harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{r(r-K)}{2K}+a_2't\frac{r}{K}=y_t'=rv_{3,t+4}+y_t+u_{23}\frac{r}{K}=rv_{3,t+4}+\frac{r(r-K)}{2K}+a_2t\frac{r}{K}+u_{23}\frac{r}{K}
\int_{1\leq |\mu-x|\leq R}\frac{\rho^N(x,t)}{|\mu-x|^2}\,dx\leq & \left(\int_{|\mu-x|\leq R}\ \rho^N(x,t)^{\frac53}\,dx\right)^{\frac35}\left(\int_{1\leq |\mu-x|}\frac{1}{|\mu-x|^{5}}\,dx\right)^{\frac25}
d_{S(G,t)}(xw,xw')> d_{S(G,t-1)}(w,(v_1)^{t-1})+\sum_{i=0}^{\frac{r-4}{2}}d_{S(G,t-1)}((v_{2i+1})^{t-1},(v_{2i+3})^{t-1}) &+d_{S(G,t-1)}((v_{r-1})^{t-1},w') \ge d_{S(G,t-1)}(w,w') ,
\langle 1 | \sigma | 1 \rangle = - \langle 2 | \sigma | 2 \rangle = - m \sum_{j = 0}^{\infty}{\frac{( - 1 )^{j + 1}\Gamma ( j - 1 / 2 )}{2 \sqrt{\pi}j !}}\left({\frac{\Lambda}{m}}\right)^{2 j}\cdotp
\sup_{B_R}\left(\frac{\left\vert \nabla u \right\vert^{2}}{u^{2}}+\lambda a (\log u)^{p}+2\left\vert b\right\vert \right)\leq \,\, n\bigg(\frac{2\lambda}{\lambda-1}K+A+\frac{n\lambda^{2}C_{1}^{2}}{4(\lambda-1)^{2}R^{2}}+\frac{2(5\lambda+4)\left\vert b\right\vert}{\lambda n}& -\frac{4(\lambda-1)aV^{p}}{n}\bigg).
\begin{array}{rcl}\exp\left\{\frac12z\left(\frac1{\sqrt{1-4y}}-1\right)\right\}= \exp\left\{zy{B(y)\over y}\right\}=\displaystyle\sum_{k\geq 0}\frac1{k!}\left({B(y)\over y}\right)^{k}z^{k}y^{k}= \displaystyle\sum_{k,n\geq 0}\frac1{k!}{\binom{2n+1}{n+1}}^{\star k}z^{k}y^{n+k}. \end{array}
\theta(n,k):=\begin{cases}1, \mbox{if}\ \ell(n)\in[0:k-1], 0, \mbox{if}\ \ell(n)\in[k+1:p-1], 1, \mbox{if}\ \ell(n)=k,\,\,k\,\,\mbox{odd}, 0, \mbox{if}\ \ell(n)=k,\,\,k\,\,\mbox{even}, 1, \mbox{if}\ \ell(n)=p.\end{cases}
\sum_{\mu \equiv j}\sum_{\nu \equiv k}u(\xi - \mu)v(\eta - \nu)\sigma_{\mu, \nu}(\xi, \eta) &=\bigg(\sum_{\mu \equiv j}\sum_{\nu \equiv k}\sigma_{\mu, \nu}(\xi, \eta) \bigg)\bigg( \sum_{\mu^{\prime}\equiv j}u(\xi - \mu^{\prime})\bigg)\bigg(\sum_{\nu^{\prime}\equiv k}v(\eta - \nu^{\prime})\bigg)
\left[\begin{array}{l}\displaystyle \sum_{i=1}^p [y^I]_i\nabla^2c_i(x)\xi_1+A^T\xi_2 +{\cal J}c(x)^T\xi_3 begin{align*}4pt]A\xi_1+\xi_2 begin{align*}4pt]{\cal J}c(x)\xi_1+\xi_3+{\cal J}_{y^I}\Psi_{\varepsilon}(y^I,z^I) \xi_4 begin{align*}4pt]-\xi_3+{\cal J}_{z^I}\Psi_{\varepsilon}(y^I,z^I)\xi_4.\end{array}\right]=0.
\Omega_{e}= e^{i \hat{\theta}\, \sigma_{2}}= \left( \begin{array}{c c}{\mathrm{c o s}\, \hat{\theta}}{\mathrm{s i n}\, \hat{\theta}}{- \mathrm{s i n}\, \hat{\theta}}{\mathrm{c o s}\, \hat{\theta}}\end{array}\right) \, .
\underset{p_i,y_i,\tilde y_k}{}\sum^{L}_{i=1}\log(1+Q^+_i)+\sum^K_{k=1}\log\left(1-\frac{1}{Q^-_k}\right) &\quad\; 0\le p_i\le P,\;\; i=1,\ldots,L &\quad\; Q^-_k>0,\;\; k=1,\ldots,K,
x(b) =\left(\frac{C_p}{b_p}\right)^{\frac{2p}{p-2}}\frac{2-b_p^2 p}{p-2}, y(b)=\frac{p\,C_p^2}{p-2}\left(\frac{C_p}{b_p}\right)^{\frac{4}{p-2}}, 0<b_p<1.
F_T(x)=\frac{(1 - x)^2 (1 - 4 x + 6 x^2 - 5 x^3 + x^4)\, C(x)- 1 + 6 x - 14 x^2 + 15 x^3 - 8 x^4 + x^5}{x (1 - 3 x + x^2) (1 - x + x^3)}\,.
\frac{e^{2\alpha t}}{2}\frac{d}{dt}\|\mathbf{u}_t\|_0^2+\mu e^{2\alpha t}|\mathbf{u}_t|_1^2+e^{2\alpha t}a_1(\mathbf{u}_t,\mathbf{u},\mathbf{u}_t)+e^{2\alpha t}J(t;\mathbf{u}_t,\mathbf{u}_t) &\qquad=-\rho t^{-\beta}e^{-(\delta-2\alpha) t}a(\mathbf{u}(0),\mathbf{u}_t)+e^{2\alpha t}(\mathbf{f}_t(t),\mathbf{u}_t).
\left<W(C)\right>_{\rm weak-field}=\left<W(C)\right>_{a_\mu}\int{\cal D}h_{\mu\nu}\exp\left[-\int d^4x\left(\frac{\Lambda^2}{24\zeta}H_{\mu\nu\lambda}^2+\frac14 h_{\mu\nu}^2\right)+\frac{ig}{4}\int\limits_{\Sigma_{\rm min.}}^{}d\sigma_{\mu\nu}h_{\mu\nu}\right],
\nabla^2\ln|\Psi_p|^2=\sigma_p\frac{2}{\hbar c}\left\{\sum_{p'}\beta_{pp'}|\Psi_{p'}|^2-e_pB^{\rm ex}\right\}+4\pi\sum_{r=1}^{n_p}\delta^2({\bf r}-{\bf R}_r^{(p)})\,, \hspace{3mm}(p=1,\ldots,M)
\#\{|\underline{a}_2|<|e_2k'|\colon \underline{a}_2/r_2' \in L(d_2\underline{c})\}\leq \left(\frac{|e_2k'|}{|r_2'd_2^{-1}|}\right)^{2}|r_2'd_2^{-1}| = |e_2k'|^2|r_2'|^{-1}|d_2|.
1+\overline{w}z \overline{f(w,\mu)}f(z,\lambda)+\overline{\mu}\lambda\langle v_{z,\lambda}, v_{w,\mu}\rangle_{\cal{H}_M}= \left\langle\begin{pmatrix}1 zf(z,\lambda) \lambda v_{z,\lambda}\end{pmatrix},\begin{pmatrix}1 wf(w,\mu) \mu v_{w,\mu}\end{pmatrix}\right\rangle_{\mathbb{C}^2\oplus\cal{H}_M}
f_1^{\prime}(\alpha)= \frac{1}{2}\textbf{Tr}[\psi^{\prime}(v+\alpha d_x) \diamond d_x+ \psi^{\prime}(v+\alpha d_s) \diamond d_s] , & f_1^{\prime\prime}(\alpha)= \frac{1}{2}\textbf{Tr}[(d_x\circ d_x) \diamond \psi^{\prime\prime}(v+\alpha d_x) + (d_s \circ d_s) \diamond \psi^{\prime\prime}(v+\alpha d_s)].
\underline{{h s_{\bf C}( 2 , 2 )}}: f = i \alpha + \alpha_{i \bar{j}}a^{i}\bar{a}^{\bar{j}}+ i \alpha_{i_{1}i_{2}\bar{j}_{1}\bar{j}_{2}}a^{i_{1}}a^{i_{2}}\bar{a}^{\bar{j}_{1}}\bar{a}^{\bar{j}_{2}}+ \ldots
d s_{1 1}^{2}= H^{- 2 / 3}( r ) \left( - f ( r ) d t^{2}+ d x_{1}^{2}+ d x_{2}^{2}\right) + H^{1 / 3}( r ) \left( \frac{d r^{2}}{f ( r )}+ r^{2}d \Omega_{7}^{2}\right) \, ,
\overline{A}^{(N-1,M)}=\Bigl(A^{(N-1,M)}_{mn}\Bigr)_{N-1\ge m,n\ge M-1},\quad\underline{A}^{(N,M+1)}=\Bigl(A^{(N,M+1)}_{mn}\Bigr)_{N+1\ge m,n\ge M+1}.
\mathbf{E}_\mu \big\|A-\widetilde{\sigma}\circ\xi\big\|^2_F = \mathbf{E}_\mu \big\|A - (\mathbf{E}_{\mu_z}A)\circ\xi\big\|_F^2 + \mathbf{E}_\mu \big\| \big(\widetilde{\sigma}- \mathbf{E}_{\mu_z}A\big)\circ \xi\big\|_F^2 \,,
\int_T^{2T}\prod_{i=1}^r (\cos( (t+\gamma)\log p_i))^{a_i}\prod_{r+1\le i\le s}(\cos(2(t+\gamma)\log p_i))^{a_i}dt &=(T+\gamma)g(n) +O(|\gamma|) + O( (p_1^{a_1}\cdots p_r^{a_r}) \cdot (p_{r+1}^{2a_{r+1}}\cdots p_{s}^{2a_{s}})),
f(x) = q \Big( q \big(\sqrt{\gamma}x - 2 \big) \big( 2 \gamma x^2 + \sqrt{\gamma}x - 9 \gamma - 1 \big) + 9 ( \gamma^{\frac{3}{2}}x + \gamma ) \Big)^2 - 4 \Big( q \gamma (x^2 -3) - q \sqrt{\gamma}x + q + 3 \gamma \big)^3\Big)
\int_0^{1/2}|g(x)|^Kdx= \int_{\mathcal{A}}|g(x)|^Kdx+\sum_{j\leq j_0}\int_{\mathcal{E}_1(K,j,n)}|g(x)|^Kdx &+\sum_{j\leq j_0}\int_{\mathcal{E}_2(K,j,n)}|g(x)|^Kdx+\int_{x_0}^{1/2}|g(x)|^Kdx.
{\mathrm d}\overline{X}_{t, j}\, =\, b( t, \overline{X}_{t, j}, u \cdot \delta_{\overline{X}_{t, j+1}}+ ( 1- u) \cdot \mathcal L_{\overline{X}_{t, j}}){\mathrm d}t +{\mathrm d}W_{t, j}\, ; t \ge 0 \,
4\frac{(q)_\infty^4(-q)_\infty^2}{[x, -xq, -x]_\infty}= \frac{(q)_\infty^2}{(-q)_\infty^2}\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}q^{n(3n+1)/2}}{1-xq^n}-\sum_{n=-\infty}^{\infty}\frac{q^{n(3n+1)/2}[1+3n(1+xq^n)]}{(1+xq^n)^2}&-\sum_{n=-\infty}^{\infty}\frac{q^{n(3n+1)/2}[1+(3n-1)(1+xq^n)]}{(1+xq^n)^2}.
\lim_{t\rightarrow\infty}\frac{\ell_R(R/I(1)_{tn_1}\cdots I(r)_{tn_r})}{t^d}= \lim_{t\rightarrow\infty}\frac{\ell_R(R/I(1)_{tn_1}\cdots I(s)_{tn_s})}{t^d}
V(\alpha_2,\alpha_3,\beta_2,\beta_3)=\begin{pmatrix}1 x_1 x_2 x_3 0 1 0 0 0 \alpha_2 x_2+\alpha_3 x_3 1 0 0 \beta_2 x_2+\beta_3 x_3 0 1\end{pmatrix}.
|N_h(t)| = y^{-1}\int_{t-y}^t |N_h(u)| du + O_B\left(\frac{M_{|g|}(t)}{\log^2 t}\right) &\leq y^{-1}\int_{t-y}^t du \left(\sum_{d \leq u}\Lambda(d)|g(d)||M_h(u/d)| + |A|\mu \sum_{d \leq u}\Lambda(d)|g(d)|M_{|g|}(u/d)\right) + O_B\left(\frac{M_{|g|}(t)}{\log^2 t}\right).
\begin{aligned}g(u) = \g(u_0) + \int_{0}^\infty g'(u_{<k}) u_k \, dk = \g(u_0) + \int_{0}^\infty g'(u_{0}) u_k \, dk + \iint_{0<k_2<k_1}g''(u_{<k_2}) u_{k_2}u_{k_1}\, dk_1 dk_2.\end{aligned}
{\frac{M^{2}\pi}{g^{2}N}}= \left( \begin{array}{c c c c}{{9 x}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{{\frac{8 1}{4}}+ 3 x + 6 y}}&{{- 3 i \sqrt{{\frac{y}{2}}}}}&{{-{\frac{3 i}{2}}\sqrt{y}}}\\{{0}}&{{3 i \sqrt{{\frac{y}{2}}}}}&{{{\frac{9}{2}}x}}&{{0}}\\{{0}}&{{{\frac{3 i}{2}}\sqrt{y}}}&{{0}}&{{{\frac{8 1}{4}}+{\frac{9}{2}}y}}\\ \end{array}\right)
\mathcal{N}_p^{\alpha}= \begin{cases}\mathcal{N}_p \alpha = 0 \mathbb{C}T_pS_{\mathcal{N}^{\alpha - 1}}\cap \mathcal{N}_p \alpha \hbox{successor ordinal}\bigcap\limits_{\beta < \alpha}\mathbb{C}T_pS_{\mathcal{N}^{\beta}}\cap \mathcal{N}_p \alpha \hbox{limit ordinal}\end{cases}
\xi_m(t) e^{t/2}- \frac 1 2 \int_0^t \xi_m(\tau) e^{\tau/2}d \tau +\int_0^t \chi_m(\tau) e^{\tau/2}d \tau\leq C |\epsilon| \int_0^t \chi(\tau) e^{\tau/2}d \tau + \xi_m(0).
\delta \left[ f \left[ A \right] \right]{\cal M}\left( x , y ; a , b \right) = - \delta \left[ f \left[ A \right] \right] \left( \partial_{3}^{y}\delta^{a b}- g \epsilon^{c}f^{a b c}A_{3}^{c}\left( y_{\perp}\right) \right) \delta^{4}\left( x - y \right) \ .
\tilde{\rho}_{\mathrm{s}}( x ) \; = \;{\frac{1}{8 \pi^{2}\beta}}\int_{a^{2}}^{b^{2}}{\frac{d y}{x + y}}\sqrt{{\frac{x ( b^{2}- x ) ( y + a^{2})}{y ( b^{2}+ y ) ( x - a^{2})}}}\; .
\mathtt{X}(Q,\mathtt{P})\;=\;Q^{\perp}\mathtt{P}Q+Q\mathtt{P}^*Q^{\perp}\,, &\mathtt{Y}(Q,\mathtt{P})\;=\;Q^{\perp}\mathtt{P}Q\mathtt{P}^*Q^{\perp}-Q\mathtt{P}^*Q^{\perp}\mathtt{P}Q+\frac{1}{2}\left[Q^{\perp}\mathtt{P}(Q^{\perp}-Q)\mathtt{P}Q+Q\mathtt{P}^*(Q^{\perp}-Q)\mathtt{P}^*Q^{\perp}\right]\,.
\begin{array}{ll}\sum_{j+k=s}F_j(F_k(X), \alpha(y_1), \cdots, \alpha(y_{n-1})) & = \sum_{i=1}^n \sum_{j+k=s}F_j(\alpha(x_1), \ldots, \alpha(x_{i-1}), F_k(x_i, y_1, \ldots, y_{n-1}), \alpha(x_{i+1}), \ldots, \alpha(x_{n})))\end{array}
\eta(\eta(\tau_j, \tau_i),\eta(\tau_j, \tau_\ell)) =\eta((\tau_n^q\tau_1)^{n-i+1}, (\tau_n^q\tau_1)^{n-j}\tau_n^{q-1+D(n-j+\ell)}\tau_{n-j+\ell+1}) =\eta(1,(\tau_n^q\tau_1)^{i-j-1}\tau_n^{q-1+D(n-j+\ell)}\tau_{n-j+\ell+1}) =(\tau_n^q\tau_1)^{i-j-1}\tau_n^{q-1+D(n-j+\ell)}\tau_{n-j+\ell+1}.
\left\{(t_1,\ldots, t_{n-1}, g, \bar{g})~\begin{tabular}{|l}$g\bar{g}=g_{t_i}, h\bar{h}=h_i$ $ h\in c(g, X), \bar{h}\in c(\bar{g}, X), g_{t_i}\in c'(h_i, Y)$ $[t_1,\ldots, t_{i-1}, g,\bar{g}, t_{i+1},\ldots, t_{n-1}]\neq \emptyset$\end{tabular}\right\}.
{\cal L}= \sum_{k = 1}^{2}[ \overline{{\psi}}_{k}{\gamma}^{\mu}( i{\hbar}c{\partial}_{\mu}- e_{k}A_{\mu}){\psi}_{k}- m_{k}c^{2}\overline{{\psi}}_{k}{\psi}_{k}] - \frac{1}{4}F_{\mu \nu}F^{\mu \nu},
2 i ( U^{\mathrm{T}}W )_{I J}+ ( U^{\mathrm{T}}V )_{I}{}^{K}F_{K J}+ F_{I K}( Z^{\mathrm{T}}W )^{K}{}_{J}- \textstyle{\frac{i}{2}}F_{I K}( Z^{\mathrm{T}}V )^{K L}F_{L J}
\sum_{n=1}^{\infty}\sigma_{k}(n) e^{-n y}= \left( \frac{2\pi}{y}\right)^{k+1}\sum_{n=1}^{\infty}(-1)^{\frac{k+1}{2}}\sigma_{k}(n) e^{-\frac{4 \pi^2 n}{y}}+ \mathcal{P}(y),
T:= \lim_{\epsilon\to 0}M_{\epsilon}(z^{\ell},w^m)\wedge\bigwedge_{j=1}^m \check{k}_j(w^{j-1},w^j) \wedge \omega(w^0) \wedge \bigwedge_{j=1}^{\ell}k_j(z^j,z^{j-1}),
M_{r-1}^{r-1}=\int_{\Omega}(-\ln u)^{r-1}\leq\left( \int_{\Omega}1\right)^{\frac{1}{r}}\left( \int_{\Omega}(-\ln u)^{\left( r-1\right)\frac{r}{r-1}}\right)^{\frac{r-1}{r}}& =\left\vert \Omega\right\vert^{\frac{1}{r}}M_{r}^{r-1}
_{+}^{-}{\cal D}_{n n^{\prime}}= \int{}\{}_{+}^{-}\phi_{p_{-}, n , + 1 , r}^{\dagger}( x ) \{}_{+}^{-}\phi_{p_{-}, n^{\prime}, + 1 , r}( x ){}d x^{2}d x^{4}. . . d x^{2 \left[ ( d - 1 ) / 2 \right]}, \; \mp \pi_{-}< 0 .
\left(\alpha(q)\alpha(q^{11})\right)^{1/4}+\left((1-\alpha(q))(1-\alpha(q^{11}))\right)^{1/4}&+2\left(16\alpha(q)\alpha(q^{11})(1-\alpha(q))(1-\alpha(q^{11}))\right)^{1/12}=1.
0 = 3cws + 3swc +3cw - 3wc + 3c + \frac{1}{2}x^3wu + \frac{1}{2}uwx^3w + \frac{1}{2}u^{-1}wx^3 & - \frac{1}{2}wx^3wu^{-1}+ \frac{1}{4}x^4wu + \frac{1}{4}wx^4wu^{-1}- \frac{1}{4}u^{-1}wx^4 + \frac{1}{4}uwx^4w
\lambda_{(j-1)k+1}-\lambda_{(j-1)k+2}=\phi(a_{1,j})-\phi(a_{2,j}) &=\sum\limits_{s=1}^ka_{s,j}-\left(\sum\limits_{s=2}^ka_{s,j}+a_{1,j+1}\right) &=a_{1,j}-a_{1,j+1}.
|K_1(t;x_j)+\phi_1(t;x_j)| \leq\epsilon^2m^2(t)\leq4\epsilon^2v_1^2(t;x_j) &<\frac{\gamma}{2}v_1^2(t;x_j)~~{\rm for~any}~~t\in[t_1,t_2]~~{\rm and}~~j=1,2.
\bar{E}_{n}\Bigl |_{n \to \infty}\sim \frac{c \xi^{2}}{4 \pi a^{2}}\left[ \frac{1 0 - 3 \xi^{2}}{9 6 0 n^{2}}- \frac{2 8 2 2 4 - 7 3 4 4 \xi^{2}+ 7 2 0 \xi^{4}}{1 5 4 8 2 8 8 0 n^{4}}+ O \left( \frac{1}{n^{6}}\right) \right] \,{.}
\langle \hat{\bf p},j_3,[{\mathsf s},j],n |S| \hat{\bf p}',j_3',[{\mathsf s}',j'],n' \rangle = 2\hat{E}(\hat{p})\delta^3 (\hat{p}-\hat{p}') \delta({\mathsf s}-{\mathsf s}') \delta_{j_3j_3'}\delta_{jj'}\langle n \| S_j({\mathsf s})\| n '\rangle \, ,
W_{A B C}= \frac{1}{2}T_{\underline{{{A}}}\ \ \ \underline{{{B}}}\dot{A}\, u n d e r l i n e{C}}^{\ \ \dot{A}}, \, \, W_{\dot{A}\dot{B}\dot{C}}= - \frac{1}{2}T_{\ \ \underline{{{\dot{A}}}}A \underline{{{\dot{B}}}}\underline{{{\dot{C}}}}}^{A}
\lambda_{1,m}= \frac{2}{\pi}\sum_{j=1}^{m-1}\int_{-\pi/2}^{\pi/2}\left(\frac{(n-1)(n-3)}{4}a_{m-j-1}x^{2(m-j-1)}-\lambda_{1,m-j}\right)\cos(x)y_{1,j}.
A^{-1}=[\begin{matrix}a&b&c\\d&e&f\\g&h&i\\ \end{matrix}]^{-1}=\frac{1}{det(A)}[\begin{matrix}A&B&C\\D&E&F\\G&H&I\\ \end{matrix}]^{T}=\frac{1}{det(A)}[\begin{matrix}A&D&G\\B&E&H\\C&F&I\\ \end{matrix}]
\begin{cases}\ddot{\mu_\delta}+\lambda_\delta^{(2)}(x(t))\mu_\delta=0; \mu_\delta(0)=0; \dot{\mu_\delta}(0)=1, \ddot{\nu_\delta}+\lambda_\delta^{(2)}(x(t))\nu_\delta=0; \nu_\delta(0)=1; \dot{\nu_\delta}(0)=0. \end{cases}
\widetilde A \widetilde D - \widetilde B \widetilde C = 1, \widetilde{E}<\widetilde{A}+\widetilde{C}, \widetilde{F}<\widetilde{B}+\widetilde{D}, -\widetilde C \leq \widetilde C \widetilde F - \widetilde D \widetilde E < \widetilde D.
I = \frac{1}{1 6 \pi G}\int d^{1 0}x \sqrt{g}\left[ e^{- 2 \phi}\Big ( R + 4 \partial_{\mu}\phi \partial^{\mu}\phi - \frac{1}{1 2}H_{3}^{2}\Big ) - \sum_{p}\frac{1}{2 ( p + 2 ) !}F_{p + 2}^{2}\right] + I_{W Z}
q^{\alpha^{2}/2}S_{n}\left(xq^{\alpha-1/2};q\right)=\int_{-\infty}^{\infty}\frac{\left(xe^{iy};q\right)_{n}}{\left(q;q\right)_{n}}\frac{\exp\left(y^{2}/\log q^{2}+i\alpha y\right)dy}{\sqrt{\pi\log q^{-2}}}.
\frac{n\mathrm{Ric}(\omega_{\phi})\wedge \omega_{\phi}^{n-1}}{\omega_{\phi}^n}=\mathrm{Tr}((\tilde{u}_{l,m}(a))(u^{l,m}(a))) +\sum_{\alpha\in \Phi_{Q^u}\cup\Phi_s^+}\frac{-\tilde{u}_l(a)\alpha^{\vee,l}}{(2\chi-p)(\alpha^{\vee})}+\sum_{\alpha\in \Phi_{Q^u}}\frac{2\chi^{ac}(\alpha^{\vee})}{(2\chi-p)(\alpha^{\vee})}
\Psi^{(1)}(\eta_1,\eta_2):=\lim_{M\to\infty}N_1^{1/3}Q_1'(0) &=\sum\limits_{r,s,t=0}^\infty\frac 1{(r!)^2s!t!}\int\limits_{(-\infty,0]^r}d^rx\int\limits_{(-\infty,0]^s}d^sx'\int\limits_{[0,\infty)^r}d^ry\int\limits_{[0,\infty)^t}d^ty' W_{r,s,r,t}^{(1)}(\mathbf{x},\mathbf{x'},\mathbf{y},\mathbf{y'}),
L_{(\tilde u,0)}(u,v)\, =\left(\begin{array}{cc}\Delta +(\sigma-2\tilde u) -\tilde u 0 -(-\Delta)^s +(\sigma-\tilde u) \end{array}\right)\left(\begin{array}{c}u v\end{array}\right) &=\left(\begin{array}{c}\Delta u+(\sigma-2\tilde u)u -\tilde u v -(-\Delta)^s v +(\sigma-\tilde u)v\end{array}\right)\,,
{\cal{T}}_{P_{n+n+1}}(X_{n+k+1}) = \beta^{k(n-k)}\prod_{a=1}^{n}x_a^{\min(n-a+1,n-k)}\mathfrak{G}_{w_{k}^{n}}^{\beta^{-1}}\big(x_1^{-1},\dots,x_{n}^{-1}\big).
E(Az)v^c=\left(\begin{array}{ccc}E(\lambda_{1}z)v_{1}\cdots E(\lambda_{n}z)v_n\end{array}\right)\left(\begin{array}{ccc}\widetilde{v}_{1}\cdots \widetilde{v}_n\end{array}\right)\left(\begin{array}{c}v_{1}^c \vdots v_n^c\end{array}\right),
G^{q}( r , \tau ; r^{\prime}, \tau^{\prime}) = \frac 1{8 \pi^{2}r r^{\prime}}\sum_{l = | q |}^{\infty}( 2 l + 1 ) Q_{\lambda_{l}- 1 / 2}\left( \frac{\alpha^{2}\Delta \tau^{2}+ r^{2}+ r^{2}}{2 r r^{\prime}}\right) \ ,
\left( \begin{array}{l}{{T_{\alpha{\dot{\alpha}}}}}\\{{W_{\alpha}}}\\{{\tau}}\\ \end{array}\right) \to \left( \begin{array}{r l}{{T_{\alpha{\dot{\alpha}}}^{n e w}=}}&{{{\widetilde T}_{\alpha{\dot{\alpha}}}- \frac{1}{3}[ D_{\alpha}, \bar{D}_{\dot{\alpha}}] U}}\\{{W_{\alpha}^{n e w}=}}&{{\frac{1}{2}{\bar{D}}^{2}D_{\alpha}\U}}\\{{}}&{{0}}\\ \end{array}\right) \ ,
d_\pm= \frac{i}{k_z r_0}\left[ a_\pm -\left(\omega \pm \frac12 \right) \right], \quad a_\pm =\textrm{sg}\left(\omega \pm \frac12 \right) \sqrt{\left( k_z r_0 \right)^2 +\left(\omega \pm \frac12 \right)^2}
\lambda \psi_{\kappa}= \ \inf_{v_2 \in V_2}\Big[ \langle b(x, \hat v_1(\theta, x), v_2) , \nabla_x \psi_{\kappa}\rangle+ r_{\kappa}(\theta, x, v_2) \Big] + \frac{1}{2}{\rm trace}(a(x) \nabla^2_x \psi_{\kappa}),
s_{ij}^{2}=1, s_{ij}s_{kl}=s_{kl}s_{ij}{\rm if}\{i,j\}\cap \{k,l \}= \varnothing, s_{ij}s_{ik}=s_{jk}s_{ij}=s_{ik}s_{jk}, s_{ik}s_{ij}=s_{ij}s_{jk}= s_{jk}s_{ik}, i < j < k.
\lim_{\lambda \rightarrow \pm \infty}e^{\mp (2\mathsf{N}+4)\lambda}\mathbf{A}(\lambda ) =2^{-2(\mathsf{N}+1)}\frac{\kappa_{+}\kappa_{-}\exp \pm(\alpha_{+}+\alpha_{-}-\beta_{+}+\beta_{-}+(\mathsf{N}-1)\eta )}{\sinh\zeta_{+}\sinh \zeta_{-}}, \lim_{\lambda \rightarrow \pm \infty}e^{\mp 2\mathsf{N}\lambda}Q(\lambda) =1.
\partial_{t}^{2}f - r^{- 1}\partial_{r^{*}}\left( r^{2}\partial_{r^{*}}{\frac{f}{r}}\right) + \left( 1 -{\frac{2 M}{r}}\right) \left( \frac{l \left( l + 1 \right)}{r^{2}}+ m^{2}\right) f = 0 \; .
\begin{gathered}S(t,x,y;u_0) \equiv \sum_{l=1}^{+\infty}\frac{1}{2\pi}\int_{\mathbb R}\, e^{it(\xi^3-b\xi+\lambda_l\xi)}e^{i\xi x}\widehat{u}_0(\xi,l)\,d\xi \psi_l(y), K(t,x,y;f)\equiv\int^t_0 S(t-\tau,x,y;f(\tau,\cdot,\cdot))\,d\tau,\end{gathered}
s_{b,k}= \sum_{a=1}^{b-1}\sum_{a_1,a_2,\ldots,a_k}\frac{1}{(aa_1a_2\cdots a_{k-2}a)_b}\leq \sum_{a=1}^{b-1}\frac{b^{\left\lceil \frac{k-2}{2}\right\rceil}}{a(b^{k-1}+1)}= \frac{b^{\left\lceil \frac{k-2}{2}\right\rceil}x}{b^{k-1}+1}.
\| Q_{<k-Cm}P_k \mathcal N(Q_{> k-Cm}\phi_{k_1},\bar \phi_{k_2})\|_{N[I]}\lesssim \ 2^{Cm}\| \chi_I^k \mathcal N(Q_{> k-Cm}\phi_{k_1},\bar \phi_{k_2})\|_{L^1 L^2}\lesssim \ 2^{Cm}\| \Box \phi_{k_1}\|_{L^2}\| \chi_I^k \phi_{k_2}\|_{L^2 L^\infty}
f ( u ) \! \! = \! \! \prod_{j = 1}^{N}\frac{\sin ( u - \lambda \theta_{j})}{\sin ( u - \lambda \theta_{j}- \mu )}, \qquad Q ( u ) \! \! = \! \! \prod_{k = 1}^{N / 2}\sin \left( u - \frac{\mu}{2}- \lambda \alpha_{k}\right)
u(x,t)= \sum_{p=0}^\infty \Big[ e^{(T-t)p^{2\beta}}\Big<u_T, \phi_p \Big> -\int_t^T e^{(s-t)p^{2\beta}}g_p(s)ds-\int_t^T e^{(s-t)p^{2\beta}}F_p(u)(s)ds \Big] \phi_p(x)
\max_{P(t), \t \in [t_1,t_2]}\int_{t_1}^{t_2}\frac{1}{2}\log\left(1 + P(\tau)\right)d\tau \mbox{s.t.}\int_{t_1}^{t_2}a e^{b \tau}P(\tau) d \tau = T_{\delta}e^{bt_2}- T_g &\int_{0}^{t}P(\tau) d\tau \leq \sum_{i=0}^{\tilde{h}(t)}\tilde{E}_i, \forall t \in [t_1,t_2]
x_{-}= \left( \begin{array}{c c c}{{0}}&{{0}}&{{0}}\\{{X_{-}}}&{{0}}&{{0}}\\{{0}}&{{- X_{-}^{T}}}&{{0}}\\ \end{array}\right) , \qquad x_{+}= \left( \begin{array}{c c c}{{0}}&{{X_{+}}}&{{0}}\\{{0}}&{{0}}&{{- X_{+}^{T}}}\\{{0}}&{{0}}&{{0}}\\ \end{array}\right) ,
H \frac{R^{(1 - s) p}}{(R - r)^p}\| (u - k)_- \|_{L^p(x_0 - R, x_0 + R)}^p = H \frac{R^{(1 - s) p}}{(R - r)^p}k^p \int_{x_0 - R}^0 dx \ge H (R - |x_0|)^{1 - s p}k^p.
\sup_{k\in\mathbb{N}}\|\sigma_{1-1/k}g\|_k=\sup_{k\in\mathbb{N}}\biggl(\frac1k\sum_{j=0}^{k-1}|g(\lambda_{k,j})|^2\biggr)^{1/2}\asymp \|\{\langle\widehat{K}_{\lambda_n},g\rangle\}_{n=1}^{\infty}\|_{\infty,2}.
x_{3}^{3}- \tfrac{3}{2}x_{3}(x_{1}^{2}+ x_{2}^{2}) + \tfrac{1}{\sqrt{2}}\cos \theta \left(x_{1}^{3}- 3x_{1}x_{2}^{2}\right)+\tfrac{1}{\sqrt{2}}\sin \theta \left(x_{2}^{3}- 3x_{2}x_{1}^{2}\right).
\big\|u\big\|_{\dot{H}^{\tilde s}_{\mathcal{L}^{\tilde q, r}}}= \big\|\dot{\Lambda}^{\tilde s-s}\dot{\Lambda}^su\big\|_{\mathcal{L}^{\tilde{q},r}}= \big\| |\xi|^{\tilde s-s}\widehat{\dot{\Lambda}^su}(\xi)\big\|_{L^{\tilde{q}',r}},
\omega = \frac{\kappa}{2 l}\biggl ( \frac{l^{2}- r^{2}}{( l^{4}+ r^{4}- 2 l^{2}r^{2}\cos 2 \theta )^{1 / 2}}+ 1 - \cos \theta_{1}+ \cos \theta_{2}\biggr ) d \phi \, .
\frac{\left(\lambda_{1,d}\lambda_{2,d}\right)^{-\frac{1}{2}n\left(n-1\right)}\left(\lambda_{3,d}\lambda_{4,d}\right)^{-\frac{1}{2}n}}{\sqrt{\det\ \bold{C}_{n,d}}}t^{2}= d^{-\frac{n}{2}}\, t^{2-n}+ O\left(t^{4-n}\right).
\dot{x}^{A A^{\prime}}= o^{A}\bar{o}^{A^{\prime}}, \acute{x}^{A A^{\prime}}= o^{A}\bar{\iota}^{A^{\prime}}+ \iota^{A}\bar{o}^{A^{\prime}}, ( \rho^{\tau}o^{A}\bar{o}^{A^{\prime}}) \dot{}={\cal F}^{A A^{\prime}},
C^{'}_{FTN}(\alpha,B) = (1-\alpha)W \log_{2}\left( 1 + \frac{P}{WN_{0}}\right) +2 \int_{0}^{B-W}\log_{2}\left( 1 + \frac{P}{2WN_{0}}\left[1 + \cos\left(\frac{f\pi}{\alpha W}\right)\right]\right) df
-\langle \phi,f'(\phi)\Phi_{\ell}^{0}+ \mathcal{K}*\Phi_{\ell}^{0}- k\mathcal{K}_{1}*\partial_{\theta}\Phi_{\ell}^{0}\rangle = \langle \Psi_{3}^{0},(\mathcal{A}_{1}- c)\Phi_{\ell}^{0}\rangle &= \langle\Psi_{3}^{0},\mathcal{A}_{1}\Phi_{\ell}^{0}\rangle - c\delta_{3\ell},
n ={\frac{\alpha}{\pi}}( x_{0}\Omega )^{2}\int \int d t d t^{\prime}\bigl [{\frac{( s i n t - s i n t^{\prime})^{2}}{( t - t^{\prime}+ i \epsilon )^{4}}}-{\frac{c o s t c o s t^{\prime}}{( t - t^{\prime}+ i \epsilon )^{2}}}\bigr ]
\varepsilon^{( 1 )}( \vec{x}) = \frac{g_{\varepsilon}\lambda^{2}}{4 \pi c \tau_{S}}\, \frac{\cos ( | \vec{x}- \vec{x}_{S}| / \lambda )}{| \vec{x}- \vec{x}_{S}|}\sum_{l \, m_{l}}c_{l \, m_{l}}Y_{l \, m_{l}}( \theta , \phi ) \, .
\int d q_{1}^{\, 3}\, d q_{2}^{\, 3}\left( \frac{\left| \vec{\nabla}\psi \right|^{2}}{q^{2}}+ \frac{\mathrm{i}}{q^{2}}\left( \vec{\nabla}_{2}\, \psi \, \vec{\nabla}_{1}\, \psi^{\ast}- \vec{\nabla}_{1}\, \psi \, \vec{\nabla}_{2}\, \psi^{\ast}\right) \right) < \infty \hspace{1 c m}.
a^{-1}(sin{\theta_n^0})^{m_1+l_1}C_{N-m_1}^{m_1+\frac{n-1}{2}}(cos{\theta_n^0})C_{N'-l_1}^{l_1+\frac{n-1}{2}}(cos{\theta_n^0})\oint_{\partial C}d^{n-1}{\eta}Y_{a}^{m_1}(\eta^{\theta_n^0})Y_{a'}^{l_1}(\eta^{\theta_n^0})
d s^{2}\sim d x_{1 , 3}^{2}+ \alpha^{\prime}g_{s}N \left[ d \rho^{2}+ \rho^{2}\left( d \theta_{1}^{2}+ \sin^{2}\theta_{1}d \phi_{1}^{2}\right) + \sum_{a = 1}^{3}\left( \sigma^{a}- A^{a}\right)^{2}\right]
\mathrm{T r}e^{- \beta P \bar{\cal K}}= - 1 + \left[ \frac{m L}{\sqrt{8 \pi \beta}}+ \frac{1}{\sqrt{\pi}}\sum_{n = 1}^{\infty}\frac{2^{n}( 1 + 2^{2 n - 1})}{( 2 n - 1 ) ! !}\beta^{n - \frac{1}{2}}\right] e^{- 4 \beta}
\mathcal{W}_{x_0}(x)=a(x_0)w_{3/2}\left(\frac{(x-x_0)\cdot \nu_{x_0}}{(\nu_{x_0}\cdot A(x_0)\nu_{x_0})^{1/2}}, \frac{x_{n+1}}{(a^{n+1,n+1}(x_0))^{1/2}}\right),
\Pi_{\infty}\bigl( \langle \mu,m_{t+h}- m_{t}\rangle > 1 + Ch^{1/16}\bigr) \leq C h, &\Pi_{\infty}\biggl( \forall \lambda \leq \langle \mu,m_{t+h}- m_{t}\rangle - Ch^{1/16}, \ \mu \bigl( z_{t-}- m_{t-}\geq 1- \alpha \lambda -C h^{1/16}\bigr)\geq \lambda \biggr) \geq 1 - Ch.
\begin{aligned}13^n_{141}\to/<->/<1em> 14^n_{2551}13^n_{1002}\to/<->/<1em> 14^n_{6487}14^n_{1346}\to/<->/<1em> 14^n_{7711}14^n_{5293}\to/<->/<1em> 14^n_{12516}14^n_{5373}\to/<->/<1em> 14^n_{12516}14^n_{6632}\to/<->/<1em> \overline{14}^n_{21021}14^n_{12393}\to/<->/<1em> 14^n_{12532}\end{aligned}
B(0)^{-\ell}B_1 = \underbrace{(B_1 B_2(0))\cdots (B_1 B_2(0))}_{\ell\ \mbox{times}}\cdot B_1 & = B_1\cdot \underbrace{(B_2(0) B_1)\cdots (B_2(0) B_1)}_{\ell\ \mbox{times}}&= B_1 \cdot B(0)^{\ell}\,.
\begin{array}{c}{{{\hat{x}}}_{j}}={{{\hat{x}}}_{j-1}}+h\cdot(1-{{{\hat{v}}}_{j}})\cdot{{f}_{0}}\left(\frac{{{{\hat{x}}}_{j-1}}+{{{\hat{x}}}_{j}}}{2},{{{\hat{u}}}_{0j}}\right) \ \ \ +h\cdot{{{\hat{v}}}_{j}}\cdot{{f}_{1}}\left(\frac{{{{\hat{x}}}_{j-1}}+{{{\hat{x}}}_{j}}}{2},{{{\hat{u}}}_{1j}}\right)\end{array}
R \ll x^{2r-n}\exp\left\{\sum_{j=1}^r\frac{x^j-1}{j}-\frac{u^{1-4/r}}{4\pi^2(\log u)^2}\left(1-\frac{\pi^2}{\sqrt{u}}-\frac{\pi^2 x^{3r/2+2}}{u^2}\right)_+\right\}.
(\tilde{B}\Sigma_\theta,\Phi) :=b({\Sigma}_\theta, \Phi), (\tilde{A}\Sigma_\theta,\Phi) := a(\Sigma_\theta, \Phi), (\tilde{f},\Phi) :=\left( f(t), \Phi\right) + c\left( \theta, \Phi\right),
\nabla\otimes\nabla := \left( \frac{\partial\}{\partial x_1}\nabla, ..., \frac{\partial\}{\partial x_n}\nabla\right) = \left( \frac{\partial^2\}{\partial x_1^2}, \frac{\partial^2\}{\partial x_1\partial x_2}, ..., \frac{\partial^2\}{\partial x_{n-1}\partial x_n}, \frac{\partial^2\}{\partial x_n^2}\right).
g_{L}^{2}V_{L}( U \pm \omega_{3}) \approx m_{L}^{2}\left( e^{U}+ e^{- U}\right) , \qquad g_{S}^{2}V_{S}( U \pm \omega_{3}) \approx 4 m_{S}^{2}\left( e^{2 U}+ e^{- 2 U}\right) .
s_{4}(x) ={Q'}_{4}(x;-1), s_{31}(x) ={Q'}_{31}(x;-1)+{Q'}_{4}(x;-1), s_{22}(x) ={Q'}_{22}(x;-1)+{Q'}_{31}(x;-1), s_{211}(x) ={Q'}_{211}(x;-1)+{Q'}_{22}(x;-1)+{Q'}_{31}(x;-1)+{Q'}_{4}(x;-1), s_{1111}(x) ={Q'}_{1111}(x;-1)+{Q'}_{211}(x;-1)-{Q'}_{22}(x;-1)+{Q'}_{4}(x;-1).
P_{n}(z;\alpha,\beta)=\frac{\Gamma(2\alpha+1)}{\Gamma(\alpha+\beta+1)\Gamma(\alpha-\beta)}\int_{0}^{1}\left(\frac{z^u-1}{z-1}\right)^{\alpha+\beta}\left(\frac{z-z^u}{z-1}\right)^{\alpha-\beta-1}\frac{e^{n u \ln z}z^u}{z-1}\ln z du.
\overline{\mathcal{L}}(y, p_{_1}(y)) = p_{_1}(y) \log \frac{1}{p_{_1}(y)}+ \eta_0 p_{_0}(y) \log\frac{p_{_0}(y)}{p_{_1}(y)}&~~~~+ \eta_1 p_{_1}(y) + \eta_2 y^2 p_{_1}(y),
\begin{cases}\frac{\partial \varphi}{\partial t}=\log\frac{\omega_{\varphi}^{n}}{\omega_{0}^{n}}+F_{0}+\beta(k|s|_{h}^{2}+\varphi)+\log|s|_{h}^{2(1-\beta)}\varphi|_{t=0}=c_{0}, \end{cases}
\frac{1+\epsilon_1}{\epsilon_1\phi_i}= \frac{1+\epsilon_1}{\epsilon_1\min\left\{\frac{\bar{\xi}_2}{\epsilon_2}, \cdots, \frac{\bar{\xi}_{M_s}}{\epsilon_{M_s}}\right\}}\geq \frac{1+\epsilon_1}{\epsilon_1 \frac{\bar{\xi}_{M_s}}{\epsilon_{M_s}}}= \frac{\epsilon_{M_s}2^{R_1}}{\epsilon_1 \bar{\xi}_{M_s}},