coil_transition_model.html

  1. Δ G f o l d i n g = ( m - 2 ) Δ H α - m T Δ S \Delta G_{folding}=(m-2)\Delta H_{\alpha}-mT\Delta S

Hemi-icosahedron.html

  1. K 6 K_{6}

Henryk_Iwaniec.html

  1. a 2 + b 4 a^{2}+b^{4}

Heparosan-N-sulfate-glucuronate_5-epimerase.html

  1. \rightleftharpoons

Hepatic_fructokinase.html

  1. \longrightarrow

Herbrand_structure.html

  1. \mathcal{L}
  2. \mathcal{L}
  3. 𝔐 \mathfrak{M}
  4. \mathcal{L}
  5. 𝔐 \mathfrak{M}
  6. \mathcal{L}
  7. 𝔐 \mathfrak{M}
  8. 𝔐 \mathfrak{M}
  9. T T
  10. T . T.

Hermite_constant.html

  1. γ n \gamma_{n}
  2. γ n \sqrt{\gamma_{n}}
  3. γ n \gamma_{n}
  4. γ 2 = 2 3 \gamma_{2}=\tfrac{2}{\sqrt{3}}
  5. γ n ( 4 3 ) ( n - 1 ) / 2 . \gamma_{n}\leq\left(\frac{4}{3}\right)^{(n-1)/2}.
  6. γ n ( 2 π ) Γ ( 2 + n 2 ) 2 / n . \gamma_{n}\leq\left(\frac{2}{\pi}\right)\Gamma\left(2+\frac{n}{2}\right)^{2/n}.

Hermite_normal_form.html

  1. A = ( 3 3 1 4 0 1 0 0 0 0 19 16 0 0 0 3 ) H = ( 3 0 1 1 0 1 0 0 0 0 19 1 0 0 0 3 ) A=\begin{pmatrix}3&3&1&4\\ 0&1&0&0\\ 0&0&19&16\\ 0&0&0&3\end{pmatrix}\qquad H=\begin{pmatrix}3&0&1&1\\ 0&1&0&0\\ 0&0&19&1\\ 0&0&0&3\end{pmatrix}
  2. A = ( 0 0 5 0 1 4 0 0 0 - 1 - 4 99 0 0 0 20 19 16 0 0 0 0 2 1 0 0 0 0 0 3 0 0 0 0 0 0 ) H = ( 0 0 5 0 0 2 0 0 0 1 0 1 0 0 0 0 1 2 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 ) A=\begin{pmatrix}0&0&5&0&1&4\\ 0&0&0&-1&-4&99\\ 0&0&0&20&19&16\\ 0&0&0&0&2&1\\ 0&0&0&0&0&3\\ 0&0&0&0&0&0\end{pmatrix}\qquad H=\begin{pmatrix}0&0&5&0&0&2\\ 0&0&0&1&0&1\\ 0&0&0&0&1&2\\ 0&0&0&0&0&3\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ \end{pmatrix}

Heston_model.html

  1. d S t = μ S t d t + ν t S t d W t S dS_{t}=\mu S_{t}\,dt+\sqrt{\nu_{t}}S_{t}\,dW^{S}_{t}\,
  2. ν t \nu_{t}
  3. d ν t = κ ( θ - ν t ) d t + ξ ν t d W t ν d\nu_{t}=\kappa(\theta-\nu_{t})\,dt+\xi\sqrt{\nu_{t}}\,dW^{\nu}_{t}\,
  4. d W t S , d W t ν {\scriptstyle dW^{S}_{t},dW^{\nu}_{t}}
  5. ν t \nu_{t}
  6. 2 κ θ > ξ 2 . 2\kappa\theta>\xi^{2}\,.
  7. d S t = μ S t d t + ν t S t d W t S . dS_{t}=\mu S_{t}\,dt+\sqrt{\nu_{t}}S_{t}\,dW^{S}_{t}\,.
  8. ν t \nu_{t}
  9. d ν t = κ t ( θ t - ν t ) d t + ξ t ν t d W t ν d\nu_{t}=\kappa_{t}(\theta_{t}-\nu_{t})\,dt+\xi_{t}\sqrt{\nu_{t}}\,dW^{\nu}_{t}\,
  10. d W t S , d W t ν {\scriptstyle dW^{S}_{t},dW^{\nu}_{t}}
  11. d S t = μ S t d t + ν t 1 S t d W t S , 1 + ν t 2 S t d W t S , 2 dS_{t}=\mu S_{t}\,dt+\sqrt{\nu^{1}_{t}}S_{t}\,dW^{S,1}_{t}+\sqrt{\nu^{2}_{t}}S% _{t}\,dW^{S,2}_{t}\,
  12. d ν t 1 = κ 1 ( θ 1 - ν t 1 ) d t + ξ 1 ν t 1 d W t ν 1 d\nu^{1}_{t}=\kappa^{1}(\theta^{1}-\nu^{1}_{t})\,dt+\xi^{1}\sqrt{\nu^{1}_{t}}% \,dW^{\nu^{1}}_{t}\,
  13. d ν t 2 = κ 2 ( θ 2 - ν t 2 ) d t + ξ 2 ν t 2 d W t ν 2 d\nu^{2}_{t}=\kappa^{2}(\theta^{2}-\nu^{2}_{t})\,dt+\xi^{2}\sqrt{\nu^{2}_{t}}% \,dW^{\nu^{2}}_{t}\,
  14. d r t = ( θ t - r t ) d t + r t σ t d W t , dr_{t}=(\theta_{t}-r_{t})\,dt+\sqrt{r_{t}}\,\sigma_{t}\,dW_{t},
  15. d α t = ( ζ t - α t ) d t + α t σ t d W t , d\alpha_{t}=(\zeta_{t}-\alpha_{t})\,dt+\sqrt{\alpha_{t}}\,\sigma_{t}\,dW_{t},
  16. d σ t = ( β t - σ t ) d t + σ t η t d W t . d\sigma_{t}=(\beta_{t}-\sigma_{t})\,dt+\sqrt{\sigma_{t}}\,\eta_{t}\,dW_{t}.
  17. n n
  18. m m
  19. M M
  20. M M
  21. M M
  22. m - n m-n
  23. e - ρ t S t e^{-\rho t}S_{t}

Heun's_method.html

  1. y ( t ) = f ( t , y ( t ) ) , y ( t 0 ) = y 0 , y^{\prime}(t)=f(t,y(t)),\qquad\qquad y(t_{0})=y_{0},
  2. y ~ i + 1 \tilde{y}_{i+1}
  3. y i + 1 y_{i+1}
  4. y ~ i + 1 = y i + h f ( t i , y i ) \tilde{y}_{i+1}=y_{i}+hf(t_{i},y_{i})
  5. y i + 1 = y i + h 2 [ f ( t i , y i ) + f ( t i + 1 , y ~ i + 1 ) ] , y_{i+1}=y_{i}+\frac{h}{2}[f(t_{i},y_{i})+f(t_{i+1},\tilde{y}_{i+1})],
  6. h h
  7. t i + 1 = t i + h t_{i+1}=t_{i}+h
  8. f ( x , y ) \textstyle f(x,y)
  9. ( x , y ) \textstyle(x,y)
  10. Slope left = f ( x i , y i ) \textstyle\,\text{Slope}_{\,\text{left}}=f(x_{i},y_{i})
  11. Slope right = f ( x i + h , y i + h f ( x i , y i ) ) \textstyle\,\text{Slope}_{\,\text{right}}=f(x_{i}+h,y_{i}+hf(x_{i},y_{i}))
  12. Slope ideal = ( 1 / 2 ) ( Slope left + Slope right ) \textstyle\,\text{Slope}_{\,\text{ideal}}=(1/2)(\,\text{Slope}_{\,\text{left}}% +\,\text{Slope}_{\,\text{right}})
  13. Slope ideal = ( Δ y / h ) \textstyle\,\text{Slope}_{\,\text{ideal}}=(\Delta y/h)
  14. Δ y = h ( Slope ideal ) \textstyle\Delta y=h(\,\text{Slope}_{\,\text{ideal}})
  15. x i + 1 = x i + h \textstyle x_{i+1}=x_{i}+h
  16. y i + 1 = y i + Δ y \textstyle y_{i+1}=y_{i}+\Delta y
  17. y i + 1 = y i + h Slope ideal \textstyle y_{i+1}=y_{i}+h\,\text{Slope}_{\,\text{ideal}}
  18. y i + 1 = y i + 1 2 h ( Slope left + Slope right ) y_{i+1}=y_{i}+\frac{1}{2}h(\,\text{Slope}_{\,\text{left}}+\,\text{Slope}_{\,% \text{right}})
  19. y i + 1 = y i + h 2 ( f ( x i , y i ) + f ( x i + h , y i + h f ( x i , y i ) ) ) y_{i+1}=y_{i}+\frac{h}{2}(f(x_{i},y_{i})+f(x_{i}+h,y_{i}+hf(x_{i},y_{i})))
  20. f ( t i + 1 , y i + 1 ) f(t_{i+1},y_{i+1})
  21. f ( t i + 1 , y ~ i + 1 ) f(t_{i+1},\tilde{y}_{i+1})
  22. y ~ i + 1 \tilde{y}_{i+1}

Hexacode.html

  1. G F ( 4 ) = { 0 , 1 , ω , ω 2 } GF(4)=\{0,1,\omega,\omega^{2}\}
  2. H = { ( a , b , c , f ( 1 ) , f ( ω ) , f ( ω 2 ) ) : f ( x ) := a x 2 + b x + c ; a , b , c G F ( 4 ) } . H=\{(a,b,c,f(1),f(\omega),f(\omega^{2})):f(x):=ax^{2}+bx+c;a,b,c\in GF(4)\}.
  3. G F ( 4 ) GF(4)
  4. H H
  5. 3. S 6 3.S_{6}

Hexacontagon.html

  1. A = 15 t 2 cot π 60 A=15t^{2}\cot\frac{\pi}{60}
  2. r = 1 2 t cot π 60 r=\frac{1}{2}t\cot\frac{\pi}{60}
  3. R = 1 2 t csc π 60 R=\frac{1}{2}t\csc\frac{\pi}{60}
  4. sin π 60 = sin 3 = 1 16 [ 2 ( 1 - 3 ) 5 + 5 + 2 ( 5 - 1 ) ( 3 + 1 ) ] \sin\frac{\pi}{60}=\sin 3^{\circ}=\tfrac{1}{16}\left[2(1-\sqrt{3})\sqrt{5+% \sqrt{5}}+\sqrt{2}(\sqrt{5}-1)(\sqrt{3}+1)\right]\,
  5. cos π 60 = cos 3 = 1 16 [ 2 ( 1 + 3 ) 5 + 5 + 2 ( 5 - 1 ) ( 3 - 1 ) ] \cos\frac{\pi}{60}=\cos 3^{\circ}=\tfrac{1}{16}\left[2(1+\sqrt{3})\sqrt{5+% \sqrt{5}}+\sqrt{2}(\sqrt{5}-1)(\sqrt{3}-1)\right]\,
  6. tan π 60 = tan 3 = 1 4 [ ( 2 - 3 ) ( 3 + 5 ) - 2 ] [ 2 - 2 ( 5 - 5 ) ] \tan\frac{\pi}{60}=\tan 3^{\circ}=\tfrac{1}{4}\left[(2-\sqrt{3})(3+\sqrt{5})-2% \right]\left[2-\sqrt{2(5-\sqrt{5})}\right]\,
  7. cot π 60 = cot 3 = 1 4 [ ( 2 + 3 ) ( 3 + 5 ) - 2 ] [ 2 + 2 ( 5 - 5 ) ] \cot\frac{\pi}{60}=\cot 3^{\circ}=\tfrac{1}{4}\left[(2+\sqrt{3})(3+\sqrt{5})-2% \right]\left[2+\sqrt{2(5-\sqrt{5})}\right]\,

Hexagonal_pyramidal_number.html

  1. n ( n + 1 ) ( 4 n - 1 ) 6 . \frac{n(n+1)(4n-1)}{6}.

Hierarchical_clustering_of_networks.html

  1. W i j W_{ij}
  2. ( i , j ) (i,j)

Higgs_prime.html

  1. ϕ ( H p n ) | i = 1 n - 1 H p i a and H p n > H p n - 1 \phi(Hp_{n})|\prod_{i=1}^{n-1}{Hp_{i}}^{a}\mbox{ and }~{}Hp_{n}>Hp_{n-1}
  2. 2 2 n + 1 2^{2^{n}}+1

High_harmonic_generation.html

  1. E m a x = I p + 3.17 U p E_{max}=I_{p}+3.17U_{p}

Hilbert's_arithmetic_of_ends.html

  1. Π \scriptstyle\Pi
  2. Π \scriptstyle\Pi
  3. ( 0 , ) \scriptstyle(0,\,\infty)
  4. x H \scriptstyle x\,\in\,H^{\prime}
  5. x = - x . x^{\prime}=-x.\,
  6. x = 1 x . x^{\prime}={1\over x}.\,
  7. ( 0 , ) \scriptstyle(0,\,\infty)
  8. a H \scriptstyle a\,\in\,H
  9. x = a x . x^{\prime}=ax.\,
  10. a H \scriptstyle a\,\in\,H
  11. ( 0 , ) \scriptstyle(0,\infty)
  12. ( ( 1 / 2 ) a , ) \scriptstyle((1/2)a,\,\infty)
  13. \scriptstyle\infty
  14. x = x + a . x^{\prime}=x+a.\,
  15. a H \scriptstyle a\,\in\,H
  16. x = x + a 1 - a x x^{\prime}=\frac{x+a}{1-ax}
  17. \scriptstyle\infty
  18. x = - 1 x . x^{\prime}=-{1\over x}.

Hilbert's_theorem_(differential_geometry).html

  1. S S
  2. K K
  3. 3 \mathbb{R}^{3}
  4. 3 \mathbb{R}^{3}
  5. φ = ψ exp p : S 3 \varphi=\psi\circ\exp_{p}:S^{\prime}\longrightarrow\mathbb{R}^{3}
  6. S S^{\prime}
  7. 3 \mathbb{R}^{3}
  8. K = - 1 K=-1
  9. 3 \mathbb{R}^{3}
  10. K K
  11. exp p : T p ( S ) S \exp_{p}:T_{p}(S)\longrightarrow S
  12. S S
  13. p p
  14. T p ( S ) T_{p}(S)
  15. S S^{\prime}
  16. T p ( S ) T_{p}(S)
  17. ψ : S 3 \psi:S\longrightarrow\mathbb{R}^{3}
  18. φ = ψ exp o : S 3 \varphi=\psi\circ\exp_{o}:S^{\prime}\longrightarrow\mathbb{R}^{3}
  19. S S^{\prime}
  20. H H
  21. S S^{\prime}
  22. H H
  23. S S^{\prime}
  24. H H
  25. q 2 q\in\mathbb{R}^{2}
  26. ( u , v ) (u,v)
  27. E = u , u = 1 F = u , v = v , u = 0 G = v , v = e u E=\left\langle\frac{\partial}{\partial u},\frac{\partial}{\partial u}\right% \rangle=1\qquad F=\left\langle\frac{\partial}{\partial u},\frac{\partial}{% \partial v}\right\rangle=\left\langle\frac{\partial}{\partial v},\frac{% \partial}{\partial u}\right\rangle=0\qquad G=\left\langle\frac{\partial}{% \partial v},\frac{\partial}{\partial v}\right\rangle=e^{u}
  28. - - e u d u d v = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{u}dudv=\infty
  29. S S^{\prime}
  30. φ : H S \varphi:H\rightarrow S^{\prime}
  31. S S^{\prime}
  32. S S
  33. φ \varphi
  34. ψ : T p ( H ) T p ( S ) \psi:T_{p}(H)\rightarrow T_{p^{\prime}}(S^{\prime})
  35. φ = exp p ψ exp p - 1 \varphi=\exp_{p^{\prime}}\circ\psi\circ\exp_{p}^{-1}
  36. p H , p S p\in H,p^{\prime}\in S^{\prime}
  37. p H p\in H
  38. H H
  39. ψ \psi
  40. S S^{\prime}
  41. ( ρ , θ ) (\rho,\theta)
  42. ( ρ , θ ) (\rho^{\prime},\theta^{\prime})
  43. p p
  44. p p^{\prime}
  45. θ = 0 \theta=0
  46. θ = 0 \theta^{\prime}=0
  47. φ \varphi
  48. K K
  49. K = - ( G ) ρ ρ G K=-\frac{(\sqrt{G})_{\rho\rho}}{\sqrt{G}}
  50. ( G ) ρ ρ + K G = 0 (\sqrt{G})_{\rho\rho}+K\cdot\sqrt{G}=0
  51. H H
  52. S S^{\prime}
  53. φ \varphi
  54. H H
  55. S S^{\prime}
  56. φ \varphi
  57. S S^{\prime}
  58. φ \varphi
  59. H H
  60. S S^{\prime}
  61. H H
  62. S = T p ( S ) S^{\prime}=T_{p}(S)
  63. \square
  64. p S p\in S^{\prime}
  65. x : U 2 S , p x ( U ) x:U\subset\mathbb{R}^{2}\longrightarrow S^{\prime},\qquad p\in x(U)
  66. x x
  67. x ( U ) = V x(U)=V^{\prime}
  68. V S V^{\prime}\subset S^{\prime}
  69. S S^{\prime}
  70. V V^{\prime}
  71. 2 π 2\pi
  72. x x
  73. S S^{\prime}
  74. t t
  75. x ( s , t ) , - < s < + x(s,t),-\infty<s<+\infty
  76. s s
  77. x : 2 S x:\mathbb{R}^{2}\longrightarrow S^{\prime}
  78. x x
  79. x x
  80. S S^{\prime}
  81. S S^{\prime}
  82. x x
  83. S S
  84. ψ : S 3 \psi:S\longrightarrow\mathbb{R}^{3}
  85. T p ( S ) T_{p}(S)
  86. exp p : T p ( S ) S \exp_{p}:T_{p}(S)\longrightarrow S
  87. φ = ψ exp p : S 3 \varphi=\psi\circ\exp_{p}:S^{\prime}\longrightarrow\mathbb{R}^{3}
  88. x : 2 S x:\mathbb{R}^{2}\longrightarrow S^{\prime}
  89. S S^{\prime}
  90. x x
  91. S S^{\prime}
  92. S S^{\prime}
  93. Q n Q_{n}
  94. Q n Q n + 1 Q_{n}\subset Q_{n+1}
  95. 2 π 2\pi
  96. S S^{\prime}
  97. \square

Hilbert_projection_theorem.html

  1. x x
  2. H H
  3. C H C\subset H
  4. y C y\in C
  5. x - y \lVert x-y\rVert
  6. C C
  7. M M
  8. C C
  9. y y
  10. x - y x-y
  11. M M
  12. y n - y m 2 = y n - x 2 + y m - x 2 - 2 y n - x , y m - x \|y_{n}-y_{m}\|^{2}=\|y_{n}-x\|^{2}+\|y_{m}-x\|^{2}-2\langle y_{n}-x\,,\,y_{m}% -x\rangle
  13. 4 y n + y m 2 - x 2 = y n - x 2 + y m - x 2 + 2 y n - x , y m - x 4\left\|\frac{y_{n}+y_{m}}{2}-x\right\|^{2}=\|y_{n}-x\|^{2}+\|y_{m}-x\|^{2}+2% \langle y_{n}-x\,,\,y_{m}-x\rangle
  14. y n - y m 2 = 2 y n - x 2 + 2 y m - x 2 - 4 y n + y m 2 - x 2 \|y_{n}-y_{m}\|^{2}=2\|y_{n}-x\|^{2}+2\|y_{m}-x\|^{2}-4\left\|\frac{y_{n}+y_{m% }}{2}-x\right\|^{2}
  15. y n - y m 2 2 ( δ 2 + 1 n ) + 2 ( δ 2 + 1 m ) - 4 δ 2 = 2 ( 1 n + 1 m ) \|y_{n}-y_{m}\|^{2}\;\leq\;2\left(\delta^{2}+\frac{1}{n}\right)+2\left(\delta^% {2}+\frac{1}{m}\right)-4\delta^{2}=2\left(\frac{1}{n}+\frac{1}{m}\right)
  16. y 2 - y 1 2 = 2 y 1 - x 2 + 2 y 2 - x 2 - 4 y 1 + y 2 2 - x 2 \|y_{2}-y_{1}\|^{2}=2\|y_{1}-x\|^{2}+2\|y_{2}-x\|^{2}-4\left\|\frac{y_{1}+y_{2% }}{2}-x\right\|^{2}
  17. y 1 + y 2 2 \frac{y_{1}+y_{2}}{2}
  18. y 1 + y 2 2 - x 2 δ 2 \left\|\frac{y_{1}+y_{2}}{2}-x\right\|^{2}\geq\delta^{2}
  19. y 2 - y 1 2 2 δ 2 + 2 δ 2 - 4 δ 2 = 0 \|y_{2}-y_{1}\|^{2}\leq 2\delta^{2}+2\delta^{2}-4\delta^{2}=0\,
  20. y 1 = y 2 y_{1}=y_{2}
  21. z M z\in M
  22. z - x , a = 0 \langle z-x,a\rangle=0
  23. a M a\in M
  24. x - a 2 = z - x 2 + a - z 2 + 2 z - x , a - z = z - x 2 + a - z 2 \|x-a\|^{2}=\|z-x\|^{2}+\|a-z\|^{2}+2\langle z-x,a-z\rangle=\|z-x\|^{2}+\|a-z% \|^{2}
  25. z z
  26. y M y\in M
  27. a M a\in M
  28. t t\in\mathbb{R}
  29. ( y + t a ) - x 2 - y - x 2 = 2 t y - x , a + t 2 a 2 = 2 t y - x , a + O ( t 2 ) \|(y+ta)-x\|^{2}-\|y-x\|^{2}=2t\langle y-x,a\rangle+t^{2}\|a\|^{2}=2t\langle y% -x,a\rangle+O(t^{2})
  30. y - x , a = 0. \langle y-x,a\rangle=0.

Hilbert_system.html

  1. Γ \Gamma
  2. Γ \Gamma
  3. Γ ϕ \Gamma\vdash\phi
  4. ϕ \phi
  5. Γ \Gamma
  6. Γ ϕ \Gamma\vdash\phi
  7. ϕ \phi
  8. Γ \Gamma
  9. y ( x P x y P t y ) \forall y(\forall xPxy\to Pty)
  10. x P x y P t y \forall xPxy\to Pty
  11. ¬ \lnot
  12. \to
  13. \forall
  14. \land
  15. \lor
  16. ϕ ϕ \phi\to\phi
  17. ϕ ( ψ ϕ ) \phi\to\left(\psi\to\phi\right)
  18. ( ϕ ( ψ ξ ) ) ( ( ϕ ψ ) ( ϕ ξ ) ) \left(\phi\to\left(\psi\rightarrow\xi\right)\right)\to\left(\left(\phi\to\psi% \right)\to\left(\phi\to\xi\right)\right)
  19. ( ¬ ϕ ¬ ψ ) ( ψ ϕ ) \left(\lnot\phi\to\lnot\psi\right)\to\left(\psi\to\phi\right)
  20. ¬ ϕ ( ϕ ψ ) \lnot\phi\to\left(\phi\to\psi\right)
  21. p p p\to p
  22. ( p q ) ( p q ) \left(p\to q\right)\to\left(p\to q\right)
  23. ϕ \phi
  24. ϕ ( p ) \phi(p)
  25. p p
  26. ψ \psi
  27. ϕ ( p ) \phi(p)
  28. ϕ ( ψ ) \phi(\psi)
  29. x ( ϕ ) ϕ [ x := t ] \forall x\left(\phi\right)\to\phi[x:=t]
  30. ϕ \,\!\phi
  31. x ( ϕ ψ ) ( x ( ϕ ) x ( ψ ) ) \forall x\left(\phi\to\psi\right)\to\left(\forall x\left(\phi\right)\to\forall x% \left(\psi\right)\right)
  32. ϕ x ( ϕ ) \phi\to\forall x\left(\phi\right)
  33. ϕ \,\!\phi
  34. x = x x=x
  35. ( x = y ) ( ϕ [ z := x ] ϕ [ z := y ] ) \left(x=y\right)\to\left(\phi[z:=x]\to\phi[z:=y]\right)
  36. x ( ϕ y ( ϕ [ x := y ] ) ) \forall x(\phi\to\exists y(\phi[x:=y]))
  37. x ( ϕ ψ ) x ( ϕ ) ψ \forall x(\phi\to\psi)\to\exists x(\phi)\to\psi
  38. x x
  39. ψ \psi
  40. α β α β \alpha\to\beta\to\alpha\land\beta
  41. α β α \alpha\wedge\beta\to\alpha
  42. α β β \alpha\wedge\beta\to\beta
  43. α α β \alpha\to\alpha\vee\beta
  44. β α β \beta\to\alpha\vee\beta
  45. ( α γ ) ( β γ ) α β γ (\alpha\to\gamma)\to(\beta\to\gamma)\to\alpha\vee\beta\to\gamma
  46. Γ ; ϕ ψ \Gamma;\phi\vdash\psi
  47. Γ ϕ ψ \Gamma\vdash\phi\to\psi
  48. Γ ϕ ψ \Gamma\vdash\phi\leftrightarrow\psi
  49. Γ ϕ ψ \Gamma\vdash\phi\to\psi
  50. Γ ψ ϕ \Gamma\vdash\psi\to\phi
  51. Γ ; ϕ ψ \Gamma;\phi\vdash\psi
  52. Γ ; ¬ ψ ¬ ϕ \Gamma;\lnot\psi\vdash\lnot\phi
  53. Γ ϕ \Gamma\vdash\phi
  54. Γ \Gamma
  55. Γ x ϕ \Gamma\vdash\forall x\phi

Hilbert–Schmidt_integral_operator.html

  1. Ω Ω | k ( x , y ) | 2 d x d y < \int_{\Omega}\int_{\Omega}|k(x,y)|^{2}\,dx\,dy<\infty
  2. ( K u ) ( x ) = Ω k ( x , y ) u ( y ) d y . (Ku)(x)=\int_{\Omega}k(x,y)u(y)\,dy.
  3. K HS = k L 2 . \|K\|_{\mathrm{HS}}=\|k\|_{L^{2}}.
  4. ( K f ) ( x ) = X k ( x , y ) f ( y ) d y (Kf)(x)=\int_{X}k(x,y)f(y)\,dy
  5. k ( x , y ) = k ( y , x ) ¯ k(x,y)=\overline{k(y,x)}

Hilbert–Schmidt_theorem.html

  1. lim i + λ i = 0. \lim_{i\to+\infty}\lambda_{i}=0.
  2. A φ i = λ i φ i for i = 1 , , N . A\varphi_{i}=\lambda_{i}\varphi_{i}\mbox{ for }~{}i=1,\dots,N.
  3. A u = i = 1 N λ i φ i , u φ i for all u H . Au=\sum_{i=1}^{N}\lambda_{i}\langle\varphi_{i},u\rangle\varphi_{i}\mbox{ for % all }~{}u\in H.

Hirschberg's_algorithm.html

  1. ( Z , W ) = NW ( X , Y ) (Z,W)=\operatorname{NW}(X,Y)
  2. ( X , Y ) (X,Y)
  3. X = X l + X r X=X^{l}+X^{r}
  4. X X
  5. Y l + Y r Y^{l}+Y^{r}
  6. Y Y
  7. NW ( X , Y ) = NW ( X l , Y l ) + NW ( X r , Y r ) \operatorname{NW}(X,Y)=\operatorname{NW}(X^{l},Y^{l})+\operatorname{NW}(X^{r},% Y^{r})
  8. X i X_{i}
  9. X X
  10. 1 < i length ( X ) 1<i\leqslant\operatorname{length}(X)
  11. X i : j X_{i:j}
  12. j - i + 1 j-i+1
  13. X X
  14. rev ( X ) \operatorname{rev}(X)
  15. X X
  16. X X
  17. Y Y
  18. x x
  19. X X
  20. y y
  21. Y Y
  22. Del ( x ) \operatorname{Del}(x)
  23. Ins ( y ) \operatorname{Ins}(y)
  24. Sub ( x , y ) \operatorname{Sub}(x,y)
  25. x x
  26. y y
  27. x x
  28. y y
  29. NWScore ( X , Y ) \operatorname{NWScore}(X,Y)
  30. Score ( i , j ) \mathrm{Score}(i,j)
  31. NWScore \operatorname{NWScore}
  32. NWScore \operatorname{NWScore}
  33. O ( min { length ( X ) , length ( Y ) } ) O(\operatorname{min}\{\operatorname{length}(X),\operatorname{length}(Y)\})
  34. X l + X r X^{l}+X^{r}
  35. X X
  36. PartitionY \mathrm{PartitionY}
  37. ymid \mathrm{ymid}
  38. Y l = Y 1 : ymid Y^{l}=Y_{1:\mathrm{ymid}}
  39. Y r = Y ymid + 1 : length ( Y ) Y^{r}=Y_{\mathrm{ymid}+1:\operatorname{length}(Y)}
  40. PartitionY \mathrm{PartitionY}
  41. X = AGTACGCA , Y = TATGC , Del ( x ) = - 2 , Ins ( y ) = - 2 , Sub ( x , y ) = { + 2 , if x = y - 1 , if x y . \begin{aligned}\displaystyle X&\displaystyle=\mathrm{AGTACGCA},\\ \displaystyle Y&\displaystyle=\mathrm{TATGC},\\ \displaystyle\operatorname{Del}(x)&\displaystyle=-2,\\ \displaystyle\operatorname{Ins}(y)&\displaystyle=-2,\\ \displaystyle\operatorname{Sub}(x,y)&\displaystyle=\begin{cases}+2,&\mbox{if }% ~{}x=y\\ -1,&\mbox{if }~{}x\neq y.\end{cases}\end{aligned}
  42. Hirschberg ( AGTACGCA , TATGC ) \operatorname{Hirschberg}(\mathrm{AGTACGCA},\mathrm{TATGC})
  43. NWScore ( AGTA , Y ) \operatorname{NWScore}(\mathrm{AGTA},Y)
  44. NWScore ( rev ( CGCA ) , rev ( Y ) ) \operatorname{NWScore}(\operatorname{rev}(\mathrm{CGCA}),\operatorname{rev}(Y))
  45. X = AGTA + CGCA X=\mathrm{AGTA}+\mathrm{CGCA}
  46. Y = TA + TGC Y=\mathrm{TA}+\mathrm{TGC}

History_monoid.html

  1. A = ( Σ 1 , Σ 2 , , Σ n ) A=(\Sigma_{1},\Sigma_{2},\ldots,\Sigma_{n})
  2. Σ k \Sigma_{k}
  3. P ( A ) P(A)
  4. P ( A ) = Σ 1 * × Σ 2 * × × Σ n * P(A)=\Sigma_{1}^{*}\times\Sigma_{2}^{*}\times\cdots\times\Sigma_{n}^{*}
  5. P ( A ) P(A)
  6. Σ k \Sigma_{k}
  7. u = ( u 1 , u 2 , , u n ) {u}=(u_{1},u_{2},\ldots,u_{n})\,
  8. v = ( v 1 , v 2 , , v n ) {v}=(v_{1},v_{2},\ldots,v_{n})\,
  9. u v = ( u 1 v 1 , u 2 v 2 , , u n v n ) {uv}=(u_{1}v_{1},u_{2}v_{2},\ldots,u_{n}v_{n})\,
  10. u , v {u},{v}
  11. P ( A ) P(A)
  12. Σ = Σ 1 Σ 2 Σ n . \Sigma=\Sigma_{1}\cup\Sigma_{2}\cup\cdots\cup\Sigma_{n}.\,
  13. w Σ * w\in\Sigma^{*}
  14. Σ k * \Sigma_{k}^{*}
  15. π k : Σ * Σ k * \pi_{k}:\Sigma^{*}\to\Sigma_{k}^{*}
  16. π : Σ * P ( A ) \pi:\Sigma^{*}\to P(A)
  17. w Σ * w\in\Sigma^{*}
  18. π k \pi_{k}
  19. π ( w ) ( π 1 ( w ) , π 2 ( w ) , , π n ( w ) ) . \pi(w)\mapsto(\pi_{1}(w),\pi_{2}(w),\ldots,\pi_{n}(w)).\,
  20. a Σ a\in\Sigma
  21. π ( a ) \pi(a)
  22. Σ k \Sigma_{k}
  23. π ( a ) = ( a 1 , a 2 , , a n ) \pi(a)=(a_{1},a_{2},\ldots,a_{n})
  24. a k = { a if a Σ k ε otherwise . a_{k}=\begin{cases}a\mbox{ if }~{}a\in\Sigma_{k}\\ \varepsilon\mbox{ otherwise }.\end{cases}
  25. ε \varepsilon
  26. H ( A ) H(A)
  27. P ( A ) P(A)
  28. H ( A ) H(A)
  29. Σ k \Sigma_{k}
  30. Σ 1 = { a , b , c } \Sigma_{1}=\{a,b,c\}
  31. Σ 2 = { a , d , e } \Sigma_{2}=\{a,d,e\}
  32. Σ = { a , b , c , d , e } \Sigma=\{a,b,c,d,e\}
  33. ( a , a ) (a,a)
  34. ( b , ε ) (b,\varepsilon)
  35. ( c , ε ) (c,\varepsilon)
  36. ( ε , d ) (\varepsilon,d)
  37. ( ε , e ) (\varepsilon,e)
  38. b c b c c bcbcc
  39. d d d e d ddded
  40. b c b d d d c c e d bcbdddcced
  41. b b
  42. d d
  43. e e
  44. a a
  45. b b
  46. c c
  47. d d
  48. e e
  49. a a
  50. b c d a b e bcdabe
  51. b d c a e b bdcaeb
  52. b c a b bcab
  53. d a e dae
  54. d d
  55. c c
  56. a a
  57. e e
  58. c c
  59. e e
  60. c c
  61. H ( Σ 1 , Σ 2 , , Σ n ) H(\Sigma_{1},\Sigma_{2},\ldots,\Sigma_{n})
  62. 𝕄 ( D ) \mathbb{M}(D)
  63. D = ( Σ 1 × Σ 1 ) ( Σ 2 × Σ 2 ) ( Σ n × Σ n ) . D=\left(\Sigma_{1}\times\Sigma_{1}\right)\cup\left(\Sigma_{2}\times\Sigma_{2}% \right)\cup\cdots\cup\left(\Sigma_{n}\times\Sigma_{n}\right).
  64. Σ k \Sigma_{k}
  65. Σ j \Sigma_{j}

History_of_algebra.html

  1. x 2 + p x = q x^{2}+px=q
  2. x 2 = p x + q x^{2}=px+q
  3. x 2 + q = p x x^{2}+q=px
  4. x 2 + p x + q = 0 x^{2}+px+q=0
  5. x 2 = A x^{2}=A
  6. x + a x = b x+ax=b
  7. x + a x + b x = c x+ax+bx=c
  8. x = ( m 1 + m 2 + + m n - 1 ) - s n - 2 = ( i = 1 n - 1 m i ) - s n - 2 x=\cfrac{(m_{1}+m_{2}+...+m_{n-1})-s}{n-2}=\cfrac{(\sum_{i=1}^{n-1}m_{i})-s}{n% -2}
  9. a ( b + c + d ) = a b + a c + a d a(b+c+d)=ab+ac+ad
  10. a 2 - b 2 = ( a + b ) ( a - b ) a^{2}-b^{2}=(a+b)(a-b)
  11. ( a + b ) 2 = a 2 + 2 a b + b 2 (a+b)^{2}=a^{2}+2ab+b^{2}
  12. a x + x 2 = b 2 ax+x^{2}=b^{2}
  13. a x + x 2 = a 2 ax+x^{2}=a^{2}
  14. d x 2 - a d x + b 2 c = 0 dx^{2}-adx+b^{2}c=0
  15. x y = a 2 xy=a^{2}
  16. 1 2 + 2 2 + 3 2 + + n 2 = n ( n + 1 ) ( 2 n + 1 ) 3 ! 1^{2}+2^{2}+3^{2}+\cdots+n^{2}={n(n+1)(2n+1)\over 3!}
  17. 1 + 8 + 30 + 80 + + n 2 ( n + 1 ) ( n + 2 ) 3 ! = n ( n + 1 ) ( n + 2 ) ( n + 3 ) ( 4 n + 1 ) 5 ! 1+8+30+80+\cdots+{n^{2}(n+1)(n+2)\over 3!}={n(n+1)(n+2)(n+3)(4n+1)\over 5!}
  18. x 3 - 2 x 2 + 10 x - 1 = 5 x^{3}-2x^{2}+10x-1=5
  19. x 3 1 x 10 - x 2 2 x 0 1 = x 0 5 {x^{3}}1{x}10-{x^{2}}2{x^{0}}1={x^{0}}5
  20. ( x 3 1 + x 10 ) - ( x 2 2 + x 0 1 ) = x 0 5 ({x^{3}}1+{x}10)-({x^{2}}2+{x^{0}}1)={x^{0}}5
  21. ( a 2 + b 2 ) ( c 2 + d 2 ) (a^{2}+b^{2})(c^{2}+d^{2})
  22. = ( a c + d b ) 2 + ( b c - a d ) 2 =(ac+db)^{2}+(bc-ad)^{2}
  23. = ( a d + b c ) 2 + ( a c - b d ) 2 =(ad+bc)^{2}+(ac-bd)^{2}
  24. 1 2 + 2 2 + + n 2 = n ( n + 1 ) ( 2 n + 1 ) 6 1^{2}+2^{2}+\cdots+n^{2}={n(n+1)(2n+1)\over 6}
  25. 1 3 + 2 3 + + n 3 = ( 1 + 2 + + n ) 2 1^{3}+2^{3}+\cdots+n^{3}=(1+2+\cdots+n)^{2}
  26. m m
  27. 1 2 ( m 2 n - n ) {1\over 2}({m^{2}\over n}-n)
  28. 1 2 ( m 2 n + n ) {1\over 2}({m^{2}\over n}+n)
  29. ( - x - 1 ) + ( 2 x - 8 ) = x - 9 (-x-1)+(2x-8)=x-9
  30. x 2 + 10 x = 39 x^{2}+10x=39
  31. x 3 + d = b x 2 \ x^{3}+d=bx^{2}
  32. x 2 ( b - x ) = d \ x^{2}(b-x)=d
  33. d \ d
  34. x = 2 b 3 x=\frac{2b}{3}
  35. 4 b 3 27 \frac{4b^{3}}{27}
  36. d \ d
  37. d \ d
  38. x = 2 b 3 x=\frac{2b}{3}
  39. d \ d
  40. 0 \ 0
  41. 2 b 3 \frac{2b}{3}
  42. 2 b 3 \frac{2b}{3}
  43. b \ b
  44. x P - N = 0 \ x^{P}-N=0
  45. N \ N
  46. x \mathit{x}
  47. x \mathit{x}
  48. x \mathit{x}
  49. x x\!
  50. x 2 + p x + q = 0 x^{2}+px+q=0
  51. x 2 + p x = q x^{2}+px=q
  52. x 2 = p x + q x^{2}=px+q
  53. x 2 + q = p x x^{2}+q=px
  54. x + a x = b x+ax=b
  55. x + a x + b x = c x+ax+bx=c
  56. ( a + b ) 2 = a 2 + 2 a b + b 2 (a+b)^{2}=a^{2}+2ab+b^{2}
  57. a 2 - b 2 = ( a + b ) ( a - b ) a^{2}-b^{2}=(a+b)(a-b)
  58. a x + x 2 = b 2 ax+x^{2}=b^{2}
  59. a x + x 2 = a 2 ax+x^{2}=a^{2}
  60. ( a - x ) d x = b 2 c (a-x)dx=b^{2}c
  61. d x 2 - a d x + b 2 c = 0 dx^{2}-adx+b^{2}c=0
  62. x = a / 2 + / - s q r t ( ( a / 2 ) 2 - b 2 c / d ) x=a/2+/-sqrt((a/2)^{2}-b^{2}c/d)
  63. x y = a 2 xy=a^{2}

History_of_Grandi's_series.html

  1. ( 1 - 1 ) + ( 1 - 1 ) + = 0 (1-1)+(1-1)+\cdots=0
  2. 1 + ( - 1 + 1 ) + ( - 1 + 1 ) + = 1. 1+(-1+1)+(-1+1)+\cdots=1.
  3. n = 0 ( - 1 ) n x 2 n a 2 n - 1 = a - x 2 a + x 4 a 3 - x 6 a 5 + \sum_{n=0}^{\infty}\frac{(-1)^{n}x^{2n}}{a^{2n-1}}=a-\frac{x^{2}}{a}+\frac{x^{% 4}}{a^{3}}-\frac{x^{6}}{a^{5}}+\cdots
  4. 1 1 + x = 1 - x + x 2 - x 3 + \frac{1}{1+x}=1-x+x^{2}-x^{3}+\cdots
  5. 1 1 + 1 = 1 1 - 1 1 + 1 . Ergo 1 1 + 1 = 1 - 1 + 1 - 1 + 1 - 1 etc . \frac{1}{1+1}=\frac{1}{1}-\frac{1}{1+1}.\;\mathrm{Ergo}\;\frac{1}{1+1}=1-1+1-1% +1-1\;\mathrm{etc.}
  6. 1 1 + 1 = 1 - ( 1 - 1 1 + 1 ) \frac{1}{1+1}=1-(1-\frac{1}{1+1})
  7. 1 1 + 1 = 1 - ( 1 - ( 1 - ( 1 - 1 1 + 1 ) ) ) \frac{1}{1+1}=1-(1-(1-(1-\frac{1}{1+1})))
  8. 1 1 + 1 = 1 - ( 1 - ( 1 - ( 1 - ( 1 - ( 1 - 1 1 + 1 ) ) ) ) ) \frac{1}{1+1}=1-(1-(1-(1-(1-(1-\frac{1}{1+1})))))
  9. k m + n = k m - k n m 2 + k n 2 m 3 - . \frac{k}{m+n}=\frac{k}{m}-\frac{kn}{m^{2}}+\frac{kn^{2}}{m^{3}}-\cdots.
  10. k 2 m = k m - k m + k m - . \frac{k}{2m}=\frac{k}{m}-\frac{k}{m}+\frac{k}{m}-\cdots.
  11. 1 1 + a = 1 - a + a 2 - a 3 + ± a n a n + 1 1 + a , \frac{1}{1+a}=1-a+a^{2}-a^{3}+\cdots\pm a^{n}\mp\frac{a^{n+1}}{1+a},
  12. 1 + x 1 + x + x 2 = 1 - x 2 + x 3 - x 5 + x 6 - x 8 + ; \frac{1+x}{1+x+x^{2}}=1-x^{2}+x^{3}-x^{5}+x^{6}-x^{8}+\cdots;
  13. 1 + 0 - x 2 + x 3 + 0 - x 5 + x 6 + 0 - x 8 + , 1+0-x^{2}+x^{3}+0-x^{5}+x^{6}+0-x^{8}+\cdots,
  14. 1 1 + 1 , \frac{1}{1+1},
  15. x = A + f ( A + f ( A + f ( ) ) ) x=A+f(A+f(A+f(\cdots)))
  16. x = a - x , x=a-x,
  17. x = a - a + a - a + = a - ( a - a + a - ) , x=a-a+a-a+\cdots=a-(a-a+a-\cdots),

History_of_quantum_mechanics.html

  1. ϵ = h ν \epsilon=h\nu\,
  2. I ( ν , T ) = 2 h ν 3 c 2 1 e h ν k T - 1 , I(\nu,T)=\frac{2h\nu^{3}}{c^{2}}\frac{1}{e^{\frac{h\nu}{kT}}-1},
  3. h ν k T h\nu\gg kT

Hitachi_DX07.html

  1. p = 250 s i n 1 60 0.0727 m m = 72.7 μ m p=250\cdot sin\tfrac{1}{60}\approx 0.0727mm=72.7\mu m

Hochschild_homology.html

  1. H H n ( A , M ) = Tor n A e ( A , M ) HH_{n}(A,M)=\,\text{Tor}_{n}^{A^{e}}(A,M)
  2. H H n ( A , M ) = Ext A e n ( A , M ) HH^{n}(A,M)=\,\text{Ext}^{n}_{A^{e}}(A,M)
  3. C n ( A , M ) := M A n C_{n}(A,M):=M\otimes A^{\otimes n}
  4. d 0 ( m a 1 a n ) = m a 1 a 2 a n d_{0}(m\otimes a_{1}\otimes\cdots\otimes a_{n})=ma_{1}\otimes a_{2}\cdots% \otimes a_{n}
  5. d i ( m a 1 a n ) = m a 1 a i a i + 1 a n d_{i}(m\otimes a_{1}\otimes\cdots\otimes a_{n})=m\otimes a_{1}\otimes\cdots% \otimes a_{i}a_{i+1}\otimes\cdots\otimes a_{n}
  6. d n ( m a 1 a n ) = a n m a 1 a n - 1 d_{n}(m\otimes a_{1}\otimes\cdots\otimes a_{n})=a_{n}m\otimes a_{1}\otimes% \cdots\otimes a_{n-1}
  7. b = i = 0 n ( - 1 ) i d i , b=\sum_{i=0}^{n}(-1)^{i}d_{i},
  8. Δ o S 1 Fin * 𝐹 k - mod . \Delta^{o}\overset{S^{1}}{\longrightarrow}\,\text{Fin}_{*}\overset{F}{% \longrightarrow}k\,\text{-}\operatorname{mod}.
  9. n + = { 0 , 1 , , n } , n_{+}=\{0,1,\dots,n\},\,
  10. n + M A n . n_{+}\mapsto M\otimes A^{\otimes n}.\,
  11. f : m + n + f:m_{+}\rightarrow n_{+}
  12. f * ( a 0 a n ) = ( b 0 b m ) f_{*}(a_{0}\otimes\cdots\otimes a_{n})=(b_{0}\otimes\cdots\otimes b_{m})
  13. b j = f ( i ) = j a i , j = 0 , , n , b_{j}=\prod_{f(i)=j}a_{i},\,\,j=0,\dots,n,
  14. Δ o S 1 Fin * ( A , M ) k - mod , \Delta^{o}\overset{S^{1}}{\longrightarrow}\,\text{Fin}_{*}\overset{\mathcal{L}% (A,M)}{\longrightarrow}k\,\text{-}\operatorname{mod},

Hodge_structure.html

  1. H := H 𝐙 𝐙 𝐂 = p + q = n H p , q , H:=H_{\mathbf{Z}}\otimes_{\mathbf{Z}}{\mathbf{C}}=\bigoplus\nolimits_{p+q=n}H^% {p,q},
  2. H p , q ¯ = H q , p . \overline{H^{p,q}}=H^{q,p}.
  3. p , q : p + q = n + 1 F p H F q H ¯ = 0 and F p H F q H ¯ = H \forall p,q\ :\ p+q=n+1\ \ F^{p}H\cap\overline{F^{q}H}=0\,\text{ and }F^{p}H% \oplus\overline{F^{q}H}=H
  4. H p , q = F p H F q H ¯ , H^{p,q}=F^{p}H\cap\overline{F^{q}H},
  5. F p H = i p H i , n - i . F^{p}H=\bigoplus\nolimits_{i\geq p}H^{i,n-i}.
  6. Q ( φ , ψ ) \displaystyle Q(\varphi,\psi)
  7. Q ( F p , F n - p + 1 ) = 0 , Q ( C φ , φ ¯ ) > 0 for φ 0 , \begin{aligned}\displaystyle Q\left(F^{p},F^{n-p+1}\right)&\displaystyle=0,\\ \displaystyle Q\left(C\varphi,\bar{\varphi}\right)&\displaystyle>0&&% \displaystyle\,\text{ for }\varphi\neq 0,\end{aligned}
  8. z p z ¯ q . z^{p}\overline{z}^{q}.
  9. P X ( t ) = rank ( H n ( X ) ) t n P_{X}(t)=\sum\,\text{rank}(H^{n}(X))t^{n}
  10. P X ( t ) = P Y ( t ) + P U ( t ) P_{X}(t)=P_{Y}(t)+P_{U}(t)
  11. 0 W 0 W 1 W 2 = H 1 ( X ) , 0\subset W_{0}\subset W_{1}\subset W_{2}=H^{1}(X),\,
  12. gr n W H = W n 𝐂 / W n - 1 𝐂 \operatorname{gr}_{n}^{W}H=W_{n}\otimes\mathbf{C}/W_{n-1}\otimes\mathbf{C}
  13. F p gr n W H = ( F p W n 𝐂 + W n - 1 𝐂 ) / W n - 1 𝐂 . F^{p}\operatorname{gr}_{n}^{W}H=(F^{p}\cap W_{n}\otimes\mathbf{C}+W_{n-1}% \otimes\mathbf{C})/W_{n-1}\otimes\mathbf{C}.
  14. R 𝐂 / 𝐑 𝐂 * R_{\mathbf{C/R}}{\mathbf{C}}^{*}
  15. E 1 p , q = H p + q ( g r n W H ) H p + q , E^{p,q}_{1}=H^{p+q}(gr^{W}_{n}H)\Rightarrow H^{p+q},

Holding_value.html

  1. H V [ i , n ] = k = 0 n - i d i v ( i + k ) ( 1 + r ) n - i - k {HV}_{[i,n]}=\sum_{k=0}^{n-i}\frac{div(i+k)}{{(1+r)}^{n-i-k}}

Holm–Bonferroni_method.html

  1. H 1 , , H m H_{1},...,H_{m}
  2. P 1 , , P m P_{1},...,P_{m}
  3. P ( 1 ) P ( m ) P_{(1)}\ldots P_{(m)}
  4. H ( 1 ) H ( m ) H_{(1)}\ldots H_{(m)}
  5. α \alpha
  6. k k
  7. P ( k ) > α m + 1 - k P_{(k)}>\frac{\alpha}{m+1-k}
  8. H ( 1 ) H ( k - 1 ) H_{(1)}\ldots H_{(k-1)}
  9. H ( k ) H ( m ) H_{(k)}\ldots H_{(m)}
  10. k = 1 k=1
  11. k k
  12. F W E R α FWER\leq\alpha
  13. F W E R FWER
  14. H ( 1 ) H ( m ) H_{(1)}\ldots H_{(m)}
  15. P ( 1 ) P ( 2 ) P ( m ) P_{(1)}\leq P_{(2)}\leq\ldots\leq P_{(m)}
  16. I 0 I_{0}
  17. m 0 m_{0}
  18. α \alpha
  19. h h
  20. h - 1 h-1
  21. h - 1 + m 0 m h-1+m_{0}\leq m
  22. 1 m - h + 1 1 m 0 \frac{1}{m-h+1}\leq\frac{1}{m_{0}}
  23. h h
  24. P ( h ) α m + 1 - h P_{(h)}\leq\frac{\alpha}{m+1-h}
  25. α m 0 \frac{\alpha}{m_{0}}
  26. α m 0 \frac{\alpha}{m_{0}}
  27. A = { P i α m 0 for some i I 0 } A=\left\{P_{i}\leq\frac{\alpha}{m_{0}}\,\text{ for some }i\in I_{0}\right\}
  28. I 0 I_{0}
  29. Pr ( A ) α \Pr(A)\leq\alpha
  30. α \alpha
  31. m m
  32. 2 m 2^{m}
  33. H i H_{i}
  34. H 1 , , H m H_{1},...,H_{m}
  35. α \alpha
  36. H i H_{i}
  37. α \alpha
  38. H ( 1 ) H_{(1)}
  39. i = 1 m < m t p l > H i \bigcap\nolimits_{i=1}^{m}<mtpl>{{H_{i}}}
  40. H 1 , , H m H_{1},...,H_{m}
  41. H ( 1 ) H_{(1)}
  42. α / m \alpha/m
  43. H ( 1 ) H_{(1)}
  44. P ( 1 ) P_{(1)}
  45. m m
  46. α / m \alpha/m
  47. H ( 2 ) H_{(2)}
  48. H ( 1 ) H_{(1)}
  49. H ( 2 ) H_{(2)}
  50. H ( 1 ) H_{(1)}
  51. P ( 2 ) α / ( m - 1 ) P_{(2)}\leq\alpha/(m-1)
  52. H ( 2 ) H_{(2)}
  53. 1 i m 1\leq i\leq m
  54. H 1 , , H 4 H_{1},...,H_{4}
  55. p 1 = 0.01 p_{1}=0.01
  56. p 2 = 0.04 p_{2}=0.04
  57. p 3 = 0.03 p_{3}=0.03
  58. p 4 = 0.005 p_{4}=0.005
  59. α = 0.05 \alpha=0.05
  60. H 4 = H ( 1 ) H_{4}=H_{(1)}
  61. p 4 = p ( 1 ) = 0.005 p_{4}=p_{(1)}=0.005
  62. α / 4 = 0.0125 \alpha/4=0.0125
  63. p 1 = p ( 2 ) = 0.01 < 0.0167 = α / 3 p_{1}=p_{(2)}=0.01<0.0167=\alpha/3
  64. H 1 = H ( 2 ) H_{1}=H_{(2)}
  65. H 3 H_{3}
  66. p 3 = p ( 3 ) = 0.03 > 0.025 = α / 2 p_{3}=p_{(3)}=0.03>0.025=\alpha/2
  67. H 1 H_{1}
  68. H 4 H_{4}
  69. H 2 H_{2}
  70. H 3 H_{3}
  71. α = 0.05 \alpha=0.05
  72. p 2 = p ( 4 ) = 0.04 < 0.05 = α p_{2}=p_{(4)}=0.04<0.05=\alpha
  73. H 2 H_{2}
  74. p ~ ( i ) = max j i { ( m - j + 1 ) p ( j ) } 1 \widetilde{p}_{(i)}=\max_{j\leq i}\left\{(m-j+1)p_{(j)}\right\}_{1}
  75. { x } 1 min ( x , 1 ) \{x\}_{1}\equiv\min(x,1)
  76. p ~ 1 = 0.03 \widetilde{p}_{1}=0.03
  77. p ~ 2 = 0.06 \widetilde{p}_{2}=0.06
  78. p ~ 3 = 0.06 \widetilde{p}_{3}=0.06
  79. p ~ 4 = 0.02 \widetilde{p}_{4}=0.02
  80. H 1 H_{1}
  81. H 4 H_{4}
  82. α = 0.05 \alpha=0.05
  83. α m , α m - 1 , , α 1 \frac{\alpha}{m},\frac{\alpha}{m-1},...,\frac{\alpha}{1}
  84. 1 - ( 1 - α ) 1 / m , 1 - ( 1 - α ) 1 / ( m - 1 ) , , 1 - ( 1 - α ) 1 1-(1-\alpha)^{1/m},1-(1-\alpha)^{1/(m-1)},...,1-(1-\alpha)^{1}
  85. P ( 1 ) , , P ( m ) P_{(1)},...,P_{(m)}
  86. H ( i ) H_{(i)}
  87. 0 w ( i ) 0\leq w_{(i)}
  88. P ( i ) P_{(i)}
  89. H ( i ) H_{(i)}
  90. P ( j ) w ( j ) k = j m w ( k ) α , j = 1 , , i P_{(j)}\leq\frac{w_{(j)}}{\sum^{m}_{k=j}{w_{(k)}}}\alpha,\quad j=1,...,i
  91. p ~ ( i ) = max j i { k = j m w ( k ) w ( j ) p ( j ) } 1 \widetilde{p}_{(i)}=\max_{j\leq i}\left\{\frac{\sum^{m}_{k=j}{w_{(k)}}}{w_{(j)% }}p_{(j)}\right\}_{1}
  92. { x } 1 min ( x , 1 ) \{x\}_{1}\equiv\min(x,1)
  93. H ( 1 ) H ( k ) H_{(1)}\ldots H_{(k)}
  94. k k
  95. P ( k ) α m + 1 - k P_{(k)}\leq\frac{\alpha}{m+1-k}

Holonomic_constraints.html

  1. f ( q 1 , q 2 , q 3 , , q n , t ) = 0 f(q_{1},q_{2},q_{3},\ldots,q_{n},t)=0
  2. { q 1 , q 2 , q 3 , , q n } \{q_{1},q_{2},q_{3},\ldots,q_{n}\}
  3. f ( q 1 , q 2 , , q n , q ˙ 1 , q ˙ 2 , , q ˙ n , t ) = 0 f(q_{1},q_{2},...,q_{n},\dot{q}_{1},\dot{q}_{2},...,\dot{q}_{n},t)=0
  4. f ( x 1 , x 2 , x 3 , , x N , t ) = 0 , f(x_{1},\ x_{2},\ x_{3},\ \dots,\ x_{N},\ t)=0,\,
  5. x j x_{j}\,\!
  6. t t\,\!
  7. x d x_{d}\,\!
  8. f i f_{i}\,\!
  9. x d = g i ( x 1 , x 2 , x 3 , , x d - 1 , x d + 1 , , x N , t ) , x_{d}=g_{i}(x_{1},\ x_{2},\ x_{3},\ \dots,\ x_{d-1},\ x_{d+1},\ \dots,\ x_{N},% \ t),\,
  10. x d x_{d}\,\!
  11. f i f_{i}\,\!
  12. C 1 C^{1}\,\!
  13. g i g_{i}\,
  14. x d x_{d}\,\!
  15. N N\,\!
  16. h h\,\!
  17. m = N - h m=N-h\,\!
  18. m m\,\!
  19. q j q_{j}\,\!
  20. x i = x i ( q 1 , q 2 , , q m , t ) , i = 1 , 2 , N . x_{i}=x_{i}(q_{1},\ q_{2},\ \dots,\ q_{m},\ t)\ ,\qquad\qquad\qquad i=1,\ 2,\ % \dots N.\,
  21. j c i j d q j + c i d t = 0 ; \sum_{j}\ c_{ij}dq_{j}+c_{i}dt=0;\,
  22. f i ( q 1 , q 2 , q 3 , , q N , t ) = 0 f_{i}(q_{1},\ q_{2},\ q_{3},\ \dots,\ q_{N},\ t)=0\,\!
  23. d f i = j c i j d q j + c i d t = 0 , df_{i}=\sum_{j}\ c_{ij}dq_{j}+c_{i}dt=0,\,
  24. x 2 + y 2 - L 2 = 0 , {x^{2}+y^{2}}-L^{2}=0,
  25. ( x , y ) (x,\ y)\,\!
  26. L L\,\!
  27. ( 𝐫 i - 𝐫 j ) 2 - L i j 2 = 0 , (\mathbf{r}_{i}-\mathbf{r}_{j})^{2}-L_{ij}^{2}=0,\,
  28. 𝐫 i \mathbf{r}_{i}\,\!
  29. 𝐫 j \mathbf{r}_{j}\,\!
  30. P i P_{i}\,\!
  31. P j P_{j}\,\!
  32. L i j L_{ij}\,\!

Holstein–Primakoff_transformation.html

  1. S x S_{x}
  2. S y S_{y}
  3. S z S_{z}
  4. [ S x , S y ] = i S z \left[S_{x},S_{y}\right]=i\hbar S_{z}
  5. S 2 S^{2}
  6. S z S_{z}
  7. | s , m s \left|s,m_{s}\right\rangle
  8. S 2 | s , m s = 2 s ( s + 1 ) | s , m s , S^{2}\left|s,m_{s}\right\rangle=\hbar^{2}s(s+1)\left|s,m_{s}\right\rangle,
  9. S z | s , m s = m s | s , m s . S_{z}\left|s,m_{s}\right\rangle=\hbar m_{s}\left|s,m_{s}\right\rangle.
  10. m s m_{s}
  11. - s , - s + 1 , , s - 1 , s -s,-s+1,\ldots,s-1,s
  12. s s
  13. | s , m s = + s \left|s,m_{s}=+s\right\rangle
  14. | s , s - n 1 n ! ( a ) n | 0 ) B \left|s,s-n\right\rangle\mapsto\frac{1}{\sqrt{n!}}\left(a^{\dagger}\right)^{n}% |0)_{B}
  15. \hbar
  16. S + = S x + i S y S_{+}=S_{x}+iS_{y}
  17. S - = S x - i S y S_{-}=S_{x}-iS_{y}
  18. S + = 2 s 1 - a a 2 s a S_{+}=\hbar\sqrt{2s}\sqrt{1-\frac{a^{\dagger}a}{2s}}\,a
  19. S - = 2 s a 1 - a a 2 s S_{-}=\hbar\sqrt{2s}a^{\dagger}\,\sqrt{1-\frac{a^{\dagger}a}{2s}}
  20. S z = ( s - a a ) S_{z}=\hbar(s-a^{\dagger}a)
  21. s s
  22. s s
  23. 2 s 2s
  24. S ± S_{\pm}
  25. S ± S_{\pm}

Homogeneous_tree.html

  1. Y × Z Y\times Z
  2. μ s s Y < ω \langle\mu_{s}\mid s\in{}^{<\omega}Y\rangle
  3. μ s \mu_{s}
  4. { t s , t T } \{t\mid\langle s,t\rangle\in T\}
  5. s 1 s 2 s_{1}\subseteq s_{2}
  6. μ s 1 ( X ) = 1 μ s 2 ( { t t l h ( s 1 ) X } ) = 1 \mu_{s_{1}}(X)=1\iff\mu_{s_{2}}(\{t\mid t\upharpoonright lh(s_{1})\in X\})=1
  7. x x
  8. T T
  9. μ x n n ω \langle\mu_{x\upharpoonright n}\mid n\in\omega\rangle
  10. μ s s Y ω \langle\mu_{s}\mid s\in{}^{\omega}Y\rangle
  11. x x
  12. [ T ] [T]
  13. n ω μ x n ( X n ) = 1 \forall n\in\omega\,\mu_{x\upharpoonright n}(X_{n})=1
  14. f Z ω f\in{}^{\omega}Z
  15. n ω f n X n \forall n\in\omega\,f\upharpoonright n\in X_{n}
  16. T T
  17. κ \kappa
  18. μ s \mu_{s}
  19. κ \kappa

Homogeneously_Suslin_set.html

  1. S S
  2. S S
  3. κ \kappa
  4. κ \kappa
  5. A ω ω A\subseteq{}^{\omega}\omega
  6. 𝚷 1 1 \mathbf{\Pi}_{1}^{1}
  7. κ \kappa
  8. A A
  9. κ \kappa
  10. 𝚷 1 1 \mathbf{\Pi}_{1}^{1}

Homomorphic_secret_sharing.html

  1. k k
  2. P ( x ) P(x)

Homotopy_category_of_chain_complexes.html

  1. h n : A n B n - 1 h^{n}\colon A^{n}\to B^{n-1}
  2. f n - g n = d B n - 1 h n + h n + 1 d A n , f^{n}-g^{n}=d_{B}^{n-1}h^{n}+h^{n+1}d_{A}^{n},
  3. f - g = d B h + h d A . f-g=d_{B}h+hd_{A}.
  4. f - g f-g
  5. f g f\sim g
  6. Hom K ( A ) ( A , B ) = Hom K o m ( A ) ( A , B ) / \operatorname{Hom}_{K(A)}(A,B)=\operatorname{Hom}_{Kom(A)}(A,B)/\sim
  7. f : A B f:A\rightarrow B
  8. g : B A g:B\rightarrow A
  9. f g I d B f\circ g\sim Id_{B}
  10. g f I d A g\circ f\sim Id_{A}
  11. K ( A ) D ( A ) K(A)\rightarrow D(A)
  12. A [ 1 ] : A n + 1 d A [ 1 ] n A n + 2 A[1]:...\to A^{n+1}\xrightarrow{d_{A[1]}^{n}}A^{n+2}\to...
  13. ( A [ 1 ] ) n = A n + 1 (A[1])^{n}=A^{n+1}
  14. d A [ 1 ] n := - d A n + 1 d_{A[1]}^{n}:=-d_{A}^{n+1}
  15. A 𝑓 B C ( f ) A [ 1 ] A\xrightarrow{f}B\to C(f)\to A[1]
  16. X i d X 0 X\xrightarrow{id}X\to 0\to
  17. C ( i d ) 0 C(id)\to 0
  18. H o m H o C ( X , Y ) = H 0 H o m C ( X , Y ) Hom_{HoC}(X,Y)=H^{0}Hom_{C}(X,Y)

Horocycle.html

  1. y = y=\infty

Hot_band.html

  1. ν 1 \nu_{1}
  2. ν 2 \nu_{2}
  3. ν 3 \nu_{3}
  4. 101 101
  5. 001 001
  6. ν 3 \nu_{3}
  7. N N 0 = e - E / k B T {{N}\over{N_{0}}}={{e^{-E/k_{B}T}}}
  8. N N 0 = e - ν / 0.6952 T {{N}\over{N_{0}}}={{e^{-\nu/0.6952T}}}
  9. 010 010
  10. 100 100
  11. 100 100
  12. 020 020
  13. 101 101
  14. 000 000
  15. 012 012
  16. 001 001

Hot_chocolate_effect.html

  1. f = 0.25 v h f=0.25\frac{v}{h}
  2. ρ \rho
  3. K K
  4. c = K ρ c=\sqrt{\frac{K}{\rho}}

HPN.html

  1. H p n Hp_{n}

Hubbert_linearization.html

  1. d Q d t = P = K Q ( 1 - Q U R R ) (1) \frac{dQ}{dt}=P=KQ\left(1-\frac{Q}{URR}\right)\qquad\mbox{(1)}~{}\!
  2. P Q = K ( 1 - Q U R R ) (2) \frac{P}{Q}=K\left(1-\frac{Q}{URR}\right)\qquad\mbox{(2)}~{}\!
  3. d P d t 1 P = K ( 1 - 2 Q U R R ) (3) \frac{dP}{dt}\frac{1}{P}=K\left(1-2\frac{Q}{URR}\right)\qquad\mbox{(3)}~{}\!
  4. P = K Q - K U R R Q 2 (4) P=KQ-\frac{K}{URR}Q^{2}\qquad\mbox{(4)}~{}\!

Huber's_equation.html

  1. σ r e d = ( σ 2 ) + 3 ( τ 2 ) \sigma_{red}=\sqrt{({\sigma}^{2})+3({\tau}^{2})}
  2. σ \sigma
  3. τ \tau
  4. σ r e d \sigma_{red}

Hutchinson_metric.html

  1. X X\,
  2. P ( X ) P(X)\,
  3. X X\,
  4. δ : X P ( X ) \delta:X\rightarrow P(X)\,
  5. x X x\in X
  6. δ x \delta_{x}\,
  7. | μ | |\mu|\,
  8. f : X 1 X 2 f:X_{1}\rightarrow X_{2}\,
  9. f * : P ( X 1 ) P ( X 2 ) f_{*}:P(X_{1})\rightarrow P(X_{2})\,
  10. μ \mu\,
  11. f * ( μ ) f_{*}(\mu)\,
  12. f * ( μ ) ( B ) = μ ( f - 1 ( B ) ) f_{*}(\mu)(B)=\mu(f^{-1}(B))\,
  13. B B\,
  14. X 2 X_{2}\,
  15. d ( μ 1 , μ 2 ) = sup { u ( x ) μ 1 ( d x ) - u ( x ) μ 2 ( d x ) } d(\mu_{1},\mu_{2})=\sup\left\{\int u(x)\,\mu_{1}(dx)-\int u(x)\,\mu_{2}(dx)\right\}
  16. sup \sup
  17. 1 . \leq 1\,.
  18. δ \delta\,
  19. X X\,
  20. P ( X ) P(X)\,
  21. f : X 1 X 2 f:X_{1}\rightarrow X_{2}\,
  22. f * : P ( X 1 ) P ( X 2 ) f_{*}:P(X_{1})\rightarrow P(X_{2})\,

Hybrid_functional.html

  1. E x HF E_{x}^{\rm HF}
  2. E x HF = - 1 2 i , j ψ i * ( 𝐫 𝟏 ) ψ j * ( 𝐫 𝟏 ) 1 r 12 ψ i ( 𝐫 𝟐 ) ψ j ( 𝐫 𝟐 ) d 𝐫 𝟏 d 𝐫 𝟐 E_{\rm x}^{\rm HF}=-\frac{1}{2}\sum_{i,j}\int\int\psi_{i}^{*}(\mathbf{r_{1}})% \psi_{j}^{*}(\mathbf{r_{1}})\frac{1}{r_{12}}\psi_{i}(\mathbf{r_{2}})\psi_{j}(% \mathbf{r_{2}})d\mathbf{r_{1}}d\mathbf{r_{2}}
  3. E xc B3LYP = E x LDA + a 0 ( E x HF - E x LDA ) + a x ( E x GGA - E x LDA ) + E c LDA + a c ( E c GGA - E c LDA ) , E_{\rm xc}^{\rm B3LYP}=E_{\rm x}^{\rm LDA}+a_{0}(E_{\rm x}^{\rm HF}-E_{\rm x}^% {\rm LDA})+a_{\rm x}(E_{\rm x}^{\rm GGA}-E_{\rm x}^{\rm LDA})+E_{\rm c}^{\rm LDA% }+a_{\rm c}(E_{\rm c}^{\rm GGA}-E_{\rm c}^{\rm LDA}),
  4. a 0 = 0.20 a_{0}=0.20\,\;
  5. a x = 0.72 a_{\rm x}=0.72\,\;
  6. a c = 0.81 a_{\rm c}=0.81\,\;
  7. E x GGA E_{\rm x}^{\rm GGA}
  8. E c GGA E_{\rm c}^{\rm GGA}
  9. E c LDA E_{\rm c}^{\rm LDA}
  10. E xc PBE0 = 1 4 E x HF + 3 4 E x PBE + E c PBE , E_{\rm xc}^{\rm PBE0}=\frac{1}{4}E_{\rm x}^{\rm HF}+\frac{3}{4}E_{\rm x}^{\rm PBE% }+E_{\rm c}^{\rm PBE},
  11. E x HF E_{\rm x}^{\rm HF}
  12. E x PBE E_{\rm x}^{\rm PBE}
  13. E c PBE E_{\rm c}^{\rm PBE}
  14. E xc ω PBEh = a E x HF , SR ( ω ) + ( 1 - a ) E x PBE , SR ( ω ) + E x PBE , LR ( ω ) + E c PBE , E_{\rm xc}^{\rm\omega PBEh}=aE_{\rm x}^{\rm HF,SR}(\omega)+(1-a)E_{\rm x}^{\rm PBE% ,SR}(\omega)+E_{\rm x}^{\rm PBE,LR}(\omega)+E_{\rm c}^{\rm PBE},
  15. a a
  16. ω \omega
  17. a = 1 4 a=\frac{1}{4}
  18. ω = 0.2 \omega=0.2
  19. ω = 0 \omega=0
  20. E x HF , SR ( ω ) E_{\rm x}^{\rm HF,SR}(\omega)
  21. E x PBE , SR ( ω ) E_{\rm x}^{\rm PBE,SR}(\omega)
  22. E x PBE , LR ( ω ) E_{\rm x}^{\rm PBE,LR}(\omega)
  23. E c PBE ( ω ) E_{\rm c}^{\rm PBE}(\omega)

Hydraulic_pump.html

  1. η v 90 % \eta_{v}\approx 90\%
  2. Q = n V s t r o k e η v o l Q=n\cdot V_{stroke}\cdot\eta_{vol}
  3. Q = Flow in cubic meter per second [ m 3 s ] n = revolution per second [ r e v s ] V s t r o k e = Swept volume in cubic meters [ m 3 r e v ] η v o l = Volumetric efficiency [ ] \begin{aligned}\displaystyle Q&\displaystyle=\,\text{Flow in cubic meter per % second }&\displaystyle\left[\frac{m^{3}}{s}\right]\\ \displaystyle n&\displaystyle=\,\text{revolution per second}&\displaystyle% \left[\frac{rev}{s}\right]\\ \displaystyle V_{stroke}&\displaystyle=\,\text{Swept volume in cubic meters}&% \displaystyle\left[\frac{m^{3}}{rev}\right]\\ \displaystyle\eta_{vol}&\displaystyle=\,\text{Volumetric efficiency}&% \displaystyle\left[\right]\end{aligned}
  4. P = n V s t r o k e Δ p η m e c h P={n\cdot V_{stroke}\cdot\Delta p\over~{}\eta_{mech}}
  5. P = Power in Watt [ N m s ] n = Revolution per second [ r e v s ] V s t r o k e = swept volume [ m 3 r e v ] Δ p = pressure difference over pump [ N m 2 ] η m e c h , h y d r = Mechanical/hydraulic efficiency [ ] \begin{aligned}\displaystyle P&\displaystyle=\,\text{Power in Watt}&% \displaystyle\left[\frac{Nm}{s}\right]\\ \displaystyle n&\displaystyle=\,\text{Revolution per second}&\displaystyle% \left[\frac{rev}{s}\right]\\ \displaystyle V_{stroke}&\displaystyle=\,\text{swept volume}&\displaystyle% \left[\frac{m^{3}}{rev}\right]\\ \displaystyle\Delta p&\displaystyle=\,\text{pressure difference over pump}&% \displaystyle\left[\frac{N}{m^{2}}\right]\\ \displaystyle\eta_{mech,hydr}&\displaystyle=\,\text{Mechanical/hydraulic % efficiency}&\displaystyle\left[\right]\end{aligned}
  6. n m e c h = T a c t u a l 100 T t h e o r e t i c a l n_{mech}={T_{actual}\cdot 100\over T_{theoretical}}
  7. n m e c h = Mechanical pump efficiency percent T t h e o r e t i c a l = Theoretical torque to drive T a c t u a l = Actual torque to drive \begin{aligned}\displaystyle n_{mech}&\displaystyle=\text{Mechanical pump % efficiency percent}\\ \displaystyle T_{theoretical}&\displaystyle=\text{Theoretical torque to drive}% \\ \displaystyle T_{actual}&\displaystyle=\text{Actual torque to drive}\\ \end{aligned}
  8. n h y d r = Q a c t u a l 100 Q t h e o r e t i c a l n_{hydr}={Q_{actual}\cdot 100\over Q_{theoretical}}
  9. n h y d r = Hydraulic pump efficiency percent Q t h e o r e t i c a l = Theoretical flow rate output Q a c t u a l = Actual flow rate output \begin{aligned}\displaystyle n_{hydr}&\displaystyle=\text{Hydraulic pump % efficiency percent}\\ \displaystyle Q_{theoretical}&\displaystyle=\text{Theoretical flow rate output% }\\ \displaystyle Q_{actual}&\displaystyle=\text{Actual flow rate output}\\ \end{aligned}

Hydraulophone.html

  1. ( Δ w ) (\Delta w)
  2. ( Δ t ) (\Delta t)
  3. 𝐱 ¯ = Δ 𝐰 Δ t . \bar{\mathbf{x}}=\frac{\Delta\mathbf{w}}{\Delta t}.

Hydrogen-like_atom.html

  1. e ( Z - 1 ) e(Z-1)
  2. Z Z
  3. V ( r ) = - 1 4 π ϵ 0 Z e 2 r V(r)=-\frac{1}{4\pi\epsilon_{0}}\frac{Ze^{2}}{r}
  4. ψ ( r , θ , ϕ ) = R n l ( r ) Y l m ( θ , ϕ ) \psi(r,\theta,\phi)=R_{nl}(r)Y_{lm}(\theta,\phi)\,
  5. Y l m Y_{lm}
  6. [ - 2 2 μ ( 1 r 2 r ( r 2 R ( r ) r ) - l ( l + 1 ) R ( r ) r 2 ) + V ( r ) R ( r ) ] = E R ( r ) , \left[-\frac{\hbar^{2}}{2\mu}\left({1\over r^{2}}{\partial\over\partial r}% \left(r^{2}{\partial R(r)\over\partial r}\right)-{l(l+1)R(r)\over r^{2}}\right% )+V(r)R(r)\right]=ER(r),
  7. μ \mu
  8. \hbar
  9. - l m l -l\leq m\leq l
  10. ψ n l m = R n l ( r ) Y l m ( θ , ϕ ) \psi_{nlm}=R_{nl}(r)\,Y_{lm}(\theta,\phi)
  11. R n l ( r ) = ( 2 Z n a μ ) 3 ( n - l - 1 ) ! 2 n ( n + l ) ! e - Z r / n a μ ( 2 Z r n a μ ) l L n - l - 1 2 l + 1 ( 2 Z r n a μ ) R_{nl}(r)=\sqrt{{\left(\frac{2Z}{na_{\mu}}\right)}^{3}\frac{(n-l-1)!}{2n(n+l)!% }}e^{-Zr/{na_{\mu}}}\left(\frac{2Zr}{na_{\mu}}\right)^{l}L_{n-l-1}^{2l+1}\left% (\frac{2Zr}{na_{\mu}}\right)
  12. L n - l - 1 2 l + 1 L_{n-l-1}^{2l+1}
  13. a μ = 4 π ε 0 2 μ e 2 = c α μ c 2 = m e μ a 0 a_{\mu}={{4\pi\varepsilon_{0}\hbar^{2}}\over{\mu e^{2}}}=\frac{\hbar c}{\alpha% \mu c^{2}}={{m_{\mathrm{e}}}\over{\mu}}a_{0}
  14. μ \mu
  15. μ = m N m e m N + m e \mu={{m_{\mathrm{N}}m_{\mathrm{e}}}\over{m_{\mathrm{N}}+m_{\mathrm{e}}}}
  16. m N m_{\mathrm{N}}
  17. μ m e . \mu\approx m_{\mathrm{e}}.
  18. μ = m e / 2. \mu=m_{\mathrm{e}}/2.
  19. E n = - ( Z 2 μ e 4 32 π 2 ϵ 0 2 2 ) 1 n 2 = - ( Z 2 2 2 μ a μ 2 ) 1 n 2 = - μ c 2 Z 2 α 2 2 n 2 . E_{n}=-\left(\frac{Z^{2}\mu e^{4}}{32\pi^{2}\epsilon_{0}^{2}\hbar^{2}}\right)% \frac{1}{n^{2}}=-\left(\frac{Z^{2}\hbar^{2}}{2\mu a_{\mu}^{2}}\right)\frac{1}{% n^{2}}=-\frac{\mu c^{2}Z^{2}\alpha^{2}}{2n^{2}}.
  20. Y l m ( θ , ϕ ) Y_{lm}(\theta,\phi)\,
  21. ( - 1 ) l {\left({-1}\right)}^{l}
  22. n = 1 , 2 , 3 , 4 , n=1,2,3,4,\dots\,
  23. l = 0 , 1 , 2 , , n - 1 l=0,1,2,\dots,n-1\,
  24. m = - l , - l + 1 , , 0 , , l - 1 , l m=-l,-l+1,\ldots,0,\ldots,l-1,l\,
  25. l < n l<n\,
  26. - l m l -l\leq m\leq\,l
  27. L 2 Y l m = 2 l ( l + 1 ) Y l m L^{2}Y_{lm}=\hbar^{2}l(l+1)Y_{lm}
  28. L z Y l m = m Y l m , L_{\mathrm{z}}Y_{lm}=\hbar mY_{lm},
  29. k = { - j - 1 2 if j = + 1 2 j + 1 2 if j = - 1 2 k=\begin{cases}-j-\tfrac{1}{2}&\,\text{if }j=\ell+\tfrac{1}{2}\\ j+\tfrac{1}{2}&\,\text{if }j=\ell-\tfrac{1}{2}\end{cases}
  30. E n j = μ c 2 ( 1 + [ Z α n - | k | + k 2 - Z 2 α 2 ] 2 ) - 1 / 2 μ c 2 { 1 - Z 2 α 2 2 n 2 [ 1 + Z 2 α 2 n ( 1 | k | - 3 4 n ) ] } \begin{array}[]{rl}E_{n\,j}&=\mu c^{2}\left(1+\left[\dfrac{Z\alpha}{n-|k|+% \sqrt{k^{2}-Z^{2}\alpha^{2}}}\right]^{2}\right)^{-1/2}\\ &\\ &\approx\mu c^{2}\left\{1-\dfrac{Z^{2}\alpha^{2}}{2n^{2}}\left[1+\dfrac{Z^{2}% \alpha^{2}}{n}\left(\dfrac{1}{|k|}-\dfrac{3}{4n}\right)\right]\right\}\end{array}
  31. Ψ = ( g n , k ( r ) r - 1 Ω k , m ( θ , ϕ ) i f n , k ( r ) r - 1 Ω - k , m ( θ , ϕ ) ) = ( g n , k ( r ) r - 1 ( k + 1 2 - m ) / ( 2 k + 1 ) Y k , m - 1 / 2 ( θ , ϕ ) - g n , k ( r ) r - 1 sgn k ( k + 1 2 + m ) / ( 2 k + 1 ) Y k , m + 1 / 2 ( θ , ϕ ) i f n , k ( r ) r - 1 ( - k + 1 2 - m ) / ( - 2 k + 1 ) Y - k , m - 1 / 2 ( θ , ϕ ) - i f n , k ( r ) r - 1 sgn k ( - k + 1 2 - m ) / ( - 2 k + 1 ) Y - k , m + 1 / 2 ( θ , ϕ ) ) \Psi=\begin{pmatrix}g_{n,k}(r)r^{-1}\Omega_{k,m}(\theta,\phi)\\ if_{n,k}(r)r^{-1}\Omega_{-k,m}(\theta,\phi)\end{pmatrix}=\begin{pmatrix}g_{n,k% }(r)r^{-1}\sqrt{(k+\tfrac{1}{2}-m)/(2k+1)}Y_{k,m-1/2}(\theta,\phi)\\ -g_{n,k}(r)r^{-1}\operatorname{sgn}k\sqrt{(k+\tfrac{1}{2}+m)/(2k+1)}Y_{k,m+1/2% }(\theta,\phi)\\ if_{n,k}(r)r^{-1}\sqrt{(-k+\tfrac{1}{2}-m)/(-2k+1)}Y_{-k,m-1/2}(\theta,\phi)\\ -if_{n,k}(r)r^{-1}\operatorname{sgn}k\sqrt{(-k+\tfrac{1}{2}-m)/(-2k+1)}Y_{-k,m% +1/2}(\theta,\phi)\end{pmatrix}
  32. Y a , b ( θ , ϕ ) Y_{a,b}(\theta,\phi)
  33. Y a , b = { ( - 1 ) b 2 a + 1 4 π ( a - b ) ! ( a + b ) ! P a b ( cos θ ) e i b ϕ if a > 0 Y - a - 1 , b if a < 0 Y_{a,b}=\begin{cases}(-1)^{b}\sqrt{\frac{2a+1}{4\pi}\frac{(a-b)!}{(a+b)!}}P_{a% }^{b}(\cos\theta)e^{ib\phi}&\,\text{if }a>0\\ Y_{-a-1,b}&\,\text{if }a<0\end{cases}
  34. P a b P_{a}^{b}
  35. Y 0 , 1 Y_{0,1}
  36. Ω - 1 , - 1 / 2 ( 0 1 ) \Omega_{-1,-1/2}\propto{\left({{0}\atop{1}}\right)}
  37. Ω - 1 , 1 / 2 ( 1 0 ) \Omega_{-1,1/2}\propto{\left({{1}\atop{0}}\right)}
  38. Ω 1 , - 1 / 2 ( ( x - i y ) / r z / r ) \Omega_{1,-1/2}\propto{\left({{(x-iy)/r}\atop{z/r}}\right)}
  39. Ω 1 , 1 / 2 ( z / r ( x + i y ) / r ) \Omega_{1,1/2}\propto{\left({{z/r}\atop{(x+iy)/r}}\right)}
  40. Ω k , m = ( z / r ( x - i y ) / r ( x + i y ) / r - z / r ) Ω - k , m \Omega_{k,m}=\begin{pmatrix}z/r&(x-iy)/r\\ (x+iy)/r&-z/r\end{pmatrix}\Omega_{-k,m}
  41. g n , k ( r ) g_{n,k}(r)
  42. f n , k ( r ) f_{n,k}(r)
  43. ρ 2 C r \rho\equiv 2Cr
  44. C = μ 2 c 4 - E 2 c C=\frac{\sqrt{\mu^{2}c^{4}-E^{2}}}{\hbar c}
  45. E n j E_{n\,j}
  46. γ k 2 - Z 2 α 2 \gamma\equiv\sqrt{k^{2}-Z^{2}\alpha^{2}}
  47. g n , k ( r ) g_{n,k}(r)
  48. f n , k ( r ) f_{n,k}(r)
  49. g n , - n ( r ) = A ( n + γ ) ρ γ e - ρ / 2 g_{n,-n}(r)=A(n+\gamma)\rho^{\gamma}e^{-\rho/2}
  50. f n , - n ( r ) = A Z α ρ γ e - ρ / 2 f_{n,-n}(r)=AZ\alpha\rho^{\gamma}e^{-\rho/2}
  51. A = 1 2 n ( n + γ ) C γ Γ ( 2 γ ) A=\frac{1}{\sqrt{2n(n+\gamma)}}\sqrt{\frac{C}{\gamma\Gamma(2\gamma)}}
  52. E n , n - 1 / 2 = γ n μ c 2 = 1 - Z 2 α 2 n 2 μ c 2 E_{n,n-1/2}=\frac{\gamma}{n}\mu c^{2}=\sqrt{1-\frac{Z^{2}\alpha^{2}}{n^{2}}}\,% \mu c^{2}
  53. C = Z α n μ c 2 c . C=\frac{Z\alpha}{n}\frac{\mu c^{2}}{\hbar c}.
  54. g n , k ( r ) and f n , k ( r ) g_{n,k}(r)\,\text{ and }f_{n,k}(r)
  55. g n , k ( r ) = A ρ γ e - ρ / 2 ( Z α ρ L n - | k | - 1 2 γ + 1 ( ρ ) + ( γ - k ) γ μ c 2 - k E c C L n - | k | 2 γ - 1 ( ρ ) ) g_{n,k}(r)=A\rho^{\gamma}e^{-\rho/2}\left(Z\alpha\rho L_{n-|k|-1}^{2\gamma+1}(% \rho)+(\gamma-k)\frac{\gamma\mu c^{2}-kE}{\hbar cC}L_{n-|k|}^{2\gamma-1}(\rho)\right)
  56. f n , k ( r ) = A ρ γ e - ρ / 2 ( ( γ - k ) ρ L n - | k | - 1 2 γ + 1 ( ρ ) + Z α γ μ c 2 - k E c C L n - | k | 2 γ - 1 ( ρ ) ) f_{n,k}(r)=A\rho^{\gamma}e^{-\rho/2}\left((\gamma-k)\rho L_{n-|k|-1}^{2\gamma+% 1}(\rho)+Z\alpha\frac{\gamma\mu c^{2}-kE}{\hbar cC}L_{n-|k|}^{2\gamma-1}(\rho)\right)
  57. A = 1 2 k ( k - γ ) C n - | k | + γ ( n - | k | - 1 ) ! Γ ( n - | k | + 2 γ + 1 ) 1 2 ( ( E k γ μ c 2 ) 2 + E k γ μ c 2 ) A=\frac{1}{\sqrt{2k(k-\gamma)}}\sqrt{\frac{C}{n-|k|+\gamma}\frac{(n-|k|-1)!}{% \Gamma(n-|k|+2\gamma+1)}\frac{1}{2}\left(\left(\frac{Ek}{\gamma\mu c^{2}}% \right)^{2}+\frac{Ek}{\gamma\mu c^{2}}\right)}
  58. Ψ ( ( 1 + γ ) r γ - 1 e - C r 0 i Z α r γ - 1 e - C r z / r i Z α r γ - 1 e - C r ( x + i y ) / r ) \Psi\propto\begin{pmatrix}(1+\gamma)r^{\gamma-1}e^{-Cr}\\ 0\\ iZ\alpha r^{\gamma-1}e^{-Cr}z/r\\ iZ\alpha r^{\gamma-1}e^{-Cr}(x+iy)/r\end{pmatrix}
  59. Ψ ( 0 ( 1 + γ ) r γ - 1 e - C r i Z α r γ - 1 e - C r ( x - i y ) / r - i Z α r γ - 1 e - C r z / r ) \Psi\propto\begin{pmatrix}0\\ (1+\gamma)r^{\gamma-1}e^{-Cr}\\ iZ\alpha r^{\gamma-1}e^{-Cr}(x-iy)/r\\ -iZ\alpha r^{\gamma-1}e^{-Cr}z/r\end{pmatrix}
  60. Ψ ( ( 1 + γ ) r γ - 1 e - C r ( 1 + γ ) r γ - 1 e - C r i Z α r γ - 1 e - C r ( x - i y + z ) / r i Z α r γ - 1 e - C r ( x + i y - z ) / r ) \Psi\propto\begin{pmatrix}(1+\gamma)r^{\gamma-1}e^{-Cr}\\ (1+\gamma)r^{\gamma-1}e^{-Cr}\\ iZ\alpha r^{\gamma-1}e^{-Cr}(x-iy+z)/r\\ iZ\alpha r^{\gamma-1}e^{-Cr}(x+iy-z)/r\end{pmatrix}
  61. Ψ ( ρ γ - 1 e - ρ / 2 ( Z α ρ + ( γ - 1 ) γ μ c 2 - E c C ( - ρ + 2 γ ) ) z / r ρ γ - 1 e - ρ / 2 ( Z α ρ + ( γ - 1 ) γ μ c 2 - E c C ( - ρ + 2 γ ) ) ( x + i y ) / r i ρ γ - 1 e - ρ / 2 ( ( γ - 1 ) ρ + Z α γ μ c 2 - E c C ( - ρ + 2 γ ) ) 0 ) \Psi\propto\begin{pmatrix}\rho^{\gamma-1}e^{-\rho/2}\left(Z\alpha\rho+(\gamma-% 1)\frac{\gamma\mu c^{2}-E}{\hbar cC}(-\rho+2\gamma)\right)z/r\\ \rho^{\gamma-1}e^{-\rho/2}\left(Z\alpha\rho+(\gamma-1)\frac{\gamma\mu c^{2}-E}% {\hbar cC}(-\rho+2\gamma)\right)(x+iy)/r\\ i\rho^{\gamma-1}e^{-\rho/2}\left((\gamma-1)\rho+Z\alpha\frac{\gamma\mu c^{2}-E% }{\hbar cC}(-\rho+2\gamma)\right)\\ 0\end{pmatrix}
  62. ρ = 2 r C \rho=2rC
  63. c / ( μ c 2 ) \hbar c/(\mu c^{2})
  64. Ψ ( ρ γ - 1 e - ρ / 2 ( Z α ρ + ( γ + 1 ) γ μ c 2 + E c C ( - ρ + 2 γ ) ) 0 i ρ γ - 1 e - ρ / 2 ( ( γ + 1 ) ρ + Z α γ μ c 2 + E c C ( - ρ + 2 γ ) ) z / r i ρ γ - 1 e - ρ / 2 ( ( γ + 1 ) ρ + Z α γ μ c 2 + E c C ( - ρ + 2 γ ) ) ( x + i y ) / r ) \Psi\propto\begin{pmatrix}\rho^{\gamma-1}e^{-\rho/2}\left(Z\alpha\rho+(\gamma+% 1)\frac{\gamma\mu c^{2}+E}{\hbar cC}(-\rho+2\gamma)\right)\\ 0\\ i\rho^{\gamma-1}e^{-\rho/2}\left((\gamma+1)\rho+Z\alpha\frac{\gamma\mu c^{2}+E% }{\hbar cC}(-\rho+2\gamma)\right)z/r\\ i\rho^{\gamma-1}e^{-\rho/2}\left((\gamma+1)\rho+Z\alpha\frac{\gamma\mu c^{2}+E% }{\hbar cC}(-\rho+2\gamma)\right)(x+iy)/r\end{pmatrix}
  65. x x
  66. y y
  67. z z

HydroGeoSphere.html

  1. - ( w m 𝐪 ) + Γ e q ± Q = w m t ( θ s S w ) -\nabla\cdot(w_{m}\,\textbf{q})+\sum\Gamma_{eq}\pm Q=w_{m}\frac{\partial}{% \partial t}(\theta_{s}S_{w})
  2. 𝐪 \,\textbf{q}
  3. 𝐪 = - 𝐊 k r ( ψ + z ) \,\textbf{q}=-\,\textbf{K}\cdot k_{r}\nabla(\psi+z)
  4. w m w_{m}
  5. Γ e x \Gamma_{ex}
  6. Q Q
  7. θ s \theta_{s}
  8. S w S_{w}
  9. 𝐊 \,\textbf{K}
  10. k r k_{r}
  11. ψ \psi
  12. z z
  13. ϕ o h o t + v ¯ x o d o x + v ¯ y o d o y + d o Γ o ± Q o = 0 \frac{\partial\phi_{o}h_{o}}{\partial t}+\frac{\partial\bar{v}_{xo}d_{o}}{% \partial x}+\frac{\partial\bar{v}_{yo}d_{o}}{\partial y}+d_{o}\Gamma_{o}\pm Q_% {o}=0
  14. v ¯ o x = - K o x h o x \bar{v}_{ox}=-K_{ox}\frac{\partial h_{o}}{\partial x}
  15. v ¯ o y = - K o y h o y \bar{v}_{oy}=-K_{oy}\frac{\partial h_{o}}{\partial y}
  16. ϕ o \phi_{o}
  17. h o h_{o}
  18. v ¯ x o \bar{v}_{xo}
  19. v ¯ y o \bar{v}_{yo}
  20. d o d_{o}
  21. Γ o \Gamma_{o}
  22. Q o Q_{o}
  23. K o x K_{ox}
  24. K o y K_{oy}
  25. - w m ( 𝐪 C - θ s S w 𝐃 C ) + [ w m θ s S w R λ C ] p a r + Ω e x + Q c = w m [ ( θ s S w R C ) t + θ s S w R λ C ] -\nabla\cdot w_{m}(\,\textbf{q}C-\theta_{s}S_{w}\,\textbf{D}\nabla C)+[w_{m}% \theta_{s}S_{w}R\lambda C]_{par}+\sum{\Omega_{ex}+Q_{c}}=w_{m}\left[\frac{% \partial(\theta_{s}S_{w}RC)}{\partial t}+\theta_{s}S_{w}R\lambda C\right]
  26. C C
  27. λ \lambda
  28. Q c Q_{c}
  29. Ω \Omega
  30. R R
  31. 𝐃 \,\textbf{D}
  32. p a r par
  33. - ( 𝐪 ρ w c w T - ( k b + c b ρ b 𝐃 ) T ) + Ω o ± Q T = ρ b c b T t -\nabla\cdot\Big(\,\textbf{q}\rho_{w}c_{w}T-(k_{b}+c_{b}\rho_{b}\,\textbf{D})% \nabla T\Big)+\Omega_{o}\pm Q_{T}=\frac{\partial\rho_{b}c_{b}T}{\partial t}
  34. ρ \rho
  35. c c
  36. T T
  37. k k
  38. D D
  39. Q T Q_{T}
  40. Ω o \Omega_{o}
  41. Q T Q_{T}

Hylomorphism_(computer_science).html

  1. h : A C h:A\rightarrow C
  2. g : A B × A g:A\rightarrow B\times A
  3. B B
  4. p : A Boolean p:A\rightarrow\,\text{Boolean}
  5. c C c\in C
  6. : B × C C \oplus:B\times C\rightarrow C
  7. h a = { c if p a b h a otherwise h\,a=\begin{cases}c&\mbox{if }~{}p\,a\\ b\oplus ha^{\prime}&\mbox{otherwise}\end{cases}
  8. ( b , a ) = g a (b,a^{\prime})=ga
  9. p p
  10. g g
  11. h = [ [ ( c , ) , ( g , p ) ] ] h=[\![(c,\oplus),(g,p)]\!]
  12. factorial = [ [ ( 1 , × ) , ( g , p ) ] ] \,\text{factorial}=[\![(1,\times),(g,p)]\!]
  13. g n = ( n , n - 1 ) g\ n=(n,n-1)
  14. p n = ( n = 0 ) p\ n=(n=0)

Hyperarithmetical_theory.html

  1. Σ 1 1 \Sigma^{1}_{1}
  2. Π 1 1 \Pi^{1}_{1}
  3. Δ 1 1 \Delta^{1}_{1}
  4. Σ 1 1 \Sigma^{1}_{1}
  5. Π 1 1 \Pi^{1}_{1}
  6. Δ 1 1 \Delta^{1}_{1}
  7. Δ 1 1 \Delta^{1}_{1}
  8. , \langle\cdot,\cdot\rangle
  9. 1 , n \langle 1,n\rangle
  10. 2 , e \langle 2,e\rangle
  11. ϕ e \phi_{e}
  12. ϕ e ( n ) \phi_{e}(n)
  13. { λ n n } \{\lambda_{n}\mid n\in\mathbb{N}\}
  14. ω 1 C K \omega^{CK}_{1}
  15. ω 1 \omega_{1}
  16. 𝒪 \mathcal{O}
  17. 𝒪 \mathcal{O}
  18. 0 ( δ ) 0^{(\delta)}
  19. δ < ω 1 C K \delta<\omega^{CK}_{1}
  20. 0 ( δ ) 0^{(\delta)}
  21. 0 ( δ ) = 0 0^{(\delta)}=0
  22. 0 ( δ ) 0^{(\delta)}
  23. 0 ( λ ) 0^{(\lambda)}
  24. 0 0^{\prime}
  25. 0 ′′ 0^{\prime\prime}
  26. 0 ( 1 ) 0^{(1)}
  27. 0 ( 2 ) 0^{(2)}
  28. λ n n \langle\lambda_{n}\mid n\in\mathbb{N}\rangle
  29. 0 ( δ ) 0^{(\delta)}
  30. 0 ( δ ) = { n , i i 0 ( λ n ) } 0^{(\delta)}=\{\langle n,i\rangle\mid i\in 0^{(\lambda_{n})}\}
  31. 0 ( λ n ) 0^{(\lambda_{n})}
  32. 0 ( δ ) 0^{(\delta)}
  33. 0 ( δ ) 0^{(\delta)}
  34. 0 ( δ ) 0^{(\delta)}
  35. δ < ω 1 C K \delta<\omega^{CK}_{1}
  36. 0 ( δ ) 0^{(\delta)}
  37. E 2 : {}^{2}E\colon\mathbb{N}^{\mathbb{N}}\to\mathbb{N}
  38. E 2 ( f ) = 1 {}^{2}E(f)=1\quad
  39. E 2 ( f ) = 0 {}^{2}E(f)=0\quad
  40. E 2 {}^{2}E
  41. \mathbb{N}
  42. 0 ( ω ) 0^{(\omega)}
  43. Π 1 1 \Pi^{1}_{1}
  44. Π 1 1 \Pi^{1}_{1}
  45. Π 1 1 \Pi^{1}_{1}
  46. Π 1 1 \Pi^{1}_{1}
  47. \mathbb{N}^{\mathbb{N}}
  48. Π 1 1 \Pi^{1}_{1}
  49. 𝒪 \mathcal{O}
  50. ϕ e ( x , y ) \phi_{e}(x,y)
  51. × ) \mathbb{N}^{\mathbb{N}}\cong\mathbb{N}^{\mathbb{N}\times\mathbb{N}})
  52. Σ 1 1 \Sigma^{1}_{1}
  53. Σ 1 1 \Sigma^{1}_{1}
  54. α < ω 1 C K \alpha<\omega^{CK}_{1}
  55. α \alpha
  56. α < ω 1 C K \alpha<\omega^{CK}_{1}
  57. α \alpha
  58. 𝒪 \mathcal{O}
  59. 𝒪 X \mathcal{O}^{X}
  60. 𝒪 X \mathcal{O}^{X}
  61. ω 1 X \omega^{X}_{1}
  62. ω 1 C K \omega^{CK}_{1}
  63. 0 ( δ ) 0^{(\delta)}
  64. X X
  65. X ( 0 ) X^{(0)}
  66. X ( 1 ) = X X^{(1)}=X^{\prime}
  67. ω 1 C K \omega^{CK}_{1}
  68. ω 1 X \omega^{X}_{1}
  69. X H Y P Y X\leq_{HYP}Y
  70. δ < ω 1 Y \delta<\omega^{Y}_{1}
  71. Y ( δ ) Y^{(\delta)}
  72. X H Y P Y X\leq_{HYP}Y
  73. Y H Y P X Y\leq_{HYP}X
  74. X H Y P Y X\equiv_{HYP}Y
  75. 𝒪 X \mathcal{O}^{X}
  76. X < H Y P Y < H Y P 𝒪 X X<_{HYP}Y<_{HYP}\mathcal{O}^{X}
  77. ω 1 C K \omega^{CK}_{1}

Hypergeometric_function_of_a_matrix_argument.html

  1. p 0 p\geq 0
  2. q 0 q\geq 0
  3. X X
  4. m × m m\times m
  5. X X
  6. α > 0 \alpha>0
  7. F q ( α ) p ( a 1 , , a p ; b 1 , , b q ; X ) = k = 0 κ k 1 k ! ( a 1 ) κ ( α ) ( a p ) κ ( α ) ( b 1 ) κ ( α ) ( b q ) κ ( α ) C κ ( α ) ( X ) , {}_{p}F_{q}^{(\alpha)}(a_{1},\ldots,a_{p};b_{1},\ldots,b_{q};X)=\sum_{k=0}^{% \infty}\sum_{\kappa\vdash k}\frac{1}{k!}\cdot\frac{(a_{1})^{(\alpha)}_{\kappa}% \cdots(a_{p})_{\kappa}^{(\alpha)}}{(b_{1})_{\kappa}^{(\alpha)}\cdots(b_{q})_{% \kappa}^{(\alpha)}}\cdot C_{\kappa}^{(\alpha)}(X),
  8. κ k \kappa\vdash k
  9. κ \kappa
  10. k k
  11. ( a i ) κ ( α ) (a_{i})^{(\alpha)}_{\kappa}
  12. C κ ( α ) ( X ) C_{\kappa}^{(\alpha)}(X)
  13. X X
  14. Y Y
  15. m × m m\times m
  16. F q ( α ) p ( a 1 , , a p ; b 1 , , b q ; X , Y ) = k = 0 κ k 1 k ! ( a 1 ) κ ( α ) ( a p ) κ ( α ) ( b 1 ) κ ( α ) ( b q ) κ ( α ) C κ ( α ) ( X ) C κ ( α ) ( Y ) C κ ( α ) ( I ) , {}_{p}F_{q}^{(\alpha)}(a_{1},\ldots,a_{p};b_{1},\ldots,b_{q};X,Y)=\sum_{k=0}^{% \infty}\sum_{\kappa\vdash k}\frac{1}{k!}\cdot\frac{(a_{1})^{(\alpha)}_{\kappa}% \cdots(a_{p})_{\kappa}^{(\alpha)}}{(b_{1})_{\kappa}^{(\alpha)}\cdots(b_{q})_{% \kappa}^{(\alpha)}}\cdot\frac{C_{\kappa}^{(\alpha)}(X)C_{\kappa}^{(\alpha)}(Y)% }{C_{\kappa}^{(\alpha)}(I)},
  17. I I
  18. m m
  19. α \alpha
  20. α \alpha
  21. α \alpha
  22. α = 2 \alpha=2
  23. α = 1 \alpha=1
  24. β \beta
  25. α \alpha
  26. α = 2 β . \alpha=\frac{2}{\beta}.
  27. α \alpha
  28. β \beta
  29. α = 2 \alpha=2
  30. β = 1 \beta=1
  31. α = 1 \alpha=1
  32. β = 2 \beta=2

Hypertranscendental_function.html

  1. F ( x , y , y , , y ( n ) ) = 0 F\left(x,y,y^{\prime},\cdots,y^{(n)}\right)=0
  2. F F

Hypertranscendental_number.html

  1. y = y y^{\prime}=y

Hypoelliptic_operator.html

  1. P P
  2. U n U\subset{\mathbb{R}}^{n}
  3. u u
  4. V U V\subset U
  5. P u Pu
  6. C C^{\infty}
  7. u u
  8. C C^{\infty}
  9. C C^{\infty}
  10. P P
  11. C C^{\infty}
  12. P ( u ) = u t - k Δ u P(u)=u_{t}-k\Delta u\,
  13. k > 0 k>0
  14. P ( u ) = u t t - c 2 Δ u P(u)=u_{tt}-c^{2}\Delta u\,
  15. c 0 c\neq 0

Hypsometric_equation.html

  1. h = z 2 - z 1 = R T ¯ g ln ( p 1 p 2 ) \ h=z_{2}-z_{1}=\frac{R\cdot\bar{T}}{g}\cdot\ln\left(\frac{p_{1}}{p_{2}}\right)
  2. h \ h
  3. z \ z
  4. R \ R
  5. T ¯ \ \bar{T}
  6. g \ g
  7. p \ p
  8. p 1 p_{1}
  9. p 2 p_{2}
  10. p = ρ g z \ p=\rho\cdot g\cdot z
  11. ρ \ \rho
  12. d p = - ρ g d z . dp=-\rho\cdot g\cdot dz.
  13. p = ρ R T \ p=\rho\cdot R\cdot T
  14. ρ \ \rho
  15. d p p = - g R T d z . \frac{\mathrm{d}p}{p}=\frac{-g}{R\cdot T}\,\mathrm{d}z.
  16. z 1 \ z_{1}
  17. z 2 \ z_{2}
  18. p ( z 1 ) p ( z 2 ) d p p = z 1 z 2 - g R T d z . \ \int_{p(z_{1})}^{p(z_{2})}\frac{\mathrm{d}p}{p}=\int_{z_{1}}^{z_{2}}\frac{-g% }{R\cdot T}\,\mathrm{d}z.
  19. T ¯ \bar{T}
  20. z 1 z_{1}
  21. z 2 z_{2}
  22. p ( z 1 ) p ( z 2 ) d p p = - g R T ¯ z 1 z 2 d z . \ \int_{p(z_{1})}^{p(z_{2})}\frac{\mathrm{d}p}{p}=\frac{-g}{R\cdot\bar{T}}\int% _{z_{1}}^{z_{2}}\,\mathrm{d}z.
  23. ln ( p ( z 2 ) p ( z 1 ) ) = - g R T ¯ ( z 2 - z 1 ) \ln\left(\frac{p(z_{2})}{p(z_{1})}\right)=\frac{-g}{R\cdot\bar{T}}(z_{2}-z_{1})
  24. ln ( p 1 p 2 ) = g R T ¯ ( z 2 - z 1 ) . \ln\left(\frac{p_{1}}{p_{2}}\right)=\frac{g}{R\cdot\bar{T}}(z_{2}-z_{1}).
  25. ( z 2 - z 1 ) = R T ¯ g ln ( p 1 p 2 ) (z_{2}-z_{1})=\frac{R\cdot\bar{T}}{g}\ln\left(\frac{p_{1}}{p_{2}}\right)
  26. p 1 p 2 = e g R T ¯ ( z 2 - z 1 ) . \frac{p_{1}}{p_{2}}=e^{{g\over R\cdot\bar{T}}\cdot(z_{2}-z_{1})}.

Ideal_(set_theory).html

  1. 1 / n \mathcal{I}_{1/n}
  2. n A 1 n + 1 \sum_{n\in A}\frac{1}{n+1}
  3. 𝒵 0 \mathcal{Z}_{0}
  4. A I × J { x X | { y | x , y A } J } I A\in I\times J\iff\{x\in X|\{y|\langle x,y\rangle\in A\}\notin J\}\in I

IK_Pegasi.html

  1. μ = μ δ 2 + μ α 2 cos 2 δ = 77.63 \begin{smallmatrix}\mu=\sqrt{{\mu_{\delta}}^{2}+{\mu_{\alpha}}^{2}\cdot\cos^{2% }\delta}=77.63\end{smallmatrix}
  2. μ α \mu_{\alpha}
  3. μ δ \mu_{\delta}
  4. V = V r 2 + V t 2 = 11.4 2 + 16.9 2 = 20.4 \begin{smallmatrix}V=\sqrt{{V_{r}}^{2}+{V_{t}}^{2}}=\sqrt{11.4^{2}+16.9^{2}}=2% 0.4\end{smallmatrix}
  5. V r V_{r}
  6. V t V_{t}
  7. log g = log 978.0 = 2.99 \begin{smallmatrix}\log\ \operatorname{g}=\log\ 978.0=2.99\end{smallmatrix}
  8. 10 5.96 912 , 000 \begin{smallmatrix}10^{5.96}\approx 912,000\end{smallmatrix}
  9. λ b = ( 2.898 × 10 6 nm K ) / ( 35 , 500 K ) 82 \begin{smallmatrix}\lambda_{b}=(2.898\times 10^{6}\operatorname{nm\ K})/(35,50% 0\ \operatorname{K})\approx 82\end{smallmatrix}

Illness_rate.html

  1. Illness rate = Illness-related Absence Times in Days Planned Working Times in Days \textstyle{\mbox{Illness rate }~{}=\frac{\sum{\mbox{Illness-related Absence % Times in Days}~{}}}{\sum{\mbox{Planned Working Times in Days}~{}}}}

Image_rectification.html

  1. M = K [ I 0 ] M=K[I~{}0]
  2. M = K [ R T ] M^{\prime}=K^{\prime}[R~{}T]
  3. e = M [ O 1 ] = M [ - R T T 1 ] = K [ I 0 ] [ - R T T 1 ] = - K R T T e=M\begin{bmatrix}O^{\prime}\\ 1\end{bmatrix}=M\begin{bmatrix}-R^{T}T\\ 1\end{bmatrix}=K[I~{}0]\begin{bmatrix}-R^{T}T\\ 1\end{bmatrix}=-KR^{T}T
  4. e = M [ O 1 ] = M [ 0 1 ] = K [ R T ] [ 0 1 ] = K T e^{\prime}=M^{\prime}\begin{bmatrix}O\\ 1\end{bmatrix}=M^{\prime}\begin{bmatrix}0\\ 1\end{bmatrix}=K^{\prime}[R~{}T]\begin{bmatrix}0\\ 1\end{bmatrix}=K^{\prime}T
  5. H = H 2 H 1 H=H_{2}H_{1}
  6. H = H R T H^{\prime}=HR^{T}

Image_sensor_format.html

  1. DOF 2 DOF 1 d 1 d 2 \frac{\mathrm{DOF}_{2}}{\mathrm{DOF}_{1}}\approx\frac{d_{1}}{d_{2}}
  2. d 1 d_{1}
  3. d 2 d_{2}
  4. DOF 2 DOF 1 l 1 l 2 \frac{\mathrm{DOF}_{2}}{\mathrm{DOF}_{1}}\approx\frac{l_{1}}{l_{2}}
  5. l 1 l_{1}
  6. l 2 l_{2}
  7. l 1 / l 2 l_{1}/l_{2}
  8. DOF 2 DOF 1 l 2 l 1 \frac{\mathrm{DOF}_{2}}{\mathrm{DOF}_{1}}\approx\frac{l_{2}}{l_{1}}
  9. SNR = P Q e t ( P Q e t ) 2 + ( D t ) 2 + N r 2 = P Q e t P Q e t + D t + N r 2 \mathrm{SNR}=\frac{PQ_{e}t}{\sqrt{(\sqrt{PQ_{e}t})^{2}+(\sqrt{Dt})^{2}+N_{r}^{% 2}}}=\frac{PQ_{e}t}{\sqrt{PQ_{e}t+Dt+N_{r}^{2}}}
  10. P P
  11. Q e Q_{e}
  12. t t
  13. D D
  14. N r N_{r}
  15. P Q e t P Q e t = P Q e t \frac{PQ_{e}t}{\sqrt{PQ_{e}t}}=\sqrt{PQ_{e}t}
  16. C G = V r t / Q r t CG=V_{rt}/Q_{rt}
  17. C = Q / V C=Q/V
  18. C G = 1 / C r t CG=1/C_{rt}
  19. D R = Q m a x / σ r e a d o u t DR=Q_{max}/\sigma_{readout}
  20. σ r e a d o u t \sigma_{readout}
  21. N r N_{r}
  22. Q m a x Q_{max}
  23. σ r e a d o u t \sigma_{readout}
  24. ξ cutoff = 1 λ N \xi_{\mathrm{cutoff}}=\frac{1}{\lambda N}
  25. MTF ( ξ / ξ cutoff ) = 2 π { cos - 1 ( ξ / ξ cutoff ) - ( ξ / ξ cutoff ) [ 1 - ( ξ / ξ cutoff ) 2 ] 1 / 2 } \mathrm{MTF}(\xi/\xi_{\mathrm{cutoff}})=\frac{2}{\pi}\left\{\cos^{-1}(\xi/\xi_% {\mathrm{cutoff}})-(\xi/\xi_{\mathrm{cutoff}})\left[1-(\xi/\xi_{\mathrm{cutoff% }})^{2}\right]^{1/2}\right\}
  26. ξ < ξ cutoff \xi<\xi_{\mathrm{cutoff}}
  27. 0
  28. ξ ξ cutoff \xi\geq\xi_{\mathrm{cutoff}}
  29. ξ cutoff \xi_{\mathrm{cutoff}}
  30. 1 / N 1/N
  31. 1 / C 1/{C}
  32. C {C}
  33. 1 / ( N C ) 1/(NC)
  34. 1 / C 1/C
  35. G objective w pixel 2 ( f / # ) objective G_{\mathrm{objective}}\simeq\frac{w_{\mathrm{pixel}}}{2{(f/\#)}_{\mathrm{% objective}}}
  36. G pixel w photoreceptor 2 ( f / # ) microlens G_{\mathrm{pixel}}\simeq\frac{w_{\mathrm{photoreceptor}}}{2{(f/\#)}_{\mathrm{% microlens}}}
  37. G pixel G objective G_{\mathrm{pixel}}\geq G_{\mathrm{objective}}
  38. w photoreceptor ( f / # ) microlens w pixel ( f / # ) objective \frac{w_{\mathrm{photoreceptor}}}{{(f/\#)}_{\mathrm{microlens}}}\geq\frac{w_{% \mathrm{pixel}}}{{(f/\#)}_{\mathrm{objective}}}
  39. ( f / # ) microlens ( f / # ) objective × 𝑓𝑓 {(f/\#)}_{\mathrm{microlens}}\leq{(f/\#)}_{\mathrm{objective}}\times\mathit{ff}

Imaginary_curve.html

  1. ( x , y ) (x,y)
  2. x 2 + y 2 = - 1 x^{2}+y^{2}=-1
  3. ( i , 0 ) (i,0)
  4. ( 5 i 3 , 4 3 ) (\frac{5i}{3},\frac{4}{3})

IMP_dehydrogenase.html

  1. \rightleftharpoons

Impact_parameter.html

  1. b b
  2. U ( r ) U(r)
  3. θ \theta
  4. θ = π - 2 b r min d r r 2 1 - ( b / r ) 2 - 2 U / m v 2 \theta=\pi-2b\int_{r_{\mathrm{min}}}^{\infty}\frac{dr}{r^{2}\sqrt{1-(b/r)^{2}-% 2U/mv_{\infty}^{2}}}
  5. v v_{\infty}
  6. r min r_{\mathrm{min}}

Impedance_parameters.html

  1. I n I_{n}\,
  2. V n V_{n}\,
  3. V = Z I V=ZI\,
  4. ( V 1 V 2 ) = ( Z 11 Z 12 Z 21 Z 22 ) ( I 1 I 2 ) {V_{1}\choose V_{2}}=\begin{pmatrix}Z_{11}&Z_{12}\\ Z_{21}&Z_{22}\end{pmatrix}{I_{1}\choose I_{2}}
  5. Z 11 = V 1 I 1 | I 2 = 0 Z 12 = V 1 I 2 | I 1 = 0 Z_{11}={V_{1}\over I_{1}}\bigg|_{I_{2}=0}\qquad Z_{12}={V_{1}\over I_{2}}\bigg% |_{I_{1}=0}
  6. Z 21 = V 2 I 1 | I 2 = 0 Z 22 = V 2 I 2 | I 1 = 0 Z_{21}={V_{2}\over I_{1}}\bigg|_{I_{2}=0}\qquad Z_{22}={V_{2}\over I_{2}}\bigg% |_{I_{1}=0}
  7. Z n m = V n I m | I k = 0 for k m Z_{nm}={V_{n}\over I_{m}}\bigg|_{I_{k}=0\,\text{ for }k\neq m}
  8. Z in = Z 11 - Z 12 Z 21 Z 22 + Z L Z\text{in}=Z_{11}-\frac{Z_{12}Z_{21}}{Z_{22}+Z_{L}}
  9. Z out = Z 22 - Z 12 Z 21 Z 11 + Z S Z\text{out}=Z_{22}-\frac{Z_{12}Z_{21}}{Z_{11}+Z_{S}}
  10. Z = z ( 1 N + S ) ( 1 N - S ) - 1 z = z ( 1 N - S ) - 1 ( 1 N + S ) z \begin{aligned}\displaystyle Z&\displaystyle=\sqrt{z}(1_{\!N}+S)(1_{\!N}-S)^{-% 1}\sqrt{z}\\ &\displaystyle=\sqrt{z}(1_{\!N}-S)^{-1}(1_{\!N}+S)\sqrt{z}\\ \end{aligned}
  11. S = ( y Z y - 1 N ) ( y Z y + 1 N ) - 1 = ( y Z y + 1 N ) - 1 ( y Z y - 1 N ) \begin{aligned}\displaystyle S&\displaystyle=(\sqrt{y}Z\sqrt{y}\,-1_{\!N})(% \sqrt{y}Z\sqrt{y}\,+1_{\!N})^{-1}\\ &\displaystyle=(\sqrt{y}Z\sqrt{y}\,+1_{\!N})^{-1}(\sqrt{y}Z\sqrt{y}\,-1_{\!N})% \\ \end{aligned}
  12. 1 N 1_{\!N}
  13. z \sqrt{z}
  14. z = ( z 01 z 02 z 0 N ) \sqrt{z}=\begin{pmatrix}\sqrt{z_{01}}&\\ &\sqrt{z_{02}}\\ &&\ddots\\ &&&\sqrt{z_{0N}}\end{pmatrix}
  15. y = ( z ) - 1 \sqrt{y}=(\sqrt{z})^{-1}
  16. z 01 = z 02 = Z 0 z_{01}=z_{02}=Z_{0}
  17. Z 11 = ( ( 1 + S 11 ) ( 1 - S 22 ) + S 12 S 21 ) Δ S Z 0 Z_{11}={((1+S_{11})(1-S_{22})+S_{12}S_{21})\over\Delta_{S}}Z_{0}\,
  18. Z 12 = 2 S 12 Δ S Z 0 Z_{12}={2S_{12}\over\Delta_{S}}Z_{0}\,
  19. Z 21 = 2 S 21 Δ S Z 0 Z_{21}={2S_{21}\over\Delta_{S}}Z_{0}\,
  20. Z 22 = ( ( 1 - S 11 ) ( 1 + S 22 ) + S 12 S 21 ) Δ S Z 0 Z_{22}={((1-S_{11})(1+S_{22})+S_{12}S_{21})\over\Delta_{S}}Z_{0}\,
  21. Δ S = ( 1 - S 11 ) ( 1 - S 22 ) - S 12 S 21 \Delta_{S}=(1-S_{11})(1-S_{22})-S_{12}S_{21}\,
  22. S 11 = ( Z 11 - Z 0 ) ( Z 22 + Z 0 ) - Z 12 Z 21 Δ S_{11}={(Z_{11}-Z_{0})(Z_{22}+Z_{0})-Z_{12}Z_{21}\over\Delta}
  23. S 12 = 2 Z 0 Z 12 Δ S_{12}={2Z_{0}Z_{12}\over\Delta}\,
  24. S 21 = 2 Z 0 Z 21 Δ S_{21}={2Z_{0}Z_{21}\over\Delta}\,
  25. S 22 = ( Z 11 + Z 0 ) ( Z 22 - Z 0 ) - Z 12 Z 21 Δ S_{22}={(Z_{11}+Z_{0})(Z_{22}-Z_{0})-Z_{12}Z_{21}\over\Delta}
  26. Δ = ( Z 11 + Z 0 ) ( Z 22 + Z 0 ) - Z 12 Z 21 \Delta=(Z_{11}+Z_{0})(Z_{22}+Z_{0})-Z_{12}Z_{21}\,
  27. S i j S_{ij}\,
  28. Z i j Z_{ij}\,
  29. Δ \Delta\,
  30. Z i j Z_{ij}\,
  31. Δ \Delta\,
  32. S i j S_{ij}\,
  33. Z 11 = Y 22 Δ Y Z_{11}={Y_{22}\over\Delta_{Y}}\,
  34. Z 12 = - Y 12 Δ Y Z_{12}={-Y_{12}\over\Delta_{Y}}\,
  35. Z 21 = - Y 21 Δ Y Z_{21}={-Y_{21}\over\Delta_{Y}}\,
  36. Z 22 = Y 11 Δ Y Z_{22}={Y_{11}\over\Delta_{Y}}\,
  37. Δ Y = Y 11 Y 22 - Y 12 Y 21 \Delta_{Y}=Y_{11}Y_{22}-Y_{12}Y_{21}\,

Implicit_certificate.html

  1. G G\,
  2. n n\,
  3. c c\,
  4. Q C A = c G Q_{CA}=cG\,
  5. α \alpha\,
  6. α G \alpha\,G\,
  7. k k\,
  8. [ 1 , n - 1 ] [1,n-1]\,
  9. k G kG\,
  10. γ = α G + k G \gamma=\alpha\,G+kG\,
  11. e = H ( γ ID A ) e=\textrm{H}(\gamma\parallel\textrm{ID}_{A})\,
  12. H \textrm{H}\,
  13. ID A \textrm{ID}_{A}\,
  14. s = e k + c ( mod n ) s=ek+c\;\;(\mathop{{\rm mod}}n)\,
  15. ( s , γ ) (s,\gamma)\,
  16. a = e α + s ( mod n ) a=e\alpha+s\;\;(\mathop{{\rm mod}}n)\,
  17. Q A = e γ + Q C A Q_{A}=e\gamma+Q_{CA}\,
  18. Q A Q_{A}\,
  19. γ \gamma\,
  20. ID A \textrm{ID}_{A}\,
  21. Q C A Q_{CA}\,
  22. γ \gamma\,
  23. Q A Q_{A}\,

Implicit_curve.html

  1. F ( x , y ) = 0. F(x,y)=0.
  2. F ( x , y ) F(x,y)
  3. y = f ( x ) y=f(x)
  4. ( x ( t ) , y ( t ) ) (x(t),y(t))
  5. x ( t ) , y ( t ) x(t)\,,y(t)
  6. t . t.
  7. y = f ( x ) y=f(x)
  8. y - f ( x ) = 0 y-f(x)=0
  9. ( t , f ( t ) ) (t,f(t))
  10. x + 2 y - 3 = 0 , x+2y-3=0\ ,
  11. x 2 + y 2 - 4 = 0 , x^{2}+y^{2}-4=0\ ,
  12. x 3 - y 2 = 0 , x^{3}-y^{2}=0\ ,
  13. ( x 2 + y 2 ) 2 - 2 c 2 ( x 2 - y 2 ) - ( a 4 - c 4 ) = 0 (x^{2}+y^{2})^{2}-2c^{2}(x^{2}-y^{2})-(a^{4}-c^{4})=0
  14. sin ( x + y ) - cos ( x y ) + 1 = 0 \sin(x+y)-\cos(xy)+1=0
  15. F ( x , y ) = 0 F(x,y)=0
  16. F ( x , y ) = 0 F(x,y)=0
  17. z = F ( x , y ) z=F(x,y)
  18. F ( x , y ) = 0 F(x,y)=0
  19. F F
  20. F F
  21. F x F_{x}
  22. F y F_{y}
  23. F x x F_{xx}
  24. ( x 0 , y 0 ) (x_{0},y_{0})
  25. ( F x ( x 0 , y 0 ) , F y ( x 0 , y 0 ) ) ( 0 , 0 ) (F_{x}(x_{0},y_{0}),F_{y}(x_{0},y_{0}))\neq(0,0)
  26. ( x 0 , y 0 ) (x_{0},y_{0})
  27. F x ( x 0 , y 0 ) ( x - x 0 ) + F y ( x 0 , y 0 ) ( y - y 0 ) = 0 F_{x}(x_{0},y_{0})(x-x_{0})+F_{y}(x_{0},y_{0})(y-y_{0})=0
  28. 𝐧 ( x 0 , y 0 ) = ( F x ( x 0 , y 0 ) , F y ( x 0 , y 0 ) ) T \mathbf{n}(x_{0},y_{0})=(F_{x}(x_{0},y_{0}),F_{y}(x_{0},y_{0}))^{T}
  29. ( x 0 , y 0 ) (x_{0},y_{0})
  30. κ = - F y 2 F x x + 2 F x F y F x y - F x 2 F y y ( F x 2 + F y 2 ) 3 / 2 \kappa=\frac{-F_{y}^{2}F_{xx}+2F_{x}F_{y}F_{xy}-F_{x}^{2}F_{yy}}{(F_{x}^{2}+F_% {y}^{2})^{3/2}}
  31. ( x 0 , y 0 ) (x_{0},y_{0})
  32. f f
  33. F ( x , f ( x ) ) = 0 F(x,f(x))=0
  34. f f
  35. f ( x ) = - F x ( x , f ( x ) ) F y ( x , f ( x ) ) f^{\prime}(x)=-\frac{F_{x}(x,f(x))}{F_{y}(x,f(x))}
  36. f ′′ = - F y 2 F x x + 2 F x F y F x y - F x 2 F y y F y 3 f^{\prime\prime}=\frac{-F_{y}^{2}F_{xx}+2F_{x}F_{y}F_{xy}-F_{x}^{2}F_{yy}}{F_{% y}^{3}}
  37. f f
  38. ( x , f ( x ) ) (x,f(x))
  39. y = f ( x 0 ) + f ( x 0 ) ( x - x 0 ) y=f(x_{0})+f^{\prime}(x_{0})(x-x_{0})
  40. κ ( x 0 ) = f ′′ ( x 0 ) ( 1 + f ( x 0 ) 2 ) 3 / 2 \kappa(x_{0})=\frac{f^{\prime\prime}(x_{0})}{(1+f^{\prime}(x_{0})^{2})^{3/2}}
  41. F ( x , y ) = 0 F(x,y)=0
  42. F ( x , y ) F(x,y)
  43. F ( x , y ) = 0 F(x,y)=0
  44. F ( x , y ) - c = 0 F(x,y)-c=0
  45. g i ( x , y ) = a i x + b i y + c i = 0 , i = 1 , , n g_{i}(x,y)=a_{i}x+b_{i}y+c_{i}=0,\ i=1,\ldots,n
  46. g i g_{i}
  47. F ( x , y ) = g 1 ( x , y ) g n ( x , y ) - c = 0 F(x,y)=g_{1}(x,y)\cdots g_{n}(x,y)-c=0
  48. c c
  49. F ( x , y ) = ( x + 1 ) ( - x + 1 ) y ( - x - y + 2 ) ( x - y + 2 ) - c = 0 F(x,y)=(x+1)(-x+1)y(-x-y+2)(x-y+2)-c=0
  50. c = 0.03 , , 0.6 c=0.03,\ldots,0.6
  51. F ( x , y ) = g 1 ( x , y ) g 2 ( x , y ) - c = 0 F(x,y)=g_{1}(x,y)g_{2}(x,y)-c=0
  52. x y - c = 0 , c 0 xy-c=0,\ c\neq 0
  53. F ( x , y ) = y ( - x 2 - y 2 + 1 ) - c = 0 F(x,y)=y(-x^{2}-y^{2}+1)-c=0
  54. F ( x , y ) = ( - x 2 - ( y + 1 ) 2 + 4 ) ( - x 2 - ( y - 1 ) 2 + 4 ) - c = 0 F(x,y)=(-x^{2}-(y+1)^{2}+4)(-x^{2}-(y-1)^{2}+4)-c=0
  55. F ( x , y ) = ( 1 - μ ) f 1 f 2 - μ ( g 1 g 2 ) 3 = 0 F(x,y)=(1-\mu)f_{1}f_{2}-\mu(g_{1}g_{2})^{3}=0
  56. f 1 ( x , y ) = ( x - x 1 ) 2 + y 2 - r 1 2 = 0 f_{1}(x,y)=(x-x_{1})^{2}+y^{2}-r_{1}^{2}=0
  57. f 2 ( x , y ) = ( x - x 2 ) 2 + y 2 - r 2 2 = 0 f_{2}(x,y)=(x-x_{2})^{2}+y^{2}-r_{2}^{2}=0
  58. g 1 ( x , y ) = x - x 1 = 0 , g 2 ( x , y ) = x - x 2 = 0 g_{1}(x,y)=x-x_{1}=0,\ g_{2}(x,y)=x-x_{2}=0
  59. μ \mu
  60. μ = 0.05 , , 0.2 \mu=0.05,\ldots,0.2
  61. grad F ( 0 , 0 ) \operatorname{grad}F\neq(0,0)
  62. 𝖢𝖯𝗈𝗂𝗇𝗍 \mathsf{CPoint}
  63. Q 0 = ( x 0 , y 0 ) Q_{0}=(x_{0},y_{0})
  64. P P
  65. j = 0 j=0
  66. ( x j + 1 , y j + 1 ) = ( x j , y j ) - F ( x j , y j ) F x ( x j , y j ) 2 + F y ( x j , y j ) 2 ( F x ( x j , y j ) , F y ( x j , y j ) ) (x_{j+1},y_{j+1})=(x_{j},y_{j})-\frac{F(x_{j},y_{j})}{F_{x}(x_{j},y_{j})^{2}+F% _{y}(x_{j},y_{j})^{2}}\,\left(F_{x}(x_{j},y_{j}),F_{y}(x_{j},y_{j})\right)
  67. g ( t ) = F ( x j + t F x ( x j , y j ) , y j + t F y ( x j , y j ) ) . g(t)=F\left(x_{j}+tF_{x}(x_{j},y_{j}),y_{j}+tF_{y}(x_{j},y_{j})\right)\ .
  68. ( x j + 1 , y j + 1 ) , ( x j , y j ) (x_{j+1},y_{j+1}),\,(x_{j},y_{j})
  69. P = ( x j + 1 , y j + 1 ) P=(x_{j+1},y_{j+1})
  70. Q 0 Q_{0}
  71. s s
  72. P 1 P_{1}
  73. 𝖢𝖯𝗈𝗂𝗇𝗍 \mathsf{CPoint}
  74. s s
  75. P 2 P_{2}
  76. 𝖢𝖯𝗈𝗂𝗇𝗍 \mathsf{CPoint}
  77. \cdots
  78. 𝖢𝖯𝗈𝗂𝗇𝗍 \mathsf{CPoint}
  79. F ( x , y ) = ( 3 x 2 - y 2 ) 2 y 2 - ( x 2 + y 2 ) 4 = 0 F(x,y)=(3x^{2}-y^{2})^{2}y^{2}-(x^{2}+y^{2})^{4}=0
  80. F ( x , y , z ) = 0 , G ( x , y , z ) = 0 \begin{matrix}F(x,y,z)=0,\\ G(x,y,z)=0\end{matrix}
  81. ( x 0 , y 0 , z 0 ) (x_{0},y_{0},z_{0})
  82. F F
  83. G G
  84. ( 0 , 0 , 0 ) (0,0,0)
  85. 𝐭 ( x 0 , y 0 , z 0 ) = grad F ( x 0 , y 0 , z 0 ) × grad G ( x 0 , y 0 , z 0 ) ( 0 , 0 , 0 ) \mathbf{t}(x_{0},y_{0},z_{0})=\operatorname{grad}F(x_{0},y_{0},z_{0})\times% \operatorname{grad}G(x_{0},y_{0},z_{0})\neq(0,0,0)
  86. 𝐭 ( x 0 , y 0 , z 0 ) \mathbf{t}(x_{0},y_{0},z_{0})
  87. ( x 0 , y 0 , z 0 ) (x_{0},y_{0},z_{0})
  88. ( 1 ) x + y + z - 1 = 0 , x - y + z - 2 = 0 (1)\quad x+y+z-1=0\ ,\ x-y+z-2=0
  89. ( 2 ) x 2 + y 2 + z 2 - 4 = 0 , x + y + z - 1 = 0 (2)\quad x^{2}+y^{2}+z^{2}-4=0\ ,\ x+y+z-1=0
  90. ( 3 ) x 2 + y 2 - 1 = 0 , x + y + z - 1 = 0 (3)\quad x^{2}+y^{2}-1=0\ ,\ x+y+z-1=0
  91. ( 4 ) x 2 + y 2 + z 2 - 16 = 0 , ( y - y 0 ) 2 + z 2 - 9 = 0 (4)\quad x^{2}+y^{2}+z^{2}-16=0\ ,\ (y-y_{0})^{2}+z^{2}-9=0

Implicit_k-d_tree.html

  1. O ( k n ) \mathrm{O}(kn)
  2. O ( n ) \mathrm{O}(n)
  3. O ( log ( n ) ) \mathrm{O}(\log(n))

Impulse_invariance.html

  1. h c ( t ) h_{c}(t)
  2. T T
  3. h [ n ] h[n]
  4. h [ n ] = T h c ( n T ) h[n]=Th_{c}(nT)\,
  5. H ( e j ω ) = k = - H c ( j ω T + j 2 π T k ) H(e^{j\omega})=\sum_{k=-\infty}^{\infty}{H_{c}\left(j\frac{\omega}{T}+j\frac{2% {\pi}}{T}k\right)}\,
  6. H c ( j Ω ) < δ H_{c}(j\Omega)<\delta
  7. | Ω | π / T |\Omega|\geq\pi/T
  8. H ( e j ω ) = H c ( j ω / T ) H(e^{j\omega})=H_{c}(j\omega/T)\,
  9. | ω | π |\omega|\leq\pi\,
  10. s = s k s=s_{k}
  11. H c ( s ) = k = 1 N A k s - s k H_{c}(s)=\sum_{k=1}^{N}{\frac{A_{k}}{s-s_{k}}}\,
  12. h c ( t ) = { k = 1 N A k e s k t , t 0 0 , otherwise h_{c}(t)=\begin{cases}\sum_{k=1}^{N}{A_{k}e^{s_{k}t}},&t\geq 0\\ 0,&\mbox{otherwise}\end{cases}
  13. h [ n ] = T h c ( n T ) h[n]=Th_{c}(nT)\,
  14. h [ n ] = T k = 1 N A k e s k n T u [ n ] h[n]=T\sum_{k=1}^{N}{A_{k}e^{s_{k}nT}u[n]}\,
  15. H ( z ) = T k = 1 N A k 1 - e s k T z - 1 H(z)=T\sum_{k=1}^{N}{\frac{A_{k}}{1-e^{s_{k}T}z^{-1}}}\,
  16. t = 0 t=0
  17. h c ( 0 ) h_{c}(0)
  18. h [ 0 ] h[0]
  19. h [ n ] = T ( h c ( n T ) - 1 2 h c ( 0 ) δ [ n ] ) h[n]=T\left(h_{c}(nT)-\frac{1}{2}h_{c}(0)\delta[n]\right)\,
  20. h [ n ] = T k = 1 N A k e s k n T ( u [ n ] - 1 2 δ [ n ] ) h[n]=T\sum_{k=1}^{N}{A_{k}e^{s_{k}nT}}\left(u[n]-\frac{1}{2}\delta[n]\right)\,
  21. H ( z ) = T k = 1 N A k 1 - e s k T z - 1 - T 2 k = 1 N A k . H(z)=T\sum_{k=1}^{N}{\frac{A_{k}}{1-e^{s_{k}T}z^{-1}}-\frac{T}{2}\sum_{k=1}^{N% }A_{k}}.

In-place_matrix_transposition.html

  1. [ 10 11 12 13 14 15 16 17 ] . \begin{bmatrix}10&11&12&13\\ 14&15&16&17\end{bmatrix}.
  2. [ 10 14 11 15 12 16 13 17 ] \begin{bmatrix}10&14\\ 11&15\\ 12&16\\ 13&17\end{bmatrix}
  3. N m + n = P ( M n + m ) Nm+n=P(Mn+m)\,
  4. ( n , m ) [ 0 , N - 1 ] × [ 0 , M - 1 ] . (n,m)\in[0,N-1]\times[0,M-1]\,.
  5. a = 0 , , M N - 1 a=0,\ldots,MN-1
  6. P ( a ) = { M N - 1 if a = M N - 1 , N a mod M N - 1 otherwise , P(a)=\left\{\begin{matrix}MN-1&\mbox{if }~{}a=MN-1,\\ Na\mod MN-1&\mbox{otherwise}~{},\end{matrix}\right.
  7. P - 1 ( a ) = { M N - 1 if a = M N - 1 , M a mod M N - 1 otherwise . P^{-1}(a^{\prime})=\left\{\begin{matrix}MN-1&\mbox{if }~{}a^{\prime}=MN-1,\\ Ma^{\prime}\mod MN-1&\mbox{otherwise}~{}.\end{matrix}\right.
  8. 1 k d | k μ ( k / d ) gcd ( N d - 1 , M N - 1 ) , \frac{1}{k}\sum_{d|k}\mu(k/d)\gcd(N^{d}-1,MN-1),
  9. x C x\in C
  10. d = gcd ( x , M N - 1 ) d=\gcd(x,MN-1)
  11. d = gcd ( s , M N - 1 ) d=\gcd(s,MN-1)
  12. d = gcd ( x , M N - 1 ) d=\gcd(x,MN-1)
  13. s N k = s mod ( M N - 1 ) sN^{k}=s\mod(MN-1)
  14. ( - s ) N k = - s mod ( M N - 1 ) (-s)N^{k}=-s\mod(MN-1)
  15. M N - 1 - s = - s mod ( M N - 1 ) MN-1-s=-s\mod(MN-1)
  16. N M / d 2 NM/d^{2}

Incomplete_Cholesky_factorization.html

  1. i i
  2. 1 1
  3. N N
  4. L i i = ( a i i - k = 1 i - 1 L i k 2 ) < m t p l > 1 2 L_{ii}=\left({a_{ii}-\sum\limits_{k=1}^{i-1}{L_{ik}^{2}}}\right)^{<}mtpl>{{1% \over 2}}
  5. j j
  6. i + 1 i+1
  7. N N
  8. L j i = 1 L i i ( a i j - k = 1 i - 1 L i k L j k ) L_{ji}={1\over{L_{ii}}}\left({a_{ij}-\sum\limits_{k=1}^{i-1}{L_{ik}L_{jk}}}\right)

Incomplete_LU_factorization.html

  1. A x = b Ax=b
  2. A = L U A=LU
  3. L y = b Ly=b
  4. U x = y Ux=y
  5. A L U A\approx LU
  6. A = L U A=LU
  7. L U x = b LUx=b
  8. A x = b Ax=b
  9. M = L U M=LU

Increment_theorem.html

  1. Δ y = f ( x ) Δ x + ε Δ x \Delta y=f^{\prime}(x)\,\Delta x+\varepsilon\,\Delta x\,
  2. Δ y = f ( x + Δ x ) - f ( x ) . \Delta y=f(x+\Delta x)-f(x).\,
  3. Δ x 0 \scriptstyle\Delta x\not=0
  4. Δ y Δ x = f ( x ) + ε , \frac{\Delta y}{\Delta x}=f^{\prime}(x)+\varepsilon,
  5. Δ y Δ x f ( x ) \scriptstyle\frac{\Delta y}{\Delta x}\approx f^{\prime}(x)
  6. Δ y Δ x \scriptstyle\frac{\Delta y}{\Delta x}
  7. f ( x ) \scriptstyle f^{\prime}(x)\,
  8. f ( x ) \scriptstyle f^{\prime}(x)\,
  9. Δ y Δ x \scriptstyle\frac{\Delta y}{\Delta x}

Indexed_grammar.html

  1. X [ σ ] α Y [ σ ] β X[\sigma]\to\alpha Y[\sigma]\beta
  2. α \alpha
  3. β \beta
  4. α \alpha
  5. β \beta
  6. X [ σ ] Y [ ] Z [ σ f ] X[\sigma]\to Y[]Z[\sigma f]
  7. Y [ ] Y[]
  8. Y [ σ ] Y^{\prime}[\sigma]
  9. Y [ ] Y[]
  10. σ \sigma
  11. Y [ σ ] Y^{\prime}[\sigma]
  12. σ \sigma
  13. Y [ ] Y[]
  14. Y [ σ f ] Y [ σ ] Y^{\prime}[\sigma f]\to Y^{\prime}[\sigma]
  15. Y [ ] Y [ ] Y^{\prime}[]\to Y[]
  16. { a n b n c n d m | n 1 , m 1 } \{a^{n}b^{n}c^{n}d^{m}|n\geq 1,m\geq 1\}
  17. S [ σ ] T [ σ ] V [ ] S[\sigma]\to T[\sigma]V[]
  18. V [ ] d | d V [ ] V[]\to d~{}|~{}dV[]
  19. T [ σ ] a T [ σ f ] c | U [ σ ] T[\sigma]\to aT[\sigma f]c~{}|~{}U[\sigma]
  20. U [ σ f ] b U [ σ ] U[\sigma f]\to bU[\sigma]
  21. U [ ] ϵ U[]\to\epsilon
  22. V V^{\prime}
  23. S [ σ ] T [ σ ] V [ σ ] S[\sigma]\to T[\sigma]V^{\prime}[\sigma]
  24. V [ σ f ] V [ σ ] V^{\prime}[\sigma f]\to V^{\prime}[\sigma]
  25. V [ ] V [ ] V^{\prime}[]\to V[]
  26. V [ ] d | d V [ ] V[]\to d~{}|~{}dV[]
  27. T [ σ ] a T [ σ f ] c | U [ σ ] T[\sigma]\to aT[\sigma f]c~{}|~{}U[\sigma]
  28. U [ σ f ] b U [ σ ] U[\sigma f]\to bU[\sigma]
  29. U [ ] ϵ U[]\to\epsilon

Indicators_of_spatial_association.html

  1. I = N S 0 i j W i j Z i Z j i Z i 2 I=\frac{\frac{N}{S_{0}}\sum_{i}{\sum_{j}{W_{ij}Z_{i}Z_{j}}}}{\sum_{i}{Z_{i}^{2% }}}
  2. Z i Z_{i}
  3. W i j W_{ij}
  4. S 0 = i j W i j S_{0}=\sum_{i}{\sum_{j}{W_{ij}}}
  5. I i = Z i m 2 j W i j Z j I_{i}=\frac{Z_{i}}{m_{2}}\sum_{j}W_{ij}Z_{j}
  6. m 2 = i Z i 2 N m_{2}=\frac{\sum_{i}Z_{i}^{2}}{N}
  7. I = i I i N I=\sum_{i}\frac{I_{i}}{N}

Induction_generator.html

  1. S 2 - P 2 \sqrt{S^{2}-P^{2}}

Infinite_alleles_model.html

  1. F = 1 4 N e u + 1 F={1\over 4N_{e}u+1}
  2. n = 1 F = 4 N e u + 1 n={1\over F}=4N_{e}u+1

Infinite_arithmetic_series.html

  1. n = 0 ( a n + b ) . \sum_{n=0}^{\infty}(an+b).
  2. n = 0 ( n + β ) = ζ H ( - 1 ; β ) . \sum_{n=0}^{\infty}(n+\beta)=\zeta_{H}(-1;\beta).
  3. - 1 12 - β 2 . -\frac{1}{12}-\frac{\beta}{2}.

Infinitely_near_point.html

  1. x infinitely near p m x ( C ) m x ( D ) \sum_{x\,\text{ infinitely near }p}m_{x}(C)m_{x}(D)
  2. g ( C ) = g ( N ) + infinitely near points x m x ( m x - 1 ) / 2 g(C)=g(N)+\sum_{\,\text{infinitely near points }x}m_{x}(m_{x}-1)/2

Infinitesimal_generator_(stochastic_processes).html

  1. d X t = b ( X t ) d t + σ ( X t ) d B t , \mathrm{d}X_{t}=b(X_{t})\,\mathrm{d}t+\sigma(X_{t})\,\mathrm{d}B_{t},
  2. A f ( x ) = lim t 0 𝐄 x [ f ( X t ) ] - f ( x ) t . Af(x)=\lim_{t\downarrow 0}\frac{\mathbf{E}^{x}[f(X_{t})]-f(x)}{t}.
  3. A f ( x ) = i b i ( x ) f x i ( x ) + 1 2 i , j ( σ ( x ) σ ( x ) ) i , j 2 f x i x j ( x ) , Af(x)=\sum_{i}b_{i}(x)\frac{\partial f}{\partial x_{i}}(x)+\frac{1}{2}\sum_{i,% j}\big(\sigma(x)\sigma(x)^{\top}\big)_{i,j}\frac{\partial^{2}f}{\partial x_{i}% \,\partial x_{j}}(x),
  4. A f ( x ) = b ( x ) x f ( x ) + 1 2 ( σ ( x ) σ ( x ) ) : x x f ( x ) . Af(x)=b(x)\cdot\nabla_{x}f(x)+\frac{1}{2}\big(\sigma(x)\sigma(x)^{\top}\big):% \nabla_{x}\nabla_{x}f(x).
  5. d Y t = ( d t d B t ) , \mathrm{d}Y_{t}={\mathrm{d}t\choose\mathrm{d}B_{t}},
  6. A f ( t , x ) = f t ( t , x ) + 1 2 2 f x 2 ( t , x ) . Af(t,x)=\frac{\partial f}{\partial t}(t,x)+\frac{1}{2}\frac{\partial^{2}f}{% \partial x^{2}}(t,x).
  7. A f ( x ) = θ ( μ - x ) f ( x ) + σ 2 2 f ′′ ( x ) . Af(x)=\theta(\mu-x)f^{\prime}(x)+\frac{\sigma^{2}}{2}f^{\prime\prime}(x).
  8. A f ( t , x ) = f t ( t , x ) + θ ( μ - x ) f x ( t , x ) + σ 2 2 2 f x 2 ( t , x ) . Af(t,x)=\frac{\partial f}{\partial t}(t,x)+\theta(\mu-x)\frac{\partial f}{% \partial x}(t,x)+\frac{\sigma^{2}}{2}\frac{\partial^{2}f}{\partial x^{2}}(t,x).
  9. A f ( x ) = r x f ( x ) + 1 2 α 2 x 2 f ′′ ( x ) . Af(x)=rxf^{\prime}(x)+\frac{1}{2}\alpha^{2}x^{2}f^{\prime\prime}(x).

Infinity_(philosophy).html

  1. 0 \aleph_{0}
  2. n ( m [ m > n P ( m ) ] ) \forall n\in\mathbb{Z}(\exists m\in\mathbb{Z}[m>n\wedge P(m)])

Institute_of_Medicine_Equation.html

  1. E E R = ( 662 - ( 9.53 * A g e ) ) + P A * ( ( 15.91 * w t ) + ( 539.6 * h t ) ) EER=(662-(9.53*Age))+PA*((15.91*wt)+(539.6*ht))
  2. E E R = ( 354 - ( 6.91 * A g e ) ) + P A * ( ( 9.36 * w t ) + ( 726 * h t ) ) EER=(354-(6.91*Age))+PA*((9.36*wt)+(726*ht))
  3. E E R = ( 88.5 - ( 61.9 * A g e ) ) + P A * ( ( 26.7 * w t ) + ( 903 * h t ) ) EER=(88.5-(61.9*Age))+PA*((26.7*wt)+(903*ht))
  4. E E R = ( 135.3 - ( 30.8 * A g e ) ) + P A * ( ( 10 * w t ) + ( 934 * h t ) ) EER=(135.3-(30.8*Age))+PA*((10*wt)+(934*ht))
  5. E E R = ( 89 * w t ) - 80 EER=(89*wt)-80
  6. E E R = ( 389 - ( 41.2 * a g e ) ) + P A * ( ( 15.0 * w t ) + ( 701.6 * h t ) ) EER=(389-(41.2*age))+PA*((15.0*wt)+(701.6*ht))
  7. E E R = ( - 114 - ( 50.9 * a g e ) ) + P A * ( ( 19.5 * w t ) + ( 1161.4 * h t ) ) EER=(-114-(50.9*age))+PA*((19.5*wt)+(1161.4*ht))

Integral_element.html

  1. a j A a_{j}\in A
  2. b n + a n - 1 b n - 1 + + a 1 b + a 0 = 0. b^{n}+a_{n-1}b^{n-1}+\cdots+a_{1}b+a_{0}=0.
  3. 2 \sqrt{2}
  4. a + b - 1 , a , b 𝐙 a+b\sqrt{-1},a,b\in\mathbf{Z}
  5. 𝐙 [ - 1 ] \mathbf{Z}[\sqrt{-1}]
  6. 𝐐 ( - 1 ) \mathbf{Q}(\sqrt{-1})
  7. 𝐐 ( 5 ) \mathbf{Q}(\sqrt{5})
  8. ( a + b 5 ) / 2 (a+b\sqrt{5})/2
  9. a 2 - 5 b 2 a^{2}-5b^{2}
  10. k ¯ \overline{k}
  11. k ¯ [ x 1 , , x n ] \overline{k}[x_{1},\dots,x_{n}]
  12. k [ x 1 , , x n ] . k[x_{1},\dots,x_{n}].
  13. R [ u ] R [ u - 1 ] R[u]\cap R[u^{-1}]
  14. 𝐂 [ [ x 1 / n ] ] \mathbf{C}[[x^{1/n}]]
  15. n 0 H 0 ( X , 𝒪 X ( n ) ) . \bigoplus\nolimits_{n\geq 0}\operatorname{H}^{0}(X,\mathcal{O}_{X}(n)).
  16. u ( M ) I M u(M)\subset IM
  17. u n + a 1 u n - 1 + + a n - 1 u + a n = 0 , a i I i . u^{n}+a_{1}u^{n-1}+\cdots+a_{n-1}u+a_{n}=0,a_{i}\in I^{i}.
  18. 𝔭 1 𝔭 n \mathfrak{p}_{1}\subset\cdots\subset\mathfrak{p}_{n}
  19. 𝔭 1 𝔭 n \mathfrak{p}^{\prime}_{1}\subset\cdots\subset\mathfrak{p}^{\prime}_{n}
  20. 𝔭 i = 𝔭 i A \mathfrak{p}_{i}=\mathfrak{p}^{\prime}_{i}\cap A
  21. 𝔮 \mathfrak{q}
  22. 𝔮 \mathfrak{q}
  23. 𝔮 A \mathfrak{q}\cap A
  24. f : A k f:A\to k
  25. f : A B f:A\to B
  26. f # : Spec B Spec A , p f - 1 ( p ) f^{\#}:\operatorname{Spec}B\to\operatorname{Spec}A,\quad p\mapsto f^{-1}(p)
  27. f # ( V ( I ) ) = V ( f - 1 ( I ) ) f^{\#}(V(I))=V(f^{-1}(I))
  28. f # f^{\#}
  29. B A R B\otimes_{A}R
  30. Spec ( B A R ) Spec R \operatorname{Spec}(B\otimes_{A}R)\to\operatorname{Spec}R
  31. Spec ( B A R ) Spec R \operatorname{Spec}(B\otimes_{A}R)\to\operatorname{Spec}R
  32. G = Gal ( L / K ) G=\operatorname{Gal}(L/K)
  33. Spec B Spec A \operatorname{Spec}B\to\operatorname{Spec}A
  34. 𝔭 2 σ ( 𝔭 1 ) \mathfrak{p}_{2}\neq\sigma(\mathfrak{p}_{1})
  35. σ \sigma
  36. 𝔭 2 \mathfrak{p}_{2}
  37. σ ( x ) 𝔭 1 \sigma(x)\not\in\mathfrak{p}_{1}
  38. σ \sigma
  39. y = σ σ ( x ) y=\prod_{\sigma}\sigma(x)
  40. y e y^{e}
  41. y e y^{e}
  42. 𝔭 2 A \mathfrak{p}_{2}\cap A
  43. 𝔭 1 A \mathfrak{p}_{1}\cap A
  44. 𝔭 1 A 𝔭 2 A \mathfrak{p}_{1}\cap A\neq\mathfrak{p}_{2}\cap A
  45. L / K L/K
  46. Spec B Spec A \operatorname{Spec}B\to\operatorname{Spec}A
  47. Spec B Spec A \operatorname{Spec}B\to\operatorname{Spec}A
  48. 𝔭 1 𝔭 n - 1 𝔭 n = 𝔭 n A \mathfrak{p}_{1}\subset\cdots\subset\mathfrak{p}_{n-1}\subset\mathfrak{p}_{n}=% \mathfrak{p}^{\prime}_{n}\cap A
  49. 𝔭 1 𝔭 n \mathfrak{p}^{\prime}_{1}\subset\cdots\subset\mathfrak{p}^{\prime}_{n}
  50. 𝔭 i ′′ \mathfrak{p}^{\prime\prime}_{i}
  51. 𝔭 i \mathfrak{p}^{\prime}_{i}
  52. σ G \sigma\in G
  53. σ ( 𝔭 n ′′ ) = 𝔭 n \sigma(\mathfrak{p}^{\prime\prime}_{n})=\mathfrak{p}^{\prime}_{n}
  54. 𝔭 i = σ ( 𝔭 i ′′ ) \mathfrak{p}^{\prime}_{i}=\sigma(\mathfrak{p}^{\prime\prime}_{i})
  55. Spec A \operatorname{Spec}A
  56. Spec B Spec A \operatorname{Spec}B\to\operatorname{Spec}A
  57. Spec A \operatorname{Spec}A
  58. A [ t ] A^{\prime}[t]
  59. A [ t ] A[t]
  60. B [ t ] B[t]
  61. A i A_{i}
  62. B i , 1 i n B_{i},1\leq i\leq n
  63. A i \prod A_{i}
  64. B i \prod B_{i}
  65. A i \prod{A_{i}}^{\prime}
  66. A i {A_{i}}^{\prime}
  67. A i A_{i}
  68. B i B_{i}
  69. \mathbb{N}
  70. \mathbb{N}
  71. \mathbb{N}
  72. I R I\subset R
  73. I ¯ \overline{I}
  74. r R r\in R
  75. x n + a 1 x n - 1 + + a n - 1 x 1 + a n x^{n}+a_{1}x^{n-1}+\cdots+a_{n-1}x^{1}+a_{n}
  76. a i I i a_{i}\in I^{i}
  77. r I ¯ r\in\overline{I}
  78. c R c\in R
  79. c r n I n cr^{n}\in I^{n}
  80. n 1 n\geq 1
  81. r I ¯ r\in\overline{I}
  82. 𝔣 = 𝔣 ( B / A ) \mathfrak{f}=\mathfrak{f}(B/A)
  83. 𝔣 \mathfrak{f}
  84. a B A aB\subset A
  85. S - 1 𝔣 ( B / A ) = 𝔣 ( S - 1 B / S - 1 A ) S^{-1}\mathfrak{f}(B/A)=\mathfrak{f}(S^{-1}B/S^{-1}A)
  86. 𝔣 ( B / A ) = Hom A ( B , A ) \ \mathfrak{f}(B/A)=\operatorname{Hom}_{A}(B,A)
  87. A = k [ t 2 , t 3 ] B = k [ t ] A=k[t^{2},t^{3}]\subset B=k[t]
  88. x 2 = y 3 x^{2}=y^{3}
  89. k ( t ) k(t)
  90. ( t 2 , t 3 ) A (t^{2},t^{3})A
  91. A = k [ [ t a , t b ] ] A=k[[t^{a},t^{b}]]
  92. ( t c , t c + 1 , ) A (t^{c},t^{c+1},\dots)A
  93. c = ( a - 1 ) ( b - 1 ) c=(a-1)(b-1)
  94. B / A B/A
  95. 𝔣 \mathfrak{f}
  96. B / A B/A
  97. V ( 𝔣 ) V(\mathfrak{f})
  98. Spec A \operatorname{Spec}A
  99. { 𝔭 Spec A | A 𝔭 is integrally closed } \{\mathfrak{p}\in\operatorname{Spec}A|A_{\mathfrak{p}}\,\text{ is integrally % closed}\}
  100. V ( 𝔣 ) V(\mathfrak{f})
  101. A A^{\prime}
  102. A A^{\prime}
  103. S = k [ x 1 , , x d ] S=k[x_{1},...,x_{d}]
  104. k k^{\prime}
  105. L k ( x 1 1 / q , , x d 1 / q ) . L\subset k^{\prime}(x_{1}^{1/q},...,x_{d}^{1/q}).
  106. k [ x 1 1 / q , , x d 1 / q ] k^{\prime}[x_{1}^{1/q},...,x_{d}^{1/q}]
  107. A A^{\prime}
  108. A A^{\prime}
  109. \Rightarrow
  110. \Rightarrow
  111. \Rightarrow
  112. A ^ \widehat{A}
  113. A ^ \widehat{A}
  114. \Rightarrow
  115. x + y , x y , - x x+y,xy,-x
  116. A [ x ] A [ y ] A[x]A[y]
  117. ϕ : B B [ t ] \phi:B\to B[t]
  118. ϕ ( b n ) = b n t n \phi(b_{n})=b_{n}t^{n}
  119. b n b_{n}
  120. A [ t ] A[t]
  121. B [ t ] B[t]
  122. A [ t ] A^{\prime}[t]
  123. A A^{\prime}
  124. ϕ ( b ) \phi(b)
  125. A [ t ] A[t]
  126. A [ t ] A^{\prime}[t]
  127. b n b_{n}
  128. ϕ ( b ) \phi(b)

Integral_of_secant_cubed.html

  1. sec 3 x d x = 1 2 sec x tan x + 1 2 ln | sec x + tan x | + C . \int\sec^{3}x\,dx=\frac{1}{2}\sec x\tan x+\frac{1}{2}\ln|\sec x+\tan x|+C.
  2. a 2 + x 2 d x , \int\sqrt{a^{2}+x^{2}}\,dx,
  3. sec 3 x d x = u d v \int\sec^{3}x\,dx=\int u\,dv
  4. d v = sec 2 x d x , v = tan x , u = sec x , d u = sec x tan x d x . \begin{aligned}\displaystyle dv&\displaystyle{}=\sec^{2}x\,dx,\\ \displaystyle v&\displaystyle{}=\tan x,\\ \displaystyle u&\displaystyle{}=\sec x,\\ \displaystyle du&\displaystyle{}=\sec x\tan x\,dx.\end{aligned}
  5. sec 3 x d x = u d v = u v - v d u = sec x tan x - sec x tan 2 x d x = sec x tan x - sec x ( sec 2 x - 1 ) d x = sec x tan x - ( sec 3 x d x - sec x d x . ) = sec x tan x - sec 3 x d x + sec x d x . \begin{aligned}\displaystyle\int\sec^{3}x\,dx&\displaystyle{}=\int u\,dv\\ &\displaystyle{}=uv-\int v\,du\\ &\displaystyle{}=\sec x\tan x-\int\sec x\tan^{2}x\,dx\\ &\displaystyle{}=\sec x\tan x-\int\sec x\,(\sec^{2}x-1)\,dx\\ &\displaystyle{}=\sec x\tan x-\left(\int\sec^{3}x\,dx-\int\sec x\,dx.\right)\\ &\displaystyle{}=\sec x\tan x-\int\sec^{3}x\,dx+\int\sec x\,dx.\end{aligned}
  6. sec 3 x d x \scriptstyle{}\int\sec^{3}x\,dx
  7. 2 sec 3 x d x \displaystyle 2\int\sec^{3}x\,dx
  8. sec 3 x d x = 1 2 sec x tan x + 1 2 ln | sec x + tan x | + C 1 . \int\sec^{3}x\,dx=\frac{1}{2}\sec x\tan x+\frac{1}{2}\ln|\sec x+\tan x|+C_{1}.
  9. sec 3 x d x = d x cos 3 x = cos x d x cos 4 x = cos x d x ( 1 - sin 2 x ) 2 = d u ( 1 - u 2 ) 2 \int\sec^{3}x\,dx=\int\frac{dx}{\cos^{3}x}=\int\frac{\cos x\,dx}{\cos^{4}x}=% \int\frac{\cos x\,dx}{(1-\sin^{2}x)^{2}}=\int\frac{du}{(1-u^{2})^{2}}
  10. u = sin x u=\sin x
  11. d u = cos x d x du=\cos x\,dx
  12. 1 ( 1 - u 2 ) 2 = 1 / 4 1 - u + 1 / 4 ( 1 - u ) 2 + 1 / 4 1 + u + 1 / 4 ( 1 + u ) 2 . \frac{1}{(1-u^{2})^{2}}=\frac{1/4}{1-u}+\frac{1/4}{(1-u)^{2}}+\frac{1/4}{1+u}+% \frac{1/4}{(1+u)^{2}}.
  13. - 1 4 ln ( 1 - u ) + 1 / 4 1 - u + 1 4 ln ( 1 + u ) - 1 / 4 1 + u + C = 1 4 ln 1 + u 1 - u + 1 2 u 1 - u 2 + C \displaystyle-\frac{1}{4}\ln(1-u)+\frac{1/4}{1-u}+\frac{1}{4}\ln(1+u)-\frac{1/% 4}{1+u}+C=\frac{1}{4}\ln\frac{1+u}{1-u}+\frac{1}{2}\frac{u}{1-u^{2}}+C
  14. sec n x tan m x d x \int\sec^{n}x\tan^{m}x\,dx
  15. sec x = cosh u tan x = sinh u sec 2 x d x = cosh u d u or sec x tan x d x = sinh u d u sec x d x = d u or d x = sech u d u u = arcosh ( sec x ) = arsinh ( tan x ) = ln | sec x + tan x | \begin{aligned}\displaystyle\sec x&\displaystyle{}=\cosh u\\ \displaystyle\tan x&\displaystyle{}=\sinh u\\ \displaystyle\sec^{2}x\,dx&\displaystyle{}=\cosh u\,du\,\text{ or }\sec x\tan x% \,dx=\sinh u\,du\\ \displaystyle\sec x\,dx&\displaystyle{}=\,du\,\text{ or }dx=\operatorname{sech% }u\,du\\ \displaystyle u&\displaystyle{}=\operatorname{arcosh}(\sec x)=\operatorname{% arsinh}(\tan x)=\ln|\sec x+\tan x|\end{aligned}
  16. sec x d x = ln | sec x + tan x | \int\sec x\,dx=\ln|\sec x+\tan x|
  17. sec 3 x d x = cosh 2 u d u = 1 2 ( cosh 2 u + 1 ) d u = 1 2 ( 1 2 sinh 2 u + u ) + C = 1 2 ( sinh u cosh u + u ) + C = 1 2 sec x tan x + 1 2 ln | sec x + tan x | + C \begin{aligned}\displaystyle\int\sec^{3}x\,dx&\displaystyle{}=\int\cosh^{2}u\,% du\\ &\displaystyle{}=\frac{1}{2}\int(\cosh 2u+1)\,du\\ &\displaystyle{}=\frac{1}{2}\left(\frac{1}{2}\sinh 2u+u\right)+C\\ &\displaystyle{}=\frac{1}{2}(\sinh u\cosh u+u)+C\\ &\displaystyle{}=\frac{1}{2}\sec x\tan x+\frac{1}{2}\ln|\sec x+\tan x|+C\end{aligned}
  18. sec n x d x = sec n - 2 x tan x n - 1 + n - 2 n - 1 sec n - 2 x d x (for n 1 ) \int\sec^{n}x\,dx=\frac{\sec^{n-2}x\tan x}{n-1}\,+\,\frac{n-2}{n-1}\int\sec^{n% -2}x\,dx\qquad\,\text{ (for }n\neq 1\,\text{)}\,\!
  19. sec n x d x = sec n - 1 x sin x n - 1 + n - 2 n - 1 sec n - 2 x d x (for n 1 ) \int\sec^{n}x\,dx=\frac{\sec^{n-1}x\sin x}{n-1}\,+\,\frac{n-2}{n-1}\int\sec^{n% -2}x\,dx\qquad\,\text{ (for }n\neq 1\,\text{)}\,\!

Integro-differential_equation.html

  1. d d x u ( x ) + x 0 x f ( t , u ( t ) ) d t = g ( x , u ( x ) ) , u ( x 0 ) = u 0 , x 0 0. \frac{d}{dx}u(x)+\int_{x_{0}}^{x}f(t,u(t))\,dt=g(x,u(x)),\qquad u(x_{0})=u_{0}% ,\qquad x_{0}\geq 0.
  2. u ( x ) + 2 u ( x ) + 5 0 x u ( t ) d t = { 1 , x 0 0 , x < 0 with u ( 0 ) = 0. u^{\prime}(x)+2u(x)+5\int_{0}^{x}u(t)\,dt=\left\{\begin{array}[]{ll}1,\qquad x% \geq 0\\ 0,\qquad x<0\end{array}\right.\qquad\,\text{with}\qquad u(0)=0.
  3. U ( s ) = { u ( x ) } = 0 e - s x u ( x ) d x . U(s)=\mathcal{L}\left\{u(x)\right\}=\int_{0}^{\infty}e^{-sx}u(x)\,dx.
  4. s U ( s ) - u ( 0 ) + 2 U ( s ) + 5 s U ( s ) = 1 s . sU(s)-u(0)+2U(s)+\frac{5}{s}U(s)=\frac{1}{s}.
  5. U ( s ) = 1 s 2 + 2 s + 5 U(s)=\frac{1}{s^{2}+2s+5}
  6. u ( x ) = 1 2 e - x sin ( 2 x ) u(x)=\frac{1}{2}e^{-x}\sin(2x)

Intel_8255.html

  1. ¬ {\neg}
  2. ¬ {\neg}
  3. ¬ {\neg}
  4. ¬ {\neg}
  5. ¬ {\neg}
  6. ¬ {\neg}
  7. ¬ {\neg}
  8. ¬ {\neg}

Intelligent_driver_model.html

  1. α \alpha
  2. x α x_{\alpha}
  3. t t
  4. v α v_{\alpha}
  5. l α l_{\alpha}
  6. s α := x α - 1 - x α - l α - 1 s_{\alpha}:=x_{\alpha-1}-x_{\alpha}-l_{\alpha-1}
  7. α - 1 \alpha-1
  8. α \alpha
  9. Δ v α := v α - v α - 1 \Delta v_{\alpha}:=v_{\alpha}-v_{\alpha-1}
  10. α \alpha
  11. x ˙ α = d x α d t = v α \dot{x}_{\alpha}=\frac{\mathrm{d}x_{\alpha}}{\mathrm{d}t}=v_{\alpha}
  12. v ˙ α = d v α d t = a ( 1 - ( v α v 0 ) δ - ( s * ( v α , Δ v α ) s α ) 2 ) \dot{v}_{\alpha}=\frac{\mathrm{d}v_{\alpha}}{\mathrm{d}t}=a\,\left(1-\left(% \frac{v_{\alpha}}{v_{0}}\right)^{\delta}-\left(\frac{s^{*}(v_{\alpha},\Delta v% _{\alpha})}{s_{\alpha}}\right)^{2}\right)
  13. with s * ( v α , Δ v α ) = s 0 + v α T + v α Δ v α 2 a b \,\text{with }s^{*}(v_{\alpha},\Delta v_{\alpha})=s_{0}+v_{\alpha}\,T+\frac{v_% {\alpha}\,\Delta v_{\alpha}}{2\,\sqrt{a\,b}}
  14. v 0 v_{0}
  15. s 0 s_{0}
  16. T T
  17. a a
  18. b b
  19. v 0 v_{0}
  20. s 0 s_{0}
  21. s 0 s_{0}
  22. T T
  23. a a
  24. b b
  25. δ \delta
  26. α \alpha
  27. v ˙ free α = a ( 1 - ( v α v 0 ) δ ) v ˙ int α = - a ( s * ( v α , Δ v α ) s α ) 2 = - a ( s 0 + v α T s α + v α Δ v α 2 a b s α ) 2 \dot{v}\text{free}_{\alpha}=a\,\left(1-\left(\frac{v_{\alpha}}{v_{0}}\right)^{% \delta}\right)\qquad\dot{v}\text{int}_{\alpha}=-a\,\left(\frac{s^{*}(v_{\alpha% },\Delta v_{\alpha})}{s_{\alpha}}\right)^{2}=-a\,\left(\frac{s_{0}+v_{\alpha}% \,T}{s_{\alpha}}+\frac{v_{\alpha}\,\Delta v_{\alpha}}{2\,\sqrt{a\,b}\,s_{% \alpha}}\right)^{2}
  28. s α s_{\alpha}
  29. a a
  30. v α v_{\alpha}
  31. v 0 v_{0}
  32. v 0 v_{0}
  33. - a ( v α Δ v α ) 2 / ( 2 a b s α ) 2 = - ( v α Δ v α ) 2 / ( 4 b s α 2 ) -a\,(v_{\alpha}\,\Delta v_{\alpha})^{2}\,/\,(2\,\sqrt{a\,b}\,s_{\alpha})^{2}=-% (v_{\alpha}\,\Delta v_{\alpha})^{2}\,/\,(4\,b\,s_{\alpha}^{2})
  34. b b
  35. - a ( s 0 + v α T ) 2 / s α 2 -a\,(s_{0}+v_{\alpha}\,T)^{2}\,/\,s_{\alpha}^{2}
  36. x ˙ = d x d t = v \dot{x}=\frac{\mathrm{d}x}{\mathrm{d}t}=v
  37. v ˙ = d v d t = a ( 1 - ( v v 0 ) δ - ( s * ( v , Δ v ) s ) 2 ) \dot{v}=\frac{\mathrm{d}v}{\mathrm{d}t}=a\,\left(1-\left(\frac{v}{v_{0}}\right% )^{\delta}-\left(\frac{s^{*}(v,\Delta v)}{s}\right)^{2}\right)
  38. with s * ( v , Δ v ) = s 0 + v T + v Δ v 2 a b \,\text{with }s^{*}(v,\Delta v)=s_{0}+v\,T+\frac{v\,\Delta v}{2\,\sqrt{a\,b}}

Intercept_theorem.html

  1. | S A | : | A B | = | S C | : | C D | |SA|:|AB|=|SC|:|CD|
  2. | S B | : | A B | = | S D | : | C D | |SB|:|AB|=|SD|:|CD|
  3. | S A | : | S B | = | S C | : | S D | |SA|:|SB|=|SC|:|SD|
  4. | S A | : | S B | = | S C | : | S D | = | A C | : | B D | |SA|:|SB|=|SC|:|SD|=|AC|:|BD|
  5. | S A | : | A B | = | S C | : | C D | |SA|:|AB|=|SC|:|CD|
  6. λ ( a + b ) = λ a + λ b \lambda\cdot(\vec{a}+\vec{b})=\lambda\cdot\vec{a}+\lambda\cdot\vec{b}
  7. λ a = | λ | a \|\lambda\vec{a}\|=|\lambda|\cdot\ \|\vec{a}\|
  8. λ a a = λ b b = λ ( a + b ) a + b = | λ | \frac{\|\lambda\cdot\vec{a}\|}{\|\vec{a}\|}=\frac{\|\lambda\cdot\vec{b}\|}{\|% \vec{b}\|}=\frac{\|\lambda\cdot(\vec{a}+\vec{b})\|}{\|\vec{a}+\vec{b}\|}=|\lambda|
  9. d d
  10. d - 1 d^{-1}
  11. A B ¯ \overline{AB}
  12. m : n m:n
  13. A B ¯ \overline{AB}
  14. m + n m+n
  15. A B ¯ \overline{AB}
  16. A B ¯ \overline{AB}
  17. 5 : 3 5:3
  18. C = 65 m + 230 m 2 = 180 m C=65m+\frac{230m}{2}=180m
  19. D = C A B = 1.63 m 180 m 2 m = 146.7 m D=\frac{C\cdot A}{B}=\frac{1.63m\cdot 180m}{2m}=146.7m
  20. | C F | |CF|
  21. | C A | |CA|
  22. | F E | |FE|
  23. | A B | = | A C | | F E | | F C | |AB|=\frac{|AC||FE|}{|FC|}
  24. C A B D CA\parallel BD
  25. C D A \triangle CDA
  26. C B A \triangle CBA
  27. | C D A | = | C B A | |\triangle CDA|=|\triangle CBA|
  28. | S C B | = | S D A | |\triangle SCB|=|\triangle SDA|
  29. | S C A | | C D A | = | S C A | | C B A | \frac{|\triangle SCA|}{|\triangle CDA|}=\frac{|\triangle SCA|}{|\triangle CBA|}
  30. | S C A | | S D A | = | S C A | | S C B | \frac{|\triangle SCA|}{|\triangle SDA|}=\frac{|\triangle SCA|}{|\triangle SCB|}
  31. baseline altitude 2 \tfrac{\,\text{baseline}\cdot\,\text{altitude}}{2}
  32. | S C | | A F | | C D | | A F | = | S A | | E C | | A B | | E C | \frac{|SC||AF|}{|CD||AF|}=\frac{|SA||EC|}{|AB||EC|}
  33. | S C | | A F | | S D | | A F | = | S A | | E C | | S B | | E C | \frac{|SC||AF|}{|SD||AF|}=\frac{|SA||EC|}{|SB||EC|}
  34. | S C | | C D | = | S A | | A B | \,\frac{|SC|}{|CD|}=\frac{|SA|}{|AB|}
  35. | S C | | S D | = | S A | | S B | \,\frac{|SC|}{|SD|}=\frac{|SA|}{|SB|}
  36. | S A | |SA|
  37. | S C | |SC|
  38. | S A | | S D | | S B | | C D | = | S B | | S C | | S D | | A B | \frac{\frac{|SA||SD|}{|SB|}}{|CD|}=\frac{\frac{|SB||SC|}{|SD|}}{|AB|}
  39. | S D | | C D | = | S B | | A B | \,\frac{|SD|}{|CD|}=\frac{|SB|}{|AB|}
  40. \,\square
  41. S D SD
  42. B D BD
  43. | A C | = | D G | |AC|=|DG|
  44. | S A | | S B | = | D G | | B D | \frac{|SA|}{|SB|}=\frac{|DG|}{|BD|}
  45. | S A | | S B | = | A C | | B D | \frac{|SA|}{|SB|}=\frac{|AC|}{|BD|}
  46. \square
  47. A C AC
  48. B D BD
  49. A C AC
  50. D D
  51. S A SA
  52. B 0 B B_{0}\neq B
  53. | S B | : | S A | = | S D | : | S C | |SB|:|SA|=|SD|:|SC|
  54. | S B | = | S D | | S A | | S C | |SB|=\frac{|SD||SA|}{|SC|}
  55. | S B 0 | = | S D | | S A | | S C | |SB_{0}|=\frac{|SD||SA|}{|SC|}
  56. B B
  57. B 0 B_{0}
  58. S S
  59. S S
  60. B = B 0 B=B_{0}
  61. A C AC
  62. B D BD
  63. \square

Interest_Equalization_Tax.html

  1. BOP = current account - capital account ± balancing item \,\text{BOP}=\,\text{current account}-\,\text{capital account}\pm\,\text{% balancing item}\,
  2. = Change in foreign ownership of domestic assets \displaystyle=\mbox{Change in foreign ownership of domestic assets}

Interval_order.html

  1. P = ( X , ) P=(X,\leq)
  2. X X
  3. x i ( i , r i ) x_{i}\mapsto(\ell_{i},r_{i})
  4. x i , x j X x_{i},x_{j}\in X
  5. x i < x j x_{i}<x_{j}
  6. P P
  7. r i < j r_{i}<\ell_{j}
  8. ( 2 + 2 ) (2+2)
  9. ( i , i + 1 ) (\ell_{i},\ell_{i}+1)
  10. X X
  11. ( X , ) (X,\cap)
  12. ( 2 + 2 ) (2+2)
  13. [ n ] [n]
  14. 2 n 2n
  15. f f
  16. [ 2 n ] [2n]
  17. i [ 2 n ] i\in[2n]
  18. i a n d a r i g h t n e s t i n g i s a n < m a t h > i [ 2 n ] iandarightnestingisan<math>i\in[2n]
  19. f ( i ) < f ( i + 1 ) < i < i + 1 f(i)<f(i+1)<i<i+1
  20. F ( t ) = n 0 i = 1 n ( 1 - ( 1 - t ) i ) F(t)=\sum_{n\geq 0}\prod_{i=1}^{n}(1-(1-t)^{i})
  21. n n
  22. t n t^{n}
  23. F ( t ) F(t)

Intraclass_correlation.html

  1. r = 1 N s 2 n = 1 N ( x n , 1 - x ¯ ) ( x n , 2 - x ¯ ) r=\frac{1}{Ns^{2}}\sum_{n=1}^{N}(x_{n,1}-\bar{x})(x_{n,2}-\bar{x})
  2. x ¯ = 1 2 N n = 1 N ( x n , 1 + x n , 2 ) \bar{x}=\frac{1}{2N}\sum_{n=1}^{N}(x_{n,1}+x_{n,2})
  3. s 2 = 1 2 N { n = 1 N ( x n , 1 - x ¯ ) 2 + n = 1 N ( x n , 2 - x ¯ ) 2 } s^{2}=\frac{1}{2N}\left\{\sum_{n=1}^{N}(x_{n,1}-\bar{x})^{2}+\sum_{n=1}^{N}(x_% {n,2}-\bar{x})^{2}\right\}
  4. r = 1 3 N s 2 n = 1 N { ( x n , 1 - x ¯ ) ( x n , 2 - x ¯ ) + ( x n , 1 - x ¯ ) ( x n , 3 - x ¯ ) + ( x n , 2 - x ¯ ) ( x n , 3 - x ¯ ) } r=\frac{1}{3Ns^{2}}\sum_{n=1}^{N}\left\{(x_{n,1}-\bar{x})(x_{n,2}-\bar{x})+(x_% {n,1}-\bar{x})(x_{n,3}-\bar{x})+(x_{n,2}-\bar{x})(x_{n,3}-\bar{x})\right\}
  5. x ¯ = 1 3 N n = 1 N ( x n , 1 + x n , 2 + x n , 3 ) \bar{x}=\frac{1}{3N}\sum_{n=1}^{N}(x_{n,1}+x_{n,2}+x_{n,3})
  6. s 2 = 1 3 N { n = 1 N ( x n , 1 - x ¯ ) 2 + n = 1 N ( x n , 2 - x ¯ ) 2 + n = 1 N ( x n , 3 - x ¯ ) 2 } s^{2}=\frac{1}{3N}\left\{\sum_{n=1}^{N}(x_{n,1}-\bar{x})^{2}+\sum_{n=1}^{N}(x_% {n,2}-\bar{x})^{2}+\sum_{n=1}^{N}(x_{n,3}-\bar{x})^{2}\right\}
  7. r = K K - 1 N - 1 n = 1 N ( x ¯ n - x ¯ ) 2 s 2 - 1 K - 1 , r=\frac{K}{K-1}\cdot\frac{N^{-1}\sum_{n=1}^{N}(\bar{x}_{n}-\bar{x})^{2}}{s^{2}% }-\frac{1}{K-1},
  8. x ¯ n \bar{x}_{n}
  9. r - 1 / ( K - 1 ) r\geq-1/(K-1)
  10. N - 1 n = 1 N ( x ¯ n - x ¯ ) 2 s 2 , \frac{N^{-1}\sum_{n=1}^{N}(\bar{x}_{n}-\bar{x})^{2}}{s^{2}},
  11. Y i j = μ + α j + ϵ i j , Y_{ij}=\mu+\alpha_{j}+\epsilon_{ij},
  12. σ α 2 σ α 2 + σ ϵ 2 . \frac{\sigma_{\alpha}^{2}}{\sigma_{\alpha}^{2}+\sigma_{\epsilon}^{2}}.

Intravascular_volume_status.html

  1. ± % Desirable Weight = Actual Weight - Desirable Weight Desirable Weight × 100 \pm\%\,\text{ Desirable Weight}=\frac{\,\text{Actual Weight}-\,\text{Desirable% Weight}}{\,\text{Desirable Weight}}\times 100

Invariance_mechanics.html

  1. x x
  2. x x^{\prime}
  3. R = | x - x | . R=|x-x^{\prime}|.\,
  4. G F ( R ) = J 1 ( R m ) m R G_{F}(R)=\frac{J_{1}(Rm)m}{R}
  5. ρ μ \rho_{\mu}
  6. R = | x - x | R=|x-x^{\prime}|\,
  7. S = ρ μ ρ μ S=\rho_{\mu}\rho^{\prime}_{\mu}\,
  8. T = ( x - x ) μ ρ μ T=(x-x^{\prime})_{\mu}\rho^{\prime}_{\mu}\,
  9. U = ρ μ ( x - x ) μ . U=\rho_{\mu}(x-x^{\prime})_{\mu}.\,
  10. ρ μ ν \rho^{\nu}_{\mu}
  11. R = | x - x | R=|x-x^{\prime}|\,
  12. S = ρ μ ν ρ μ ν S=\rho^{\nu}_{\mu}\rho^{\nu}_{\mu}
  13. T = ρ μ ν ( x - x ) μ ( x - x ) τ ρ τ ν . T=\rho^{\nu}_{\mu}(x-x^{\prime})_{\mu}(x-x^{\prime})_{\tau}\rho^{\nu}_{\tau}.
  14. R = | x - x | , R=|x-x^{\prime}|,\,
  15. S = u α γ 5 α β u ¯ β , S=u^{\alpha}\gamma^{\alpha\beta}_{5}\overline{u}^{\beta},
  16. T = u α γ μ α β ( x - x ) μ u ¯ β . T=u^{\alpha}\gamma^{\alpha\beta}_{\mu}(x-x^{\prime})_{\mu}\overline{u}^{\beta}.
  17. G ψ ( R , S , T , m ) = ( T + T 2 R R - m S ) G F ( R , m ) . G^{\psi}(R,S,T,m)=\left(\frac{\partial}{\partial T}+\frac{T}{2R}\frac{\partial% }{\partial R}-mS\right)G^{F}(R,m).
  18. V = u α γ μ α β ρ μ u ¯ β . V=u^{\alpha}\gamma^{\alpha\beta}_{\mu}\rho_{\mu}\overline{u}^{\beta}.
  19. 1 / R 2 1/R^{2}
  20. S 1 = a 2 + b 2 + c 2 S_{1}=a^{2}+b^{2}+c^{2}
  21. S 2 = a 4 + b 4 + c 4 S_{2}=a^{4}+b^{4}+c^{4}
  22. S 3 = a 6 + b 6 + c 6 S_{3}=a^{6}+b^{6}+c^{6}
  23. m = 1 m=1
  24. A 3 ( a , b , c ) = S 1 - S 1 2 24 - < m t p l > S 2 24 + S 1 S 3 6912 - S 3 288 - 43 S 1 2 S 2 230400 + S 1 S 2 192 + S 2 2 460800 + 37 S 1 4 1382400 + A_{3}(a,b,c)={\it S_{1}}-{{{\it S_{1}}^{2}}\over{24}}-{<mtpl>{{\it S_{2}}}% \over{24}}+{{{\it S_{1}}\,{\it S_{3}}}\over{6912}}-{{{\it S_{3}}}\over{288}}-{% {43\,{\it S_{1}}^{2}\,{\it S_{2}}}\over{230400}}+{{{\it S_{1}}\,{\it S_{2}}}% \over{192}}+{{{\it S_{2}}^{2}}\over{460800}}+{{37\,{\it S_{1}}^{4}}\over{13824% 00}}+...
  25. ( 2 a 2 + 3 a a + 2 b 2 + 3 a b + a 2 + b 2 - c 2 a b 2 a b + m 2 ) A 3 ( a , b , c ) = G F ( a ) G F ( b ) \left(\frac{\partial^{2}}{\partial a^{2}}+\frac{3}{a}\frac{\partial}{\partial a% }+\frac{\partial^{2}}{\partial b^{2}}+\frac{3}{a}\frac{\partial}{\partial b}+% \frac{a^{2}+b^{2}-c^{2}}{ab}\frac{\partial^{2}}{\partial a\partial b}+m^{2}% \right)A_{3}(a,b,c)=G_{F}(a)G_{F}(b)
  26. A m p ( e - e - + γ ) = [ u ( γ . x + m γ 5 ) ( γ . ρ ) ( γ . y + m γ 5 ) u ¯ ] A 3 ( a , b , c ) Amp(e^{-}\rightarrow e^{-}+\gamma)=\left[u(\gamma.\partial_{x}+m\gamma_{5})(% \gamma.\rho)(\gamma.\partial_{y}+m\gamma_{5})\overline{u}\right]A_{3}(a,b,c)
  27. m 1 m_{1}
  28. m 2 m_{2}
  29. m 3 m_{3}
  30. A ( a , b , c ; m 1 , m 2 , m 3 ) A(a,b,c;m_{1},m_{2},m_{3})
  31. E 6 E_{6}
  32. 1 / V 6 1/V_{6}

Invariant_factor.html

  1. R R
  2. M M
  3. R R
  4. M R r R / ( a 1 ) R / ( a 2 ) R / ( a m ) M\cong R^{r}\oplus R/(a_{1})\oplus R/(a_{2})\oplus\cdots\oplus R/(a_{m})
  5. r 0 r\geq 0
  6. a 1 , , a m R a_{1},\ldots,a_{m}\in R
  7. a 1 a 2 a m a_{1}\mid a_{2}\mid\cdots\mid a_{m}
  8. r r
  9. M M
  10. a 1 , , a m a_{1},\ldots,a_{m}
  11. M M

Inverse_problem_for_Lagrangian_mechanics.html

  1. S ( u ) = 0 T L ( t , u ( t ) , u ˙ ( t ) ) d t , S(u)=\int_{0}^{T}L(t,u(t),\dot{u}(t))\,\mathrm{d}t,
  2. d d t L u ˙ i - L u i = 0 for 1 i n , \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial\dot{u}^{i}}-\frac{% \partial L}{\partial u^{i}}=0\quad\,\text{for }1\leq i\leq n,
  3. T ( u ˙ ) = 1 2 m | u ˙ | 2 , T(\dot{u})=\frac{1}{2}m|\dot{u}|^{2},
  4. V : [ 0 , T ] × n , V:[0,T]\times\mathbb{R}^{n}\to\mathbb{R},
  5. L ( t , u , u ˙ ) = T ( u ˙ ) - V ( t , u ) , L(t,u,\dot{u})=T(\dot{u})-V(t,u),
  6. u ¨ i = - V ( t , u ) u i for 1 i n , \ddot{u}^{i}=-\frac{\partial V(t,u)}{\partial u^{i}}\quad\,\text{for }1\leq i% \leq n,
  7. i.e. u ¨ = - u V ( t , u ) . \mbox{i.e. }~{}\ddot{u}=-\nabla_{u}V(t,u).
  8. u ¨ i = f i ( u j , u ˙ j ) for 1 i , j n , (E) \ddot{u}^{i}=f^{i}(u^{j},\dot{u}^{j})\quad\,\text{for }1\leq i,j\leq n,\quad% \mbox{(E)}~{}
  9. v i = u ˙ i v^{i}=\dot{u}^{i}
  10. Φ j i = 1 2 d d t f i v j - f i u j - 1 4 f i v k f k v j . \Phi_{j}^{i}=\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial f^{i}}{% \partial v^{j}}-\frac{\partial f^{i}}{\partial u^{j}}-\frac{1}{4}\frac{% \partial f^{i}}{\partial v^{k}}\frac{\partial f^{k}}{\partial v^{j}}.
  11. g Φ = ( g Φ ) , (H1) g\Phi=(g\Phi)^{\top},\quad\mbox{(H1)}~{}
  12. d g i j d t + 1 2 f k v i g k j + 1 2 f k v j g k i = 0 for 1 i , j n , (H2) \frac{\mathrm{d}g_{ij}}{\mathrm{d}t}+\frac{1}{2}\frac{\partial f^{k}}{\partial v% ^{i}}g_{kj}+\frac{1}{2}\frac{\partial f^{k}}{\partial v^{j}}g_{ki}=0\mbox{ for% }~{}1\leq i,j\leq n,\quad\mbox{(H2)}~{}
  13. g i j v k = g i k v j for 1 i , j , k n . (H3) \frac{\partial g_{ij}}{\partial v^{k}}=\frac{\partial g_{ik}}{\partial v^{j}}% \mbox{ for }~{}1\leq i,j,k\leq n.\quad\mbox{(H3)}~{}
  14. 2 ( n + 1 3 ) 2\left(\begin{matrix}n+1\\ 3\end{matrix}\right)
  15. ( n k ) \left(\begin{matrix}n\\ k\end{matrix}\right)
  16. Ψ j k i = 1 3 ( Φ j i v k - Φ k i v j ) . \Psi_{jk}^{i}=\frac{1}{3}\left(\frac{\partial\Phi_{j}^{i}}{\partial v^{k}}-% \frac{\partial\Phi_{k}^{i}}{\partial v^{j}}\right).
  17. g m i Ψ j k m + g m k Ψ i j m + g m j Ψ k i m = 0 for 1 i , j n . (A) g_{mi}\Psi_{jk}^{m}+g_{mk}\Psi_{ij}^{m}+g_{mj}\Psi_{ki}^{m}=0\mbox{ for }~{}1% \leq i,j\leq n.\quad\mbox{(A)}~{}
  18. ( n 3 ) \left(\begin{matrix}n\\ 3\end{matrix}\right)
  19. ( n 2 ) \left(\begin{matrix}n\\ 2\end{matrix}\right)

Irrational_rotation.html

  1. T θ : [ 0 , 1 ] [ 0 , 1 ] , T θ ( x ) x + θ mod 1 T_{\theta}:[0,1]\rightarrow[0,1],\quad T_{\theta}(x)\triangleq x+\theta\mod 1
  2. T θ : S 1 S 1 , T θ ( x ) = x e 2 π i θ T_{\theta}:S^{1}\to S^{1},\quad\quad\quad T_{\theta}(x)=xe^{2\pi i\theta}
  3. ϕ : ( [ 0 , 1 ] , + ) ( S 1 , ) ϕ ( x ) = x e 2 π i θ \phi:([0,1],+)\to(S^{1},\cdot)\quad\phi(x)=xe^{2\pi i\theta}
  4. θ = a b \theta=\frac{a}{b}
  5. gcd ( a , b ) = 1 \gcd(a,b)=1
  6. T θ b ( x ) = x T_{\theta}^{b}(x)=x
  7. x \isin [ 0 , 1 ] x\isin[0,1]
  8. T θ i ( x ) x T_{\theta}^{i}(x)\neq x
  9. 1 i < b 1\leq i<b
  10. 0 , 11 0,11
  11. 0 , 11 0,11
  12. lim N 1 N n = 0 N - 1 χ [ a , b ) ( T θ n ( t ) ) = b - a \,\text{lim}_{N\to\infty}\frac{1}{N}\sum_{n=0}^{N-1}\chi_{[a,b)}(T_{\theta}^{n% }(t))=b-a
  13. F : F:\mathbb{R}\to\mathbb{R}
  14. π F = f π \pi\circ F=f\circ\pi
  15. π ( t ) = t mod 1 \pi(t)=t\bmod 1
  16. < v a r > J <var>J

Irregularity_of_distributions.html

  1. x 1 , , x N x_{1},\ldots,x_{N}
  2. x 1 , , x N x_{1},\ldots,x_{N}
  3. k - 1 n x i < k n . \frac{k-1}{n}\leq x_{i}<\frac{k}{n}.
  4. 0 < x 1 < 1 5 < x 5 < 2 5 < x 3 < 3 5 < x 4 < 4 5 < x 2 < 1. 0<x_{1}<\frac{1}{5}<x_{5}<\frac{2}{5}<x_{3}<\frac{3}{5}<x_{4}<\frac{4}{5}<x_{2% }<1.

ISO_217.html

  1. 1 : 2 1:\sqrt{2}
  2. 1.05 * 2 - 0.25 m × 1.05 * 2 0.25 m 861.7 m m × 1218.6 m m \sqrt{1.05}*2^{-0.25}m\times\sqrt{1.05}*2^{0.25}m\approx 861.7mm\times 1218.6mm
  3. 860 m m × 1220 m m 860mm\times 1220mm
  4. 43 : 61 1 : 1.4186 43:61\approx 1:1.4186
  5. 61 : 86 1 : 1.4098 61:86\approx 1:1.4098
  6. 45 : 64 1 : 1.4222 45:64\approx 1:1.4222
  7. 32 : 45 = 1 : 1.40625 32:45=1:1.40625

Isomorphism-closed_subcategory.html

  1. 𝒜 \mathcal{A}
  2. \mathcal{B}
  3. \mathcal{B}
  4. h : A B h:A\to B
  5. A 𝒜 A\in\mathcal{A}
  6. 𝒜 . \mathcal{A}.
  7. B B
  8. h - 1 : B A h^{-1}:B\to A
  9. 𝒜 \mathcal{A}
  10. \mathcal{B}
  11. 𝒜 \mathcal{A}
  12. 𝒜 \mathcal{A}
  13. 𝐓𝐨𝐩 . \mathbf{Top}.

Isomorphism_extension_theorem.html

  1. F F
  2. E E
  3. F F
  4. ϕ \phi
  5. F F
  6. F F^{\prime}
  7. ϕ \phi
  8. τ \tau
  9. E E
  10. E E^{\prime}
  11. F F^{\prime}
  12. F F^{\prime}

Isothermal_coordinates.html

  1. g = e φ ( d x 1 2 + + d x n 2 ) , g=e^{\varphi}(dx_{1}^{2}+\cdots+dx_{n}^{2}),
  2. φ \varphi
  3. d s 2 = E d x 2 + 2 F d x d y + G d y 2 , ds^{2}=E\,dx^{2}+2F\,dx\,dy+G\,dy^{2},
  4. d s 2 = λ | d z + μ d z ¯ | 2 , ds^{2}=\lambda|\,dz+\mu\,d\overline{z}|^{2},
  5. d s 2 = ρ ( d u 2 + d v 2 ) ds^{2}=\rho(du^{2}+dv^{2})
  6. ρ | d w | 2 = ρ | w z | 2 | d z + w z ¯ w z d z ¯ | 2 , \rho\,|dw|^{2}=\rho|w_{z}|^{2}|\,dz+{w_{\overline{z}}\over w_{z}}\,d\overline{% z}|^{2},
  7. w z ¯ = μ w z {\partial w\over\partial\overline{z}}=\mu{\partial w\over\partial z}
  8. \star
  9. Δ = d * d \Delta=d^{*}d
  10. d u = d v \star du=dv
  11. d d u = 0 d\star du=0
  12. d = d * , \star d\star=d^{*},
  13. K = - 1 2 e - φ ( 2 φ u 2 + 2 φ v 2 ) , K=-\frac{1}{2}e^{-\varphi}\left(\frac{\partial^{2}\varphi}{\partial u^{2}}+% \frac{\partial^{2}\varphi}{\partial v^{2}}\right),
  14. ρ = e φ \rho=e^{\varphi}

Isotopy_of_loops.html

  1. ( Q , ) (Q,\cdot)
  2. ( P , ) (P,\circ)
  3. α ( x ) β ( y ) = γ ( x y ) \alpha(x)\circ\beta(y)=\gamma(x\cdot y)\,
  4. ( Q , ) (Q,\cdot)
  5. ( L , ) (L,\cdot)
  6. ( K , ) (K,\circ)
  7. ( α , β , γ ) : L K (\alpha,\beta,\gamma):L\to K
  8. ( α 0 , β 0 , i d ) (\alpha_{0},\beta_{0},id)
  9. ( L , ) (L,\cdot)
  10. ( L , * ) (L,*)
  11. γ \gamma
  12. ( L , * ) (L,*)
  13. ( K , ) (K,\circ)
  14. α 0 = γ - 1 α \alpha_{0}=\gamma^{-1}\alpha
  15. β 0 = γ - 1 β \beta_{0}=\gamma^{-1}\beta
  16. x * y = α ( x ) β ( y ) x*y=\alpha(x)\cdot\beta(y)
  17. ( L , ) (L,\cdot)
  18. ( L , ) (L,\circ)
  19. ( L , ) (L,\cdot)
  20. ( α , β , i d ) (\alpha,\beta,id)
  21. ( L , ) (L,\cdot)
  22. ( L , ) (L,\circ)
  23. α = R b - 1 \alpha=R_{b}^{-1}
  24. β = L a - 1 \beta=L_{a}^{-1}
  25. a = α ( e ) a=\alpha(e)
  26. b = β ( e ) b=\beta(e)
  27. α ( x y ) c = α ( x ) ( α ( y ) c ) \alpha(xy)c=\alpha(x)(\alpha(y)c)

Isotropic_quadratic_form.html

  1. 1 - 1 \scriptstyle\langle 1\rangle\oplus\langle-1\rangle

IT_Grade.html

  1. T = 10 0.2 × ( I T G - 1 ) ( 0.45 × D 3 + 0.001 × D ) T=10^{0.2\times(ITG-1)}\cdot(0.45\times\sqrt[3]{D}+0.001\times D)
  2. D D
  3. T T
  4. I = 0.004 D + 2.1 I=0.004\cdot D+2.1
  5. T = k I T=k\cdot I

Iteratively_reweighted_least_squares.html

  1. arg min s y m b o l β i = 1 n | y i - f i ( s y m b o l β ) | p , \underset{symbol\beta}{\operatorname{arg\,min}}\sum_{i=1}^{n}\big|y_{i}-f_{i}(% symbol\beta)\big|^{p},
  2. s y m b o l β ( t + 1 ) = arg min s y m b o l β i = 1 n w i ( s y m b o l β ( t ) ) | y i - f i ( s y m b o l β ) | 2 . symbol\beta^{(t+1)}=\underset{symbol\beta}{\operatorname{arg\,min}}\sum_{i=1}^% {n}w_{i}(symbol\beta^{(t)})\big|y_{i}-f_{i}(symbol\beta)\big|^{2}.
  3. \ell
  4. \ell
  5. \ell
  6. arg min s y m b o l β 𝐲 - X s y m b o l β p = arg min s y m b o l β i = 1 n | y i - X i s y m b o l β | p , \underset{symbol\beta}{\operatorname{arg\,min}}\big\|\mathbf{y}-Xsymbol\beta\|% _{p}=\underset{symbol\beta}{\operatorname{arg\,min}}\sum_{i=1}^{n}\left|y_{i}-% X_{i}symbol\beta\right|^{p},
  7. s y m b o l β ( t + 1 ) = arg min s y m b o l β i = 1 n w i ( t ) | y i - X i s y m b o l β | 2 = ( X T W ( t ) X ) - 1 X T W ( t ) 𝐲 , symbol\beta^{(t+1)}=\underset{symbol\beta}{\operatorname{arg\,min}}\sum_{i=1}^% {n}w_{i}^{(t)}\left|y_{i}-X_{i}symbol\beta\right|^{2}=(X^{\rm T}W^{(t)}X)^{-1}% X^{\rm T}W^{(t)}\mathbf{y},
  8. w i ( 0 ) = 1 w_{i}^{(0)}=1
  9. w i ( t ) = | y i - X i s y m b o l β ( t ) | p - 2 . w_{i}^{(t)}=\big|y_{i}-X_{i}symbol\beta^{(t)}\big|^{p-2}.
  10. w i ( t ) = 1 | y i - X i s y m b o l β ( t ) | . w_{i}^{(t)}=\frac{1}{\big|y_{i}-X_{i}symbol\beta^{(t)}\big|}.
  11. w i ( t ) = 1 max ( δ , | y i - X i s y m b o l β ( t ) | ) . w_{i}^{(t)}=\frac{1}{\,\text{max}(\delta,\big|y_{i}-X_{i}symbol\beta^{(t)}\big% |)}.
  12. δ \delta

Itō_diffusion.html

  1. d X t = b ( X t ) d t + σ ( X t ) d B t , \mathrm{d}X_{t}=b(X_{t})\,\mathrm{d}t+\sigma(X_{t})\,\mathrm{d}B_{t},
  2. | b ( x ) - b ( y ) | + | σ ( x ) - σ ( y ) | C | x - y | |b(x)-b(y)|+|\sigma(x)-\sigma(y)|\leq C|x-y|
  3. 𝐏 [ X t = Y t ] = 1 for all t . \mathbf{P}[X_{t}=Y_{t}]=1\mbox{ for all }~{}t.
  4. u ( x ) = 𝐄 x [ f ( X t ) ] . u(x)=\mathbf{E}^{x}[f(X_{t})].
  5. Σ t = Σ t B = σ { B s - 1 ( A ) Ω : 0 s t , A 𝐑 n Borel } . \Sigma_{t}=\Sigma_{t}^{B}=\sigma\left\{B_{s}^{-1}(A)\subseteq\Omega\ :\ 0\leq s% \leq t,A\subseteq\mathbf{R}^{n}\mbox{ Borel}~{}\right\}.
  6. 𝐄 x [ f ( X t + h ) | Σ t ] ( ω ) = 𝐄 X t ( ω ) [ f ( X h ) ] . \mathbf{E}^{x}\big[f(X_{t+h})\big|\Sigma_{t}\big](\omega)=\mathbf{E}^{X_{t}(% \omega)}[f(X_{h})].
  7. 𝐄 x [ f ( X t + h ) | F t ] \displaystyle\mathbf{E}^{x}\left[f(X_{t+h})\big|F_{t}\right]
  8. A f ( x ) = lim t 0 𝐄 x [ f ( X t ) ] - f ( x ) t . Af(x)=\lim_{t\downarrow 0}\frac{\mathbf{E}^{x}[f(X_{t})]-f(x)}{t}.
  9. A f ( x ) = i b i ( x ) f x i ( x ) + 1 2 i , j ( σ ( x ) σ ( x ) ) i , j 2 f x i x j ( x ) , Af(x)=\sum_{i}b_{i}(x)\frac{\partial f}{\partial x_{i}}(x)+\tfrac{1}{2}\sum_{i% ,j}\left(\sigma(x)\sigma(x)^{\top}\right)_{i,j}\frac{\partial^{2}f}{\partial x% _{i}\,\partial x_{j}}(x),
  10. A f ( x ) = b ( x ) x f ( x ) + 1 2 ( σ ( x ) σ ( x ) ) : x x f ( x ) . Af(x)=b(x)\cdot\nabla_{x}f(x)+\tfrac{1}{2}\left(\sigma(x)\sigma(x)^{\top}% \right):\nabla_{x}\nabla_{x}f(x).
  11. A f ( x ) = 1 2 i , j δ i j 2 f x i x j ( x ) = 1 2 i 2 f x i 2 ( x ) Af(x)=\tfrac{1}{2}\sum_{i,j}\delta_{ij}\frac{\partial^{2}f}{\partial x_{i}\,% \partial x_{j}}(x)=\tfrac{1}{2}\sum_{i}\frac{\partial^{2}f}{\partial x_{i}^{2}% }(x)
  12. u ( t , x ) = 𝐄 x [ f ( X t ) ] , u(t,x)=\mathbf{E}^{x}[f(X_{t})],
  13. { u t ( t , x ) = A u ( t , x ) , t > 0 , x 𝐑 n ; u ( 0 , x ) = f ( x ) , x 𝐑 n . \begin{cases}\dfrac{\partial u}{\partial t}(t,x)=Au(t,x),&t>0,x\in\mathbf{R}^{% n};\\ u(0,x)=f(x),&x\in\mathbf{R}^{n}.\end{cases}
  14. 𝐏 [ X t S ] = S ρ ( t , x ) d x . \mathbf{P}\left[X_{t}\in S\right]=\int_{S}\rho(t,x)\,\mathrm{d}x.
  15. { ρ t ( t , x ) = A * ρ ( t , x ) , t > 0 , x 𝐑 n ; ρ ( 0 , x ) = ρ 0 ( x ) , x 𝐑 n . \begin{cases}\dfrac{\partial\rho}{\partial t}(t,x)=A^{*}\rho(t,x),&t>0,x\in% \mathbf{R}^{n};\\ \rho(0,x)=\rho_{0}(x),&x\in\mathbf{R}^{n}.\end{cases}
  16. v ( t , x ) = 𝐄 x [ exp ( - 0 t q ( X s ) d s ) f ( X t ) ] . v(t,x)=\mathbf{E}^{x}\left[\exp\left(-\int_{0}^{t}q(X_{s})\,\mathrm{d}s\right)% f(X_{t})\right].
  17. { v t ( t , x ) = A v ( t , x ) - q ( x ) v ( t , x ) , t > 0 , x 𝐑 n ; v ( 0 , x ) = f ( x ) , x 𝐑 n . \begin{cases}\dfrac{\partial v}{\partial t}(t,x)=Av(t,x)-q(x)v(t,x),&t>0,x\in% \mathbf{R}^{n};\\ v(0,x)=f(x),&x\in\mathbf{R}^{n}.\end{cases}
  18. 𝒜 \mathcal{A}
  19. 𝒜 f ( x ) = lim U x 𝐄 x [ f ( X τ U ) ] - f ( x ) 𝐄 x [ τ U ] , \mathcal{A}f(x)=\lim_{U\downarrow x}\frac{\mathbf{E}^{x}\left[f(X_{\tau_{U}})% \right]-f(x)}{\mathbf{E}^{x}[\tau_{U}]},
  20. U k + 1 U k and k = 1 U k = { x } , U_{k+1}\subseteq U_{k}\mbox{ and }~{}\bigcap_{k=1}^{\infty}U_{k}=\{x\},
  21. τ U = inf { t 0 : X t U } \tau_{U}=\inf\{t\geq 0\ :\ X_{t}\not\in U\}
  22. D 𝒜 D_{\mathcal{A}}
  23. 𝒜 f ( x ) = 0. \mathcal{A}f(x)=0.
  24. D A D 𝒜 D_{A}\subseteq D_{\mathcal{A}}
  25. A f = 𝒜 f for all f D A . Af=\mathcal{A}f\mbox{ for all }~{}f\in D_{A}.
  26. 𝒜 f ( x ) = i b i ( x ) f x i ( x ) + 1 2 i , j ( σ ( x ) σ ( x ) ) i , j 2 f x i x j ( x ) . \mathcal{A}f(x)=\sum_{i}b_{i}(x)\frac{\partial f}{\partial x_{i}}(x)+\tfrac{1}% {2}\sum_{i,j}\left(\sigma(x)\sigma(x)^{\top}\right)_{i,j}\frac{\partial^{2}f}{% \partial x_{i}\,\partial x_{j}}(x).
  27. 𝒜 \mathcal{A}
  28. Δ LB = 1 det ( g ) i = 1 m x i ( det ( g ) j = 1 m g i j x j ) , \Delta_{\mathrm{LB}}=\frac{1}{\sqrt{\det(g)}}\sum_{i=1}^{m}\frac{\partial}{% \partial x_{i}}\left(\sqrt{\det(g)}\sum_{j=1}^{m}g^{ij}\frac{\partial}{% \partial x_{j}}\right),
  29. R α g ( x ) = 𝐄 x [ 0 e - α t g ( X t ) d t ] . R_{\alpha}g(x)=\mathbf{E}^{x}\left[\int_{0}^{\infty}e^{-\alpha t}g(X_{t})\,% \mathrm{d}t\right].
  30. R α ( α 𝐈 - A ) f = f ; R_{\alpha}(\alpha\mathbf{I}-A)f=f;
  31. ( α 𝐈 - A ) R α g = g . (\alpha\mathbf{I}-A)R_{\alpha}g=g.
  32. A * ρ ( x ) = 0 , x 𝐑 n . A^{*}\rho_{\infty}(x)=0,\quad x\in\mathbf{R}^{n}.
  33. d X t = - Ψ ( X t ) d t + 2 β - 1 d B t , \mathrm{d}X_{t}=-\nabla\Psi(X_{t})\,\mathrm{d}t+\sqrt{2\beta^{-1}}\,\mathrm{d}% B_{t},
  34. ρ ( x ) = Z - 1 exp ( - β Ψ ( x ) ) , \rho_{\infty}(x)=Z^{-1}\exp(-\beta\Psi(x)),
  35. Z = 𝐑 n exp ( - β Ψ ( x ) ) d x . Z=\int_{\mathbf{R}^{n}}\exp(-\beta\Psi(x))\,\mathrm{d}x.
  36. F [ ρ ] = E [ ρ ] + 1 β S [ ρ ] , F[\rho]=E[\rho]+\frac{1}{\beta}S[\rho],
  37. E [ ρ ] = 𝐑 n Ψ ( x ) ρ ( x ) d x E[\rho]=\int_{\mathbf{R}^{n}}\Psi(x)\rho(x)\,\mathrm{d}x
  38. S [ ρ ] = 𝐑 n ρ ( x ) log ρ ( x ) d x S[\rho]=\int_{\mathbf{R}^{n}}\rho(x)\log\rho(x)\,\mathrm{d}x
  39. d X t = - κ ( X t - m ) d t + 2 β - 1 d B t , \mathrm{d}X_{t}=-\kappa(X_{t}-m)\,\mathrm{d}t+\sqrt{2\beta^{-1}}\,\mathrm{d}B_% {t},
  40. Ψ ( x ) = 1 2 κ | x - m | 2 , \Psi(x)=\tfrac{1}{2}\kappa|x-m|^{2},
  41. ρ ( x ) = ( β κ 2 π ) n 2 exp ( - β κ | x - m | 2 2 ) \rho_{\infty}(x)=\left(\frac{\beta\kappa}{2\pi}\right)^{\frac{n}{2}}\exp\left(% -\frac{\beta\kappa|x-m|^{2}}{2}\right)
  42. M t = f ( X t ) - 0 t A f ( X s ) d s , M_{t}=f(X_{t})-\int_{0}^{t}Af(X_{s})\,\mathrm{d}s,
  43. f ( X t ) = f ( x ) + 0 t A f ( X s ) d s + 0 t f ( X s ) σ ( X s ) d B s . f(X_{t})=f(x)+\int_{0}^{t}Af(X_{s})\,\mathrm{d}s+\int_{0}^{t}\nabla f(X_{s})^{% \top}\sigma(X_{s})\,\mathrm{d}B_{s}.
  44. 𝐄 x [ M t | Σ s ] = M s . \mathbf{E}^{x}\big[M_{t}\big|\Sigma_{s}\big]=M_{s}.
  45. 𝐄 x [ M t | F s ] = 𝐄 x [ 𝐄 x [ M t | Σ s ] | F s ] = 𝐄 x [ M s | F s ] = M s , \mathbf{E}^{x}[M_{t}|F_{s}]=\mathbf{E}^{x}\left[\mathbf{E}^{x}\big[M_{t}\big|% \Sigma_{s}\big]\big|F_{s}\right]=\mathbf{E}^{x}\big[M_{s}\big|F_{s}\big]=M_{s},
  46. 𝐄 x [ f ( X τ ) ] = f ( x ) + 𝐄 x [ 0 τ A f ( X s ) d s ] . \mathbf{E}^{x}[f(X_{\tau})]=f(x)+\mathbf{E}^{x}\left[\int_{0}^{\tau}Af(X_{s})% \,\mathrm{d}s\right].
  47. 𝐄 0 [ τ R ] = R 2 . \mathbf{E}^{0}[\tau_{R}]=R^{2}.
  48. τ H ( ω ) = inf { t 0 | X t H } . \tau_{H}(\omega)=\inf\{t\geq 0|X_{t}\not\in H\}.
  49. μ G x ( F ) = 𝐏 x [ X τ G F ] \mu_{G}^{x}(F)=\mathbf{P}^{x}\left[X_{\tau_{G}}\in F\right]
  50. φ ( x ) = 𝐄 x [ f ( X τ H ) ] , \varphi(x)=\mathbf{E}^{x}\left[f(X_{\tau_{H}})\right],
  51. φ ( x ) = G φ ( y ) d μ G x ( y ) . \varphi(x)=\int_{\partial G}\varphi(y)\,\mathrm{d}\mu_{G}^{x}(y).
  52. G ( x , H ) = 𝐄 x [ 0 τ D χ H ( X s ) d s ] , G(x,H)=\mathbf{E}^{x}\left[\int_{0}^{\tau_{D}}\chi_{H}(X_{s})\,\mathrm{d}s% \right],
  53. D f ( y ) G ( x , d y ) = 𝐄 x [ 0 τ D f ( X s ) d s ] . \int_{D}f(y)\,G(x,\mathrm{d}y)=\mathbf{E}^{x}\left[\int_{0}^{\tau_{D}}f(X_{s})% \,\mathrm{d}s\right].
  54. G ( x , H ) = H G ( x , y ) d y , G(x,H)=\int_{H}G(x,y)\,\mathrm{d}y,
  55. f ( x ) = 𝐄 x [ f ( X τ D ) ] - D A f ( y ) G ( x , d y ) . f(x)=\mathbf{E}^{x}\left[f\left(X_{\tau_{D}}\right)\right]-\int_{D}Af(y)\,G(x,% \mathrm{d}y).
  56. f ( x ) = - D A f ( y ) G ( x , d y ) . f(x)=-\int_{D}Af(y)\,G(x,\mathrm{d}y).

Jack_function.html

  1. J κ ( α ) ( x 1 , x 2 , , x m ) J_{\kappa}^{(\alpha)}(x_{1},x_{2},\ldots,x_{m})
  2. κ \kappa
  3. α \alpha
  4. x 1 , x 2 , , x_{1},x_{2},\ldots,
  5. J k ( α ) ( x 1 ) = x 1 k ( 1 + α ) ( 1 + ( k - 1 ) α ) J_{k}^{(\alpha)}(x_{1})=x_{1}^{k}(1+\alpha)\cdots(1+(k-1)\alpha)
  6. J κ ( α ) ( x 1 , x 2 , , x m ) = μ J μ ( α ) ( x 1 , x 2 , , x m - 1 ) x m | κ / μ | β κ μ , J_{\kappa}^{(\alpha)}(x_{1},x_{2},\ldots,x_{m})=\sum_{\mu}J_{\mu}^{(\alpha)}(x% _{1},x_{2},\ldots,x_{m-1})x_{m}^{|\kappa/\mu|}\beta_{\kappa\mu},
  7. μ \mu
  8. κ / μ \kappa/\mu
  9. κ 1 μ 1 κ 2 μ 2 κ n - 1 μ n - 1 κ n \kappa_{1}\geq\mu_{1}\geq\kappa_{2}\geq\mu_{2}\geq\cdots\geq\kappa_{n-1}\geq% \mu_{n-1}\geq\kappa_{n}
  10. μ n \mu_{n}
  11. J μ ( x 1 , , x n - 1 ) = 0 J_{\mu}(x_{1},\ldots,x_{n-1})=0
  12. β κ μ = ( i , j ) κ B κ μ κ ( i , j ) ( i , j ) μ B κ μ μ ( i , j ) , \beta_{\kappa\mu}=\frac{\prod_{(i,j)\in\kappa}B_{\kappa\mu}^{\kappa}(i,j)}{% \prod_{(i,j)\in\mu}B_{\kappa\mu}^{\mu}(i,j)},
  13. B κ μ ν ( i , j ) B_{\kappa\mu}^{\nu}(i,j)
  14. κ j - i + α ( κ i - j + 1 ) \kappa_{j}^{\prime}-i+\alpha(\kappa_{i}-j+1)
  15. κ j = μ j \kappa_{j}^{\prime}=\mu_{j}^{\prime}
  16. κ j - i + 1 + α ( κ i - j ) \kappa_{j}^{\prime}-i+1+\alpha(\kappa_{i}-j)
  17. κ \kappa^{\prime}
  18. μ \mu^{\prime}
  19. κ \kappa
  20. μ \mu
  21. ( i , j ) κ (i,j)\in\kappa
  22. ( i , j ) (i,j)
  23. κ \kappa
  24. J μ ( α ) J_{\mu}^{(\alpha)}
  25. J μ ( α ) = T d T ( α ) s T x T ( s ) J_{\mu}^{(\alpha)}=\sum_{T}d_{T}(\alpha)\prod_{s\in T}x_{T(s)}
  26. λ \lambda
  27. d T ( α ) = s T critical d λ ( α ) ( s ) d_{T}(\alpha)=\prod_{s\in T\,\text{ critical}}d_{\lambda}(\alpha)(s)
  28. d λ ( α ) ( s ) = α ( a λ ( s ) + 1 ) + ( l λ ( s ) + 1 ) d_{\lambda}(\alpha)(s)=\alpha(a_{\lambda}(s)+1)+(l_{\lambda}(s)+1)
  29. λ \lambda
  30. λ \lambda
  31. s = ( i , j ) λ s=(i,j)\in\lambda
  32. T ( i , j ) = T ( i , j - 1 ) T(i,j)=T(i,j-1)
  33. f , g = [ 0 , 2 π ] n f ( e i θ 1 , , e i θ n ) g ( e i θ 1 , , e i θ n ) ¯ 1 j < k n | e i θ j - e i θ k | 2 / α d θ 1 d θ n \langle f,g\rangle=\int_{[0,2\pi]^{n}}f(e^{i\theta_{1}},\cdots,e^{i\theta_{n}}% )\overline{g(e^{i\theta_{1}},\cdots,e^{i\theta_{n}})}\prod_{1\leq j<k\leq n}|e% ^{i\theta_{j}}-e^{i\theta_{k}}|^{2/\alpha}d\theta_{1}\cdots d\theta_{n}
  34. C κ ( α ) ( x 1 , x 2 , , x n ) = α | κ | ( | κ | ) ! j κ J κ ( α ) ( x 1 , x 2 , , x n ) , C_{\kappa}^{(\alpha)}(x_{1},x_{2},\ldots,x_{n})=\frac{\alpha^{|\kappa|}(|% \kappa|)!}{j_{\kappa}}J_{\kappa}^{(\alpha)}(x_{1},x_{2},\ldots,x_{n}),
  35. j κ = ( i , j ) κ ( κ j - i + α ( κ i - j + 1 ) ) ( κ j - i + 1 + α ( κ i - j ) ) . j_{\kappa}=\prod_{(i,j)\in\kappa}(\kappa_{j}^{\prime}-i+\alpha(\kappa_{i}-j+1)% )(\kappa_{j}^{\prime}-i+1+\alpha(\kappa_{i}-j)).
  36. α = 2 , C κ ( 2 ) ( x 1 , x 2 , , x n ) \alpha=2,\;C_{\kappa}^{(2)}(x_{1},x_{2},\ldots,x_{n})
  37. C κ ( x 1 , x 2 , , x n ) C_{\kappa}(x_{1},x_{2},\ldots,x_{n})
  38. J λ = H λ P λ J_{\lambda}=H^{\prime}_{\lambda}P_{\lambda}
  39. H λ = s λ ( α a λ ( s ) + l λ ( s ) + 1 ) H^{\prime}_{\lambda}=\prod_{s\in\lambda}(\alpha a_{\lambda}(s)+l_{\lambda}(s)+1)
  40. a λ a_{\lambda}
  41. l λ l_{\lambda}
  42. α = 1 \alpha=1
  43. P λ P_{\lambda}
  44. P λ P_{\lambda}
  45. α \alpha
  46. P λ P_{\lambda}
  47. P λ = T ψ T ( α ) s λ x T ( s ) P_{\lambda}=\sum_{T}\psi_{T}(\alpha)\prod_{s\in\lambda}x_{T(s)}
  48. λ \lambda
  49. T ( s ) T(s)
  50. ψ T ( α ) \psi_{T}(\alpha)
  51. λ \lambda
  52. = ν 1 ν 2 ν n = λ \emptyset=\nu_{1}\to\nu_{2}\to\dots\to\nu_{n}=\lambda
  53. ν i + 1 / ν i \nu_{i+1}/\nu_{i}
  54. ψ T ( α ) = i ψ ν i + 1 / ν i ( α ) \psi_{T}(\alpha)=\prod_{i}\psi_{\nu_{i+1}/\nu_{i}}(\alpha)
  55. ψ λ / μ ( α ) = s R λ / μ - C λ / μ ( α a μ ( s ) + l μ ( s ) + 1 ) ( α a μ ( s ) + l μ ( s ) + α ) ( α a λ ( s ) + l λ ( s ) + α ) ( α a λ ( s ) + l λ ( s ) + 1 ) \psi_{\lambda/\mu}(\alpha)=\prod_{s\in R_{\lambda/\mu}-C_{\lambda/\mu}}\frac{(% \alpha a_{\mu}(s)+l_{\mu}(s)+1)}{(\alpha a_{\mu}(s)+l_{\mu}(s)+\alpha)}\frac{(% \alpha a_{\lambda}(s)+l_{\lambda}(s)+\alpha)}{(\alpha a_{\lambda}(s)+l_{% \lambda}(s)+1)}
  56. λ \lambda
  57. λ / μ \lambda/\mu
  58. α = 1 \alpha=1
  59. J κ ( 1 ) ( x 1 , x 2 , , x n ) = H κ s κ ( x 1 , x 2 , , x n ) , J^{(1)}_{\kappa}(x_{1},x_{2},\ldots,x_{n})=H_{\kappa}s_{\kappa}(x_{1},x_{2},% \ldots,x_{n}),
  60. H κ = ( i , j ) κ h κ ( i , j ) = ( i , j ) κ ( κ i + κ j - i - j + 1 ) H_{\kappa}=\prod_{(i,j)\in\kappa}h_{\kappa}(i,j)=\prod_{(i,j)\in\kappa}(\kappa% _{i}+\kappa_{j}^{\prime}-i-j+1)
  61. κ \kappa
  62. J κ ( α ) ( x 1 , x 2 , , x m ) = 0 , if κ m + 1 > 0. J_{\kappa}^{(\alpha)}(x_{1},x_{2},\ldots,x_{m})=0,\mbox{ if }~{}\kappa_{m+1}>0.
  63. X X
  64. x 1 , x 2 , , x m x_{1},x_{2},\ldots,x_{m}
  65. J κ ( α ) ( X ) = J κ ( α ) ( x 1 , x 2 , , x m ) . J_{\kappa}^{(\alpha)}(X)=J_{\kappa}^{(\alpha)}(x_{1},x_{2},\ldots,x_{m}).

Jackknife_resampling.html

  1. N N
  2. N - 1 N-1
  3. x ¯ i \bar{x}_{i}
  4. x ¯ i = 1 n - 1 j i n ( x ¯ j ) \bar{x}_{\mathrm{i}}=\frac{1}{n-1}\sum_{j\neq i}^{n}(\bar{x}_{j})
  5. Var ( jackknife ) = n - 1 n i = 1 n ( x ¯ i - x ¯ ( . ) ) 2 \operatorname{Var}_{\mathrm{(jackknife)}}=\frac{n-1}{n}\sum_{i=1}^{n}(\bar{x}_% {i}-\bar{x}_{\mathrm{(.)}})^{2}
  6. x ¯ i \bar{x}_{i}
  7. x ¯ ( . ) \bar{x}_{\mathrm{(.)}}
  8. θ ^ \hat{\theta}
  9. n {n}
  10. θ ^ ( . ) = 1 n i = 1 n θ ^ ( i ) \hat{\theta}_{\mathrm{(.)}}=\frac{1}{n}\sum_{i=1}^{n}\hat{\theta}_{\mathrm{(i)}}
  11. θ ^ ( i ) \hat{\theta}_{\mathrm{(i)}}
  12. θ ^ ( . ) \hat{\theta}_{\mathrm{(.)}}
  13. B I A S ^ ( θ ) = ( n - 1 ) ( θ ^ ( . ) - θ ^ ) \hat{BIAS}_{\mathrm{(\theta)}}=(n-1)(\hat{\theta}_{\mathrm{(.)}}-\hat{\theta})
  14. O ( N - 1 ) O(N^{-1})
  15. O ( N - 2 ) O(N^{-2})

Jackson's_inequality.html

  1. f : [ 0 , 2 π ] f:[0,2\pi]\to\mathbb{C}
  2. r r
  3. | f ( r ) ( x ) | 1 , 0 x 2 π , |f^{(r)}(x)|\leq 1,\quad 0\leq x\leq 2\pi,
  4. n n
  5. T n - 1 T_{n-1}
  6. n - 1 n-1
  7. | f ( x ) - T n - 1 ( x ) | C ( r ) n r , 0 x 2 π , |f(x)-T_{n-1}(x)|\leq\frac{C(r)}{n^{r}},\quad 0\leq x\leq 2\pi,
  8. C ( r ) C(r)
  9. r r
  10. C ( r ) C(r)
  11. C ( r ) = 4 π k = 0 ( - 1 ) k ( r + 1 ) ( 2 k + 1 ) r + 1 . C(r)=\frac{4}{\pi}\sum_{k=0}^{\infty}\frac{(-1)^{k(r+1)}}{(2k+1)^{r+1}}~{}.
  12. ω ( δ , f ( r ) ) \omega(\delta,f^{(r)})
  13. r r
  14. f f
  15. δ \delta
  16. T n T_{n}
  17. n \leq n
  18. | f ( x ) - T n ( x ) | C 1 ( r ) ω ( 1 / n , f ( r ) ) n r , 0 x 2 π . |f(x)-T_{n}(x)|\leq\frac{C_{1}(r)\omega(1/n,f^{(r)})}{n^{r}},\quad 0\leq x\leq 2\pi.
  19. n n
  20. f f
  21. 2 π 2\pi
  22. T n T_{n}
  23. n \leq n
  24. | f ( x ) - T n ( x ) | c ( k ) ω k ( 1 n , f ) , x [ 0 , 2 π ] , |f(x)-T_{n}(x)|\leq c(k)\omega_{k}\left(\frac{1}{n},f\right),\quad x\in[0,2\pi],
  25. c ( k ) c(k)
  26. k k\in\mathbb{N}
  27. ω k \omega_{k}
  28. k k
  29. k = 1 k=1
  30. k = 2 , ω 2 ( t , f ) c t , t > 0 k=2,\omega_{2}(t,f)\leq ct,t>0
  31. k = 2 k=2
  32. k > 2 k>2

Jackson_integral.html

  1. f ( x ) d q x = ( 1 - q ) x k = 0 q k f ( q k x ) . \int f(x)d_{q}x=(1-q)x\sum_{k=0}^{\infty}q^{k}f(q^{k}x).
  2. f ( x ) D q g d q x = ( 1 - q ) x k = 0 q k f ( q k x ) D q g ( q k x ) = ( 1 - q ) x k = 0 q k f ( q k x ) g ( q k x ) - g ( q k + 1 x ) ( 1 - q ) q k x , \int f(x)D_{q}gd_{q}x=(1-q)x\sum_{k=0}^{\infty}q^{k}f(q^{k}x)D_{q}g(q^{k}x)=(1% -q)x\sum_{k=0}^{\infty}q^{k}f(q^{k}x)\frac{g(q^{k}x)-g(q^{k+1}x)}{(1-q)q^{k}x},
  3. f ( x ) d q g ( x ) = k = 0 f ( q k x ) ( g ( q k x ) - g ( q k + 1 x ) ) , \int f(x)d_{q}g(x)=\sum_{k=0}^{\infty}f(q^{k}x)(g(q^{k}x)-g(q^{k+1}x)),
  4. 0 < q < 1. 0<q<1.
  5. | f ( x ) x α | |f(x)x^{\alpha}|
  6. [ 0 , A ) [0,A)
  7. 0 α < 1 , 0\leq\alpha<1,
  8. F ( x ) F(x)
  9. [ 0 , A ) [0,A)
  10. f ( x ) . f(x).
  11. F ( x ) F(x)
  12. x = 0 x=0
  13. F ( 0 ) = 0 F(0)=0
  14. f ( x ) f(x)

Jacobi_theta_functions_(notational_variations).html

  1. ϑ 00 ( z ; τ ) = n = - exp ( π i n 2 τ + 2 π i n z ) \vartheta_{00}(z;\tau)=\sum_{n=-\infty}^{\infty}\exp(\pi in^{2}\tau+2\pi inz)
  2. ϑ 00 ( z , q ) = n = - q n 2 exp ( 2 π i n z ) \vartheta_{00}(z,q)=\sum_{n=-\infty}^{\infty}q^{n^{2}}\exp(2\pi inz)
  3. ϑ 0 , 0 ( x ) = n = - q n 2 exp ( 2 π i n x / a ) \vartheta_{0,0}(x)=\sum_{n=-\infty}^{\infty}q^{n^{2}}\exp(2\pi inx/a)
  4. ϑ 1 , 1 ( x ) = n = - ( - 1 ) n q ( n + 1 / 2 ) 2 exp ( π i ( 2 n + 1 ) x / a ) \vartheta_{1,1}(x)=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{(n+1/2)^{2}}\exp(\pi i(% 2n+1)x/a)
  5. ϑ 11 \vartheta_{11}
  6. ϑ 1 ( z ) = - i n = - ( - 1 ) n q ( n + 1 / 2 ) 2 exp ( ( 2 n + 1 ) i z ) \vartheta_{1}(z)=-i\sum_{n=-\infty}^{\infty}(-1)^{n}q^{(n+1/2)^{2}}\exp((2n+1)iz)
  7. ϑ 2 ( z ) = n = - q ( n + 1 / 2 ) 2 exp ( ( 2 n + 1 ) i z ) \vartheta_{2}(z)=\sum_{n=-\infty}^{\infty}q^{(n+1/2)^{2}}\exp((2n+1)iz)
  8. ϑ 3 ( z ) = n = - q n 2 exp ( 2 n i z ) \vartheta_{3}(z)=\sum_{n=-\infty}^{\infty}q^{n^{2}}\exp(2niz)
  9. ϑ 4 ( z ) = n = - ( - 1 ) n q n 2 exp ( 2 n i z ) \vartheta_{4}(z)=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{n^{2}}\exp(2niz)
  10. ϑ j \vartheta_{j}
  11. ϑ ( z ) \vartheta(z)
  12. ϑ ( z ) \vartheta(z)

James_Anderson_(computer_scientist).html

  1. lim x 0 0 x = 0 \lim_{x\to 0}\frac{0}{x}=0
  2. lim x 0 + 1 x = + \lim_{x\to 0^{+}}\frac{1}{x}=+\infty
  3. lim x 0 - 1 x = - \lim_{x\to 0^{-}}\frac{1}{x}=-\infty
  4. lim x 0 sin x x = 1 \lim_{x\to 0}\frac{\sin x}{x}=1
  5. lim x 0 1 - cos x x = 0 \lim_{x\to 0}\frac{1-\cos x}{x}=0
  6. 0 0 0^{0}
  7. 0 0 = N a N \frac{0}{0}=NaN
  8. 0 0 0^{0}
  9. 1 1
  10. f ( x ) = x 0 f(x)=x^{0}
  11. f ( x ) = 0 x f(x)=0^{x}
  12. 0 0 = Φ \frac{0}{0}=\Phi
  13. 0 0 = Φ 0^{0}=\Phi\,
  14. 0 0 = 0 0 0^{0}=\frac{0}{0}
  15. \infty
  16. - -\infty
  17. Φ \Phi
  18. a ÷ b = c a\div b=c
  19. a = b × c a=b\times c
  20. c c
  21. 0 ÷ 0 = c 0\div 0=c
  22. 0 = 0 × c 0=0\times c
  23. c c
  24. 0 ÷ 0 0\div 0
  25. 0 ÷ 0 0\div 0
  26. 0 ÷ 0 0\div 0
  27. + +\infty
  28. - -\infty
  29. N a N NaN
  30. N a N = N a N NaN=NaN
  31. 0 ÷ 0 = Φ 0\div 0=\Phi
  32. 0 ÷ 0 = N a N 0\div 0=NaN
  33. × 0 = Φ \infty\times 0=\Phi
  34. × 0 = N a N \infty\times 0=NaN
  35. - = Φ \infty-\infty=\Phi
  36. - = N a N \infty-\infty=NaN
  37. Φ + a = Φ \Phi+a=\Phi
  38. N a N + a = N a N NaN+a=NaN
  39. Φ × a = Φ \Phi\times a=\Phi
  40. N a N × a = N a N NaN\times a=NaN
  41. - Φ = Φ -\Phi=\Phi
  42. - N a N = N a N -NaN=NaN
  43. + 1 ÷ 0 = + +1\div 0=+\infty
  44. 1 ÷ + 0 = - 1 ÷ - 0 = + 1\div+0=-1\div-0=+\infty
  45. - 1 ÷ 0 = - -1\div 0=-\infty
  46. 1 ÷ - 0 = - 1 ÷ + 0 = - 1\div-0=-1\div+0=-\infty
  47. Φ = Φ T r u e \Phi=\Phi\Rightarrow True
  48. N a N = N a N F a l s e NaN=NaN\Rightarrow False
  49. a - a = 0 a-a=0
  50. a a
  51. Φ - Φ = Φ \Phi-\Phi=\Phi
  52. 1 1
  53. Φ \Phi
  54. Φ \Phi
  55. t t
  56. Φ \Phi
  57. t = Φ t=\Phi

Jamshīd_al-Kāshī.html

  1. x P - N = 0 x^{P}-N=0
  2. sin 3 ϕ = 3 sin ϕ - 4 sin 3 ϕ \sin 3\phi=3\sin\phi-4\sin^{3}\phi\,\!
  3. τ \tau
  4. n \scriptstyle n
  5. 59 / 60 n + 1 + 59 / 60 n + 2 + = 1 / 60 n \scriptstyle 59/60^{n+1}+59/60^{n+2}+\dots=1/60^{n}
  6. n = 9 \scriptstyle n=9
  7. 2 π \scriptstyle 2\pi
  8. 1 / 60 9 9.92 × 10 - 17 < 10 - 16 \scriptstyle 1/60^{9}\approx 9.92\times 10^{-17}<10^{-16}\,
  9. 2 π \scriptstyle 2\pi
  10. 2 π \scriptstyle 2\pi
  11. 6.283 185 307 179 586 476 \scriptstyle 6.283\,185\,307\,179\,586\,476

Janko_group_J1.html

  1. × 10 5 \times 10^{5}
  2. 𝐘 = ( 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 ) {\mathbf{Y}}=\left(\begin{matrix}0&1&0&0&0&0&0\\ 0&0&1&0&0&0&0\\ 0&0&0&1&0&0&0\\ 0&0&0&0&1&0&0\\ 0&0&0&0&0&1&0\\ 0&0&0&0&0&0&1\\ 1&0&0&0&0&0&0\end{matrix}\right)
  3. 𝐙 = ( - 3 + 2 - 1 - 1 - 3 - 1 - 3 - 2 + 1 + 1 + 3 + 1 + 3 + 3 - 1 - 1 - 3 - 1 - 3 - 3 + 2 - 1 - 3 - 1 - 3 - 3 + 2 - 1 - 3 - 1 - 3 - 3 + 2 - 1 - 1 + 1 + 3 + 3 - 2 + 1 + 1 + 3 + 3 + 3 - 2 + 1 + 1 + 3 + 1 ) . {\mathbf{Z}}=\left(\begin{matrix}-3&+2&-1&-1&-3&-1&-3\\ -2&+1&+1&+3&+1&+3&+3\\ -1&-1&-3&-1&-3&-3&+2\\ -1&-3&-1&-3&-3&+2&-1\\ -3&-1&-3&-3&+2&-1&-1\\ +1&+3&+3&-2&+1&+1&+3\\ +3&+3&-2&+1&+1&+3&+1\end{matrix}\right).

Janko_group_J2.html

  1. × 10 5 \times 10^{5}
  2. 𝐀 = ( w 2 w 2 0 0 0 0 1 w 2 0 0 0 0 1 1 w 2 w 2 0 0 w 1 1 w 2 0 0 0 w 2 w 2 w 2 0 w w 2 1 w 2 0 w 2 0 ) {\mathbf{A}}=\left(\begin{matrix}w^{2}&w^{2}&0&0&0&0\\ 1&w^{2}&0&0&0&0\\ 1&1&w^{2}&w^{2}&0&0\\ w&1&1&w^{2}&0&0\\ 0&w^{2}&w^{2}&w^{2}&0&w\\ w^{2}&1&w^{2}&0&w^{2}&0\end{matrix}\right)
  3. 𝐁 = ( w 1 w 2 1 w 2 w 2 w 1 w 1 1 w w w w 2 w 2 1 0 0 0 0 0 1 1 w 2 1 w 2 w 2 w w 2 w 2 1 w 2 w w 2 w ) . {\mathbf{B}}=\left(\begin{matrix}w&1&w^{2}&1&w^{2}&w^{2}\\ w&1&w&1&1&w\\ w&w&w^{2}&w^{2}&1&0\\ 0&0&0&0&1&1\\ w^{2}&1&w^{2}&w^{2}&w&w^{2}\\ w^{2}&1&w^{2}&w&w^{2}&w\end{matrix}\right).
  4. 𝐀 2 = 𝐁 3 = ( 𝐀𝐁 ) 7 = ( 𝐀𝐁𝐀𝐁𝐁 ) 12 = 1. {\mathbf{A}}^{2}={\mathbf{B}}^{3}=({\mathbf{A}}{\mathbf{B}})^{7}=({\mathbf{A}}% {\mathbf{B}}{\mathbf{A}}{\mathbf{B}}{\mathbf{B}})^{12}=1.

Janko_group_J3.html

  1. × 10 7 \times 10^{7}
  2. a 17 = b 8 = a b a - 2 = c 2 = b c b 3 = ( a b c ) 4 = ( a c ) 17 = d 2 = [ d , a ] = [ d , b ] = ( a 3 b - 3 c d ) 5 = 1. a^{17}=b^{8}=a^{b}a^{-2}=c^{2}=b^{c}b^{3}=(abc)^{4}=(ac)^{17}=d^{2}=[d,a]=[d,b% ]=(a^{3}b^{-3}cd)^{5}=1.
  3. a 19 = b 9 = a b a 2 = c 2 = d 2 = ( b c ) 2 = ( b d ) 2 = ( a c ) 3 = ( a d ) 3 = ( a 2 c a - 3 d ) 3 = 1. a^{19}=b^{9}=a^{b}a^{2}=c^{2}=d^{2}=(bc)^{2}=(bd)^{2}=(ac)^{3}=(ad)^{3}=(a^{2}% ca^{-3}d)^{3}=1.

Janko_group_J4.html

  1. × 10 1 9 \times 10^{1}9
  2. a 2 = b 3 = c 2 = ( a b ) 23 = [ a , b ] 12 = [ a , b a b ] 5 = [ c , a ] = ( a b a b a b - 1 ) 3 ( a b a b - 1 a b - 1 ) 3 = ( a b ( a b a b - 1 ) 3 ) 4 = [ c , b a b ( a b - 1 ) 2 ( a b ) 3 ] = ( b c b a b - 1 a b a b - 1 a ) 3 = ( ( b a b a b a b ) 3 c c ( a b ) 3 b ( a b ) 6 b ) 2 = 1. \begin{aligned}\displaystyle a^{2}&\displaystyle=b^{3}=c^{2}=(ab)^{23}=[a,b]^{% 12}=[a,bab]^{5}=[c,a]=\left(ababab^{-1}\right)^{3}\left(abab^{-1}ab^{-1}\right% )^{3}=\left(ab\left(abab^{-1}\right)^{3}\right)^{4}\\ &\displaystyle=\left[c,bab\left(ab^{-1}\right)^{2}(ab)^{3}\right]=\left(bc^{% bab^{-1}abab^{-1}a}\right)^{3}=\left((bababab)^{3}cc^{(ab)^{3}b(ab)^{6}b}% \right)^{2}=1.\end{aligned}

Jenkins–Traub_algorithm.html

  1. P ( z ) = i = 0 n a i z n - i , a 0 = 1 , a n 0 P(z)=\sum_{i=0}^{n}a_{i}z^{n-i},\quad a_{0}=1,\quad a_{n}\neq 0
  2. α 1 , α 2 , , α n \alpha_{1},\alpha_{2},\dots,\alpha_{n}
  3. H ¯ \bar{H}
  4. ( H ( λ ) ( z ) ) λ = 0 , 1 , 2 , \left(H^{(\lambda)}(z)\right)_{\lambda=0,1,2,\dots}
  5. ( s λ ) λ = 0 , 1 , 2 , (s_{\lambda})_{\lambda=0,1,2,\dots}
  6. H ( 0 ) ( z ) = P ( z ) H^{(0)}(z)=P^{\prime}(z)
  7. ( X - s λ ) H ( λ + 1 ) ( X ) H ( λ ) ( X ) ( mod P ( X ) ) . (X-s_{\lambda})\cdot H^{(\lambda+1)}(X)\equiv H^{(\lambda)}(X)\;\;(\mathop{{% \rm mod}}P(X))\ .
  8. H ( λ + 1 ) ( X ) = 1 X - s λ ( H ( λ ) ( X ) - H ( λ ) ( s λ ) P ( s λ ) P ( X ) ) , H^{(\lambda+1)}(X)=\frac{1}{X-s_{\lambda}}\cdot\left(H^{(\lambda)}(X)-\frac{H^% {(\lambda)}(s_{\lambda})}{P(s_{\lambda})}P(X)\right)\,,
  9. s λ s_{\lambda}
  10. P ( X ) = p ( X ) ( X - s λ ) + P ( s λ ) H ( λ ) ( X ) = h ( X ) ( X - s λ ) + H ( λ ) ( s λ ) } H ( λ + 1 ) ( z ) = h ( z ) - H ( λ ) ( s λ ) P ( s λ ) p ( z ) . \left.\begin{aligned}\displaystyle P(X)&\displaystyle=p(X)\cdot(X-s_{\lambda})% +P(s_{\lambda})\\ \displaystyle H^{(\lambda)}(X)&\displaystyle=h(X)\cdot(X-s_{\lambda})+H^{(% \lambda)}(s_{\lambda})\\ \end{aligned}\right\}\implies H^{(\lambda+1)}(z)=h(z)-\frac{H^{(\lambda)}(s_{% \lambda})}{P(s_{\lambda})}p(z).
  11. H ( λ + 1 ) ( X ) H^{(\lambda+1)}(X)
  12. - H ( λ ) ( s λ ) P ( s λ ) -\tfrac{H^{(\lambda)}(s_{\lambda})}{P(s_{\lambda})}
  13. H ¯ ( λ + 1 ) ( X ) = 1 X - s λ ( P ( X ) - P ( s λ ) H ( λ ) ( s λ ) H ( λ ) ( X ) ) = 1 X - s λ ( P ( X ) - P ( s λ ) H ¯ ( λ ) ( s λ ) H ¯ ( λ ) ( X ) ) . \begin{aligned}\displaystyle\bar{H}^{(\lambda+1)}(X)&\displaystyle=\frac{1}{X-% s_{\lambda}}\cdot\left(P(X)-\frac{P(s_{\lambda})}{H^{(\lambda)}(s_{\lambda})}H% ^{(\lambda)}(X)\right)\\ &\displaystyle=\frac{1}{X-s_{\lambda}}\cdot\left(P(X)-\frac{P(s_{\lambda})}{% \bar{H}^{(\lambda)}(s_{\lambda})}\bar{H}^{(\lambda)}(X)\right)\,.\end{aligned}
  14. λ = 0 , 1 , , M - 1 \lambda=0,1,\dots,M-1
  15. s λ = 0 s_{\lambda}=0
  16. R n + | a n - 1 | R n - 1 + + | a 1 | R = | a 0 | . R^{n}+|a_{n-1}|\,R^{n-1}+\dots+|a_{1}|\,R=|a_{0}|\,.
  17. s = R exp ( i ϕ random ) s=R\cdot\exp(i\,\phi\text{random})
  18. H ( λ + 1 ) ( z ) H^{(\lambda+1)}(z)
  19. λ = M , M + 1 , , L - 1 \lambda=M,M+1,\dots,L-1
  20. s λ = s s_{\lambda}=s
  21. t λ = s - P ( s ) H ¯ ( λ ) ( s ) t_{\lambda}=s-\frac{P(s)}{\bar{H}^{(\lambda)}(s)}
  22. | t λ + 1 - t λ | < 1 2 | t λ | |t_{\lambda+1}-t_{\lambda}|<\tfrac{1}{2}\,|t_{\lambda}|
  23. | t λ - t λ - 1 | a r e s i m u l t a n e o u s l y m e t . I f t h e r e w a s n o s u c c e s s a f t e r s o m e n u m b e r o f i t e r a t i o n s , a d i f f e r e n t r a n d o m p o i n t o n t h e c i r c l e i s t r i e d . T y p i c a l l y o n e u s e s a n u m b e r o f 9 i t e r a t i o n s f o r p o l y n o m i a l s o f m o d e r a t e d e g r e e , w i t h a d o u b l i n g s t r a t e g y f o r t h e c a s e o f m u l t i p l e f a i l u r e s . = = = = S t a g e t h r e e : v a r i a b l e - s h i f t p r o c e s s = = = = T h e < m a t h > H ( λ + 1 ) ( X ) |t_{\lambda}-t_{\lambda-1}|aresimultaneouslymet.% Iftherewasnosuccessaftersomenumberofiterations,% adifferentrandompointonthecircleistried.Typicallyoneusesanumberof9% iterationsforpolynomialsofmoderatedegree,% withadoublingstrategyforthecaseofmultiplefailures.\par ====Stagethree:variable% -shiftprocess====\par The<math>H^{(\lambda+1)}(X)
  24. s λ , λ = L , L + 1 , s_{\lambda},\quad\lambda=L,L+1,\dots
  25. s L = t L = s - P ( s ) H ¯ ( λ ) ( s ) s_{L}=t_{L}=s-\frac{P(s)}{\bar{H}^{(\lambda)}(s)}
  26. s λ + 1 = s λ - P ( s λ ) H ¯ ( λ + 1 ) ( s λ ) , λ = L , L + 1 , , s_{\lambda+1}=s_{\lambda}-\frac{P(s_{\lambda})}{\bar{H}^{(\lambda+1)}(s_{% \lambda})},\quad\lambda=L,L+1,\dots,
  27. H ¯ ( λ + 1 ) ( z ) \bar{H}^{(\lambda+1)}(z)
  28. H ( λ ) ( z ) H^{(\lambda)}(z)
  29. z i + 1 = z i - P ( z i ) P ( z i ) . z_{i+1}=z_{i}-\frac{P(z_{i})}{P^{\prime}(z_{i})}.
  30. P \scriptstyle P^{\prime}
  31. s λ + 1 = s λ - P ( s λ ) H ¯ λ + 1 ( s λ ) = s λ - W λ ( s λ ) ( W λ ) ( s λ ) s_{\lambda+1}=s_{\lambda}-\frac{P(s_{\lambda})}{\bar{H}^{\lambda+1}(s_{\lambda% })}=s_{\lambda}-\frac{W^{\lambda}(s_{\lambda})}{(W^{\lambda})^{\prime}(s_{% \lambda})}
  32. W λ ( z ) = P ( z ) H λ ( z ) . W^{\lambda}(z)=\frac{P(z)}{H^{\lambda}(z)}.
  33. λ \lambda
  34. P ( z ) H ¯ λ ( z ) = W λ ( z ) L C ( H λ ) \frac{P(z)}{\bar{H}^{\lambda}(z)}=W^{\lambda}(z)\,LC(H^{\lambda})
  35. z - α 1 , z-\alpha_{1},\,
  36. α 1 \alpha_{1}
  37. P P
  38. α 1 , , α n \alpha_{1},\dots,\alpha_{n}
  39. P m ( X ) = P ( X ) - P ( α m ) X - α m . P_{m}(X)=\frac{P(X)-P(\alpha_{m})}{X-\alpha_{m}}.
  40. H ( λ ) ( X ) = m = 1 n [ κ = 0 λ - 1 ( α m - s κ ) ] - 1 P m ( X ) . H^{(\lambda)}(X)=\sum_{m=1}^{n}\left[\prod_{\kappa=0}^{\lambda-1}(\alpha_{m}-s% _{\kappa})\right]^{-1}\,P_{m}(X)\ .
  41. H ¯ ( λ ) ( X ) = m = 1 n [ κ = 0 λ - 1 ( α m - s κ ) ] - 1 P m ( X ) m = 1 n [ κ = 0 λ - 1 ( α m - s κ ) ] - 1 = P 1 ( X ) + m = 2 n [ κ = 0 λ - 1 α 1 - s κ α m - s κ ] P m ( X ) 1 + m = 1 n [ κ = 0 λ - 1 α 1 - s κ α m - s κ ] . \bar{H}^{(\lambda)}(X)=\frac{\sum_{m=1}^{n}\left[\prod_{\kappa=0}^{\lambda-1}(% \alpha_{m}-s_{\kappa})\right]^{-1}\,P_{m}(X)}{\sum_{m=1}^{n}\left[\prod_{% \kappa=0}^{\lambda-1}(\alpha_{m}-s_{\kappa})\right]^{-1}}=\frac{P_{1}(X)+\sum_% {m=2}^{n}\left[\prod_{\kappa=0}^{\lambda-1}\frac{\alpha_{1}-s_{\kappa}}{\alpha% _{m}-s_{\kappa}}\right]\,P_{m}(X)}{1+\sum_{m=1}^{n}\left[\prod_{\kappa=0}^{% \lambda-1}\frac{\alpha_{1}-s_{\kappa}}{\alpha_{m}-s_{\kappa}}\right]}\ .
  42. | α 1 - s κ | < min | m = 2 , 3 , , n α m - s κ | |\alpha_{1}-s_{\kappa}|<\min{}_{m=2,3,\dots,n}|\alpha_{m}-s_{\kappa}|
  43. P 1 ( X ) P_{1}(X)
  44. | α 1 | < | α 2 | = min | m = 2 , 3 , , n α m | |\alpha_{1}|<|\alpha_{2}|=\min{}_{m=2,3,\dots,n}|\alpha_{m}|
  45. H ( λ ) ( X ) = P 1 ( X ) + O ( | α 1 α 2 | λ ) . H^{(\lambda)}(X)=P_{1}(X)+O\left(\left|\frac{\alpha_{1}}{\alpha_{2}}\right|^{% \lambda}\right).
  46. α 1 \alpha_{1}
  47. H ( λ ) ( X ) = P 1 ( X ) + O ( | α 1 α 2 | M | α 1 - s α 2 - s | λ - M ) H^{(\lambda)}(X)=P_{1}(X)+O\left(\left|\frac{\alpha_{1}}{\alpha_{2}}\right|^{M% }\cdot\left|\frac{\alpha_{1}-s}{\alpha_{2}-s}\right|^{\lambda-M}\right)
  48. s - P ( s ) H ¯ ( λ ) ( s ) = α 1 + O ( | α 1 - s | ) . s-\frac{P(s)}{\bar{H}^{(\lambda)}(s)}=\alpha_{1}+O\left(\ldots\cdot|\alpha_{1}% -s|\right).
  49. H ( λ ) ( X ) = P 1 ( X ) + O ( κ = 0 λ - 1 | α 1 - s κ α 2 - s κ | ) H^{(\lambda)}(X)=P_{1}(X)+O\left(\prod_{\kappa=0}^{\lambda-1}\left|\frac{% \alpha_{1}-s_{\kappa}}{\alpha_{2}-s_{\kappa}}\right|\right)
  50. s λ + 1 = s λ - P ( s ) H ¯ ( λ + 1 ) ( s λ ) = α 1 + O ( κ = 0 λ - 1 | α 1 - s κ α 2 - s κ | | α 1 - s λ | 2 | α 2 - s λ | ) s_{\lambda+1}=s_{\lambda}-\frac{P(s)}{\bar{H}^{(\lambda+1)}(s_{\lambda})}=% \alpha_{1}+O\left(\prod_{\kappa=0}^{\lambda-1}\left|\frac{\alpha_{1}-s_{\kappa% }}{\alpha_{2}-s_{\kappa}}\right|\cdot\frac{|\alpha_{1}-s_{\lambda}|^{2}}{|% \alpha_{2}-s_{\lambda}|}\right)
  51. ϕ 2 = 1 + ϕ 2.61 \phi^{2}=1+\phi\approx 2.61
  52. ϕ = 1 2 ( 1 + 5 ) \phi=\tfrac{1}{2}(1+\sqrt{5})
  53. P ( X ) = ( X - α 1 ) P 1 ( X ) P(X)=(X-\alpha_{1})\cdot P_{1}(X)
  54. α 1 \C \alpha_{1}\in\C
  55. P 1 ( X ) = P ( X ) / ( X - α 1 ) P_{1}(X)=P(X)/(X-\alpha_{1})
  56. M X ( H ) = ( X H ( X ) ) mod P ( X ) . M_{X}(H)=(X\cdot H(X))\bmod P(X)\,.
  57. 0 = ( M X - α i d ) ( H ) = ( ( X - α ) H ) mod P , 0=(M_{X}-\alpha\cdot id)(H)=((X-\alpha)\cdot H)\bmod P\,,
  58. ( X - α ) H ) = C P ( X ) (X-\alpha)\cdot H)=C\cdot P(X)
  59. ( X - α ) (X-\alpha)
  60. M X M_{X}
  61. M X ( H ) = m = 1 n - 1 ( H m - 1 - P m H n - 1 ) X m - P 0 H n - 1 , M_{X}(H)=\sum_{m=1}^{n-1}(H_{m-1}-P_{m}H_{n-1})X^{m}-P_{0}H_{n-1}\,,
  62. A = ( 0 0 0 - P 0 1 0 0 - P 1 0 1 0 - P 2 0 0 1 - P n - 1 ) . A=\begin{pmatrix}0&0&\dots&0&-P_{0}\\ 1&0&\dots&0&-P_{1}\\ 0&1&\dots&0&-P_{2}\\ \vdots&\vdots&\ddots&\vdots&\vdots\\ 0&0&\dots&1&-P_{n-1}\end{pmatrix}\,.

Johann_Nikuradse.html

  1. λ \lambda
  2. λ = 1 ( 1.74 + 2 l o g ( r k ) ) 2 \lambda=\frac{1}{(1.74+2log(\frac{r}{k}))^{2}}

Johannes_Martin_Bijvoet.html

  1. Δ \Delta
  2. Δ \Delta

John_ellipsoid.html

  1. i = 1 m c i u i = 0 \sum_{i=1}^{m}c_{i}u_{i}=0
  2. x = i = 1 m c i ( x u i ) u i . x=\sum_{i=1}^{m}c_{i}(x\cdot u_{i})u_{i}.

Johnson_circles.html

  1. u \vec{u}
  2. v \vec{v}
  3. w \vec{w}
  4. H + u H+\vec{u}
  5. H + v H+\vec{v}
  6. H + w H+\vec{w}
  7. H + u + v H+\vec{u}+\vec{v}
  8. H + u + w H+\vec{u}+\vec{w}
  9. H + v + w H+\vec{v}+\vec{w}
  10. H + u + v + w H+\vec{u}+\vec{v}+\vec{w}

Jordan_and_Einstein_frames.html

  1. g ~ μ ν = Φ - 2 / ( d - 2 ) g μ ν \tilde{g}_{\mu\nu}=\Phi^{-2/(d-2)}g_{\mu\nu}
  2. - g ~ = Φ - d / ( d - 2 ) - g \sqrt{-\tilde{g}}=\Phi^{-d/(d-2)}\sqrt{-g}
  3. R ~ = Φ 2 / ( d - 2 ) [ R + 2 d d - 2 Φ Φ - 3 ( d - 1 ) ( d - 2 ) ( Φ Φ ) 2 ] \tilde{R}=\Phi^{2/(d-2)}\left[R+\frac{2d}{d-2}\frac{\Box\Phi}{\Phi}-\frac{3(d-% 1)}{(d-2)}\left(\frac{\nabla\Phi}{\Phi}\right)^{2}\right]
  4. S = d d x - g ~ Φ R ~ = d d x - g [ R - 3 ( d - 1 ) ( d - 2 ) ( ( ln Φ ) ) 2 ] S=\int d^{d}x\sqrt{-\tilde{g}}\Phi\tilde{R}=\int d^{d}x\sqrt{-g}\left[R-\frac{% 3(d-1)}{(d-2)}\left(\nabla\left(\ln\Phi\right)\right)^{2}\right]

K-alpha.html

  1. E = ( 10.2 e V ) ( Z - 1 ) 2 E=(10.2eV)\left(Z-1\right)^{2}
  2. α \alpha
  3. β \beta
  4. α \alpha
  5. β \beta

K-mer.html

  1. L - k + 1 L-k+1
  2. n k n^{k}
  3. L L
  4. L - 1 L-1
  5. k - 1 k-1
  6. k - 1 k-1

K-space_(magnetic_resonance_imaging).html

  1. k FE k_{\mathrm{FE}}
  2. k PE k_{\mathrm{PE}}
  3. k FE = γ ¯ G FE m Δ t k_{\mathrm{FE}}=\bar{\gamma}G_{\mathrm{FE}}m\Delta t
  4. k PE = γ ¯ n Δ G PE τ k_{\mathrm{PE}}=\bar{\gamma}n\Delta G_{\mathrm{PE}}\tau
  5. Δ t \Delta t
  6. τ \tau
  7. γ ¯ \bar{\gamma}
  8. k FE k_{\mathrm{FE}}
  9. k PE k_{\mathrm{PE}}
  10. S ( - k FE , - k PE ) = S * ( k FE , k PE ) S(-k_{\mathrm{FE}},-k_{\mathrm{PE}})=S^{*}(k_{\mathrm{FE}},k_{\mathrm{PE}})\,
  11. * {}^{*}

Kabsch_algorithm.html

  1. ( x 1 y 1 z 1 x 2 y 2 z 2 x N y N z N ) \begin{pmatrix}x_{1}&y_{1}&z_{1}\\ x_{2}&y_{2}&z_{2}\\ \vdots&\vdots&\vdots\\ x_{N}&y_{N}&z_{N}\end{pmatrix}
  2. A = P T Q A=P^{T}Q\,
  3. A i j = k = 1 N P k i Q k j , A_{ij}=\sum_{k=1}^{N}P_{ki}Q_{kj},
  4. U = ( A T A ) 1 / 2 A - 1 U=(A^{T}A)^{1/2}A^{-1}
  5. A = V S W T A=VSW^{T}\,
  6. d = sign ( det ( W V T ) ) d=\operatorname{sign}(\det(WV^{T}))\,
  7. U = W ( 1 0 0 0 1 0 0 0 d ) V T U=W\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&0&d\end{pmatrix}V^{T}

Kai_(conjunction).html

  1. χ 2 \chi^{2}

Kan_fibration.html

  1. n n
  2. Δ n \Delta^{n}
  3. Δ n ( i ) = Hom 𝚫 ( [ i ] , [ n ] ) \Delta^{n}(i)=\mathrm{Hom}_{\mathbf{\Delta}}([i],[n])
  4. n n
  5. ( t 0 , , t n ) (t_{0},\dots,t_{n})
  6. Λ k n \Lambda^{n}_{k}
  7. Δ n \Delta^{n}
  8. Δ n - 1 Δ n \Delta^{n-1}\rightarrow\Delta^{n}
  9. Δ n \Delta^{n}
  10. Λ k 2 \Lambda_{k}^{2}
  11. Δ 2 \Delta^{2}
  12. X X
  13. s : Λ k n X s:\Lambda_{k}^{n}\to X
  14. n + 1 n+1
  15. n n
  16. n n
  17. ( s 0 , , s k - 1 , s k + 1 , , s n + 1 ) (s_{0},\dots,s_{k-1},s_{k+1},\dots,s_{n+1})
  18. d i s j = d j - 1 s i d_{i}s_{j}=d_{j-1}s_{i}\,
  19. i < j i<j
  20. i , j k i,j\neq k
  21. ( n - 1 ) (n-1)
  22. Λ k n \Lambda_{k}^{n}
  23. Δ n \Delta^{n}
  24. f : X Y f:X\rightarrow Y
  25. n 1 n\geq 1
  26. 0 k n 0\leq k\leq n
  27. s : Λ k n X s:\Lambda^{n}_{k}\rightarrow X
  28. y : Δ n Y y:\Delta^{n}\rightarrow Y\,
  29. f s = y i f\circ s=y\circ i
  30. x : Δ n X x:\Delta^{n}\rightarrow X
  31. s = x i s=x\circ i
  32. y = f x y=f\circ x
  33. n n
  34. X X
  35. Δ n X \Delta^{n}\to X
  36. f s : Λ k n Y fs:\Lambda_{k}^{n}\to Y
  37. f s fs
  38. y i yi
  39. n n
  40. Y Y
  41. f s fs
  42. x : Δ n X x:\Delta^{n}\to X
  43. X X
  44. s s
  45. 2 2
  46. 2 2
  47. 1 1
  48. 1 1
  49. X X
  50. n n
  51. n n
  52. X X
  53. f : Δ n X f:\Delta_{n}\to X
  54. n n
  55. S ( X ) = n S n ( X ) S(X)=\coprod_{n}S_{n}(X)
  56. d i : S n ( X ) S n - 1 ( X ) d_{i}:S_{n}(X)\to S_{n-1}(X)
  57. ( d i f ) ( t 0 , , t n - 1 ) = f ( t 0 , , t i - 1 , 0 , t i , , t n - 1 ) (d_{i}f)(t_{0},\dots,t_{n-1})=f(t_{0},\dots,t_{i-1},0,t_{i},\dots,t_{n-1})\,
  58. s i : S n ( X ) S n + 1 ( X ) s_{i}:S_{n}(X)\to S_{n+1}(X)
  59. ( s i f ) ( t 0 , , t n + 1 ) = f ( t 0 , , t i - 1 , t i + t i + 1 , t i + 2 , , t n + 1 ) (s_{i}f)(t_{0},\dots,t_{n+1})=f(t_{0},\dots,t_{i-1},t_{i}+t_{i+1},t_{i+2},% \dots,t_{n+1})\,
  60. n + 1 n+1
  61. Δ n + 1 \Delta_{n+1}
  62. Δ n + 1 \Delta_{n+1}
  63. Δ n + 1 \Delta_{n+1}
  64. S ( X ) S(X)