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"\begin{equation}\tag{BO}
\tfrac{d}{dt} q = \h q'' - 2qq' ,
\end{equation}" |
"\begin{equation}
Q_c(t,x) = \frac{2c}{c^2 (x-ct)^2 + 1} \qtq{with} c>0.
\end{equation}" |
"\begin{align}
q(t,x) \mapsto q_\lambda(t,x) = \lambda q(\lambda^2 t, \lambda x) \qquad\text{for $\lambda>0$}
\end{align}" |
"\begin{equation}
P(q) = \int \tfrac{1}{2}q^2\, dx \qtq{and}
\hbo(q) = \int {\tfrac{1}{2} q\h q' } - \tfrac{1}{3} q^3 \, dx,
\end{equation}" |
"\begin{equation}
\{ F,G \} = \int \tfrac{\del F}{\del q}(x) \cdot \big( \tfrac{\del G}{\del q} \big)'(x)\, dx ,
\end{equation}" |
"\begin{equation}
q(t,x) = \widetilde q(t,x-2ct) + c.
\end{equation}" |
"\begin{equation}
H_2 := \int \tfrac12 \big[q'\bigr]^2 - \tfrac34 q^2 \h q' + \tfrac14 q^4 \,dx .
\end{equation}" |
"\begin{equation}
\mc L f = -i f' - C_+\bigl( q f\bigr)
\end{equation}" |
"\begin{equation}
\mc P := - i\partial^2 - 2 \partial C_+q + 2 q'_+ \qtq{or}
\mc P - i\mc{L}^2 = i C_+(\h q') - iC_+qC_+q .
\end{equation}" |
"\begin{equation}
-i \partial_x \olN - C_+( q \olN) = z \olN - z \quad\text{with $\olN(x)\to 1$ as $x\to+\infty$}
\end{equation}" |
"\begin{equation}
W = 1 + (\mc L_0 - z)^{-1} C_+(q W), \qtq{where} z\in\C\setminus[0,\infty)
\end{equation}" |
"\begin{equation}
-i m' - C_+[q(m+1)] +\kappa m =0 \qtq{or equivalently,} m = (\mc L + \kappa)^{-1} C_+ q .
\end{equation}" |
"\begin{equation}
\beta(\kappa;q) := \int q(x) m(x;\kappa,q)\, dx = \langle q_+, (\mc L +\kappa)^{-1} q_+ \rangle_{L^2_+} .
\end{equation}" |
"\begin{equation}
\beta(\kappa;q) = \kappa^{-1} P(q) - \kappa^{-2} \hbo(q) + \kappa^{-3} H_2(q) + \bigO(\kappa^{-4}).
\end{equation}" |
"\begin{align}
e^{tJ\nabla \hbo}(q^0) - e^{tJ\nabla \hbo}(\tilde q^{\:\!0}) &= e^{tJ\nabla \hbo}(q^0) - e^{tJ\nabla H_\kappa}(q^0) \notag \\
&+ e^{tJ\nabla H_\kappa }(q^0) - e^{tJ\nabla H_\kappa}(\tilde q^{\:\!0}) \\
&\qquad+ e^{tJ\nabla H_\kappa}(\tilde q^{\:\!0}) - e^{tJ\nabla \hbo}(\tilde q^{\:\!0}). \notag
\end{align}" |
"\begin{align}
e^{tJ\nabla \hbo}(q^0) - e^{tJ\nabla H_\kappa}(q^0) = \bigl[ e^{tJ\nabla(\hbo - H_\kappa) } - \Id\bigr]\circ e^{tJ\nabla H_\kappa}(q^0) .
\end{align}" |
"\begin{equation}
\hk (q):= \begin{cases} \kappa P(q) - \kappa^2 \beta(\kappa;q) & \text{on $\R$,} \\
\bigl[ \kappa + \tint \bigr] P(q) - \kappa^2 \beta(\kappa;q) + \tfrac{\kappa}2 \btint^2 + \tfrac16 \btint^3 & \text{on $\T$.}
\end{cases}
\end{equation}" |
"\begin{equation}
2q= \tfrac{1}{w+\kappa} \h (w') + \h \bigl[\tfrac{w'}{w+\kappa}\bigr] +\tfrac{2\kappa w}{w+\kappa}.
\end{equation}" |
"\begin{equation}
\tfrac{d}{dt} w = \h w'' - 2qw',
\end{equation}" |
"\begin{align}
q(\vec t;q_0) = \Bigl[\exp\bigl\{ {\textstyle\sum} t_i J\nabla H_i \bigr\}q_0 \Bigr] ( x=0 )
\end{align}" |
"\begin{equation}
H_\phi(q) := \langle q_+, \phi(\mc L)q_+\rangle
\end{equation}" |
"\begin{equation}
q(\phi;q_0) = \bigl( e^{ J\nabla H_\phi} q_0\bigr)( x=0 ).
\end{equation}" |
"\begin{equation}
C_+\bigl( e^{ J\nabla H_\phi} q_0\bigr)( x+iy ) = \tfrac{1}{2\pi i} I_+ \Big( \big( X - t \psi(\mc L_{q_0}) - x - iy \big)^{-1} q^0_+\Big) \end{equation}" |
"\begin{equation}
\psi(E) = \phi(E) + E\phi'(E).
\end{equation}" |
"\begin{align*}
\hat f(\xi) = \tfrac{1}{\sqrt{2\pi}} \int_\R e^{-i\xi x} f(x)\,dx \qtq{so} f(x) = \tfrac{1}{\sqrt{2\pi}} \int_\R e^{i\xi x} \hat f(\xi)\,d\xi
\end{align*}" |
"\begin{align*}
\hat f(\xi) = \int_0^1 e^{- i\xi x} f(x)\,dx \qtq{so} f(x) = \sum_{\xi\in 2\pi\Z} \hat f(\xi) e^{i\xi x}.
\end{align*}" |
"\begin{align*}
\|f\|_{L^2(\mathbb R)}=\|\hat f\|_{L^2(\mathbb R)} \qtq{and} \|f\|_{L^2(\mathbb T)}=\sum_{\xi\in 2\pi \mathbb Z}|\hat f(\xi)|^2.
\end{align*}" |
"\begin{equation}
\wh{\h f}(\xi) = -i\sgn(\xi)\wh{f}(\xi)
\end{equation}" |
"\begin{align*}
%\end{align*}" |
"\begin{equation}
\wh{C_\pm f}(\xi) = 1_{[0,\infty)}(\pm\xi) \wh{f}(\xi)
\end{equation}" |
"\begin{align}
C_++C_-=1 \text{only on the line; on the circle,} C_+ f + C_- f = f + \textstyle \int \!f.
\end{align}" |
"\begin{align}
s\in (-\tfrac12,0) \qtq{and define} \eps:= \tfrac12( \tfrac 12-|s|)\in (0,\tfrac14).
\end{align}" |
"\begin{align*}
\norm{fg}_{H^{2r-1}}^2
&= \frac{1}{2\pi} \int_0^\infty \frac{1}{(\xi+1)^{4|r|+2}} \bigg| \int_0^\xi \wh{f}(\xi-\eta) \wh{g}(\eta)\,d\eta\bigg|^2\,d\xi.
\end{align*}" |
"\[
\limsup_{\delta\to0} \ \sup_{q\in Q} \ \sup_{|y|<\delta}\|q(\cdot +y) - q(\cdot)\|_{H^\sigma} = 0.
\]" |
"\begin{equation}
\lim_{ \kappa\to\infty}\sup_{q\in Q}\int_{|\xi|\geq \kappa} |\wh{q}(\xi)|^2 (|\xi|+1)^{2\sigma}\, d\xi =0\quad\text{on $\R$}
\end{equation}" |
"\begin{equation}
\lim_{ \kappa\to\infty}\sup_{q\in Q}\sum_{|\xi|\geq \kappa} |\wh{q}(\xi)|^2 (|\xi|+1)^{2\sigma}=0\quad\text{on $\T$}.
\end{equation}" |
"\begin{equation}
\begin{aligned}
\norm{ C_+qR_0(\kappa)C_+f }_{H^s} &\lesssim \kappa^{-2\eps} \norm{q}_{H^s} \norm{f_+}_{H^s_\kappa} ,\\
\norm{ C_+qR_0(\kappa)C_+f }_{H^s_\kappa} &\lesssim \kappa^{-2\eps} \norm{q}_{H^s_\kappa} \norm{f_+}_{H^s_\kappa} ,
\end{aligned}
\end{equation}" |
"\begin{align*}
\norm{ C_+qR_0(\kappa)C_+f }_{H^s} ^2
&= \frac{1}{2\pi} \int_0^\infty \frac{1}{(\xi+1)^{2|s|}} \bigg| \int_0^\infty \wh{q}(\xi-\eta) \frac{\wh{f}(\eta)}{\eta+\kappa}\,d\eta \bigg|^2 d\xi .
\end{align*}" |
"\begin{align}
\bigl| \langle f, qf\rangle \bigr|
\lesssim \kappa^{-2\eps} \norm{q}_{H^s_\kappa} \|f\|_{H^{1/2}_\kappa}^2
= \kappa^{-2\eps} \norm{q}_{H^s} \langle f, (\mc L_0 + \kappa) f \rangle.
\end{align}" |
"\begin{equation}
R(\kappa;q) = (\mc{L}+\kappa)^{-1} = R_0(\kappa)\sum_{\ell\geq 0}\bigl[C_+qR_0(\kappa)\bigr]^\ell,
\end{equation}" |
"\begin{equation}
R(\kappa)-R_0(\kappa) = R_0(\kappa) C_+ q \sqrt{R_0(\kappa)} \cdot \sqrt{R_0(\kappa)} \bigl[ 1 + C_+ q R(\kappa) \bigr].
\end{equation}" |
"\begin{equation}
q\in \BA:=\{ \text{real-valued }q\in H^s : \norm{q}_{H^s} \leq A \} \qtq{and} \kappa \geq C_s \bigl( 1 + A \bigr)^{\frac1{2\eps}}.
\end{equation}" |
"\begin{equation}
C_+\big( f\,\ol{\mc{L}g} - \ol{g}\,\mc{L}f \big) = iC_+\big(f\ol{g}\big)' + f[1-C_-](q_+\ol{g}) .
\end{equation}" |
"\begin{align*}
C_+\big( f\,\ol{\mc{L}g} - \ol{g}\,\mc{L}f \big)
&= C_+\big\{ if\ol{g}' - fC_-(q\ol{g}) + if'\ol{g} + \ol{g}C_+(qf) \big\} \\
&= iC_+\big(f\ol{g}\big)' + C_+\bigl\{ \ol{g}qf - fC_-(q\ol{g}) \big\}\\
&= iC_+\big(f\ol{g}\big)' + C_+\bigl\{ f [1-C_-] (q\ol{g}) \big\}\\
&= iC_+\big(f\ol{g}\big)' + f [1-C_-] (q\ol{g}).
\end{align*}" |
"\begin{align}
\lim_{y\to\infty} \pi y f(iy) = \lim_{y\to\infty} \int \frac{y^2}{x^2+y^2} f(x) \,dx = \lim_{\xi\downarrow 0} \tfrac{\sqrt{2\pi}}{2}\widehat f(\xi)
\end{align}" |
"\begin{align}
I_+(f) := \lim_{y\to\infty} 2\pi y f(iy) = \lim_{y\to\infty} \bigl\langle \chi_y, f \bigr\rangle
= \lim_{\xi\downarrow 0} \sqrt{{2\pi}} \widehat f(\xi) \ \text{with}\ \chi_y(x) = \tfrac{iy}{x+iy}.
\end{align}" |
"\begin{align}
f(z) = \tfrac{1}{2\pi i} I_+\bigl( (X-z)^{-1} f \bigr) = \lim_{y\to\infty} \tfrac{1}{2\pi i} \bigl\langle \chi_y, (X-z)^{-1} f \bigr\rangle
\end{align}" |
"\begin{equation}
[X,C_+q] f = \tfrac{i}{2\pi} q_+ I_+(f), \qtq{and} [X,\mc L ] f = i - \tfrac{i}{2\pi} q_+ I_+(f) .
\end{equation}" |
"\begin{equation}
m' = -i\kappa m + i C_+[q(m+1)],
\end{equation}" |
"\begin{equation}
m(x;\kappa,q) := R(\kappa,q) q_+ = R_0(\kappa) \sum_{\ell\geq 1} [C_+q R_0(\kappa)]^{\ell-1} q_+
\end{equation}" |
"\begin{equation}
m(x+h;\kappa,q) = m(x;\kappa,q(\cdot+h)) \quad\text{for all }h\in\R .
\end{equation}" |
"\begin{equation}
\snorm{ m^{(\sigma)} }_{H^{s+1}_\kappa}
\lesssim \sum_{\ell=1}^\infty \ell^\sigma 2^{\sigma-\ell} \Bigl( 1 + \| q\|_{H^{s}_\kappa}\Bigr)^\sigma \| q \|_{H^{s+\sigma}_\kappa} < \infty
\end{equation}" |
"\begin{equation}
dm|_q(g)= \frac{d}{d\theta} m(x;\kappa,q+\theta g) \bigg|_{\theta=0}= R(\kappa,q) \bigl[ (m+1)C_+g\bigr],
\end{equation}" |
"\begin{equation}
dm|_0(g)= R_0(\kappa)C_+ g .
\end{equation}" |
"\begin{equation}
\beta(\kappa;q): = \int q(x) m(x;\kappa,q)\,dx = \int q(x) \ol{m}(x;\kappa,q)\,dx = \bigl\langle q_+, (\mc L+\kappa)^{-1} q_+ \bigr\rangle
\end{equation}" |
"\begin{equation}
\tfrac{\del\beta}{\del q} = m + \ol m + |m|^2
\end{equation}" |
"\begin{align*}
\int q(x) m(x;\kappa,q)\,dx = \langle q, m\rangle = \langle q_+, m\rangle = \bigl\langle q_+, (\mc L+\kappa)^{-1} q_+ \bigr\rangle.
\end{align*}" |
"\begin{align*}
d\beta|_q(f) &= \int f m + q \cdot R(\kappa,q)[(m+1)f_+] \,dx \\
&= \int f m + \ol{R(\kappa,q)q_+} \cdot (m+1)f \,dx\\
&= \int [m +\ol{m}(m+1) ] f \,dx,
\end{align*}" |
"\begin{align}
\int_{\kappa}^\infty \!\vk^{2s} \langle q_+, R_0(\vk)q_+\rangle \,d\vk
&= \int_{0}^\infty \! \int_{\kappa}^\infty \! \vk^{2s} \frac{|\wh{q}(\xi)|^2}{\xi+\vk}\,d\vk\,d\xi
\simeq_s \norm{q_+}_{H^{s}_\kappa}^2.
\end{align}" |
"\begin{align*}
Q_{**}= \Bigl\{ q(b) \Big|\, q:[a,b]\to H^s \text{ is continuous, } q(a)\in Q, \text{ and } \beta(z;q(t))\equiv \beta(z;q(a))\Bigr\},
\end{align*}" |
"\begin{equation}
\kappa = C_s \bigl( 1 + 2 C_sA \bigr)^{\frac1{2\eps}}.
\end{equation}" |
"\begin{equation}
m := m(x;\kappa,q) \qtq{and} n := m(x;\vk,q)
\end{equation}" |
"\begin{equation}
\lim_{ \kappa\to\infty}\ \sup_{q\in Q}\norm{m}_{H^{s+1}_\kappa} = 0 \qtq{and}
\lim_{ \kappa\to\infty}\ \sup_{q\in Q}\norm{ \mc{L} R(\kappa,q) n}_{H^{s+1}} =0
\end{equation}" |
"\begin{align*}
\norm{ \mc{L} R(\kappa,q) n}_{H^{s+1}}
= \norm{ R(\vk,q) \mc{L} m}_{H^{s+1}}
&\lesssim \norm{ \mc{L} m }_{H^{s}}\lesssim (1+\|q\|_{H^s})\norm{ m}_{H^{s+1}} .
\end{align*}" |
"\begin{align}
\tfrac{d}{dt} q_+ &= \mc{P} q_+ = - i q_+'' - 2C_+(qq_+)' + 2q_+ q'_+ \\
\tfrac{d}{dt} m &= - im'' - 2C_+([q-q_+] m)' - 2q_+m' \\
\tfrac{d}{dt} \beta(\kappa) &= 0 .
\end{align}" |
"\begin{align}
\mc{P}q_+ = - iq''_+ - 2C_+(qq_+)' + 2q_+'q_+= C_+\bigl(\h q'' - 2qq'\bigr) = \tfrac{d}{dt} q_+\,.
\end{align}" |
"\begin{align*}
\tfrac{d}{dt} m &= [\mc{P}, R(\kappa) ] q_+ + R(\kappa) \mc P q_+ = \mc P m .
\end{align*}" |
"\begin{align*}
\tfrac{d}{dt} \beta(\kappa) &= \bigl\langle \mc P q_+, R(\kappa) q_+\bigr\rangle + \bigl\langle q_+, [\mc P, R(\kappa) \:\!] q_+\bigr\rangle
+ \bigl\langle q_+, R(\kappa) \mc P q_+\bigr\rangle \\
&=\bigl\langle \mc P q_+, R(\kappa) q_+\bigr\rangle +\bigl\langle q_+, \mc P R(\kappa) q_+\bigr\rangle.
\end{align*}" |
"\begin{align}
\int \ol{m} n \,dx = \int m \ol{n} \,dx = \bigl\langle (\mc L+\vk)^{-1} q_+, (\mc L+\kappa)^{-1} q_+ \bigr\rangle = - \frac{\beta(\kappa)-\beta(\vk)}{\kappa-\vk}
\end{align}" |
"\begin{align}
\kappa \int \ol{m} \,dx = \kappa \int m \,dx = \int qm \,dx + \int q \,dx = \beta(\kappa) + \int q \,dx
\end{align}" |
"\begin{align}
\langle 1, (\mc L + \kappa)^{-1} 1\rangle = \kappa^{-1} + \kappa^{-2} \beta(\kappa;q) + \kappa^{-2}\int q\,dx .
\end{align}" |
"\begin{align*}
(\mc L + \kappa)^{-1} 1 = (\mc L_0 + \kappa)^{-1} 1 + (\mc L + \kappa)^{-1}C_+ q(\mc L_0 + \kappa)^{-1} 1
= \kappa^{-1} (1 + m) .
\end{align*}" |
"\begin{equation}
\kappa \int m+ \ol{m} +|m|^2 \,dx = 2\beta(\kappa) - \kappa\frac{\partial\beta}{\partial\kappa}+ 2\int q \,dx,
\end{equation}" |
"\begin{equation}
\begin{aligned}
\h (m\ol{n}+m+\ol{n})' + i(m+1)\ol{n}' -im'(\ol{n}+1) - 2q (m+1)(\ol{n}+1)& \\
{}+ i(\kappa - \vk)\h (m\ol{n}+m+\ol{n}) + (\kappa + \vk)(m\ol{n}+m+\ol{n}) &= 0 ,
\end{aligned}
\end{equation}" |
"\begin{equation}
\text{\upshape LHS\eqref{m ODE 1}} = \vk \int m \,dx + \kappa\int \ol n \,dx .
\end{equation}" |
"\begin{align*}
\text{LHS}\eqref{m ODE 1}
={} &\kappa ( 1+i\h )\ol{n} + \vk ( 1 -i\h )m + (1+i\h) \big[ (\ol{n}+1)C_+(q(m+1)) \big] \\
&+ (1-i\h) \big[ (m+1)C_-(q(n+1)) \big] - 2q(m+1)(\ol{n}+1) .
\end{align*}" |
"\begin{align*}
\text{LHS}\eqref{m ODE 1}
={} &\kappa ( 1+i\h )\ol{n} + \vk ( 1 -i\h )m + (1+i\h) \big[ (\ol{n}+1)[C_+-1] (q(m+1)) \big] \\
&+ (1-i\h) \big[ (m+1)[C_- -1] (q(n+1)) \big] .
\end{align*}" |
"\begin{equation}
\{ \beta(\kappa) , \beta(\vk) \} = 0 \qtq{and} \{ P , \beta(\vk) \} = 0
\end{equation}" |
"\begin{align}
F:= \bigl[ |m|^2 + m + \ol{m}\bigr] \bigl[|n|^2 + n + \ol{n}\bigr]'
- \bigl[ |m|^2 + m + \ol{m}\bigr]' \bigl[|n|^2 + n + \ol{n}\bigr].
\end{align}" |
"\begin{align*}
F=G+\ol{G}+K+\ol{K} \qtq{where} G &= [ \ol{m}n + \ol{m} + n] \bigl[ (m+1)\ol{n}' - m'(\ol{n}+1) \bigr]\\
\qtq{and} K &= (m -n)(\ol{m}+\ol{n})' .
\end{align*}" |
"\begin{equation}
\begin{aligned}
(m+1)\ol{n}' - m'(\ol{n}+1) &= i \h (m\ol{n}+m+\ol{n})' -(\kappa - \vk)\h (m\ol{n}+m+\ol{n}) \\
&\quad{} + i (\kappa + \vk - 2 q)(m\ol{n}+m+\ol{n}) - 2 i q - i c ,
\end{aligned}
\end{equation}" |
"\begin{align}
\int\! (G + \overline G) \, dx = i\int (2q+c) [ m\ol{n} - \ol{m}n + m - \ol{m} - n + \ol{n}] \,dx.
\end{align}" |
"\begin{align}
\int\! (G + \overline G) \, dx = 2i \int q [ m\ol{n} - \ol{m}n] \,dx.
\end{align}" |
"\begin{align*}
\int (K + \ol{K}) \, dx &= 2 \int m \ol{n}' + \ol{m} n' \,dx \\
&= 2i \int (\vk - q) [m \ol{n} - \ol{m} n] \,dx - 2 i \int q[m-\ol{m}]\,dx.
\end{align*}" |
"\begin{align}
\int (K + \ol{K}) \, dx &= - 2i \int q [m \ol{n} - \ol{m} n] \,dx .
\end{align}" |
"\begin{align*}
&\{\beta(\vk),P(q)\}
= \int \big( |n|^2 + n + \ol n \big) q'\, dx
= \int - [(1+\ol n)n' + (1+n)\ol n']q\, dx.
\end{align*}" |
"\begin{align*}
\{\beta(\vk),P(q)\} &= i \int \vk[ n(\ol n+1)-\ol n(n+1) ]q \,dx \\
&- i \int (\ol n+1)q \cdot C_+ (n+1)q - (n+1)q \cdot C_- (\ol n+1)q \,dx \\
& =0.\qedhere
\end{align*}" |
"\begin{equation}
2q = \tfrac{1}{w+\kappa} \h (w') + \h \big( \tfrac{w'}{w+\kappa} \big) + \tfrac{2\kappa w}{w+ \kappa} .
\end{equation}" |
"\begin{equation}
w = \kappa \tfrac{\del\beta}{\del q} = \kappa \big( |m|^2+ m + \ol m \big)
\end{equation}" |
"\begin{align*}
2q&= \tfrac{\h(|m|^2+m+\ol m)'}{|m+1|^2} + i\Bigl[ \tfrac{\ol{m}'}{\ol m +1} - \tfrac{m'}{m+1} \Bigr] + 2\kappa \tfrac{|m|^2+m+ \ol m}{|m+1|^2}\\
&=\tfrac{\h(|m|^2+m+\ol m)'}{|m+1|^2} + i\h\Bigl[ \tfrac{\ol{m}'}{\ol m +1} + \tfrac{m'}{m+1} \Bigr]+ 2\kappa \tfrac{|m|^2+m+ \ol m}{|m+1|^2},
\end{align*}" |
"\begin{align}
1 + u(x) = [1+\mu(x)] [1+\ol\mu(x)] \qtq{with} \mu \in H^{s+1}_+(\R).
\end{align}" |
"\begin{align*}
(1+u) \h \big[ \tfrac{u'}{1+u} \big] =
|\mu+1|^2 \h \big( \tfrac{\ol\mu'}{1+\ol\mu} + \tfrac{\mu'}{1+\mu} \big) =
i(1+\mu)\ol\mu' - i\mu' (\ol\mu+1) .
\end{align*}" |
"\begin{equation}
2q [1+\mu] [1+\ol\mu] = 2C_-\bigl[ i(1+\mu)\ol\mu'\bigr] - 2C_+\bigl[ i\mu' (\ol\mu+1) \bigr] + 2\kappa [\mu+\ol\mu+|\mu|^2] .
\end{equation}" |
"\begin{equation}
C_+\Bigl[ (1+\ol\mu)\bigl( - i\mu' - C_+(q\mu) + \kappa\mu - q_+ \bigr) \Bigr] =0 .
\end{equation}" |
"\begin{equation}
- i\mu' - C_+(q\mu) + \kappa\mu - q_+ = \tfrac{f}{1+\ol\mu} \in H^s_-(\R) .
\end{equation}" |
"\begin{equation}
\int w(x;\kappa, q) \,dx = 2\beta(\kappa) - \kappa\frac{\partial\beta}{\partial\kappa}+ 2\int q \,dx .
\end{equation}" |
"\begin{align}
\alpha(\kappa;q):= \sum_{\ell\geq 2} \tfrac1\ell\tr\bigl\{
(R_0(\kappa)C_+ q)^\ell \bigr\},
\end{align}" |
"\begin{align*}
\|\sqrt{R_0(\kappa)}C_+ q\sqrt{R_0(\kappa)}\|_{\textrm{HS}}^2
&= \frac{1}{2\pi} \int_0^\infty\!\!\int_0^\infty \frac{|\widehat
q(\xi-\eta)|^2\,d\eta\,d\xi}{(\eta+\kappa)(\xi+\kappa)}\\
&=\frac{1}{2\pi} \int_\R \frac{\log(1+
\frac{|\xi|}\kappa)}{|\xi|}|\widehat q(\xi)|^2\,d\xi \lesssim
\kappa^{-4\eps} \|q\|_{H^s_\kappa}^2<1,
\end{align*}" |
"\begin{align*}
\alpha(\kappa;q)= \tfrac1{2\pi}\int_0^\infty
\tfrac{\beta(\kappa+\xi;q)}{\kappa+\xi}\, d\xi \quad\text{on $\R$}
\qquad\text{and} \alpha(\kappa;q)= \sum_{\xi\in 2\pi
\Z_+}\tfrac{\beta(\kappa+\xi;q)}{\kappa+\xi} \quad\text{on $\T$},
\end{align*}" |
"\begin{align*}
\frac1\ell &\tr\bigl\{ (R_0(\kappa)C_+ q)^\ell \bigr\}\\
&= \sum_{\xi_1, \ldots, \xi_\ell\in 2\pi\Z_+} \frac1\ell \, \frac{\widehat q(\xi_1-\xi_2)}{\kappa+\xi_1}
\frac{\widehat q(\xi_2-\xi_3)}{\kappa+\xi_2} \cdots \frac{\widehat q(\xi_\ell-\xi_1)}{\kappa+\xi_\ell}\\
&=\sum_{\substack{\xi_1\leq\min\{\xi_2, \ldots, \xi_\ell\} \\[0.5ex] \xi_1, \ldots, \xi_\ell\in 2\pi\Z_+}}
\frac{\widehat q(\xi_1-\xi_2)}{\kappa+\xi_1} \frac{\widehat q(\xi_2-\xi_3)}{\kappa+\xi_2} \cdots \frac{\widehat
q(\xi_\ell-\xi_1)}{\kappa+\xi_\ell}\\
&=\sum_{\xi\in 2\pi\Z_+} \frac1{\kappa+\xi}
\sum_{\substack{ \eta_2, \ldots, \eta_\ell \in 2\pi\Z \\[0.5ex] \eta_j+ \cdots + \eta_\ell\geq 0, \forall 2\leq j\leq \ell}}
\widehat q \bigl(-(\eta_2+\cdots\eta_\ell)\bigr) \prod_{j=2}^\ell\frac{\widehat q(\eta_j)}{\kappa+\xi + \eta_j+ \cdots + \eta_\ell }\\
&=\sum_{\xi\in 2\pi\Z_+} \frac1{\kappa+\xi} \Bigl\langle q,\, \bigl(R_0(\kappa+\xi) C_+ q\bigr)^{\ell-2}R_0(\kappa+\xi) q_+\Bigr\rangle.
\end{align*}" |
Dataset Description
English version is here
这里有两个数据集：raw_formulas和tokenized_formulas。
我们在arxiv上爬取了约100万条未经过清洗以及文本分词的latex公式的图片文本对从而得到了raw_formulas数据集。在将raw_formulas数据集进行清洗以及文本分词后得到了tokenized_formulas数据集。
渲染公式对应的图片时，需要用到以下外部包：
- amsmath
- amsfonts
- amssymb
- mathtools
有关raw_formulas数据集的一些细节
我们爬取了包含以下公式环境的latex公式:
- equation
- align
- align*
- gather
- gather*
公式中不存在以下内容：
- \label
- %
- \quad
- \qquad
- \vspace
- \hspace
- \resizebox
- \scalebox
- \rotatebox
- \parbox
- \fbox
- \makebox
- \raisebox
- \addvspace
- \hfill
- \vfill
- \textwidth
- \textheight
- \rule
tokenized_formulas数据集的一些预处理细节
清洗
- 我们删除了raw_formulas中一些无用的垃圾数据
- 我们删除了raw_formulas中过于复杂的公式：
- 公式对应的渲染图片高宽比如果大于0.8，那么公式会被删除
- 字符长度大于200的公式被删除了
- raw_formulas公式中的以下内容被删除了：
- \tag
- \text
- \begin{split}
- \end{split}
- \nonumber
- \notag
- raw_formulas中的
equation、equation*、align、\[...\]公式环境被替换成了align*公式环境，gather公式环境被替换成了gather*环境 - 我们删除了raw_formulas中包含自定义宏的公式，只有以下的常见自定义宏被保留了:
- \newcommand{\R}{\mathbb{R}}
- \newcommand{\N}{\mathbb{N}}
- \newcommand{\Z}{\mathbb{Z}}
- \newcommand{\Q}{\mathbb{Q}}
- \newcommand{\C}{\mathbb{C}}
- \newcommand{\avg}[1]{\left<#1\right>}
- \newcommand{\Deriv}[2]{\frac{\mathrm{d} #1}{\mathrm{d} #2}}
- \newcommand{\dd}{\mathrm{d}}
- \newcommand{\norm}[1]{\left\lVert#1\right\rVert}
- \newcommand{\abs}[1]{\left|#1\right|}
- \newcommand{\vect}[1]{\mathbf{#1}}
分词
符合以下pattern的子串被视为一个词：
- \begin{.*?}
- \end{.*?}
- \[A-Za-z]+
以下字符串也被视为一个词：
- \[
- \]
- \\
- \{
- \}
- \_
- \$
- \&
- \#
- \%
- \|
- '
- ''
- '''
- ''''
- '^
- ''^
- '''^
- ''''^
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