harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
f \star g = \frac{1}{(2\pi )^2}\sum_{m, n}\int_{-\pi}^{+\pi}d\theta ' d \theta '' \int_{-\infty}^{+\infty}dp' dp'' \delta (p_{\theta}- p' +\frac{\hbar m}{2}) \delta(p_{\theta}- p'' - \frac{\hbar n}{2})e^{[-\frac{4i}{\hbar}A({\bf r},{\bf r}',{\bf r}'')]}f({\bf r}') g ({\bf r}'').
\begin{aligned}s_v^\Lambda+\sum_{\substack{\emptyset \neq G\subseteq E \mu\in \mathrm{MCE}(G)}}(-1)^{|G|}s_\mu^{\Lambda}{s_\mu^{\Lambda}}^*=\phi\bigg(s_v^{\Lambda^i}+\sum_{\substack{\emptyset \neq G\subseteq E \mu\in \mathrm{MCE}(G)}}(-1)^{|G|}s_\mu^{\Lambda^i}{s_\mu^{\Lambda^i}}^*\bigg)=\phi\big(\Delta(s^{\Lambda^i})^E\big).\end{aligned}
\begin{aligned}\begin{bmatrix}i\delta_1 \lambda \overline{\lambda}\delta_2 \end{bmatrix}^{-1}=\begin{bmatrix}-i\delta_2(|\lambda|^2+\delta_1\delta_2)^{-1}\lambda (|\lambda|^2+\delta_1\delta_2)^{-1}\overline{\lambda}(|\lambda|^2+\delta_1\delta_2)^{-1}-i\delta_1(|\lambda|^2+\delta_1\delta_2)^{-1}\end{bmatrix}.\end{aligned}
|\o (t,x) - \o^\beta (t,x) | = | \o_0 (X(0; t, x) ) - \o_0^\beta (X^\beta (0; t, x)) | &\le |\o_0 (X(0; t, x)) - \o_0^\ell (X(0; t, x))| + |\o_0^\ell (X^\beta (0; t, x)) - \o_0^\beta (X^\beta (0; t, x))| & + |\o_0^\ell (X(0; t, x)) - \o_0^\ell (X^\beta (0; t, x))|,
\psi_{\kappa}(\theta, x) \ = \ \inf_{v_2 \in{\mathcal M}_2}E^{\hat v_1, v_2}_x \Big[ e^{\frac{\kappa \|r_2\|_{\infty}}{\alpha}}e^{\theta \int^{T_{\kappa}}_0 e^{-\alpha s}r_2(X(t), \hat v_1(\theta(t),X(t)), v_2(t, X(t))) dt}\Big].
G_{+ +}^{2}( x ) ={\frac{i}{1 6 \pi^{2}}}[{\frac{1}{n - 4}}+ \psi ( 1 ) - 4 \pi ] \delta^{4}( x ) + i \Sigma ( x ) - \eta ( x ) - L o g [ 4 \pi \mu^{2}] , \nonumber
\int_{\Omega}|\nabla \phi|\chi_{A}~dx \leq \liminf_{\nu\rightarrow \infty}||\nabla \phi_\nu||_{L^1(A)}=\liminf_{\nu\rightarrow \infty}\int_{\Omega}|\nabla \phi_{\nu}|\chi_{A}~dx \leq C\lim_{\nu\rightarrow \infty}\int_{\Omega}\det\nabla \phi_{\nu}\chi_{A}~dx =C\int_{\Omega}\det\nabla \phi\chi_{A}~dx,
\tilde{\Theta}_{a,k,\zeta,\mu,\sigma}(x) = \Psi_{a,\zeta,\mu,\sigma}(x_1,\dots,x_{k-1},x_{k+1},\dots,x_d), \tilde{W}_{a,k,\zeta,\mu,\sigma}(x) = \tilde{\Psi}_{a,\zeta,\mu,\sigma}(x_1,\dots,x_{k-1},x_{k+1},\dots,x_d)e_k.
\mu_t=f_t\cdot\mathcal{L}^N\mbox{and}\|f_t\|_{L^q(\Omega)}\le \left((1-t)\,\|f_0\|^q_{L^q(\Omega)}+t\,\|f_1\|^q_{L^q(\Omega)}\right)^\frac{1}{q},t\in[0,1].
\lim_{\vec{p}\rightarrow 0}\,\Pi_{1}= \lim_{\vec{p}\rightarrow0}\,\delta_{ij}\Pi_{ij}= 2e^{2}\int{d^{2}k\over(2\pi)^{2}}\,{1\over \omega_{k}}\,{\partial\over\partial\omega_{k}}\left({(\omega_{k}^{2}-m^{2})n_{F}(\omega_{k})\over\omega_{k}}\right) = 0,
\sideset{}{^+}\sum_{\chi(q)}\epsilon(\chi) \chi(m\ell_1 \ell_3) \overline{\chi}(n) = \frac{1}{q^{1/2}}\mathop{\sum \sum}_{\substack{vw = q (v,w)=1}}\mu^2(v)\varphi(w) \cos \left(\frac{2\pi n \overline{m \ell_1 \ell_3 v}}{w}\right).
E_{LJ}(\Lambda_A)\leq E_{LJ}(L) |L|=A &\iff A\leq \inf_{|L|=1 \atop L\neq \Lambda_1}\left( \frac{\zeta_{L}(12)-\zeta_{\Lambda_1}(12)}{2(\zeta_{L}(6)-\zeta_{\Lambda_1}(6))}\right)^{1/3}.
\frac{1}{N}(M_1^{(N)}, \ldots, M_K^{(N)})\cdot \vec{1}_{\{Z=d\}}\stackrel{\rm{d}}{\to}\left(\hat{M}_1,\ldots,\hat{M}_K \right)\cdot \vec{1}_{\{Z=d\}}\stackrel{\rm d}{=}\pi_d\left(\frac{\hat{\rho}_{1,\infty}}{g_1},\ldots,\frac{\hat{\rho}_{K,\infty}}{g_K}\right) \cdot X
J(nn^{\prime}mm^{\prime})=\int\int\phi_{n}^{*}(r_{1}-R)\phi_{n^{\prime}}^{*}(r_{2}-R^{\prime})\frac{e^{2}}{r_{12}}\phi_{m}(r_{2}-R)\phi_{m^{\prime}}(r_{1}-R^{\prime})dr_{1}dr_{2}
[\boldsymbol{\lambda}(a_1,a_2),\boldsymbol{\lambda}(b_1,b_2)]=2(-1)^{(|a_1|+|a_2|)|b_2|}\mathbf{J}([a_1,a_2],b_1,b_2)+\boldsymbol{\lambda}([a_1,a_2],[b_1,b_2])-\boldsymbol{\lambda}(\overline{[a_1,a_2]},[b_1,b_2]).
\left\{\widehat{A}(TX){\rm exp}(\frac{c}{2})(20+{\rm ch}(\overline{W_i})-8{\rm ch}({W_i})+B_1\wedge[{\rm ch}({W_i})-8]+B_2)\right\}^{(10)}&=61920\left\{\widehat{A}(TX){\rm exp}(\frac{c}{2})\right\}^{(10)}.
\Phi_{R,L}(x,t,k)=\Phi^{\pm\infty}(x,t,k)+\int_{\pm\infty}^{x}\Phi^{\pm\infty}(x,t,k)\left(\Phi^{\pm\infty}\right)^{-1}(y,t,k)\left(Q-Q^{\pm\infty}\right)(y,t,k)\Phi_{R,L}(y,t,k)dy.
\tilde u(t,x)= \ \mathbf E\left[\phi\left(-X^{t,\beta}(\tau_0(t)),X^{x,\alpha}(\tau_0(t))\right)\mathbf 1_{\{\tau_0(t)<\tau_\Omega(x)\}}\right] \ +\mathbf E \left[\int_0^{\tau_0(t)\wedge \tau_\Omega(x)}g \left(-X^{t,\beta}(s),X^{x,\alpha}(s)\right)ds\right],
\int_0^\infty \Psi^{(3)}(\Phi W(r\cdot))^{(3)}+ \int_0^\infty \Psi(Z) \frac{(\Phi(Z) W(rZ))'}{Z}\:dZ = \int_0^\infty \frac{S(Z)}{Z}(\Phi(Z) W(rZ))'\:dZ.
F^{s+c}\left(\sum_{i=0}^{r+s+c-1}V^i[\pi a_i]\right)=\sum_{i=0}^{c-1}p^i[\pi a_i]^{p^{s+c-i}}+p^cF^s\left(\sum_{i=c}^{r+s+c-1}V^{i-c}[\pi a_i]\right)=0.
\MoveEqLeft \int |\delta E(m)|^2 dx - \int |m \cdot \delta E(m)|^2 dx = \int |\delta E(\eta)|^2 dx & - 2 \int \beta_*^2 \mu dx - 2 \int \beta_* \eta \cdot \delta E(\eta) dx - \int |w_* \cdot \delta E(\eta)|^2 dx + O(\| \eta \|_{H^1}\| \eta \|_{H^2}^2).
y_3/ h_{3,1}= \sqrt{\rho}L_{z}(a_1, a_2) + \sqrt{\rho}L_{y}(b_1, b_2) z_3/ g_{3,1}= \sqrt{\rho^{\alpha}}L_{z}(a_1, a_2) + \sqrt{\rho^{\alpha}}L_{y}(b_1, b_2).
\mathrm{E}(X^n)= \frac{\mu^n}{\mathrm{B}(\alpha,\beta)}\sum_{j=0}^{\infty}(-1)^j\binom{\beta-1}{j}\sum_{i=0}^n\binom{n}{i}\left(\frac{\sigma}{\mu}\right)^i &\times\sum_{k=0}^\infty\left[v_k(j+\alpha-1)+(-1)^{i+k}\binom{j+\alpha-1}{k}\right]J_{i,k}^{(s)},
\lim_{\substack{\alpha\to 0 \alpha \to 0 M\to +\infty}}\sum_{j=1}^{\lfloor \frac{1}{\alpha_*^2}\rfloor}\P\left(\left\vert \xi(\alpha_*,M,\tau_j)-\xi(\alpha_*,M,\tau_{j-1})\right\vert >\delta\right)=0,
\widehat{P_Nu}\left(k\right) := 1_{N \leq \left|k\right| <2N}\left(k\right) \widehat{u}\left(k\right), \widehat{P_{\leq N}u}\left(k \right) := 1_{\left|k\right| \leq N}\left(k\right) \widehat{u}\left(k\right), \widehat{P_{\geq N}u}\left(k \right) := 1_{\left|k\right| \geq N}\left(k\right) \widehat{u}\left(k\right),
\hat{V}^{(B=0)}(x,t)={\rm diag.}\,(\overline{V}_{1},\cdots ,\overline{V}_{4})={\rm diag.}\,(\scriptstyle{{\sqrt{-{\frac{t_{3}}{t_{1}}}}{\frac{t-t_{1}}{t-t_{3}}},\sqrt{-{\frac{t_{4}}{t_{2}}}}{\frac{t-t_{2}}{t-t_{4}}},\sqrt{-{\ \frac{t_{1}}{t_{3}}}}{\frac{t-t_{3}}{t-t_{1}}},\sqrt{-{\frac{t_{2}}{t_{4}}}}{\ \frac{t-t_{4}}{t-t_{2}}}}})
{r_c}= \min \{{\left\|{\left[{\Delta A,\Delta B}\right]}\right\|_0}:\Delta A \in{{\mathbb R}^{n \times n}},\Delta B \in{{\mathbb R}^{n \times m}},(A + \Delta A,B + \Delta B){\kern 1pt}{\kern 1pt}{\kern 1pt}{\rm{is}}{\kern 1pt}{\kern 1pt}{\kern 1pt}{\rm{controllable}}\}.
\lim_{p\rightarrow\infty}\sum_{j=1}^p(\epsilon_j -\tan \epsilon_j) >\sum_{j=1}^{j^*-1}(\epsilon_j -\tan \epsilon_j)+ (1-\tan 1)\left(\frac{\chi}{\pi}\right)^3\lim_{p\rightarrow\infty}\sum_{j=j*}^p \frac{1}{(j-1)^3}.
P=\left[ \begin{array}{rr}-1 -2 -1 2 1 2 1 -2 -1 0 1 0 \end{array}\right] \mbox{and}\Omega^0 =\left[ \begin{array}{rr}1 0 1 0 0 1 0 1 -2 0 0 -2 \end{array}\right]\left[ \begin{array}{rrrrrr}1 1 0 0 -2 0 0 0 1 1 0 -2 \end{array}\right].
A: = f(1,1)^2-g(1,1)^2=\left(f(1,1)+g(1,1)\right)(f(1,1)-g(1,1)), B_1: = f(1,\omega)f(1,\omega^2)- g(1,\omega)g(1,\omega^2), B_2: = f(\omega,1)f(\omega^2,1)- g(\omega,1)g(\omega^2,1), B_3: = f(\omega,\omega)f(\omega^2,\omega^2)- g(\omega,\omega)g(\omega^2,\omega^2), B_4: = f(\omega,\omega^2)f(\omega^2,\omega)- g(\omega,\omega^2)g(\omega^2,\omega).
\begin{array}{l l c l l}{}&{V_{3}}&{\stackrel{C_{3}^{1 2}}{\longrightarrow}}&{V_{1}\otimes V_{2}}&{}\\{\Pi_{3}}&{\downarrow}&{}&{\downarrow}&{\Pi_{1}\otimes \Pi_{2}}\\{}&{V_{3}}&{\stackrel{C_{3}^{1 2}}{\longrightarrow}}&{V_{1}\otimes V_{2}}&{}\\ \end{array}
\bigl \{A_{\mu}( \vec{x}, t ) , \dot{A}^{\nu}( \vec{y}, t ) \bigr \}= - \, \Bigl ( \delta_{\mu}^{\nu}+ ( \alpha - 1 ) \, \eta_{0 \mu}\delta_{0}^{\nu}\Bigr ) \, \delta ( \vec{x}- \vec{y}) \, .
\lim\limits_{j\rightarrow\infty}\ell_j^{pb_{j-1}+1}=\lim\limits_{j\rightarrow\infty}\ell_j^{\beta_j}\leqslant\lim\limits_{j\rightarrow\infty}\exp\left(B_0\,(pq)^{j/2}\log\left(1+(pq)^{-j/2}\right)\right)=\exp\left(B_0(pq)^{1/2}\right),
\tilde L_1^{(m)}\tilde u_2=0\ \ \mbox{in}\ \ \hat\sigma_0^+, \tilde u_2\big|_{x_0=t_1+\delta}=\frac{\partial\tilde u_2}{\partial x_0}\Big|_{x_0=t_1+\delta}=0, \tilde u_2\big|_{(t_1+\delta,t_2-\delta)\times\gamma_0}=f,
2\left\langle \nabla f,\nabla \left( \nabla^{k}\mathrm{Rm}\right)\right\rangle =-\left( k+2\right) \nabla^{k}\mathrm{Rm}+\sum_{j=0}^{k}\nabla^{j}\mathrm{Rm}\ast \nabla^{k-j}\mathrm{Rm}+2\Delta\nabla^{k}\mathrm{Rm}.
\omega^\alpha_0\textbf{U}^n+\tau^\alpha\textbf{K}\textbf{U}^n=-\sum_{k=1}^{n-1}\omega^\alpha_k\textbf{U}^{n-k}+\sum_{k=0}^{n-1}\omega^\alpha_k\textbf{U}^0+\tau^\alpha\textbf{G}^n,
v^i_2(t,y)=\gamma_i\Psi^{0*}_2\big(\bar{X}(t), y\big)Q^{-1}_i\big(\bar{X}(t)\big)\bigg(\partial_t\psi(t)-A_1\psi(t)-\overline{D_xF}\big(\bar{X}(t)\big)\psi(t)\bigg)
V_1^*V_2(z^n\otimes \xi) =\overline q^{2n}z^n\otimes U^*P^\perp U^*P\xi+\overline q^{2n+2}z^{n+1}\otimes U^*P^\perp U^*P^\perp\xi &\quad+\overline q^{2n-1}z^{n-1}\otimes U^*PU^*P\xi+\overline q^{2n+1}z^n\otimes U^*PU^*P^\perp \xi.
\Psi_{k,t}(x) =P_{l,t}(x)+h_{l,t}(x),\left\vert h_{l,t}(x)\right\vert <1/4, P_{l,t}(x) ={\textstyle\sum\limits_{\left\vert p\right\vert <l}}\left( \Psi_{k,t},e^{i(2\pi p+t)x}\right) e^{i(2\pi p+t)x}
\lim_{n \to +\infty}\frac{\mathrm{Jac}\big(H_{f^{-n}(x),f^{-n}(x')}^u\big)(f^{-n}(q))\cdot \|D f^{-n}(q)|_{E^c}\|}{\mathrm{Jac}\big(H_{f^{-n}(x),f^{-n}(x')}^u\big)(f^{-n}(x))\cdot \|D f^{-n}(x)|_{E^c}\|}=\lim_{n \to +\infty}\frac{\|Df^{-n}(q)|_{E^c}\|}{\|Df^{-n}(x)|_{E^c}\|}dq.
L(|Du|^2/2) = w^{ii}(u_{ki}u_{ki}-u_ku_{kii}- D_{p_l}A_{ii}u_{k}u_{lk}) - B_{p_l}u_{k}u_{kl}= w^{ii}u_{ki}u_{ki}+ u_k[w^{ii}(u_{kii}- D_{p_l}A_{ii}u_{lk})- B_{p_l}u_{kl}].
\ \int_{H^{\pi(d(x))}}\pi_*^{d(x)}\left(x^{-1}\cdot f\middle|_{G^{r(x)}}\right) \, d \lambda_H^{\pi(d(x))}& = \int_{H^{\pi(d(x))}}\pi(x^{-1}) \cdot \pi_*^{r(x)}\left( f \middle|_{G^{r(x)}}\right) \, d \lambda_H^{\pi(d(x))}& = \int_{H^{\pi(r(x))}}\pi_*^{r(x)}\left( f \middle|_{G^{r(x)}}\right) \, d \lambda_H^{\pi(r(x))}\;.
g(R(X,Y)Z,W) = \tilde g( \tilde R(X,Y)Z,W) + \tilde g(h(X,W),h(Y,Z)) & \hskip1.6in - \tilde g(h(X,Z),h(Y,W)), & (\tilde R(X,Y)Z)^\perp =(\bar \nabla_X h)(Y,Z)-(\bar\nabla_Y h)(X,Z),
\zeta^{\prime}\left( 0|A\right) =\frac{1}{4\pi}\int \left[d^{2}z\right] \,tr'[k_{1}\left( z\right)] \qquad ,\qquad \left[ \frac{d}{ds}\zeta^{\prime}\left( s|A\right) \right]_{s=0}=\gamma \zeta^{\prime}\left(0|A\right)
E_{n+1}^{A_2^{(2)}}(x,q,0)=\sum_{k_{22}+k_{12}+k_{11}=n+1}\left(q^{2n+1}c_2(k_{22}-1,k_{12},k_{11})+c_1(k_{22},k_{12}-1,k_{11})\right)x^{k_{11}-k_{22}+1}.
\Omega = -\frac i 2 V \int \frac{d^3 p}{(2\pi)^3}\int\frac{dp_0}{2\pi}\mbox{Tr}\left [M^2_{11}\mbox{Ln}\gamma^0 S^+(p) -M_{12}M_{21}\mbox{Ln}\gamma^0 S^-(p) \right ] - \Delta\Omega.
U^{\mu}= - \Gamma_{\rho \sigma}^{\mu}( \dot{\eta}^{\rho}\dot{\eta}^{\sigma}- \eta^{\rho}\eta^{\sigma}) - 2 \Gamma_{\rho \sigma , \lambda}^{\mu}\dot{q}^{\rho}\eta^{\lambda}\dot{\eta}^{\sigma}- \frac{1}{2}\Gamma_{\rho \sigma , \lambda \delta}^{\mu}\dot{q}^{\rho}\dot{q}^{\sigma}\eta^{\lambda}\eta^{\delta}.
\displaystyle\sum\limits_{j = 0}^{\infty}p (1-p)^j \frac{1}{2}\log(1 + \tilde{g}(j)) & = \displaystyle\sum\limits_{j = 0}^{\infty}p(1-p)^j \frac{1}{2}\log(1 + p(1-p)^j B_{max})
\langle \Psi_0\left(\emph p_{J_{T^*T^*Q}^{-1}(0)}\left(d(\widehat{\varphi}\circ \emph p_{T^*Q})(\alpha_q)\right)\right), T_{\alpha_q}\emph p_{T^*Q}(v_{\alpha_q})\rangle =\langle d(\widehat\varphi \circ \emph p_{T^*Q})(\alpha_q), v_{\alpha_q}\rangle &=\langle d\widehat\varphi(p_{T^*Q}(\alpha_q)), T_{\alpha_q}\emph p_{T^*Q}(v_{\alpha_q})\rangle,
\phi_1({x}) =\int d^n\mu(\mathbf{p}) \overline{\Psi_{\mathbf{x},x_{0}}(\mathbf{p})}a_c(\mathbf{p}) &=(2\pi)^{-n/2}\,\int d^n\mu(\mathbf{p}) e^{i{p}\cdot{x}}\,(2\omega_{p})^{1/2}a_c(\mathbf{p}),
\theta_{L}^{T}\Gamma^{0}\Gamma_{1 1}\Gamma_{M}\theta_{R}= - \theta_{L}^{T}\Gamma_{1 1}\Gamma_{0}\Gamma_{1 1}\Gamma_{M}\Gamma_{1 1}\theta_{R}= - \theta_{L}^{T}\Gamma_{0}\Gamma_{1 1}\Gamma_{M}\theta_{R}
\sum_{n\ge1}\frac{\mu(n)}{n}e^{-t^2/{4n^2}}\cosh(\frac{t}{n})=\sqrt{\pi}e\int_{0}^{\infty}\sum_{n\ge1}\frac{\mu(n)}{n^2}e^{-n^2 y^2}\cos(2n y)\cos(yt)dy.
D^s(fg) =\sum_j \Bigl( \sum_{|\alpha|\le s_1}\frac 1{\alpha!}\partial^{\alpha}f_{\le j-2}D^{s,\alpha}g_j+ \sum_{|\beta|\le s_2}\frac 1{\beta!}\partial^{\beta}g_{\le j-2}D^{s,\beta}f_j \Bigr) + \operatorname{error},
G_{\theta}^{i j}( \nu ) = \delta_{i j}\left( - \frac{\bar{R}_{i}}{\bar{\gamma}_{i}}+ \nu \bar{R}_{i}^{2}\cot ( \nu R_{p}) \right) - ( 1 - \delta_{i j}) \nu \bar{R}_{i}\bar{R}_{j}\csc ( \nu R_{p}) .
\frac{D_{2}}{D_{1}}= e^{- 2 i \pi \mu}e^{4 a L \mu}\frac{\Gamma ( 2 \mu ) \Gamma ( 1 - \mu - \nu - \lambda ) \Gamma ( - \mu - \nu + \lambda )}{\Gamma ( - 2 \mu ) \Gamma ( 1 + \mu - \nu - \lambda ) \Gamma ( \mu - \nu + \lambda )}
P_{1}(x)=P_{1}(x;N,h)=P(x)+(-1)^{h+1}2^{2h-1}\pi^{2h}\sum_{j=1}^{\left\lfloor\frac{h}{N}\right\rfloor}\left(\frac{-1}{4\pi^2}\right)^{jN}\frac{B_{2j}B_{2h-2jN}x^{2j-1}}{(2j)!(2h-2jN)!}.
\langle v,K_{11}u\rangle=\; \frac{1}{2}\langle v\otimes u_1\,,\, w \,u_1\otimes u\rangle \langle v,K_{22}u\rangle=\; \frac{1}{2}\langle v\otimes u_2\,,\,w\,u_2\otimes u\rangle begin{align*}2mm]\langle v,K_{12}u\rangle=\; \langle v\otimes u_1\,,\,w\,u_2\otimes u\rangle.
\begin{array}{c}{\psi_{A}( x ) = \psi_{A}^{\left( - \right)}( x ) + \psi_{A}^{\left( + \right)}( x ) , \quad \psi_{B}( x ) = \psi_{B}^{\left( - \right)}( x ) + \psi_{B}^{\left( + \right)}( x )}\\{\psi ( x ) = \psi_{A}^{\left( - \right)}( x ) + \psi_{B}^{\left( + \right)}( x )}\\ \end{array}
\int \partial_{\lambda}\Bigl( F_1 (\lambda r, \lambda^{-2}z) \Bigr)\Bigr|_{\lambda=1}\frac 1 r \bigl( -\partial_{zz}a_0 - \partial_{rr}a_0 +\frac 1 r \partial_r a_0 \bigr) dr dz = \int F_2(r,z) a_0(r,z)dr dz,
\frac{\partial L_{d}D(p\|q)}{\partial q_{j}}= \left\{\left[\frac{a-1}{a-b}A^{a-2}-\frac{b-1}{a-b}A^{b-2}\right]\frac{\partial A}{\partial q_{j}}\right\}&-\left\{\left[\frac{a-1}{a-b}B^{a-2}-\frac{b-1}{a-b}B^{b-2}\right]\frac{\partial B}{\partial q_{j}}\right\}
\int_{0}^{1}f(t)g_k(t)\,\d t = \int_{\Omega_1}f(t)g_k(t)\,\d t+ \int_{\Omega_2}f(t)g_k(t)\,\d t & \le (i_{\pi(k)}+ \varepsilon) \int_{\Omega_1}f(t)\,\d t+ (i_{\pi(k)}- \varepsilon) \int_{\Omega_2}f(t)\,\d t & \le \frac12(i_{\pi(k)}+ \varepsilon)-\frac12(i_{\pi(k)}- \varepsilon)\le \varepsilon.
F_0(\mathbf{y})=H_0(\mathbf{y})+H_1(\mathbf{y})+H_2(\mathbf{y})+H_3(\mathbf{y}), &F_1(\mathbf{y})=H_0(\mathbf{y})+H_1(\mathbf{y})-H_2(\mathbf{y})-H_3(\mathbf{y}), &F_2(\mathbf{y})=H_0(\mathbf{y})-H_1(\mathbf{y})+H_2(\mathbf{y})-H_3(\mathbf{y}), &F_3(\mathbf{y})=H_0(\mathbf{y})-H_1(\mathbf{y})-H_2(\mathbf{y})+H_3(\mathbf{y}).
y_1'=-y_2-\frac{y_1 y_3}{\sqrt{y_1^2+y_2^2}}\,,~y_2'=y_1-\frac{y_2 y_3}{\sqrt{y_1^2+y_2^2}}\,, ~y_3'=\frac{y_1}{\sqrt{y_1^2+y_2^2}}\,,
\int_{\Omega}\varepsilon \nabla \mathbf{v}_{\varepsilon}\nabla c_{\varepsilon}\otimes \nabla c_{\varepsilon}\, \mathrm{d}x=\int_{\Omega}\Big( \frac{1}{2}\partial_{t}(\mathbf{v}_{\varepsilon})^2+(\nabla \mathbf{v}_{\varepsilon})^2 \Big)\mathrm{d}x.
\tau =\frac{1}{2}\begin{pmatrix}\sqrt{-3}\eta^{-1}-1 -1 \sqrt{-3}\eta \end{pmatrix}, \zeta =\frac{1}{2}\left((\dfrac{z_1}{1-\omega}-\dfrac{z_2}{1-\omega^2})\eta^{-1},\dfrac{z_1}{1-\omega}+\dfrac{z_2}{1-\omega^2}\right),
|\hat{\mathbb{E}}_{\tau+}[X]-\hat{\mathbb{E}}_{\sigma+}[X]| =|\sum_{i=1}^{n}\hat{\mathbb{E}}_{t_i}[X]I_{\{\tau=t_i\}}-\sum_{j=1}^{m}\hat{\mathbb{E}}_{s_j}[X]I_{\{\sigma=s_j\}}| &\leq \sum_{i=1}^{n}\sum_{j=1}^{m}|\hat{\mathbb{E}}_{t_i}[X]-\hat{\mathbb{E}}_{s_j}[X]|I_{\{\tau=t_i\}\cap \{\sigma=s_j\}}.
\left(\begin{array}{cc}L'^2\cdot D_{X^\prime}\xi \cdot L' \cdot D_{X^\prime}\xi \cdot L' \cdot D_{X^\prime}\xi^2 \cdot D_{X^\prime}\end{array}\right)=\left(\begin{array}{cc}2 c_1+3 c_1+3 c_1^2+3c_1-2c_2\end{array}\right).
\hat\jmath^V_{i,i+e_k}(\mu) = \hat\jmath^{\;\! 0}_{i,i+e_k}(\mu)\cosh\bigl(\tfrac 12 \nabla^{i,i+e_k}V(\cdot/L)\bigr) + \hat\chi_{i,i+e_k}^0(\mu)2\sinh\bigl(-\tfrac 12 \nabla^{i,i+e_k}V(\cdot/L)\bigr).
\left. \begin{array}{cc}(\hat{e}_{\hat{\mu}}^{\ \hat{a}})=\left(\begin{array}{cc}e^{-\phi/3}e_{\mu}^{\a}e^{2\phi/3}C^{(1)}_{\mu}0 e^{2\phi/3}\end{array}\right), (\hat{e}^{\ \hat{\mu}}_{\hat{a}})=\left(\begin{array}{cc}e^{\phi/3}e^{\ \mu}_{a}-e^{\phi/3}C^{(1)}_{a}0 e^{-2\phi/3}\end{array}\right).\end{array}\right.
K(x_1,x_2,x_3)^{G_{7,2,3}}=K(s_1,s_2,s_3)^{\langle-I_3\rangle}, K(x_1,x_2,x_3)^{G_{7,3,3}}=K(s_1,s_2,s_3)^{\langle-\beta_3\rangle}, K(x_1,x_2,x_3)^{G_{7,5,3}}=K(s_1,s_2,s_3)^{\langle\beta_3,-I_3\rangle}
K_{s,t,x}' := \{v : s <{\mathcal T}_{t,x_1,v_1}- 2, |v| < \frac{\delta^{-\beta}}2, \check \tau_{t,x}\langle v^\perp \rangle < \langle x^\perp \rangle /5 \}, K''_{s,t,x}:= \{v : s <{\mathcal T}_{t,x_1,v_1}- 3, |v| <\frac{\delta^{-\beta}}3, \check \tau_{t,x}\langle v^\perp \rangle < \langle x^\perp\rangle /6 \}.
z^{(p)}_m-z_m\sim C_m\frac{\Psi_p(z_m)}{\Psi_p(z_{k+1})}\quad C_m=(-1)^{k-m}\frac{T_{1,\ldots,m-1,m+1,\ldots,k+1}}{T_{1,\ldots,k}}(z_{k+1}-z_m)\prod^k_{\substack{i=1 i\neq m}}\frac{z_{k+1}-z_i}{z_m-z_i}.
\mu \left( \left \{y \in Y: \left(\int\limits_0^{T_k}F_{n(T_k)}(g_t y) \, dt \right)^2 \geqslant T_k^{2\alpha}\| f_{n(T_k)}\|_{B}^2\right\}\right) \leqslant 4Q T_k^{1-2\alpha}\log T_k = 4Q \frac{2\alpha \log k}{(2\alpha-1)k^{2\alpha}}
G_{x_{1}x_{2}, y_{1}y_{2}}^{J_{1}J_{2}, K_{1}K_{2}}\Big |_{\mathrm{f r e e}}^{q = 1}= D_{x_{1}x_{2}, y_{1}y_{2}}^{J_{1}J_{2}, K_{1}K_{2}; 0}\, \frac{4 J}{\sqrt{J_{1}J_{2}K_{1}K_{2}}}\, \frac{\sinh \frac{J J_{1}}{2 N}\, \sinh \frac{J_{2}K_{1}}{2 N}\, \sinh \frac{J_{2}K_{2}}{2 N}}{\frac{J^{2}}{2 N}}.
\frac{2 \binom{s- 2}{r- 1}e( \overline{G})}{n - s +2}\geq\frac{2 \binom{s-2}{r-1}}{n}\left( \binom{n}{2}- e(G) \right)\geq\frac{2 \binom{s-2}{r-1}}{n}\left( \frac{n^2}{2 (r - 1)}- \frac{n}{2}- \alpha_{r,s}\right).
\Phi^{-1}((\phi^{-1})^{-1}(L_{(r,A,p')}))=(\phi^{-1})^{-1}(L_{(r,A,p)}), &\Phi^* (\phi^{-1})^*\mathcal{L}_{(r,A,p',q')}\cong (\phi^{-1})^*\mathcal{L}_{(r,A,p,q)}.
\tilde{\delta}_{\mu \nu}= \delta_{\mu \nu}- u_{\mu}u_{\nu}, \qquad \tilde{p}_{\mu}= p_{\mu}- ( u \cdot p ) u_{\mu}, \qquad \overline{{u}}_{\mu}= u_{\mu}- \frac{u \cdot p}{p^{2}}p_{\mu}
\lambda_{top}\leq \tilde{c}\left( \left[ r^{n-2}e^{- \frac{1}{2 \sigma^2}\left(r^2 -1\right)^2}\right]_0^\infty- \int_0^\infty (n-2) r^{n-3}\, e^{- \frac{1}{2 \sigma^2}\left(r^2 -1\right)^2}\, dr \right) &= - \tilde{c}(n-2) \int_0^\infty r^{n-3}\, e^{- \frac{1}{2 \sigma^2}\left(r^2 -1\right)^2}\, dr <0.
{Y_i^{(k)}}(t) ={R_{m_k,i}}(t) - \frac{{{R_{m_k,i}}\left(\tau^{(k)}\right)}}{\hat G_{m_k+1,k}^{\left(\alpha_i^{(k),*}\right)}\left({{\tau^{(k)}}}\right)}\hat G_{m_k+1,k}^{\left(\alpha_i^{(k),*}\right)}\left({{t}}\right)\forall i.
\begin{aligned}u_{-1}- v_{-1}= \frac{(h_0-\delta)^2}{2}(u''_{\delta}-v''_{\delta}) - \frac{(h_0-\delta)^3}{6}(u'''_{\delta}-v'''_{\delta}) + \frac{(h_0-\delta)^4}{24}(u''''_{\delta}-v''''_{\delta}) + \mathcal O((h_0-\delta)^5),\end{aligned}
-dS_1 T^{d-1}+\dots+(-1)^d dS_d =dF =-F\sum_{1\le i\le d}\frac{d{\it pr}^\ast_i t}{T-{\it pr}^\ast_i t}=F\sum_{1\le i\le d}\frac{1}{{\it pr}^\ast_i t}\frac{d{\it pr}^\ast_i t}{1-\frac{T}{{\it pr}^\ast_i t}}=F\sum_{r\ge 0}\omega_{r+1}T^r.
z^n\frac{df^n}{dz^n}+ \left(\sum_{m \geq 0}b_{m,1}z^m\right)z^{n-1}\frac{df^{n-1}}{dz^{n-1}}+ \cdots +\left(\sum_{m \geq 0}b_{m,n}z^m\right)f = 0,
X_{1+2}= \beta_0 + \beta_1 \left( k_1 + k_2 \right)_\mu \left( k_1 + k_2 \right)^\mu + \beta_2\left( \left( k_1 + k_2 \right)_\mu \left( k_1 + k_2 \right)^\mu \right)^2+ \ldots.
\sum_{n=0}^{\infty}\frac{z^nq^{2n^2+2n}}{(q;q^2)_{n+1}(zq;q^2)_{n+1}}=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}b_\omega^{1'}(m,n)z^m q^n =\sum_{n=0}^{\infty}\frac{z^nq^n}{(q;q^2)_{n+1}},
{\left. \frac{\delta \Omega^{\dot{a}\, a}}{\delta C^{\dot{b}\, \alpha}}\right\vert}_{C ={\cal P}= \lambda = 0}= T_{\alpha}\delta^{\dot{a}}{}_{\dot{b}}\, , \quad{\left. \frac{\delta \Omega^{\dot{a}\, a}}{\delta \lambda^{b \, \alpha}}\right\vert}_{C = \lambda = 0}= -{\cal P}_{\alpha}^{\dot{a}}\delta^{a}{}_{b}= -{\cal P}_{\dot{b}\, \alpha}\epsilon^{\dot{b}\dot{a}}\, \delta^{a}{}_{b}\, ,
Q_{I}( \theta ) \rightarrow \mathrm{e}^{i \beta}Q_{I}( \mathrm{e}^{- i \beta}\theta ) , \qquad{\overline{Q}}^{I}( \theta ) \rightarrow \mathrm{e}^{i \beta}{\overline{Q}}^{I}( \mathrm{e}^{- i \beta}\theta ) .
\psi_{k}(x,\bar{x},E, \bar{E}, \mathbf{g}) = \omega^{(n-1)k/2}\psi(\omega^{-k}x,\omega^{k}\bar{x},\omega^{-nkM}E,\omega^{nkM}\bar{E},\mathbf{g}) \,, \ \omega = \mbox{e}^{\frac{2\pi i}{n(M+1)}}\,
\begin{aligned}\langle \mathfrak L_{N,M_N}\sqrt f, \sqrt f \rangle \leq - N^2 D_{M_N}( f) + A_0 |\L_{N,M_N}| \leq - N^2 D_{M_N}^{\widehat b}( f) + A_0 |\L_{N,M_N}|\end{aligned}
c_{2,\tau_2}(\widetilde{Y}_{j_1,j_2}) =\frac{1}{2}f^{(2)}(\tau_2)+c_{3,\tau_2}(\widetilde{Y}_{j_1,j_2})\widetilde{Y}_{j_1,j_2}, c_{2,\tau_3}(\widetilde{Z}_{i,j}) =\frac{1}{2}f^{(2)}(\tau_3)+c_{3,\tau_3}(\widetilde{Z}_{i,j})\widetilde{Z}_{i,j}.
\liminf_{n \to \infty}\mathbb{P}_n \left(\beta_{\hat{\mathbb{M}}_n, n}^{*, (j)}\in \mathrm{CI}_{1-\alpha, \hat{\mathbb{M}}_n}^{(j), \mathrm{hlm}}j = 1, \hdots, m(\hat{\mathbb{M}}_n) \right) \geq 1-\alpha,
[C(x),B(y)] = \mathbf{l}_{\tilde{D}\tilde{D}}(x,y)\tilde{D}(x)\tilde{D}(y)+\mathbf{l}_{AA}(x,y)A(x)A(y)+\mathbf{m}_{AA}(x,y)A(y)A(x) +\mathbf{m}_{A\tilde{D}}(x,y)A(y)\tilde{D}(x)+\mathbf{l}_{A\tilde{D}}(x,y)A(x)\tilde{D}(y)+\mathbf{l}_{\tilde{D}A}(x,y)\tilde{D}(x)A(y)\,,
\int_{\Lambda^\infty}S_\lambda f^{m,v}\overline{S_\lambda f^{m',v}}\, dM =\sum_{\mu \in D_v^J}c^{m,v}_\mu \overline{c^{m',v}_\mu}\rho(\Lambda)^{d(\lambda)}M(Z(\lambda \mu)) &= \sum_{\mu \in D_v^J}c^{m,v}_\mu \overline{c^{m', v}_\mu}\rho(\Lambda)^{-d(\mu)}x^{\Lambda}_{s(\mu)}&= \delta_{m, m'},
\boxed{a_{j_{r_0}}-\sum_t\,{\sf c}_t\,a_1^{r'_\bullet}\overline a_1^{s'_\bullet}a_{i_t}=a_1^{p_\bullet}\overline a_1^{q_\bullet}\big({\sf a}_{j_{r_0}}t_1+{\sf b}_{j_{r_0}}t_2+{\sf a'}_{j_{r_0}}\overline t_1+{\sf b'}_{j_{r_0}}\overline t_2\big)}.
\int_{|\eta| \ge 10}\partial_\zeta G(\zeta,\eta) d\eta = \int_{|\eta| \ge 10}e^{-3i \Phi(\zeta,\eta)}f(\eta) i (\partial_\zeta \Phi) g(\eta- \zeta) d\eta - \int_{|\eta| \ge 10}e^{-3i \Phi(\zeta,\eta)}f(\eta)g'(\eta-\zeta) d\eta
\begin{aligned}f(C'_{t}, h^{\Omega^{\bullet}(Z, F|_{Z})})=\int_{Z}e(TZ, \nabla^{TZ})f(\nabla^{F}, h^{F}) &+ (-1)^{{\rm \mathbf{bd}}}\frac{1}{2}\int_{Y}e(TY, \nabla^{TY})f(\nabla^{F}, h^{F}) +\mathcal{O}(\sqrt{t}).\end{aligned}
z = \frac{1}{\gamma_R}\Big( \left( \left( \alpha_R^2 + \beta_R^2 \right)\left( \gamma_R - 2 \alpha_R \right) - 4 \alpha_R^2 \gamma_R \right) x &\quad-\left( \alpha_R^2 + \beta_R^2 + 2 \alpha_R \gamma_R \right) y + \mu \Big) \;.
\langle \hat{\phi}^2 \rangle={b(m) \over 16\pi^2}~~,~~\langle \hat{\bar{\psi}}\hat{\psi}\rangle=4m\langle \hat{\phi}^2 \rangle~~,~~\langle \hat{V}_\mu\hat{V}^\mu \rangle=3\langle \hat{\phi}^2 \rangle~~~.
U(\xi) =9X, U(\xi X) =-9X+ 252X^2- 891X^3+ 729X^4, U(\xi X^2) =X -378X^2+ 8613X^3 -54675X^4 +138510X^5-150903X^6 &\qquad+ 59049X^7, U(\xi X^3) =147X^2 -14553X^3 +312255X^4 -2617839X^5 +10764414X^6 &-23914845X^7 +29288304X^8 -18600435X^9 + 4782969X^{10}.
V^{0-1-j,l}(u)(\psi):=\sum_{n=1}^{\infty}V^{0-1-j,l,(n)}(u)(\psi), (j=1,2),V^{0-1,l}:=\sum_{j=1}^2V^{0-1-j,l}, &V^{0-2,l}(u)(\psi):=\sum_{n=1}^{\infty}V^{0-2,l,(n)}(u)(\psi)
R_{1234}^2 + 2 R_{1342}R_{1423}= \frac 13 [ 2(x-y)^2 + 2(x-y)(x+2y) - (x+2y)^2 ] \ge \frac 13 [ 2(x-y)^2 + 2(x-y)(K_{14}-K_{13}) - (K_{14}-K_{13})^2 ].
\sqrt{\frac{3}{N}}\frac{1}{2\pi}=\sum_{k=0}^\infty \frac{\left(\frac{1}{2}\right)_k\left(\frac{1}{3}\right)_k\left(\frac{2}{3}\right)_k}{(k!)^3}\left(b^\dagger_N k+a^\dagger_N\right) \left(X^\dagger_N\right)^k,
V( B[f_1, \ldots, f_j]_c^{\textsf{com}}) \underset{\ne}{\subset}V(B[f_1, \ldots, f_j]_{f_j}^{\textsf{com}}) , | V( B[f_1, \ldots, f_j]_{f_j}^{\textsf{com}}) | < | V( B[f_1, \ldots, f_{j-1}]_{f_{j-1}}^{\textsf{com}}) | , | V( B[f_1, \ldots, f_j]_{f_j}^{\textsf{com}}) | = | V( B[f_1, \ldots, f_{j-1}]_{f_j}^{\textsf{com}}) | .
0 ~ = ~ \det \left( \begin{array}{c c}{{2 z - r ( \alpha - 1 ) s ( \alpha - 1 )}}&{{2 z}}\\{{4 z}}&{{-{\mathrm{\small{~ \frac{1}{N}~}}}p ( \beta ) q ( \beta ) - ( \Pi_{B 2}+ \Pi_{C 2})}}\\ \end{array}\right)
\|\rho^n\|_A \leq \mathcal{C}k\sum\limits_{n=1}^{\infty}e^{-\frac{c}{2}n}\left( \left\|R_1^{n-\frac{1}{2}}\right\| + \left\|R_2^{n-\frac{1}{2}}\right\| \right).
M \le\left(\frac{4bd}{\varepsilon}\right)^{bdk}\left( \frac{4k}{\varepsilon}\right)^k \le\left(\frac{4bdk}{\varepsilon}\right)^{bdk}\left( \frac{4bdk}{\varepsilon}\right)^{bdk}= \left( \frac{4bdk}{\varepsilon}\right)^{2bdk}.
\sum_{\boldsymbol{\nu}_a=1}^{n}\nu^m = \sum_{\boldsymbol{\nu}=1}^n \sum_k c_{mk}\psi_m(\nu)\,, = \sum_k c_{mk}\sum_{\boldsymbol{\nu}=1}^n \psi_m(\nu)\,, = \sum_k c_{mk}\psi_m^{(a)}(n)\,.