harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
F_S(x)^\dagger =\frac{1}{2\pi}\left[\int_{-\infty}^\infty (S(k)-{U_0})e^{ikx}dk\right]^\dag =\frac{1}{2\pi}\int_{-\infty}^\infty [S(k)-{U_0}]^\dag e^{-ikx}dk =\frac{1}{2\pi}\int_{-\infty}^\infty [S(-k)-{U_0}]e^{-ikx}dk =\frac{1}{2\pi}\int_{-\infty}^\infty [S(k)-{U_0}]e^{ikx}dk=F_S(x).
f^{( a , b )}\; \in \; L^{N - 1}\Big ( \mathrm{I \hspace{- 0 . 6 2 m m}R}^{D ( n_{a}+ n_{b})}\; , \; \prod_{i = 1}^{n_{a}}d^{D}z_{i}^{( a )}\prod_{j = 1}^{n_{b}}d^{D}z_{j}^{( b )}\Big ) \; \; ,
\Psi = \psi + \psi_{\alpha A}\lambda_{\alpha A}+ \frac{1}{2}\ \psi_{\alpha A \beta B}\lambda_{\alpha A}\lambda_{\beta B}+{\dots}+ \frac{1}{\Lambda !}\ \psi_{\alpha_{1}A_{1}, \dots , \alpha_{\Lambda}A_{\Lambda}}\lambda_{\alpha_{1}A_{2}}\dots \lambda_{\alpha_{\Lambda}A_{\Lambda}}\ .
dx^{1}=\frac{r}{2R_{+}}dr,\;\;\; dy^{1}=R_{-}\theta^{1}, & dx^{2}=-R_{+}\theta^{1},\;\;\; dy^{2}=\frac{r}{2R_{-}}dr, & dx^{3}=-R_{+}\theta^{2},\;\;\; dy^{3}=-R_{-}\theta^{5}, & dx^{4}=-R_{+}\theta^{3},\;\;\; dy^{4}=-R_{-}\theta^{6}, & dx^{5}=-R_{+}\theta^{4},\;\;\; dy^{5}=-R_{-}\theta^{7}.
B(c,Q,a,q,s)=\sum_{n_{1}=0}^{\infty}\ldots \sum_{n_{d}=0}^{\infty}\frac{a^{n_{1}+ \ldots +\n_{d}}}{n_{1}! \ldots n_{d}!}\exp \left( \nu_{c}Q^{d-c}A(c,q,s,\mathbf{n}) - a(c-1)\right).
\frac{q^{\sum_{\alpha < \beta}2 r_{\beta}+ \sum_{\alpha}r_{\alpha}}}{\prod_{j \geq 1}( q )_{\nu_{j}}}= \frac{q^{\sum_{\beta = 1}^{\ell}( 2 \beta - 1 ) r_{\beta}}}{\prod_{j \geq 1}( q )_{\nu_{j}}}= \frac{q^{\sum_{m = 1}^{p - 2}N_{m}^{2}}}{( q )_{N_{1}- N_{2}}( q )_{N_{2}- N_{3}}\dots},
\int_{\{R_1 \le |x| \le R_2\}}D\Phi|_{(g_{\mathbb{E}}, 0)}(g-g_{\mathbb{E}},\pi) \cdot (N, X) \,dx= B^{R_2}_{(g,\pi)}( N, X) - B^{R_1}_{(g,\pi)}(N, X).
g^{-1}ag =g^{-1}[x_1,y_1][x_2,y_2]\cdot \cdot \cdot [x_n,y_n]g = g^{-1}[x_1,y_1]e[x_2,y_2]e\cdot \cdot \cdot e[x_n,y_n]g = g^{-1}[x_1,y_1]gg^{-1}[x_2,y_2]g\cdot \cdot \cdot g^{-1}[x_n,y_n]g =[g^{-1}x_1g,g^{-1}y_1g][g^{-1}x_2g,g^{-1}y_2g]\cdot \cdot \cdot [g^{-1}x_ng,g^{-1}y_ng]\in \mathcal{G}'.
\Big((1-\lambda)I+\lambda A^{-\frac{1}{2}}BA^{-\frac{1}{2}}\Big)-\Big((1-\lambda)I+\lambda A^{\frac{1}{2}}B^{-1}A^{\frac{1}{2}}\Big)^{-1}=\lambda(1-\lambda)(I-A^{-\frac{1}{2}}BA^{-\frac{1}{2}})\Big((1-\lambda)A^{-\frac{1}{2}}BA^{-\frac{1}{2}}+\lambda I\Big)^{-1}(I-A^{-\frac{1}{2}}BA^{-\frac{1}{2}}).
[\nabla u]^p_{W^{\tau,p}(B_{R/4})}\le \frac{(2-\Gamma)^{-p}\,(\Gamma-1)^{-p}}{(\Gamma-1-\tau)\,\tau\,}\,\frac{C}{(\ell_0-p)^p}\,(1-s)\,\frac{\mathcal{A}_R(u,f)}{R^{p\,(1+\tau)}},
\Delta_g u = \nu(g)^{-1}\sum_{1\leq i,j\leq n}\delta_i \left( \sqrt{\nu(g)}g^{ij}\sqrt{\nu(g)}\delta_j(u)\right) & = \sum_{1\leq i,j\leq n}\nu(g)^{-\frac12}g^{ij}\nu(g)^{\frac12}\delta_i \delta_j (u)+ \nu(g)^{-1}\sum_{1\leq i,j\leq n}\delta_i \left( \sqrt{\nu(g)}g^{ij}\sqrt{\nu(g)}\right)\delta_j(u).
R_{\kappa, a^3e^3i^3}=\begin{bmatrix}3 &3 & 3\end{bmatrix}; R_{\kappa, a^3e^2fhi^2}=\begin{bmatrix}3 &2 1 &1 2\end{bmatrix}; R_{\kappa, a^3ef^2h^2i}=\begin{bmatrix}3 &1 2 &2 1\end{bmatrix}.
G_{1,2}^{1,1}\left(\begin{matrix}1 \frac{k}{2},\frac{k+1}{2}\end{matrix}\Big| \frac{1}{X_n}\right) = \frac{\Gamma\left( \frac{k}{2}\right)}{X_n^{\frac{k}{2}}\sqrt{\pi}}{}_1F_{1}\left( \frac{k}{2}; \frac{1}{2}; - \frac{1}{X_n}\right).
\left|\frac{(ae^{i \phi}t;q)_\infty}{(e^{i(\theta + \phi)}t;q)_\infty}\right|=\frac{(ae^{i \phi}t,ae^{-i \phi}t;q)_\infty}{(e^{i(\theta + \phi)}t,e^{-i(\theta + \phi)}t;q)_\infty}=\sum_{n=0}^{\infty}P_n(x;a,\phi|q) t^n,
\widetilde{V}(z) = \sum_{m_1, \ldots, m_{2k+2}\in \mathbb{Z}}y^{\frac{k+1}{2}}e^{2\pi i x(m_1^2 + \cdots m_{2k+1}^2 - m_{2k+2}^2)}e^{-2\pi y(m_1^2 + \cdots + m_{2k+2}^2)}
\pi_{\sigma, \sigma\cap \tau}\circ \rho_u \circ \iota_{\sigma \cap \tau, \sigma}= x^{u}\cdot \prod\limits_{i=0}^{-2u-1}(2\theta_x-i)= \pi_{\tau, \sigma\cap \tau}\circ \delta_u \circ \iota_{\sigma \cap \tau, \tau}
\sqrt{a_{c}b_{c}}\left[{\frac{1}{4 \beta_{c}}}-{\frac{d - 2}{2}}\int_{a_{c}}^{b_{c}}{\frac{d y}{y}}{\frac{\rho_{c}( y )}{\sqrt{( b_{c}+ y ) ( y + a_{c})}}}\right] \; = \; 0 \; .
f^+(x) := \!\! \int_{\underline{x}}^x \!\!| f'(t) | \,\,t = \!\! \int_{\underline{x}}^x\!\!\! | f'(t)| - f'(t) + f'(t)\,\,t & = \!\!\int_{\underline{x}}^x \!\!\!|f'(t)| - f'(t) \,\,t + f(x) - f(\underline{x}) &= f(x) - f(\underline{x}) + \Delta(x),
V \approx \frac{m}{4 \pi}g^{2}\{( 1 + \frac{m^{2}}{8 M^{2}}) \phi + \frac{m^{2}}{4 M^{2}}\mathrm{\boldmath \sigma}^{( 1 )}\cdot \mathrm{\boldmath \sigma}^{( 2 )}\phi \}
\begin{aligned}\widetilde{\Psi}^{(1,r)}(\eta) =\mathcal{L}W^{(p-r)}[\psi](\eta)\ , \widetilde{\Psi}^{(p'-1,r)}(\eta) =\mathcal{L}W^{(p-r)}[\mathcal{L}\chi](\eta)\ , \widetilde{\Psi}^{(s,1)}(\eta) =W^{(s)}[\chi](\eta)\ , \widetilde{\Psi}^{(s,p-1)}(\eta) =W^{(s)}\left[\mathcal{L}\psi\right](\eta)\ .\end{aligned}
0 \leq-\sum^{k-1}_{i=1}i(i-1)k_{i}+(2k-a'-3)\sum^{k-1}_{i=1}ik_{i}-(k-a'-1)(k-1)\sum^{k-1}_{i=1}k_{i}=-((k-1)^{2}-a')(k(k-2)-a')+(2k-a'-3)((k-1)^{2}-a')k-(k-a'-1)(k-1)\sum^{k-1}_{i=1}k_{i}=((k-1)^{2}-a')(k-1)(k-a')-(k-a'-1)(k-1)\sum^{k-1}_{i=1}k_{i}\;.
\widehat{w_h}^{\theta_1,\theta_2}= \theta_1\theta_2w_h^{-,-}+ \theta_1(1-\theta_2)w_h^{-,+}+ \theta_2(1-\theta_1)w_h^{+,-}+ (1-\theta_1)(1-\theta_2)w_h^{+,+}.
f_{R}(r;k)=k(k-1)\int_{-\infty}^{\infty}f_{X}(t+\frac{1}{2}r)f_{X}(t-\frac{1}{2}r)[\int_{t-\frac{1}{2}r}^{t+\frac{1}{2}r}f_{X}(x)dx]^{k-2}dt
\int d^{2}\vec{\vartheta}\,F(\vec{\vartheta})\,e^{i\varphi}(\partial_{\vartheta}+i\cot \vartheta\,\partial_{\varphi})\,G(\vec{\vartheta})=-\int d^{2}\vec{\vartheta}\left[e^{i\varphi}(\partial_{\vartheta}+i\cot \vartheta\,\partial_{\varphi})F(\vec{\vartheta})\right]G(\vec{\vartheta}).
a_ma_n =\frac{1}{2}\left(a_{m+n}+ (a_1^2-b_0^2p)^na_{m-n}\right), b_ma_n =\frac{1}{2}\left(b_{m+n}+ (a_1^2-b_0^2p)^nb_{m-n}\right), pb_mb_n =\frac{1}{2}\left(a_{m+n+2}-(a_1^2-b_0^2p)^{n+1}a_{m-n}\right).
A_G = \left(\begin{array}{cccccc}0 \hspace{1.7mm}1\hspace{1.7mm}\hspace{1.7mm}0\hspace{1.7mm}\hspace{1.7mm}0\hspace{1.7mm}\hspace{1.7mm}0\hspace{1.7mm}\hspace{1.7mm}0\hspace{1.7mm}\noalign{\medskip}0 0 1 0 0 0 \noalign{\medskip}0 0 0 1 0 0 \noalign{\medskip}0 0 0 0 1 0 \noalign{\medskip}0 0 0 0 0 1 \noalign{\medskip}-1 0 0 0 0 0\end{array}\right)
\log^\star(\mathbb E)(a_1\cdots a_n) =\sum\limits_{k=1}^n\frac{(-1)^{k-1}}{k}\sum\limits_{P \in OP^k_n}\mathbb E[a_{P_1}]\cdots \mathbb E[a_{P_k}] &=\sum\limits_{k=1}^n(-1)^{k-1}(k-1)! \sum\limits_{P \in P^k_n}\mathbb E[a_{P_1}]\cdots \mathbb E[a_{P_k}],
|S(\alpha)|^2\ll X^{2n-10}L^{2n-6}\int_{0}^{1}\sum_{|\mathbf{v}|\le X}\big|\mathcal{K}_{0,\mathbf{v}}(\alpha,\gamma)\big|\prod_{j=1}^4\Phi\big(g_j(\mathbf{v}\alpha)\big)\Phi(\gamma)d\gamma.
\frac{\partial v}{\partial\xi_{s+1}}\frac{\partial v}{\partial\xi_{s}}=z_j^2\frac{(1-|w_j|^2)^2}{(1+\overline{w}_jv^\prime)^4}\frac{\partial v^\prime}{\partial\xi_{s+1}}\frac{\partial v^\prime}{\partial\xi_{s}}.
\Delta\rho^\varepsilon(x,t) = \varepsilon^{-2}\partial_r^2\eta_0(r)+\varepsilon^{-1}\{-H\partial_r\eta_0(r)+\partial_r^2\eta_1(r)\}-r|W|^2\partial_r\eta_0(r)+\Delta_\Gamma\eta_0(r)-H\partial_r\eta_1(r)+\partial_r^2\eta_2(r)+O(\varepsilon).
\lim_{N\to\infty}\prod_{1\le i\le n}\frac{\Gamma_{p^{4N},q}\big(t^i\big)}{\Gamma_{p^{4N},q}\big(t^{i-1}q/c^2\big)}K^{(n)}_{p^{2N}c}\big(p^{-N+1}\vec{x};p^{-N+1}\vec{y};q,t;p^{4N}\big){}=\sum_{\mu}c^{|\mu|}\frac{C^0_\mu\big(qt^{n-1}/c^2;q,t\big)C^-_\mu(t;q,t)}{C^0_\mu\big(t^n;q,t\big) C^-_\mu(q;q,t)}P_\mu(p\vec{x};q,t)P_\mu(p\vec{y};q,t)
\mathrm{ML}(R_{n,e,P})=k[x]\hookrightarrow\mathrm{\mathrm{AL}}(R_{n,e,P})=k[x,s]\hookrightarrow\mathrm{\mathrm{AL}}_{d}(R_{n,e,P})=B_{n,P}\hookrightarrow R_{n,e,P}.
\sum_{r,s>0}rsd\overline D_{rs}z^{r+s}\zeta^{-r-s-2}+{d\over 9}{1\over{\zeta^2}}=\sum_{r,s>0}rsd{D_{rs}}^\ast z^{-r-s}\zeta^{r+s-2}+{d\over 9}{1\over{\zeta^2}}\,,
G(s,K) =\frac{\frac{1}{x_0}.\underset{k=0}{\overset{K}{\sum}}\beta_{2.k+1}.\Gamma (s+2.k+1)}{-s.\Gamma (s)}&=-\frac{1}{x_0}.\underset{k=0}{\overset{K}{\sum}}\beta_{2.k+1}.(s+1).(s+2)\ldots (s+2.k)
\langle 0 p_{1}p_{2}p_{3}p_{4}\rangle = \xi_{1 2 3}^{2}\xi_{1 2 4}^{2}F^{p_{1}- 4 \vert p_{2}- 4 \vert p_{3}- 2 \vert p_{4}- 2}( x , u ) + O ( \theta^{5}\bar{\theta}) \; .
\mathcal{Z}(\widehat{\mu}_{\{b_k\},\{D_k\}})=\left\{x \in \mathbb{R}:~ m_{D_{k}}\big(\frac{x}{b_1b_2\cdots b_k}\big)=0 ~k \in \mathbb{N}\right\}=\bigcup_{k=1}^{\infty}b_1b_2\cdots b_k\mathcal{Z}\big(m_{D_k}\big).
{}_{C_0(E^0)}\langle \delta_e,\delta_e\rangle\cdot \delta_f=\delta_{r(e)}\cdot\delta_f=\delta_f\neq 0=\delta_{e,f}\delta_e\cdot\delta_{s(e)}=\delta_e\cdot \langle \delta_e,\delta_f\rangle_{C_0(E^0)}.
\kappa (x)=\sqrt{\left\vert y^{^{\prime \prime}}(x)^{2}-z^{^{\prime \prime}}(x)^{2}\right\vert},\tau (x)=\frac{y^{^{\prime \prime}}(x)z^{^{\prime \prime \prime}}(x)-y^{^{\prime \prime \prime}}(x)z^{^{\prime \prime}}(x)}{\kappa^{2}(x)}.
\Omega(\vartheta(f_{\underline s})) =\int_{N}\Omega_M (f_{\underline s}( \theta_N( w_M^{-1})n))\chi(\theta_N(n))^{-1}\; dn &=\int_{N}\Omega_M (f_{\underline s}( \theta_N( w_M^{-1})n))\chi(n)^{-1}\; dn.
\sum_{\substack{0<\gamma,\gamma'\le T |\gamma-\gamma'|\le \frac{2\pi\lambda}{\log T}}}\!\!\!\!\!\!1 \ ge \, \sum_{0<\gamma,\gamma'\le T}r\Big((\gamma-\gamma') \frac{\log T}{2\pi}\Big) \, w(\gamma-\gamma')= N(T)\int_{\mathbb R}\widehat r(\alpha) \, F(\alpha)\,d\alpha .
J^{\mu\nu}=Eg^{\mu\nu},\ \ \j^{\mu}_1={{\nabla^{\mu}\phi}\over{(\nabla\phi)^2}},\ \ \j^{\mu}_2=j^{\mu}_{\cal R}+V{{\nabla^{\mu}\phi}\over{(\nabla\phi)^2}},\ \ \j^{\mu}_3=Ej^{\mu}_1,
\hat{I}= - \frac{1}{4}\int d^{4}x \; [ F^{\mu \nu}F_{\mu \nu}- \frac{1}{2}\theta^{\alpha \beta}F_{\alpha \beta}F^{\mu \nu}F_{\mu \nu}+ 2 \theta^{\alpha \beta}F_{\alpha \mu}F_{\beta \nu}F^{\mu \nu}] + O ( \theta^{2}) \; ,
F_2(t)= \tfrac{1}{\nu}\left\| w(t)\right\|_2^2+c(1+\alpha)\left(\left\|y_1\right\|_{L^\infty(I;H^3)}+\left\|y_2\right\|_{L^\infty(I;H^3)}\right)\left\|z_k\right\|^2_{L^\infty(I;L^2)}, &G_2=\tfrac{c(1+\alpha)}{\alpha}\left(\left\|y_1\right\|_{L^\infty(I;H^3)}+\left\|y_2\right\|_{L^\infty(I;H^3)}\right).
\begin{array}[c]{rl}\hat{Y}(r)= p(r)\hat{X}(r)+\varphi(r), \hat{Z}(r)= K_{1}(r)\hat{X}(r)+(1-p(r)\sigma_{z}(r))^{-1}[p(r)\sigma_y(r)\varphi(r)+p(r)\varepsilon_2(r)+\nu(r)],\end{array}
\hat{\Gamma}(k,v_1,v_2) = \eta(k) \hat{\Gamma}+ (1-\eta)(k) \hat{\Gamma}=: \hat{\Gamma}_1+\hat{\Gamma}_2 \hat{h}_B(k,v) = \eta(k) \hat{h}_B + (1-\eta)(k) \hat{h}_B =: \hat{h}_{B,1}+\hat{h}_{B,2}.
\gamma^{0}= \left( \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{array}\right) \, , \qquad \gamma^{1}= \left( \begin{array}{c c}{{0}}&{{1}}\\{{- 1}}&{{0}}\\ \end{array}\right) \, , \qquad \gamma^{2}= \left( \begin{array}{c c}{{0}}&{{i}}\\{{i}}&{{0}}\\ \end{array}\right)
\sum_{m}\frac{u_{m}( z ) u_{m}( z^{\prime})}{m^{2}+ p^{2}}\longrightarrow \int_{0}^{\infty}d m \lim_{z_{c}\to \infty}\frac{1}{\pi k e^{- k z_{c}}}\left( \frac{u_{m}( z ) u_{m}( z^{\prime})}{m^{2}+ p^{2}}\right)
J({\bf a}_{i}) = \max_{P_{{\bf X}_i|{\bf A}_{i-1}}({\bf x}_i| y^{i-1})}\{\xi({\bf a}_{i}, \{P_{{\bf X}_{i+1}|{\bf A}_{i}}({\bf x}_{i+1}|{\bf a}_{i}) \}, P_{\bf S}) & + \mathbb{E}\left[ J\left({\bf a}_{i-1}\right) \right] \}.
S_{\mathrm{e f f}}^{d = 2}|_{\mu = 0}={\frac{4}{\left( \beta \sqrt{\pi}\right)^{3}}}\sum_{k = 1}^{\infty}{\frac{{\cal B}_{2 k}( \beta^{2}B / 2 )^{2 k}}{( 2 k ) !}}\sum_{j = 0}^{\infty}{\frac{\left( -{\frac{\beta^{2}m^{2}}{4}}\right)^{j}\Gamma \left( \frac{3}{2}- j - 2 k \right) \eta ( 3 - 2 j - 4 k )}{j !}}
p_{\pm \i \pi}=|\alpha_{\pm \i \pi})(\beta_{\pm \i \pi}| |\alpha_{\pm \i \pi}) = \binom{1}{-\displaystyle\frac{1}{\pm\i \pi-\Omega}|E)}(\beta_{\pm \i \pi}| =\left( 1, (E|\frac{1}{\pm \i \pi-\Omega}\right)
\frac{\|\partial_{y_n}^k (u-u_h)\|_{H^1_0}}{k!}\leq \sum_{l=1}^k \theta_n^k \frac{\|\partial_{y_n}^{k-l}(u-u_h)\|_{H_0^1}}{(k-l)!}+\theta_n^k h \tilde c_4\leq (2\theta_n)^k \frac{1}{2}(\tilde c_4 h + \|u-u_h\|_{H_0^1}) &\leq h (2\theta_n)^k \frac{1}{2}\left(\tilde c_4 + \frac{C_{\mathrm{mesh}}C_{\mathrm{reg}}}{\sqrt{a_{\min}}}\right),
\alpha(\mu,\nu) =\#\{i<j | (\mu_i=0,\mu_j=2), (\nu_i=\nu_j=1) \}, \beta(\mu,\nu) =\#\{i<j | (\mu_i=1,\mu_j=2), (\nu_i=\nu_j=1) \}, \gamma(\mu,\nu) =\#\{i<j | (\mu_i=0,\mu_j=1), (\nu_i=\nu_j=1) \},
(z^2-1)^N\sum_{n\geq 0}\dot{\phi}(n)z^n = \sum_{k=0}^N\sum_{n\geq 0}\tbinom{N}{k}(-1)^k\dot{\phi}(n)z^{n+2N-2k}&= \sum_{k=0}^N\sum_{n\geq 0}\tbinom{N}{k}(-1)^k\dot{\phi}(n-2N+2k)z^{n}&= \sum_{n\geq 0}\mathfrak{d}_2^N\dot{\phi}(n-2N)z^{n}.
ds^2=e^{2A(x)}\sum_{\mu,\nu=0}^{q-1}\eta_{\mu\nu}dy^\mu dy^\nu +\sum_{i=q}^{D-s-3}e^{2F_i(x)}dy^i dy^i +e^{2B(x)}\sum_{\gamma=D-s-2}^{D-1}dx^\gamma dx^\gamma,
|\{u < \frac{M\sigma(r)}{2}\}\cap B_{r/4}(x_1)| \leq \frac{C r^n \sigma(r)}{\left((M+1)\sigma(r)-\frac{M \sigma(r)}{2}\right)^\epsilon}& \leq \frac{C r^n \sigma(r)^{(1-\epsilon)}}{\left(\frac{M}{2}\right)^\epsilon}\leq\frac{Cr^n}{M^{\epsilon}}
\eta\chi(t;\eta) =\frac{(-1)^{p'}}{\beta_{p,p'}}\mathbb{Q}^T(t;\partial_t)\chi(t;\eta)\ , \partial_\eta \chi(t;\eta) =(-1)^{p'}\beta_{p,p'}\mathbb{P}^T(t;\partial_t)\chi(t;\eta)\ ,\partial_\eta\equiv g_s\frac{\partial}{\partial \eta}
\alpha_{j}(A)\equiv \sum_{m_{1},\ldots,m_{j}\neq 0}\epsilon^{m_1+\cdots +m_j}\oint \frac{dt_{1}}{2\pi i}\cdots \oint \frac{dt_{j}}{2\pi i}{\rm e}^{-N(\widetilde{\Phi}_{m_{1}}(t_{1})+\cdots+ \widetilde{\Phi}_{m_{j}}(t_{j}))}M_{12}M_{23}\cdots M_{j1}
\alpha_{n}=\frac{1}{\Gamma(1+\omega/2)}\sqrt{\frac{\Gamma(1+n+\omega/2)}{\Gamma(1+n)}}\approx n^{\omega/4}\beta_{n}=4^{-n}\frac{1}{\sqrt{\Gamma(1+n)\Gamma(1+n+\omega/2)}}\approx \frac{n^{-\omega/4}4^{-n}}{n!}
g^* h^B_{\alpha \beta \gamma}\: = \:\left( h^B_{\alpha \beta \gamma}\right) \,\left( \nu^{F g}_{\alpha \beta \gamma}\right)^{-1}\,\left( \nu^{R g}_{\alpha \beta \gamma}\right) \,\left( \kappa^g_{\alpha \beta}\right) \,\left( \kappa^g_{\beta \gamma}\right) \,\left( \kappa^g_{\gamma \alpha}\right)
L_n\delta_0 =\frac{\lambda}{12\kappa}v^{n+2}\bigg\{n(n+1)(-1)^n\hat{I}_{n,0}\sum_{m=0}^n\binom{n}{m}(-1)^{m}&\qquad\qquad\qquad+\sum_{m=0}^n\sum_{j=1}^{n+2-m}\frac{(-1)^{n-m-j}n!(n+2-m)(n+1-m)(n-j)(n+1-j)}{m!j!(n+2-m-j)!}\hat{I}_{n,j}\bigg\}.
\Delta_{jk}=\frac{2}{3\mathfrak{s}_n(s^n_j)\mathfrak{s}_n(s^n_k)}\sum_{i=1}^nK_h(t_{i-1}-s^n_j)K_h(t_{i-1}-s^n_k)\left\{(X_{t_i}-X_{t_{i-1}})^4-E[(X_{t_i}-X_{t_{i-1}})^4]\right\},
R_0(u) := - \mathcal F_{1/2}^+ (\overline{g_\infty}\, (\mbox{Id}- \Pi ) [\, \cdot] ) + i \mathcal F_{1/2}^+ ( u \overline{g_\infty}\, \partial_x^{-1}\, [\, \cdot] ) : \, H^{s-1}_{r,0}\to \frak h^{s+\frac 12}
F_2=\, b+a_1 b+a_1^3 b+a_1^4 b+b^2+a_1 b^2+a_1^3 b^2+a_1^4 b^2+b^3+a_1^2 b^3+a_1^3b^3+b^4 &+a_1^2 b^4+a_1^3 b^4+b^5+a_1 b^5+a_1^2 b^5+b^6+a_1 b^6+a_1^2 b^6+b^7+b^8\cr +(a_1^3 +a_1^4 +a_1^3b +a_1^3 b^2 +a_1^5 b^2 +a_1^3 b^3 +a_1^4 b^3)k\cr +(a_1^4 +a_1^4 b +a_1^4 b^2 +a_1^6 b^2+a_1^4 b^3)k^2+a_1^7 k^3+a_1^8 k^4,
Q_1(\widetilde{\Sigma}) = \frac{1}{|\widetilde{\Sigma}|^{\frac{n-1}{n}}}\int_{\widetilde{\Sigma}}\widetilde{H}d\widetilde{\mu}= \frac{-\displaystyle\int_{\Sigma}\frac{H}{f^{2(n-1)}}d\mu+\displaystyle\int_{\mathbb{S}^n}\frac{2n}{f^{n-1}}d\mu_{\mathbb{S}^n}}{\left(\displaystyle\int_{\Sigma}\frac{1}{f^{2n}}d\mu\right)^{\frac{n-1}{n}}}
J^{(j)}(z) = \begin{dcases}\begin{pmatrix}1 0 s^z F^{(j)}(z)^{-1}1\end{pmatrix}, z\in\tilde\gamma, \begin{pmatrix}1 - s^{-z}F^{(j)}(z) 0 1\end{pmatrix}, z\in\gamma,\end{dcases}
\left(\rho\circ S\,\otimes\,\theta\circ S\right)(\Delta x) =(\rho\otimes\theta\otimes\omega)(W_{13}^*W_{23}^*) &=(\theta\otimes\rho\otimes\omega)(W_{23}^*W_{13}^*) =(\theta\otimes\rho)\bigl(\Delta(S(x))\bigr),
\zeta_{p}(s)=-\frac{1}{2}\,\frac{1}{1+\frac{1}{\pi p}}+\left(\frac{1}{2}C_{p}^{(2)}-\frac{\gamma}{2}-\frac{e^{2\pi p}}{1+\frac{1}{\pi p}}Q_{2\pi p}(0)-\frac{\log(2\pi)}{2}\right)\,s+O\left(s^{2}\right)
\begin{pmatrix}u^{\infty}_{pp}(\hat{x}) u^{\infty}_{sp}(\hat{x})\end{pmatrix}= \sqrt{\frac{2}{\pi}}^{i\frac{\pi}{4}}iR \sum_{n \in \mathbb{Z}}Q^{-1}\mathcal{H}_n^{-1}\begin{pmatrix}J^{\prime}_n(t_p) \frac{in}{t_p}J_n(t_p)\end{pmatrix}^{in \theta}.
\tilde{r}_{a}( \Delta , h ) = \lim_{\stackrel{N \to \infty}{h \; \mathrm{f i x e d}}}\frac{\tilde{m}_{a}}{\tilde{m}_{F}}= \lim_{\stackrel{N \to \infty}{h \; \mathrm{f i x e d}}}\frac{\tilde{G}_{a}}{\tilde{G}_{F}}\, , \qquad a \in \{B 1 , B 2 , C \}
\psi_w(t)=at^{2-q}\|w\|^2+\epsilon t^{4-q}\|w\|^{4}-t^{2^*_\alpha-q}\int_{\Omega\times\{0\}}|w(z,0)|^{2^*_\alpha}dz.\;\textrm{Then}\; &\psi_w^{\prime}(t)=a(2-q)t^{1-q}\|w\|^2+\epsilon (4-q)t^{3-q}\|w\|^{4}-(2^*_\alpha-q)t^{2^*_\alpha-1-q}\int_{\Omega\times\{0\}}|w(z,0)|^{2^*_\alpha}dz.
\hat{H}(\xi) = \frac{\hat h(\xi)}{(i\xi)^4}= \frac{1}{|\xi|^4}\left(\exp(-|\xi|) - \left(1- |\xi| +\frac{3}{2}|\xi|^2 - \frac{7}{6}|\xi|^3\right) \exp(-|\xi|^2)\right).
C.(r(h)+\frac{i}{2n}(C.r(h)))= C.r(h)+\frac{i}{2n}(C^{2}r(h)) \equiv C.r(h)+\frac{i}{2n}(-4n^{2}r(h)) \equiv (C.r(h)-i(2nr(h))) \equiv 2ni\big(r(h)+\frac{i}{2n}C.r(h)\big)
N_1(x,y) \le (xy)^{-2\beta}\sup_{0<s \le 1/4}s^{-1/2}\big(x+\sqrt{s}\big)^{2\beta-1}\exp\bigg(-\frac{(x \vee y)^2}{512s}\bigg)e^{x^2}& \lesssim (xy)^{-2\beta}(x \vee y)^{2\beta-2}e^{x^2}= (x \wedge y)^{-2\beta}(x \vee y)^{-2}e^{x^2}.
Y_0 = \begin{pmatrix}\cos \phi_{L0}\sin \phi_{L0}\end{pmatrix}s_0 \begin{pmatrix}\cos \phi_{R0}\sin \phi_{R0}\end{pmatrix}\in \pi^{-1}(\Omega), Y_1 = I(Y_0, \Phi(Y_0) - Y_0) = \begin{pmatrix}\cos \phi_{L1}\sin \phi_{L1}\end{pmatrix}s_0 \begin{pmatrix}\cos \phi_{R1}\sin \phi_{R1}\end{pmatrix}.
(\mu\hat\oplus\mu')\left(\begin{array}{c}\mathfrak c \updownarrow \mathfrak c'\end{array}\right)= \mu\left(\begin{array}{c}\mathfrak c \updownarrow \mathfrak c'\end{array}\right)\hat\oplus \mu'\left(\begin{array}{c}\mathfrak c \updownarrow \mathfrak c'\end{array}\right).
\begin{cases}-u_1'' = t \mu_1( 1 - u_1) u_1 + (1-t) \frac{\mu_1}{\alpha^2}\left(\alpha - (\alpha u_{1}- d u_2)^+\right)(\alpha u_{1}- d u_2)^+ - k \omega u_1 u_2 -d u_2'' = t \mu_2( 1 - u_2) u_2 + (1-t) \frac{\mu_2}{d^2}\left(d - (\alpha u_{1}- d u_2)^-\right)(\alpha u_{1}- d u_2)^- - \alpha k \omega u_1 u_2 \end{cases}
\chi (\pi_{\boldsymbol{\nu}})=\mathrm{tr}\,g(\mathbf{K}^{\ast}\boldsymbol{\nu}\mathbf{K}+\boldsymbol{\mu}+ \left( \mathbf{K}^{\ast}\mathbf{K}-\mathbf{I}_{B}\right) /2)-\mathrm{tr}g(\boldsymbol{\mu}+\left(\mathbf{K}^{\ast}\mathbf{K}-\mathbf{I}_{B}\right) /2).
V_a(0) \leq \int_{|x| \leq \sqrt{R}}|x|^2|u_0(x)|^2 dx + \int_{\sqrt{R}\leq |x| \leq 2R}|x|^2 |u_0(x)|^2 dx + R^2 \int_{|x|\geq 2R}|u_0(x)|^2 dx & \leq Rm_0^2+ R^2 \int_{|x| \geq \sqrt{R}}|u_0(x)|^2 dx + R^2 \int_{|x|\geq 2R}|u_0(x)|^2 dx & = o_R(1) R^2.
\liminf_{k\to\infty}\Upsilon_{\delta}(z^{\ell(k)-j})\ge\lim_{k\to\infty}\Upsilon_{\delta}(z^{\ell(k)}) \ \{\rm and}\ \ \lim_{k\rightarrow\infty}\|z^{\ell(k)-j}-z^{\ell(k)-j-1}\|=0.
E f(x) = \int_S e^{2 \pi i x \cdot \xi}f(\xi) d\sigma(\xi)\sim \int_{D(1)}e^{2 \pi i(x_1\xi_1+x_2\xi_2+x_3 \xi_1\xi_2)}\tilde f(\xi_1,\xi_2) d \xi_1 d\xi_2
S_{m a s s i v e}= S + \int d x^{+}d x^{-}\, \mathrm{t r}\left[ - \frac{1}{2}m_{b}^{2}\phi^{2}+ \frac{i}{2}m_{f}^{2}\psi \frac{1}{\partial_{-}}\psi + m_{f}g \psi \frac{1}{\partial_{-}}[ \phi , \psi ] \right] .
\int_M K_{ij}\delta(i_{\nabla f}H)_{ij}e^{-f}dV_g =-\int_M K_{ij}h_{lk}H_{kij}\nabla_lfe^{-f}dV_g +\int_M K_{ij}[\nabla_l\phi H_{lij}+\nabla_l f (dK)_{lij}]e^{-f}dV_g &=-\int_M K_{ij}h_{lk}H_{kij}\nabla_lfe^{-f}dV_g-\frac{1}{3}\int_M \langle dK,H \rangle\phi e^{-f}dV_g+\int_M (dK)_{lij}K_{ij}\nabla_l fe^{-f}dV_g
0<\frac{\bar{u}_{\bar{t}}}{\bar{u}}=\frac{1}{(1-C_3)\bar{t}+\frac{N}{l_i}-(1-C_3)b_i}&\le\frac{1}{(1-C_3)\bar{t}+\frac{N}{l_i}-\frac{2(1-C_3)}{l_i}}&\le\frac1{(1-C_3)\bar{t}}.
\mathcal{E}_{AB}=\max_{|j-k|\geq\iota}\left|\frac{1}{n}\sum^{r}_{i=1}\{(\widetilde{A}_{i,jk}-\widehat{A}_{i,jk})^2+(\widetilde{B}_{i,jk}-\widehat{B}_{i,jk})^2\}\right|,
\hat{\Omega}|\delta_x) = |\delta_x) (\delta_x|\hat{\Omega}= (\delta_x| \hat{\Omega}\frac{\mathcal{P}}{x - \Omega}|E) = - |\hat{E}) + x\frac{\mathcal{P}}{x \Omega}|E) (E|\frac{\mathcal{P}}{x - \Omega}\hat{\Omega}= - (\hat{E}| + x(E|\frac{\mathcal{P}}{x - \Omega}
t_1\bullet(t_2\bullet\Phi)\quad\stackrel{\mathcal{D}}=(t_1t_2)\bullet\Phi\quad\stackrel{\mathcal{D}}=t_2\bullet(t_1\bullet\Phi), t\bullet(\Phi_1+\Phi_2)\quad\stackrel{\mathcal{D}}=t\bullet\Phi_1+t\bullet\Phi_2,
\Pi^m_{\pm q}={\cal D}_{\pm q}X^m-i\left({\cal D}_{\pm q}\Theta^{\alpha 1}\sigma^m_{\alpha \beta}\Theta^{\beta 1}+{\cal D}_{\pm q}\Theta_{\alpha}^2\tilde\sigma^{m\alpha \beta}\Theta_{\beta}^2\right)=0,
\int_s^{t-v}w(u,u+v) du = \sum_{i=0}^{M-1}\int_{t_i}^{t_{i+1}}w(u,u+v) du \leq \sum_{i=1}^M (t_{i+1}- t_i) \sup_{u \in [t_i,t_{i+1}]}w(u,u+v) &\leq v \sum_{i=1}^M w(t_i,t_{i+2}) \leq 2 v w(s,t)
+\frac{m}{m_0^2}\left[\left( \kappa_1 -\kappa_2-\kappa_3\right){\cal F}_{\mu\nu}D_\alpha\varphi_\alpha-\left(\kappa_1 -\kappa_2 +\kappa_3\right)\widetilde{{\cal F}}_{\mu\nu}D_\alpha\widetilde{\varphi}_\alpha\right]
\mathrm{d}X_{t}=\beta X_{t}\mathrm{d}t+CC^{T}\nabla\log\phi(X_{t})\mathrm{d}t+C\mathrm{d}B_{t}, \mathrm{d}Y_{t}=GX_{t}\mathrm{d}t+\Gamma\mathrm{d}W_{t},
\dim \delta H_3^g = \; 6 + 9 k + (k-1)k +{3\,k^2}+(k-1)k^2/2 \; -{(k+2)^2(k+3)/2}= \; 0, \dim \delta E_3^g = \; I_M(V_3^g\times W_3^g)=\; 0.
\begin{array}{ll}D=1: \chi_y(E)=0\,, begin{align*}2mm]D=2: \chi_y(E)=2(1+10y+y^2)(1-y)^{r-2}\,, begin{align*}2mm]D=3: \chi_y(E)=-\chi(E)y(1+y)(1-y)^{r-3}\,, begin{align*}2mm]D=4: \chi_y(E)=(2+(2r-8-\chi(E))y begin{align*}2mm] \hspace{2cm}+(8r+12-4\chi(E))y^2+(2r-8-\chi(E))y^3+2y^4)(1-y)^{r-4}\,, begin{align*}2mm]D=5: \chi_y(E)=-\chi(E)y(1+y)(1+10y+y^2)(1-y)^{r-5}\,,\end{array}
X_3=\textbf{BR}_{\nu}\left(\textbf{g}\left(l+m\sum^{\infty}_{n=0}\frac{m^n}{(n+1)!}\left[\frac{d^n}{dh^n}\textbf{BR}_{\nu,n\lambda}(\textbf{g}(h))\right]_{h=l}\right)\right)
\tau(s_{\mu,\nu})\xi_m=\rho(a_{\mu,\nu})\xi_m+\sum_{j=1}^{r_{\mu,\nu}}t(x^{(\mu,\nu)}_{j,1}) t(x^{(\mu,\nu)}_{j,2})\cdots t(x^{(\mu,\nu)}_{j,N_j})\xi_m
\psi_{m}(x+h,t)-\psi_{m}(x,t)=\int_{x}^{x+h}u_{0}(\xi)\,d\xi-\int_{x}^{x+h}u_{m}(\xi,t)\,d\xi-\int_{0}^{t}\int_{x}^{x+h}u_{m}(\xi,\tau)^p\,d\xi d\tau
\mathcal{Z}^*_K (\underline{\mathrm{pt}^{J}}, \{\emptyset\}) (\underline{\mathrm{pt}^J}, \{\emptyset\}) =\left\{\left({\bigsqcup}_{j_i}\mathrm{pt}, \{\emptyset\}\right) \right\}_{i \in \{1, \ldots, m\}}.
\lambda_{qrr}\chi_{(i\,r)}(g_{(i\,q)}g_{(i\,p)}) =\lambda_{qrr}\xi^2\chi_{r}(g_{(i\,p)})\chi_{(i\,r)}(g_{(i\,p)}) &=\lambda_{qrr}\chi_{r}(g_{(i\,p)})\chi_{qr}(g_{(i\,p)})=\lambda_{qrr}.
c^{-1}m_{c}\leq c^{-1}E(\tilde{u})=\frac{1}{2}\|\Delta_{x,y}u\|_2^2+\frac{\beta}{2}\|\nabla_{x,y}u\|_2^2-\frac{c^{\frac{\alpha}{2}}}{\alpha+2}\|u\|_{\alpha+2}^{\alpha+2}\to-\infty
R_{-}^{+}f_{-}^{+}= e^{+ i \theta / 2}f_{-}^{+}\hspace{0 . 5 c m}R_{+}^{-}f_{+}^{-}= e^{+ i \theta / 2}f_{+}^{-}\hspace{0 . 5 c m}R_{-}^{+}g_{-}^{+}= e^{+ i \theta / 2}g_{-}^{+}\hspace{0 . 5 c m}R_{+}^{-}g_{+}^{-}= e^{+ i \theta / 2}g_{+}^{-},
D(t)=\left(\check\gamma_0\rho_0+\frac{(1-\rho_0)\check\gamma_0F(\check\gamma_0)}{1-\rho_0+\rho_0\check\gamma_0 F(\check{\gamma}_0)}t\right)\left(\check\gamma\rho+\frac{(1-\rho)\check\gamma F(\check\gamma)}{1-\rho+\rho \check\gamma F(\check{\gamma})}t\right)>0.
1 \le |\Delta(P_n)| = |a_n|^{2n-2}|V(P_n)|^2 = |a_n|^{2n-2}|V(\hat P_n)|^2 \prod_{1\le j<k \atop m_n<k\le n}|\alpha_{j,n}-\alpha_{k,n}|^2 \le |a_n|^{2n-2}|V(\hat P_n)|^2 \prod_{m_n<k\le n}(2|\alpha_{k,n}|)^{2(n-1)}\le |V(\hat P_n)|^2 4^{(n-1)(n-m_n)}\left(|a_n| \prod_{m_n<k\le n}|\alpha_{k,n}|\right)^{2(n-1)},
b_{j+1}-b_j =(x-y)^2\cdot(z-w)\cdot z^j\cdot w^{n-2-j}=(x-y)\cdot(z-w)\cdot x\cdot z^j\cdot w^{n-2-j}-(x-y)\cdot(z-w)\cdot y\cdot z^j\cdot w^{n-2-j}.
\begin{aligned}\int_{\Omega_i}e^{-\phi}|s_i|^2 \leq \int_{\Omega_i}e^{-\phi}|\overline\partial\mu_i|^2 &= \int_{B(x_i, \delta_1r_{x_i})\backslash B(x_i, \frac{\delta_1r_{x_i}}{3})}e^{-\phi}|\overline\partial\mu_i|^2 &\leq \exp(-\xi\min\limits_{y\in B(q_i, \frac{\delta_3R}{2})\backslash B(q_i, \frac{\delta_3R}{4})}v_{i, 1}(y))C(n, v).\end{aligned}
\arcsin_p(x) -{\rm arcsinh}_p(x) = \frac{2x^{p+1}}{p(1+p)}\,{}_3F_2\left(\frac{1+p}{2p},\,\frac{1+p}{2p},\,\frac1{2p}+1; \frac32,\,\frac1{2p}+\frac32; x^{2p}\right)\, .
\widetilde{\underline{D}}\;\mathbf{CK}_{x_2}^{\mu_2}\big[x_1^{k}\big]=0,\widetilde{\mathbb{E}}\;\mathbf{CK}_{x_2}^{\mu_2}\big[x_1^{k}\big]=k \;\mathbf{CK}_{x_2}^{\mu_2}\big[x_1^{k}\big],R_1R_2 \;\mathbf{CK}_{x_2}^{\mu_2}\big[x_1^{k}\big]=(-1)^{k}\;\mathbf{CK}_{x_2}^{\mu_2}\big[x_1^{k}\big].
Y_\varepsilon(t)(\omega) = A_\varepsilon(\omega) + \int_0^t \nabla_x b(s, \omega, X(s)(\omega)) Y_\varepsilon(s)(\omega) ds + \int_0^t \nabla_x \sigma(s, \omega, X(s)(\omega)) Y_\varepsilon(s)(\omega) dW(s),