harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Delta_{\underline x}\big(hP_k\big) =\big(\Delta_{\underline x}h\big)P_k+2\sum_{j=1}^p\big(\partial_{x_j}h\big)\big(\partial_{x_j}P_k\big)+h\big(\Delta_{\underline x}P_k\big) &=\left(\partial_r^2h+\frac{p-1}{r}\,\partial_rh\right)P_k+2\frac{\partial_rh}{r}\sum_{j=1}^px_j\partial_{x_j}P_k=\left(\partial_r^2h+\frac{2k+p-1}{r}\,\partial_rh\right)P_k.
\partial_r H_\phi (x,r) = \frac{m-1}{r}H_\phi (x,r) + 2 D_\phi (x,r) \partial_v H_\phi (x,r) = - 2 \int \phi' \left({\textstyle\frac{|y-x|}{r}}\right) |y-x|^{-1}\sum_{i=1}^Q u_i (y) \cdot \partial_v u_i (y)\, dy\, .
\left|\left(\sqrt{x_ix_j}-\sqrt{x_ix'_j}\right) u_{x_ix_j}(P')\right| \leq \left|\sqrt{x_ix'_j}u_{x_ix_j}(P')\right|= \frac{\left|\sqrt{x_ix'_j}u_{x_ix_j}(P') - 0\right|}{\rho^{\alpha}(P',P^{''})}\rho^{\alpha}(P',P^{''}) &\leq [\sqrt{x_ix_j}u_{x_ix_j}]_{C^{\alpha}_{WF}([0,T]\times\bar S_{n,m})}(2\eta)^{\alpha/2},
\lim_{r \searrow 0}\frac{\mathcal{N}_{q,D_{V_\omega}(b)}^\pm \big( \vert \omega \vert r,\vert \omega \vert r_0 \big)}{\textup{Tr}\, \textbf{\textup{1}}_{[r,\infty)}\big({\bf P}_{q}^\pm \vert W \vert{\bf P}_{q}^\pm \big)}= 1.
\left\{Y_{l_{1}}^{m_{1}},Y_{l_{2}}^{m_{2}}\right\}=\frac{(-1)^{l_{1}+l_{2}}}{\lambda_{l_{1},m_{1}}\lambda_{l_{2},m_{2}}}\left\{J_{-}^{l_{1}-m_{1}}(x+iy)^{l_{1}},J_{-}^{l_{2}-m_{2}}(x+iy)^{l_{2}}\right\}
\frac{\lambda_v^{K_{\ell}}}{\lambda_v^L}=\Bigl\{q_v^{(\dim M_v^{K_{\ell}}-n+1)/2}\cdot|M_v^{K_{\ell}}|^{-1}\cdot\prod_{i=2}^{n}(q_v^i-1)\Bigr\}\Bigl\{q_v^{(\dim M_v^L-n)/2}\cdot|M_v^L|^{-1}\cdot\prod_{i=2}^{n+1}(q_v^i-1)\Bigr\}^{-1}=\frac{q_v^{n_{v,\nu_v}}-1}{q_v^{n+1}-1}.
{\mathbf T}_j(E,\ell)= q^{-(j+1)(\ell+\frac{1}{2})}\mathbf{D}^{+}(E \Omega^{j+1},\ell) \mathbf{D}^{-}(E \Omega^{-(j+1)},\ell) &-q^{(j+1)(\ell+\frac{1}{2})}\mathbf{D}^{-}(E \Omega^{j+1},\ell) \mathbf{D}^{+}(E \Omega^{-(j+1)},\ell) .
-\int_{-\infty}^\infty \frac{v}{y}(c|z|^2-bx+a)e^{-\frac{\pi v}{y}(c|z|^2-bx+a)^2}e^{2\pi i \bar{\tau}(-b^2/4+ac)}e^{2\pi i wa}da &= -y e^{-\pi i \bar{\tau}b^2/2}e^{2\pi i (c\bar{\tau}+w)(bx-c|z|^2)}\,\, \int_{-\infty}^\infty t e^{-\pi t^2}e^{2\pi i t\frac{y}{\sqrt{v}}(c\bar{\tau}+w)}dt,
\left(\begin{matrix}x=0 x=\infty \, \left( \frac13 \right) \overbrace{\begin{matrix}0 0 0 0 t_1 \theta^0 \end{matrix}}\overbrace{\begin{matrix}-1 \frac{t_2}{3}\theta^\infty_1/3 -\omega \frac{\omega^2 t_2}{3}\theta^\infty_1/3 -\omega^2 \frac{\omega t_2}{3}\theta^\infty_1/3 \end{matrix}}\end{matrix}\right) ,
\begin{array}{c l}{{\cal{L}}_{F}( x , e )}&{={\cal{L}}_{F}^{r}( x , r )}\\{}&{= i \overline{{L}}\gamma^{\mu}({\partial}_{\mu}+ L_{\mu}) L + i \overline{{R}}\gamma^{\mu}({\partial}_{\mu}+ R_{\mu}) R + \lambda ( \overline{{L}}\phi{R}+ \overline{{R}}{\phi}^{\dag}L ) .}\\ \end{array}
\phi(D)g_a(z) & =(-1)^{m}\left( \frac{(4a)^{2m}\phi_1(4ai) \cos(4az)}{8}+ \frac{(2a)^{2m}\cosh(2ar)\phi_1(2ai)\cos(2az)}{2}\right) & =(-1)^m 2^{2m-1}a^{2m}\big( 4^{m-1}\phi_1(4ai) \cos(4az) + \cosh(2ar)\phi_1(2ai) \cos(2az)\big).
\begin{array}{c}{{\omega = \mathrm{d}{\bf p}\wedge \mathrm{d}{\bf x}+ \sum_{i = 1}^{N}\mathrm{d}{\bf p}_{i}\wedge \mathrm{d}{\bf e}_{i},}}\\{{{\cal H}={s}\left[{\bf p}{\bf e}_{1}+ \sum_{i = 1}^{N}{k}_{i - 1}{\phi}_{i - 1 . i}+ \frac{{k}_{N}}{2 c}({\Phi}_{N . N}- c^{2}) + \sum_{i , j = 1}^{N}d_{i j}({\bf e}_{i}{\bf e}_{j}- \delta_{i j}) \right] ,}}\\ \end{array}
\bigg\|S^{n_k}\bigg(\sum_{i=0}^{s_{2k+1}-1}x_i e_i\bigg)\bigg\|\leq \bigg\| \sum_{i=0}^{s_{2k+1}-1}\frac{x_i}{2^k}e_i\bigg\| < \frac{1}{2^k}\|x\| <\frac{\epsilon}{2},
\psi_{_{( \alpha )}}^{2}( x ) = ( \frac{\mu}{2 \pi})^{^{1 / 2}}{\cal K}( \phi_{1}) \,{\cal K}( \phi_{3}) \, : e^{- i \sqrt{\pi}\{\int_{- \infty}^{x}d z^{1}\pi_{_2}( z ) + \gamma_{_{\alpha \alpha}}^{5}\phi_{_2}( x ) \}}: ,
\int_v\phi(z,v)\omega(v) =\int_v\phi_0(z,v)\omega(v)-\int_v\int_u \phi_0(z,u)\omega(u)\omega(v)-\int_v\int_u \phi_0(v,u)\omega(u)\omega(v) &=\int_v\phi_0(z,v)\omega(v)-\int_u \phi_0(z,u)\omega(u)-0 &=0.
K_{S}^{(1)}( [\omega_1] , [ \omega_2 ] )=Res \sum_{A=1}^{d}\frac{1/2 (P_2{\partial \over \partial X^A}P_1 -P_1{\partial \over \partial X^A}P_2)dX^1 \ldots dX^d}{\frac{\partial W}{\partial X^A}\prod_{B=1}^{d}\frac{\partial W}{\partial X^B}}
F_{~ ~ ~ ~ \mu \nu ~ \alpha \beta}^{- 1 \sigma ~ ~ \rho}F_{\rho ~ ~ \omega}^{~ \alpha \beta ~ \eta \kappa}= \frac{1}{2}\delta_{\omega}^{\sigma}\left( \delta_{\mu}^{\eta}\delta_{\nu}^{\kappa}- \delta_{\mu}^{\kappa}\delta_{\nu}^{\eta}\right)
\left\Vert \widetilde{V}\right\Vert_{H^{1}\left( \Omega_{d+c}\right)}^{2}\leq C_{2}\left( 1+\left\Vert \widetilde{V}\right\Vert_{H^{2}\left(\Omega \right)}^{2}\right) \sigma^{2\rho},\rho =c/\left(m+c\right) .\square
y(x)x^{l_k+\frac{m}{2}+\lambda-i-1}= &\sum_{n=0}^{\infty}\frac{(l_k+\frac{m}{2}+\lambda-i+1+n-1)\cdots(l_k+\frac{m}{2}+\lambda+n-1)}{(l_k+\frac{m}{2}+\lambda-i)\cdots(l_k+\frac{m}{2}+\lambda-1)n!}x^{n+l_k+\frac{m}{2}+\lambda-i-1}.
[X,Y]_{x}=\frac{1}{2}\frac{d^{2}}{dt^{2}}|_{t=0}(\Phi_{-t}^{Y}\circ\Phi_{-t}^{X}\circ\Phi_{t}^{Y}\circ\Phi_{t}^{X})(x)=\frac{d}{dt}|_{t=0}(\Phi_{-\sqrt{t}}^{Y}\circ\Phi_{-\sqrt{t}}^{X}\circ\Phi_{\sqrt{t}}^{Y}\circ\Phi_{\sqrt{t}}^{X})(x).
\eta_(A) = \sum_{1 \leq i' \leq n}\sum_{j=d_{i'-1,1},d_{i'-1,2},\ldots,d_{i'-1,i'-1}}\left(\sum_{1 \leq j' \leq j}a_{i'j'}\right) &= \sum_{1 \leq \tilde{j}\leq i' \leq n-1}\left(\sum_{1 \leq j' \leq d_{i'\tilde{j}}}a_{i'+1,j'}\right) &= \sum_{1 \leq j \leq i \leq n-1}\left(\sum_{1 \leq j' \leq d_{ij}}a_{i+1,j'}\right).
\lim_{\beta \rightarrow \infty}\;\lim_{m \rightarrow 0^-}\langle{\bar{\psi}}\psi \rangle_{\beta}=\lim_{\beta \rightarrow \infty}\;\lim_{m \rightarrow 0^+}\langle \bar{\psi}\psi \rangle_{\beta}= \frac{eB}{4 \pi}\; .
f_0=1,f_{i+1}=zf_i+zg_i,i\ge0, g_i=zf_{i+1}+zg_{i+1}+zh_{i+1},i\ge1, g_0=zg_{i+1}+zh_{i+1},i\ge1, h_i=zh_{i+1}+zg_{i+1},i\ge0.
\lim_{k\to\infty}\frac{|\delta^p_{\epsilon_k}B_p\setminus E|}{|\delta^p_{\epsilon_k}|}\ge \lim_{k\to\infty}\frac{|\delta^p_{\epsilon_k}B(x_k,a)|}{|\delta^p_{\epsilon_k}B_p|}= \lim_{k\to\infty}\frac{|B(x_k,a)|}{|B_p|}>0,
0 =H^0(A,a_*(\Omega_X^1\otimes P^{\vee})\otimes F^{e,*}\widehat{L}\otimes Q_0) &=H^0(X,\Omega_X^1\otimes P^{\vee}\otimes a^*(F^{e,*}\widehat{L}\otimes Q_0)) &=H^0(X,\Omega_X^1\otimes P^{\vee}\otimes a^*(F^{e,*}\widehat{L}\otimes Q)),
\lim_{h\to\infty}\frac{P_{s_h}(E_h)}{P_{s_h}(B)}=1\,,D_{s_h}(E_h)\le\frac{|E_h\Delta B|^2}{M|B|^2}\,,\lim_{h\to\infty}{\boldsymbol\alpha}(E_h)=0\,.
\breve{M}^r(x,t,0)=I+\frac{\mathfrak{B}^r(\xi,t)}{k_0}+\frac{\overline{\mathfrak{B}^r(\xi,t)}}{k_0}+R(\xi,t), \breve{M}^r(x,t,i\kappa)=I+\frac{\mathfrak{B}^r(\xi,t)}{k_0+i\kappa}+\frac{\overline{\mathfrak{B}^r(\xi,t)}}{k_0-i\kappa}+R(\xi,t).
{\cal O}_{\pm}^{a}( \{x \}) : = \frac{1}{( N ! )^{2}}\sum_{\pi , \pi^{\prime}}\prod_{b = 1}^{N}\overline{{\psi}}^{( b )}( x^{( \pi ( b ) )}) P_{\pm}\psi^{( b )}( x^{( \pi^{\prime}( b ) )}) .
(n-1) \tau_{1}^{(n-1)}-\frac{1}{4}\sum_{l=1}^{n}\partial_{2}^{2}\tau_{2}^{(n-l-2)}(b_{2}^{(l)}-\tilde{b}_{2}^{(l)}) +\frac{1}{4}\sum_{l=2}^{n-1}\partial_{2}\tau_{2}^{(n-l-1)}\partial_{2}(b_{2}^{(l-1)}-\tilde{b}_{2}^{(l-1)}) -\frac{1}{4}\sum_{l=2}^{n-1}\tau_{2}^{(n-l-1)}\partial_{2}^{2}(b_{2}^{(l-1)}-\tilde{b}_{2}^{(l-1)})=0
\pi_*((N^{(1)})^2. N^{(2)}) = \frac1{d^2 \Delta_d}\pi_* \tilde\tau_*((\tilde\tau^*N^{(1)})^2\,.\, \tilde\tau^*N^{(2)}) & = \frac1{d^2 \Delta_d}\tau_* (\varpi \times \varpi)_*\left(\sum_{\mu \in T}\sum_{\nu \in T}(\mu_* N_{X(d)}^{(1)})^2 \,. \,\nu_* N_{X(d)}^{(2)}\right) &= \frac{d^2}{\Delta_d}\tau_* (-\lambda^{(1)}_{X(d)}) \, = \, \, - d^2\lambda^{(1)}
S_{f e r m i o n i c}= \frac{1}{2}\sqrt{d e t ( B )}( 2 \pi )^{d / 2}\int d^{d}x \, \psi \ast \sigma^{j}[ \nabla_{j}, \psi ]_{\ast}+ \psi \ast \sigma^{I}[ \phi_{I}, \psi ]_{\ast}\, .
\begin{aligned}\nabla p+2\nabla\cdot(\mu\;\varepsilon(\b u))-\rho\b u_{tt}=\b F \b u(x,t) = \b K \b u|_{\partial\Omega}= 0 \b u|_{\Omega\times\{t=0\}}= \b g \partial_t \b u|_{\Omega\times\{t=0\}}=\b h \end{aligned}
\frac{\partial U^{- 1}}{\partial z_{\alpha}}+ \frac{1}{2 \pi \kappa}\sum_{\beta \neq \alpha}\frac{\hat{Q}^{a}( \xi_{\alpha}, \bar{\xi}_{\alpha}) \hat{Q}^{a}( \xi_{\beta}, \bar{\xi}_{\beta})}{z_{\alpha}- z_{\beta}}U^{- 1}= 0
\varphi(\omega,y) = \gamma(1_G,\widehat{\phi}(\mathfrak{C}((\bar{\omega},y^{1_G}),1_{H_1},1_{H_2}))) & = \gamma(1_G,\widehat{\phi}(c^{1_G})) & = \gamma(1_G,\widetilde{\Psi}(x)) & = x.
< W ( A , \gamma ) >^{( 1 )}= \oint_{\gamma}d s \oint_{\gamma}d t \dot{\gamma}^{a}( s ) \dot{\gamma}^{b}( t ) \epsilon_{a b c}{\frac{( \gamma^{c}( s ) - \gamma^{c}( t ) )}{| \gamma ( s ) - \gamma ( t ) |^{3}}}= \mathrm{G S L}( \gamma )
- \int_{B_r(p)\setminus \{p \}}\nabla \eta \cdot (\cos R^{u}\cos R^{v}\nabla F) \, d\mu_g = - \sum_i\int_{U_i}\nabla (\varphi_i \eta) \cdot (\cos R^{u}\cos R^{v}\nabla F) \, d\mu_g \ge 0,
D_{\mu \nu}^{R}( q ) = \delta_{\mu 0}\delta_{\nu 0}{\frac{1}{( \vec{q}^{2}+ M^{2})}}+ \delta_{\mu i}\delta_{\nu j}\left( \delta_{i j}-{\frac{q_{i}q_{j}}{( \vec{q}^{2}+ M^{2})}}\right){\frac{1}{q^{2}- M^{2}}}~ ,
(\pi_*E^+)^+ =K_{E^+}^+\oplus L_{E^+}^+,\qquad(\pi_*E^+)^-=K_{E^+}^-\oplus L_{E^+}^-, (\pi_*E^-)^+ =K_{E^-}^+\oplus L_{E^-}^+,\qquad(\pi_*E^-)^-=K_{E^-}^-\oplus L_{E^-}^-,
\left\Vert e^{\gamma \left\vert x\right\vert^{p}}\tilde{u}_{k}\left(x,0\right) \right\Vert_{X}=\left\Vert e^{\gamma \left( \frac{\alpha}{\beta}\right)^{p/2}\left\vert x\right\vert^{p}}u_{k}\left( x,0\right)\right\Vert_{X}=B_{0},
W[Y]=\int\frac{d^4x}{32\pi^2}\int^\infty_0\frac{dT}{T^3}\rho (T,\Lambda^2)e^{-TK}\mbox{tr}\left(e^{-TM\tau_3}[1-f_1(T,B)+f_2(T,B) -\tilde{f}_3(T,B)+\ldots ]\right),
\pi_{1}(D^4 - S, (a_{0}, b_{0})) = \langle \gamma'_{1}, \gamma'_{2}, \dots, \gamma'_{m}| (\gamma'_{j_{i}}){w_{i}}_{*}= (\gamma'_{j_{i}+1}){w_{i}}_{*}\i = 1, 2, \dots n \rangle ,
K_{1,3}(z,z'')= \Big(\int_{2r<r'<\frac{r''}2}+\int_{\frac{r''}2\leq r'\leq2r''}+\int_{r'>2r''}\Big)G(z,z')Q(z',z'')\;d\mu(z') = K_{1,31}(z,z'')+K_{1,32}(z,z'')+K_{1,33}(z,z'').
V_{1 - \mathrm{l o o p}}( \sigma ) = -{\frac{N}{2 L}}\sum_{n}p_{n +}= -{\frac{N}{2 L}}\sum_{n}{\frac{1}{c}}\left[ - s{\frac{n \pi}{L}}+ \sqrt{\left({\frac{n \pi}{L}}\right)^{2}+ c \sigma^{2}}\right] .
\begin{array}{ll}(P-SDP): (D-SDP): \;\; \begin{array}{lll}\mbox{\rm maximize}b^Ty \mbox{\rm subject to}C - \sum^{m}_{i=1}y_i A_i \succeq 0 \end{array}\begin{array}{lll}\mbox{\rm minimize}C \cdot X \mbox{\rm subject to}A_i \cdot X = b_i \forall i \in \{1,\ldots,m\}& X \succeq 0,\end{array}\end{array}
\begin{array}{@{}l@{}}{\displaystyle \frac{1}{\tau}\left(\|{\theta}_{\Psi}^{k}\|^{2}_{\mathcal{L}^{2}}- \|{\theta}_{\Psi}^{k-1}\|^{2}_{\mathcal{L}^{2}}\right) = -\mathrm{Im}\sum_{i=1}^{5}V_i^{k}( \overline{\theta}_{\Psi}^{k}) \leq \;\sum_{i=1}^{5}\vert V_i^{k}( \overline{\theta}_{\Psi}^{k}) \vert.}begin{align*}2mm] \end{array}
\mathcal{E}_{1_\chi}(T_1, \ldots, T_n) = E(T_1\cdots T_n) = E(T_1 \cdots T_{q-1}L_b T_{q+1}\cdots T_n) = E(L_bT_1 \cdots T_{q-1}T_{q+1}\cdots T_n) = bE(T_1 \cdots T_{q-1}T_{q+1}\cdots T_n) = \sum_{\sigma \in BNC(\chi|_{\setminus q})}b\kappa_{\sigma}(T_1, \ldots, T_{q-1}, T_{q+1}, \ldots, T_n),
\mathbf{W}^{s}_{\epsilon}(\omega)=\begin{pmatrix}\mathbf{p}^{s}_{1}(\omega,\epsilon) \hdots \mathbf{p}^{s}_{k_{\max}(\omega)}(\omega,\epsilon) \mathbf{1}\hdots \mathbf{0}\vphantom{\vdots}\vdots \ddots \vdots \mathbf{0}\hdots \mathbf{1}\vphantom{\vdots}\mathbf{0}\hdots \mathbf{0}\vphantom{\vdots}\end{pmatrix}.
{\cal C}' =\Big( \displaystyle\sum_{\sigma \subset U \atop |\sigma|=\alpha}\prod_{u_i \in \sigma}(d_i)(d_i-1) \Big)\Big( \displaystyle\sum_{\sigma \subset W \atop |\sigma|=\alpha}\prod_{w_i\in \sigma}(d_i')(d_i'-1) \Big)\Big( \dfrac{\alpha!\alpha!}{2^{r_2}4^{r_4}\ldots k^{r_k}}\Big)\:,
\eta^{k+1}- \eta^k \geq \hat{Q}_k({\mu}_k)-\hat{Q}_k(0) = \int^{{\mu}_k}_0 \hat{Q}'_k(\mu)\;d\mu &={\mu}_k\left(F(z^{k+1}) - \widetilde{F}^k(z^{k+1}) - \frac{{\mu}_k}{2\rho}\big\|s^{k+1}- g^{k+1}\big\|^2\right).
\begin{aligned}\|\bar{y}^{k+1}-\bar{g}^{k+1}\|^2= (1-\alpha^k)^2\| \bar{y}^k-\bar{g}^k\|^2 +(\gamma_2^k)^2 \|\bar{\xi}_w^k\|^2 &+2\left\langle (1-\alpha^k)(\bar{y}^k-\bar{g}^k), \gamma_2^k\bar{\xi}_w^k\right\rangle\end{aligned}
=\varphi\Big(\prod_{j=1}^r h_j^{(-1)^{\dim \sigma_j}}\Big)\cdot \varphi\Big( i_*\Big(\prod_{j=r+1}^s h_j^{(-1)^{\dim \sigma_j}}\Big)\Big)\cdot \tau\left(C_*^{\varphi}(M,\overline{W});\mathfrak{f}_2',\mathfrak{h}'\right)\cdot \cdot\tau\left(C_*^{\varphi}(\overline{C},Q^+);\mathfrak{f}_2'',\mathfrak{h}''\right)=
\Pi_{3 B}= - \frac{2 \pi^{4 \mu}}{( \mu - 1 )^{3}\Gamma^{4}( \mu ) \Delta}\left[ 1 - \frac{\Delta ( \mu - 1 )}{2}\left( 3 \Theta + \frac{1}{( \mu - 1 )^{2}}\right) \right]
\lim_{t \to \infty}- \frac 1t \log \mu_Y(\tau_r > t) = \lim_{t \to \infty}\frac 1t \log \mu(Y) - \frac 1t \log \mu(\tau_r > t \wedge Y) & \ge \lim_{t \to \infty}- \frac 1t \log \mu(\tau_r > t) .
\int_{0}^{\infty}d t_{1 1}\frac{1}{t_{1 1}^{2}}\exp \left( - \frac{A}{t_{1 1}}\right) = \frac{1}{A}, \int_{0}^{\infty}d t_{2 2}\exp \left( - B t_{2 2}\right) = \frac{1}{B},
\tau\sum\limits_{i=0}^{m-1}h\bar bv_{i\bar t}^2(k) +\sum\limits_{i=0}^{m-1}hv_{ix}^2(k)-\sum\limits_{i=0}^{m-1}hv_{ix}^2(k-1)+ \tau^2\sum\limits_{i=0}^{m-1}hv_{ix\bar t}^2(k) \leq \frac1{\bar b}\tau\sum\limits_{i=0}^{m-1}hf_{ik}^2 +2\tau p_kv_{m\bar t}(k)-2\tau g_k^nv_{0\bar t}(k),\forall k =1,\ldots,n .
L^2(\Omega,\Lambda) =\overline{{\rm{\sf R}}(d)}\stackrel{\bot}{\oplus}\overline{{\rm{\sf R}}(\underline{\delta})}\stackrel{\bot}{\oplus}{\rm{\sf N}}(D_{{}^\Vert}) &=\overline{{\rm{\sf R}}(\delta)}\stackrel{\bot}{\oplus}\overline{{\rm{\sf R}}(\underline{d})}\stackrel{\bot}{\oplus}{\rm{\sf N}}(D_{{}^\bot}),
\mathcal{E}_{\phi_{\alpha}}(z) = -\frac{1}{\phi_{\alpha}'(0^+)}\frac{1}{2\pi i}\int_{a - i \infty}^{a + i \infty}\frac{\phi_{\alpha}(-\xi)}{\xi}\frac{\Gamma(-\xi)\Gamma(1-\xi)}{W_{\phi_{\alpha}}(1-\xi)}(-z)^{-\xi}d\xi
\begin{gathered}R_4 := R_3[Y_{13};\alpha_3][Y_{31};\alpha_4]; \alpha_3: Y_{11}\mapsto t^{-1}Y_{11}, \Y_{12}\mapsto t^{-1}Y_{12}; \alpha_4: Y_{11}\mapsto t^{-1}Y_{11}, \Y_{21}\mapsto t^{-1}Y_{21};\end{gathered}
\Bigl [ \prod_{a}\Delta^{2}( \varphi_{a}) \Bigr ] \times \exp \Bigl \{-{\frac{1}{4}}{\frac{N}{L}}\Bigl ( \sum_{l = 1}^{L - 1}{\frac{1}{\sin^{2}{\frac{\pi l}{L}}}}+{\frac{1}{1 2}}\Bigr ) \sum_{a , \alpha}\varphi_{a , \alpha}^{2}+ O ( \varphi^{4}) \Bigr \}.
\|\hat\delta(x)\|_{Lip_0(M)^*}= \sup_{\|g\|_{Lip_0(M)}\leq 1}\langle g, \hat\delta(x) \rangle &= \sup_{\|g\|_{Lip_0(M)}\leq 1}\langle E(g), \delta_x \rangle & \leq \|E\|d(x,\bar x).
\sum_{i=0}^ma_i\frac{d^i}{dx^i}\overline{U}(x,p,r)+\sum_{i=0}^nb_i \overline{U}(x,p,r)-\sum_{j=1}^i p^{j-1}b_i \overline{u}^{(i-j)}(x,0,r)+\overline{U}(x,p,r)+\overline{F}(x, t,r)=K(t)\overline{U}(x,p,r).
\mu\epsilon\frac{\partial^{2}\Pi_{e}}{\partial t^{2}}=\frac{\mu Il\omega cos[\omega t]_{t^{\prime}}}{4\pi}\sum_{l=0}^{\infty}\frac{r^{\prime l}}{r^{l+1}}P_{l}(cos\gamma)[cos(\theta)\hat{r}-sin(\theta)\hat{\theta}]
\| D_{i,x}[f^{=1}] \|_2 = | \mathbb{E}[D_{i,x}f] | = |\hat{f}(\{i\})| \, |\chi_i(x_i)| \leq \delta \sqrt{\frac{1-p}{p}}\leq \frac{\sqrt{2}\sigma}{\sqrt{m p}},
{k \choose 2}^4 \frac{3{(2!)}^2}{n^{4}}(\zeta_d^h)^2 +{k \choose 2}^4 \frac{48}{n^4}\eta^h + O(n^{-5}) ={k \choose 2}^4\frac{12}{n^{4}}\left\{(\zeta_2^h)^2+ 4\eta^h\right\}+ O(n^{-5}).
c_1=\frac{(n-1+p)^{\frac{p}{n-1+p}}}{p}\left(\frac{|B_p^n|_n}{|B_p^{n-1}|_{n-1}}\right)^{\frac{p}{n-1+p}}, c_2=G_{B_p^n}((n^{-1/p},\dots,n^{-1/p}))\quad.
\mathbf{E}[f(s_n);T_g>n] &=\mathbf{E}\left[f\left(\nu_{m(n)},\frac{S_{\nu_{m(n)}}}{B_n},s_n\right);T_g>n,Z^*_{\nu_{m(n)}}\in(\rho_n^{2/3}B_n,2B_{m(n)}]\right]+o\left(\mathbf{P}(T_g>n)\right).
S_{\chi}^{\rm stiff}(\varepsilon^2 z) = \Bigl(I + \varepsilon^2 z\bigl( \mathcal{A}_{0,\chi}^{\rm stiff}- \varepsilon^2 zI\bigr)^{-1}\Bigr)\Pi^{\rm stiff}_\chi = \Pi^{\rm stiff}_\chi +\mathcal{O}(\varepsilon^2),
\begin{gathered}\left(\prod_{j:r\to 1}P_j\right)e^{-rX}=\prod_{j:r\to 1}(e^{(r-j)X}P_je^{-(r-j+1)X})=\prod_{j:r\to 1}\mathrm{Ad}_{e^{(r-j)X}}(P_je^{-X}), e^{rX}\prod_{i:1\to r}P_i=\prod_{i:1\to r}(e^{(r-i+1)X}P_ie^{-(r-i)X})=\prod_{i:1\to r}\mathrm{Ad}_{e^{(r-i)X}}(e^XP_i),\end{gathered}
f(x)=\frac{1}{(2\pi)^{n/2}}\sqrt{y_1\ldots y_n}\int_{(0,\infty)^n}\left(\frac{\det(D_{u+y}^{-1}+L_W)}{\prod_{i=1}^n(u_i(u_i+y_i)))}\right)^{1/2}du_1\ldots du_n
\left|\frac{\det(\Lambda_d)n_1}{(\lambda_2-\alpha\mu_2)^2}\varepsilon B^{-\frac{1}{r}}- \frac{n_1\mu_1+n_2\mu_2}{\lambda_2-\alpha\mu_2}\varepsilon B^{-\frac{1}{r}}\right|=\left|\frac{\mu_2(n_2-\theta n_1)}{\lambda_2-\alpha\mu_2}\varepsilon B^{-\frac{1}{r}}\right| \leqslant 2^{10}K\varepsilon^2 b^2\det(\Lambda_d)^{\frac{3}{2}}B^{1-\frac{2}{r}},
C_m(\gamma_1,\gamma_2;\mathbf{t},\mathbf{c})\sim \left(\sum_{i=1}^n c_i t_i\right)^m \sum_{\sigma\in\{1,2\}^m}\prod_{j=1}^m (-\gamma_{\sigma_j}-\gamma_{\sigma_{j-1}'}-1)^{-1},
\nabla_{\mathfrak{q}^*h}^2(\mathfrak{q}^*X-X_\infty) =\nabla_{\mathfrak{q}^*h}^2\mathfrak{q}^*X-\nabla_{h_E}^2X_\infty+(\nabla_{h_E}^2-\nabla_{\mathfrak{q}^*h}^2)X_\infty &=\nabla_{\mathfrak{q}^*h}^2\mathfrak{q}^*X-\nabla_{h_E}^2X_\infty+O'(\rho^{-\tau-1})\nabla_{h_E}X_\infty+O'(\rho^{-\tau-2})X_\infty &=O'(\rho^{-\tau-1}).
\varepsilon_F^\ell(L;x) \coloneqq \sup_{m \ge 1}\frac{p^{s_F^\ell(L^m;x)}- 1}{m/(\ell+1)}= \limsup_{m \to \infty}\frac{p^{s_F^\ell(L^m;x)}- 1}{m/(\ell+1)}.
S_1 = \int_{\mathcal{X}_n^c}f(x) \int_0^1 \mathrm{B}_{k,n-k}(s) \log^2 u_{x,s}\, ds \, dx = O\biggl(\frac{k^{\frac{\alpha}{\alpha + d}- \epsilon}}{n^{\frac{\alpha}{\alpha + d}- \epsilon}}\biggr).
\varphi_\pm(x) = \frac{p_\pm \cdot x}{|x|^n}+ O\left(\frac{1}{|x|^{n-1+\varepsilon}}\right), \nabla \varphi_\pm(x) = \nabla \left( \frac{p_\pm \cdot x}{|x|^n}\right) + O\left(\frac{1}{|x|^{n+\varepsilon}}\right),
I_{n}= \frac{1}{\pi}\int_{- \infty}^{+ \infty}d \xi^{\prime}e^{( \alpha - 1 ) ( \xi^{\prime}- \xi )}\ln \coth \frac{1}{2}| \xi^{\prime}- \xi | \left( \alpha - 1 + \frac{d}{d \xi^{\prime}}\right) ( \xi^{\prime}- \xi )^{n}.
\Delta s^i\! =\! -D\bar{G}^i(w^{(l)})\Delta w^i\!- \! r^{i,(l)}_{}, \Delta \lambda^i\! =\! -(S^{i,(l)})^{-1}\!\left(\! \Lambda^{i,(l)}\Delta s^i\! - \! r^{i,(l)}_{}+\mu^{(l)}\hat{e}\right)\!,
v_{z,t}(k)+\frac{1}{\pi}\int_{\mathbb C}e^{i(\overline{\varsigma}z+\overline{z}{\varsigma})/2}\overline{v_{z,t}}(\varsigma) \Pi^o h(\varsigma,\varsigma,t) \frac{d\sigma_\varsigma}{\varsigma-k}=I, w_{z,t}(\cdot)=v_{z,t}(\cdot)-I\in\mathcal B^{s}.
z_2 = \frac{2 \gamma (p^2 - 2p + 1) + (3p - 2)(n - m - 1)}{p^2 - 3p + 2}\geq 2 \gamma, z_4 = \frac{2 \gamma (p - 1) - p - 2 + 3(n - m)}{p-2}\geq 2 \gamma - 1.
Q = \int_{x^{0}= t}\! \! d^{3}x \, \Big ( H^{\alpha \beta}( x )_{, \beta}+ \frac{1}{4}\Phi ( x )^{, \alpha}\Big ){\stackrel{\longleftrightarrow}{\partial_{0}^{x}}}u^{\gamma}( x ) \eta_{\alpha \gamma}\, .
(D^\alpha u)_x(x)=\frac{1}{\Gamma(1-\alpha)}\left(\frac{\alpha(u(0)-u(x))+(\alpha+1)u'(x)x}{x^{\alpha+1}}\right. &\quad\left.+\alpha(\alpha+1)\int_0^x[u(x-z)-u(x)+u'(x)z]\frac{dz}{z^{\alpha+2}}\right)
{\frac{\partial \psi^{\beta}}{\partial \omega_{2}}}{\frac{\partial \psi^{\beta}}{\partial \omega_{2}}}={\frac{2}{( \omega_{2})^{2}}}\operatorname{s i n}^{2}\left({\frac{\theta_{2}- 2 \theta_{1}}{2}}\right) \hspace{0 . 5 c m}.
E_{\varepsilon}(\xi)=\frac{1}{2}g_{\varepsilon}(\lvert \xi\rvert^{2})\,(\xi\in{\mathbb R}^{Nn})\textrm{with}g_{\varepsilon}(\sigma)\coloneqq 2b\sqrt{\varepsilon^{2}+\sigma}+g_{p}(\varepsilon^{2}+\sigma)(0\le \sigma<\infty)
\int_s^tr^{-\theta_1}(t-r)^{-\theta_2}dr =\int_0^{t-s}(t-r)^{-\theta_1}r^{-\theta_2}dr\le\int_0^{t-s}(t-s-r)^{-\theta_1}r^{-\theta_2}dr &\lesssim(t-s)^{1-\theta_1-\theta_2}.
R_{[c,M,s]}(x):=\sup_{\begin{array}{c}n\in\mathbb{N}, x_1=s,\x_{n+1}=x, \{(x_i,y_i)\}_{i=1}^n\subset G(M) \end{array}}\ \ \sum_{i=1}^n [c(x_{i+1},y_i)-c(x_i,y_i)].
\begin{array}{lll}\mathfrak{Q}_n^{k-1}\Big( x_{[n]}^k, \xi_m\Big) = F_{\tau_m}\Big(x_{[m]}^k, \Psi_{m}\Big)-\mathcal{Q}_m\Big( x_{[m]}^k \Big)+\mathcal{Q}_m^{k-1}\Big( x_{[m]}^k \Big) \geq \mathfrak{Q}_n \Big(x_{[n]}^k, \xi_{m}\Big)-\mathcal{Q}_m\Big( x_{[m]}^k \Big)+\mathcal{Q}_m^{k-1}\Big( x_{[m]}^k \Big) \end{array}
I_{n , k , d}( z_{1}, \dots , z_{n}; \epsilon ) = \frac{( 4 \pi )^{\hat{d}/ 2}\Gamma ( \hat{d}/ 2 )}{( 4 \pi )^{d / 2}\Gamma ( d / 2 )}\int \frac{d^{\hat{d}}p}{( 2 \pi )^{\hat{d}}}\left( p^{2}\right)^{r}\prod_{i = 1}^{n}\frac{1}{\Delta_{i}},
\tau_{\mathrm{n c}}^{\mathrm{n o n p l}}( q ) = \frac{- 1}{3 2 \pi}\left( \frac{| \theta q |^{2}}{4}+ \frac{1}{\Lambda^{2}}\right)^{- \frac 1 2}+ \frac{| q |}{1 6 \pi}\int_{0}^{1}\frac{d \alpha}{\sqrt{\alpha ( 1 - \alpha )}}\exp \left( - | \theta q | | q | \sqrt{\alpha ( 1 - \alpha )}\right)
\frac{\cal L}{N}= \frac{1}{2 N}( \partial_{\mu}\phi_{i})^{2}- \frac{1}{2}\mu^{2}\rho - \frac{\lambda_{0}}{4}\rho^{2}- \frac{\eta}{6}\rho^{3}+ \frac{\chi}{2}( \rho - \frac{\vec{\phi}\cdot \vec{\phi}}{N}) .
I = \frac{1}{2}\int d^{2}x \sqrt{- g}e^{- 2 \phi}\Big ( R + 4 ( D \phi )^{2}+ 4 \Lambda - ( D \psi )^{2}- \frac{1}{4}( D \psi_{1})^{2}- \frac{1}{4}e^{- \psi_{1}}F_{2}^{2}- \frac{1}{4}e^{\psi_{1}}F^{2}\Big )
\Big( \sum_{r,s\ge 0}- \sum_{r,s<0}\Big)(-1)^{r+s}x^ry^sq^{\binom{r}{2}+2rs+\binom{s}{2}}&=j(y;q)m\big (\frac{q^2x}{y^2},q^3,-1\big )+j(x;q)m\big (\frac{q^2y}{x^2},q^3,-1\big )- \frac{yJ_{3}^3j(-x/y;q)j(q^2xy;q^3)}{\overline{J}_{0,3}j(-qy^2/x;q^3)j(-qx^2/y;q^3)}.
\delta_{n,x}^{\ast}=\sqrt{\frac{1}{(n+\beta)^2}\left\{\beta^2x^2+\left(n-2 \alpha \beta+2n\mu\frac{e_\mu(-nr_n(x))}{e_\mu(nr_n(x))}\right)x+\alpha^2-\left(\frac{5}{12}+\mu\frac{e_\mu(-nr_n(x))}{e_\mu(nr_n(x))}\right)\right\}}.
e^{\begin{bmatrix}0 \tfrac{t}{2}\tfrac{t}{2}0\end{bmatrix}}=\begin{bmatrix}1 0 0 1\end{bmatrix}+\begin{bmatrix}0 \tfrac{t}{2}\tfrac{t}{2}0\end{bmatrix}+o(t^2), e^{\begin{bmatrix}t/2 0 0 -t/2\end{bmatrix}}=\begin{bmatrix}1 0 0 1\end{bmatrix}+ \begin{bmatrix}t/2 0 0 -t/2\end{bmatrix}+o(t^2);
\binom{c_nkn}l\frac{(k(j-1))_{c_nkn-l}}{(kn)_{c_nkn}}\le C\frac{(k(j-1))_{c_nkn-l}}{(kn)_{c_nkn-l}}\le C\frac{(kj)_{c_nkn-l}}{(kn)_{c_nkn-l}}.
||\varphi_{p_1}||^2=\displaystyle \sum_{m \in \mathbb{Z}_N^2}|\varphi_{p_1}(m)|^2= \displaystyle |\varphi_{p_1}(n_1)|^2 + \sum_{m\neq n_1 \in \mathbb{Z}_N^2}|\varphi_{p_1}(m)|^2 =1,
H_{HP}^{(1)}=\frac{g^2\sqrt{N}}{4\pi}\int_0^{\infty}\frac{dR}{\sqrt{R}}\int_0^1 dz \sum_{nml}\left( K_{nml}(z)-\tilde{K}_{nml}(z) \right)\left( b_R^{(n)\dagger}b_{Rz}^{(m)}b_{R(1-z)}^{(l)}+{\rm h.c.}\right),
F^*_{j,p}\left(\tau\right) -\frac{1}{\sqrt{-i\tau}}\sqrt{\frac{2}{p}}\sum_{k=1}^{p-1}\sin\left(\frac{\pi kj}{p}\right)F^*_{k,p}\left(-\frac{1}{\tau}\right)=i\sqrt{2p}\cdot r_{f_{j,p}}(\tau),
H^{+ +{\breve{\beta}}-}= x_{h}^{{\breve{\alpha}}-}\partial_{h{\breve{\alpha}}}^{-}H^{+ +{\breve{\beta}}+}= x_{h}^{{\breve{\alpha}}-}\partial_{h{\breve{\alpha}}}^{-}\partial_{h}^{-{\breve{\beta}}}{\cal L}^{+ 4},
\sum_{m=1}^\infty |\tilde{e}^{(k)}_{nm}|=\sum_{m=1}^{n-1}\frac{w_k(n)}{w_{k+1}(m)}|e_{nm}|\leq C\frac{1}{n}\sum_{m=1}^{n-1}\frac{r_k^{\alpha_n}}{r_{k+1}^{\alpha_m}}\frac{n^\alpha}{m^\alpha}.
L=\nabla v=[\begin{matrix}\frac{\partial v_{x}}{\partial x}&\frac{\partial v_{x}}{\partial y}&\frac{\partial v_{x}}{\partial z}\\ \frac{\partial v_{y}}{\partial x}&\frac{\partial v_{y}}{\partial y}&\frac{\partial v_{y}}{\partial z}\\ \frac{\partial v_{z}}{\partial x}&\frac{\partial v_{z}}{\partial y}&\frac{\partial v_{z}}{\partial z}\end{matrix}]
\eta_p(r)=\frac{P_p r^{-\alpha}{g_p}}{I_{\phi}+ I_{\psi\setminus P_0}}, I_{\phi}=\sum\limits_{i\in \phi}P_m{d_i}^{-\alpha}{h_i}I_{\psi\setminus P_0}=\sum\limits_{j\in (\psi \setminus P_0)}P_p{r_j}^{-\alpha}{g_j}
\begin{cases}e^{-Q}\delta_{kl}\leq \hat{h}_{\delta,kl}=\hat{h}_{\delta}(\frac{\partial}{\partial \hat{f}^{k}},\frac{\partial}{\partial \hat{f}^{l}})\leq e^{Q}\delta_{kl}; R^{1+\alpha}\left\|\frac{\partial \hat{h}_{\delta,kl}}{\partial \hat{f}^{j}}\right\|_{C^{0,\alpha}(B_{R}(\hat x,\hat h_\delta))}\leq \Psi(\delta\mid n, r,v, \rho,\alpha,Q)\end{cases}
\forall\,x:n\forall\,y:n\forall\,z:m(((x+_ny)\ast_{nm}z)=(x\ast_{mn}z+_{n+m}y\ast_{nm}z)) \forall\,x:n\forall\,y:m\forall\,z:m((x\ast_{mn}(y+_mz))=(x\ast_{nm}y+_{n+m}x\ast_{nm}z)) \forall\,x:n\forall\,y:m (h_{n+m}(x\ast_{nm}y) = h_n(x)\ast_{nm}h_m(y))
G^{(2)}_{\mathrm{cl.}}(\zeta,Q)=\begin{cases}\prod_{a=1}^{(p-1)/2}\left(T_p\left(\frac{Q}{2\cos(\pi p' a/p)}\right)-T_{p'}((-1)^a\zeta)\right)\ , p\ \mathrm{odd}\ , Q^{p/2}\prod_{a=1}^{(p-2)/2}\left(T_p\left(\frac{Q}{2\cos(\pi p' a/p)}\right)-T_{p'}((-1)^a\zeta)\right)\ , p\ \mathrm{even}\ .\end{cases}
\begin{bmatrix}\alpha\overline\alpha \alpha J(T^{-2r},T^{-3r}) \overline\alpha^2 0 -\alpha J(T^{-r},T^{-r})J(T^{-2r},T^{-3r}) \alpha^2J(T^{-3r},T^{-3r})-\overline\alpha^2J(T^{-r},T^{-r}) 0 \overline\alpha^2 J(T^{-2r},T^{-r})-\alpha^2 J(T^{-2r},T^{-3r}) 0\end{bmatrix}.