harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
X_{P_\lambda}= X_{P_{\lambda_1}}\cdots X_{P_{\lambda_{\ell(\lambda)}}}=\prod_{i=1}^{\ell(\lambda)}\left(\sum_{\beta \vDash \lambda_i}(-1)^{\lambda_i-\ell(\beta)}p_{\widetilde\beta}\right) =\sum_{\alpha\preccurlyeq\lambda}(-1)^{|\lambda| - \ell(\alpha)}p_{\widetilde\alpha}.
\|\mathcal{L}_{q,\alpha}-\mathcal{L}_{p,\alpha}\|_{\mathcal{B}(\mathcal{E}_{\alpha})}=\sup_{x\in\mathbb{R}}|\partial_{Y}R(Y_0(x-q))-\partial_{Y}R(Y_0(x-p))|\leq C|q-p|, \|\mathcal{L}_{q}-\mathcal{L}_{p}\|_{\mathcal{B}(\mathcal{E}_{0})}=\sup_{x\in\mathbb{R}}|\partial_{Y}R(Y_0(x-q))-\partial_{Y}R(Y_0(x-p))|\leq C|q-p|,
\delta \kappa =-\kappa_r-\frac{\lambda_r^2}{60\left[ \frac{45\lambda_r^2}{4\pi}\left( 2+\ln \left[ \frac{m_r^2+\frac QC}{4\Lambda^2}\right] \right) +\frac{54\kappa_r^2}{\pi (m_r^2+\frac QC)}\right]}
R^{\infty}= \int_{\, g(\omega)}g(\omega) \sqrt{1- \left(\frac{\omega}{K R^{\infty}}\right)^2}~ d\omega &\geq \sqrt{1- \left(\frac{M}{K R^{\infty}}\right)^2}\int_{\, g(\omega)}g(\omega) ~ d\omega = \sqrt{1- \left(\frac{M}{K R^{\infty}}\right)^2}.
\textup{\textbf{w}}(p \otimes \tau) \textup{\textbf{w}}^{\ast}= \textup{\textbf{w}}e_{+}(p \otimes c^{\ast}c) e_{+}\textup{\textbf{w}}^{\ast}= \big((p \otimes c) e_{+}\textup{\textbf{w}}^{\ast}\big)^{\ast}\big((p \otimes c) e_{+}\textup{\textbf{w}}^{\ast}\big).
E_{\epsilon,0}^{x_0,\gamma}\Bigl\{\tau^{\epsilon,\gamma}\Bigr\}\le T\sum\nolimits_{\ell=1}^{\infty}\Bigl[\sup_{x_0 \in D}P_{\epsilon,0}^{x_0,\gamma}\Bigl\{\tau^{\epsilon,\gamma}> T \Bigr\}\Bigr]^{\ell-1}< \infty,
\langle E_i, U_xE_i\rangle' = \langle E_i, P U_xE_i\rangle' &= \langle E_i', PU_xE_i\rangle + \langle E_i, P' U_xE_i\rangle + \langle E_i, PU_x'E_i\rangle + \langle E_i, PU_x PE_i'\rangle &= \langle E_i, PU_x'E_i\rangle.
F_{_{( 2 )}}= \left( \begin{array}{c c c}{f}&{0}&{M_{_{( 2 )}}}\\{0}&{0}&{R_{_{( 2 )}}}\\{- M_{_{( 2 )}}^{^T}}&{- R_{_{( 2 )}}^{^T}}&{0}\\ \end{array}\right) ,
V_{e f f}( \sigma ) ={\frac{\sigma^{2}}{2 \lambda}}+{\frac{1}{4 \pi^{3 / 2}}}\int_{1 / \Lambda^{2}}^{\infty}{\frac{d s}{s^{5 / 2}}}e^{- s \sigma^{2}}( e B s ) \coth ( e B s )
\int_{\Omega}F(x,t)dx=\int_{(\Omega_{q})_{\delta/2}}F(x,u)dx+\lambda\int_{\Omega_{N}\backslash (\Omega_{q})_{\delta/2}}F_{1}(x,u)dx+\int_{\Omega_{p}\backslash (\Omega_{q})_{\delta/2}}\frac{|u|^{p^{*}(x)}}{p^{*}(x)}dx
\Sigma_1 = \textbf{1}_{\left\{\substack{m_1 = n_1 m_2 = n_2}\right\}}\iint \limits_{\mathbb{R}_2^+}\left|F(y_1,y_2)\right|^2 \frac{dy_1 dy_2}{(y_1 y_2)^3}= \textbf{1}_{\left\{\substack{m_1 = n_1 m_2 = n_2}\right\}}\left\langle F, F\right\rangle,
\big|g_i(x)\big| \leq \begin{cases}L(\log x)^{-\alpha\delta}\emph{if}\i=1, L(-\log x)^{\alpha\delta}\emph{if}\i=2, L\frac{1}{x^\delta}\emph{if}\i=3, L(-\log(-x))^{-\alpha\delta}\emph{if}\i=4, L(\log(-x))^{\alpha\delta}\emph{if}\i=5, Lx^\delta \emph{if}\i=6,\end{cases}
2\kappa^2{\cal L}= \sqrt{-g}\left( R +{1\over 2}m^2 (T^2 - 2T_{ij}T^{ij}) -Tr(P_{\mu}P^{\mu}) -{1\over 2}(V_I{}^i V_J{}^j F_{\mu \nu}^{IJ})^2 \right) .
\dfrac{M_{2}(\mathcal{C}(G))}{|e(\mathcal{C}(G))|}=\dfrac{2^{6k}-6\cdot2^{5k}+14\cdot2^{4k}-9\cdot2^{3k}-15\cdot2^{2k}+15\cdot2^{k}+8}{2^{4k}-2\cdot2^{3k}-2\cdot2^{2k}+3\cdot2^{k}+2}.
\sigma_k^2 U_k(t_k + \sigma_k t) &= e^{-N(t_k + \sigma_k t)}D^2 F(\bar u_k,\bar v_k)(t) - \sigma_k^2 \beta_k^{2}\left (\begin{array}{cc}W_1 (t_k + \sigma_k t) 0 0 W_2 (t_k + \sigma_k t)\end{array}\right) &\longrightarrow\; e^{-N t_0}D^2 F(\bar u,\bar v)(t)
\lambda_{\phi}(t_n)=\!\int_{\mathbb{R}^d}\!\left(\frac12|\nabla\phi(\cdot,t_n)|^2+V|\phi(\cdot,t_n)|^2+\beta|\phi(\cdot,t_n)|^4-\Omega\overline{\phi}(\cdot,t_n)L_z\phi(\cdot,t_n)\!\right)\!\mathrm{d}\mathbf{x}-\|\dot{\phi}(\cdot,t_n)\|^2.
\begin{aligned}\delta^{\nabla y}_{j}(T) \leq C \tilde{C}_{T}\Big[(k^{q}_{j}(T)+k^{q}_{j-1}(T))+(k^{q}_{j}(T)+k^{q}_{j-1}(T))(2 k^{\infty}_{j}(T) + \lvert \overline{b}\rvert)+k^{q}_{j-1}(T)^2 \Big] \delta_{j-1}(T) &< C \tilde{C}_{T}[2 K_{1}+6 K_{1}K_{2}+ K_{1}^2] \delta_{j-1}(T).\end{aligned}
\xi^{1}(t)=\frac{1}{V^{\prime \prime}\left( x_{0}\right)}\left[ \sqrt{m}\xi_{10}\sqrt{V^{\prime \prime}\left( x_{0}\right)}\sin \left( \frac{\sqrt{V^{\prime \prime}\left( x_{0}\right)}}{\sqrt{m}}t\right) -\xi_{20}\cos \left( \frac{\sqrt{V^{\prime \prime}\left( x_{0}\right)}}{\sqrt{m}}t\right) +\xi_{20}\right] ,
{\cal P}^x_a \, =\, \frac{1}{2}tr( J^x a + J^x q t^t a) \quad ; \quad{\cal P}^x_b \, =\, -\frac{1}{2}tr (J^x q b^t) \quad ; \quad{\cal P}^x_c \, =\, -\frac{1}{2}tr (J^x q c q^t)
\begin{array}{rl}jka(2ix-(i-2))^2+ikb(2jy-(j-2))^2+ijc(2kz-(k-2))^2 &\hskip 1cm =8ijkn+jka(i-2)^2+ikb(j-2)^2+ijc(k-2)^2=: \Psi_{i,j,k}^{a,b,c}(n)\end{array}
g_n' = (e_r^{-1})^{1-t_r(n)}e_{r+1}^{t_{r+1}(n)}\dots e_k^{t_k(n)}\gamma(n) &= (e_r^{-1})^{1-t_r(n)}e_{r+1}^{t_{r+1}'(n)}\dots e_k^{t'_k(n)}\gamma'(n),
\|\overline{\Delta}p\|_{\xi_2}\leq \frac{\xi_2^d}{d}\sum_{i=1}^m (\deg_i(p))^2\leq d \xi_2^d \leq \left( \frac{\xi_1}{\xi_2}\right)^d \xi_2^d = \xi_1^d = \|p\|_{\xi_1}.
\psi_{m^{i j}}= \delta ( y - m^{1 2}) \, \delta ( x_{0}- m^{1 3}) \, \delta ( x_{1}- m^{1 4}) \, \delta ( x_{2}- m^{2 3}) \, e^{\frac{i m^{2 4}\, x_{3}}{m^{1 2}}}\ ,
A_{n,s}= \left(\frac{1}{n^s}- \frac{1}{(n+1)^s}\right) + \left(\frac{1}{(n+2)^s}- \frac{1}{(n+3)^s}\right) + \cdots \intertext{and}B_{n,s}= \left(-\frac{1}{n^s}+ \frac{1}{(n+1)^s}\right) + \left(-\frac{1}{(n+2)^s}+ \frac{1}{(n+3)^s}\right) + \cdots .
\sum_{i=1}^{q}\int_0^T \alpha (t) \theta_{i}(t) \frac{d}{dt}f_{i,\lambda}(u_{\lambda}(t)) dt \rightarrow \sum_{i=1}^{q}\int_0^T \alpha (t) \theta_{i}(t) \frac{d}{dt}f_i (u(t)) dt.
Df_{t}^{H_{0}}(f_{t}^{H_{1}^{t}}(z))X_{H_{1}^{t}}(f_{t}^{H_{1}^{t}}(z)) =Df_{t}^{H_{0}}(f_{t}^{H_{1}^{t}}(z))X_{H_{1}^{t}}((f_{t}^{H_{0}})^{-1}f_{t}^{H_{0}}f_{t}^{H_{1}^{t}}(z)) & =X_{H_{1}^{t}\circ(f_{t}^{H_{0}})^{-1}}(f_{t}^{H_{0}}f_{t}^{H_{1}^{t}}(z)) & =X_{H_{1}}(f_{t}^{H_{0}}f_{t}^{H_{1}^{t}}(z)).
\sum_{\substack{B\subset \{1,2,\cdots , n\}|B| = k}}\prod_{i\in B}\frac{b(i;n)}{1-b(i;n)}\geq \frac{\beta_n^k}{k!}\, \left(1- \, \frac{k\, \beta}{\lambda_n\, (1- \beta)}\right)^k .
\big| D_2(wH) \big| = \big|(D_2 w)^{}\big| = \big| \big( -\tfrac{1}{2}(\sinh{2x})(\sin{y}){L_w}_{*}e_1 + \tfrac{1}{2}(\sinh{2x})(\cos{y}){L_w}_{*}e_2 + (\sinh^2{x}){L_w}_{*}e_3 \big)^{}\big| = \tfrac{1}{2}\big| \sinh{2x}\big|,
1= K_2e_\lambda(I-R(\lambda,A)Le_\lambda)^{-1}e^{-\int_0^\cdot(\lambda+\mu(s))ds}= K_2e_\lambda f = \int_0^{+\infty}\beta_2(a)e^{-\lambda r}e^{-\int_0^a(\lambda+\mu(s)+e^{-\lambda r}\alpha(s))ds}da.
\Vert e_{\alpha}a - a \Vert_{\mathcal{A}}= \Vert f_{\alpha}(x_0)g(x_0)a-g( x_0)a \Vert_{\mathcal{A}}&\leq \Vert f_{\alpha}\cdot f_{a}- f_{a}\Vert_{\infty , \mathcal{A}}< \epsilon.
\Psi_n(t) =t\psi_{n-1}'[\gamma(t)]\Psi(t)e^{\psi(t)}+(1+t\psi'(t)) &=\Psi(t)\big(1+\gamma(t)\psi_{n-1}'[\gamma(t)]\big)=\Psi(t)\Psi_{n-1}[\gamma(t)].
\tilde b(x,\xi)=\varepsilon^{-2N}\sum_{j=0}^{K-1}\frac{1}{j!}\frac{(ih)^j}{2^j}\sigma(D_x,D_\xi;D_y,D_\eta)^j (b(x,\xi) \psi (y/h^\delta,\eta/h^\delta))|_{y=x,\eta=\xi}.
\theta^{*}(1^{(a)}ti_1...i_{k-1}\otimes j_1...t...j_{b-k-1}) = 1^{(a-1)}ti_1...i_{k-1}\otimes1j_1...t...j_{b-k-1}+\sum_{u=1}^{k-1}1^{(a)}ti_1...\widehat{i_u}...i_{k-1}\otimes i_uj_1...t...j_{b-k-1}),
{^t}C_N ( G_N^{n-1}({\bf x}) ) ={^t}C_N([ 1 : T_N^{n-1}(x) ]) = \left(\begin{matrix}\left\lfloor \frac{N}{T_N^{n-1}(x)}\right\rfloor 1 N 0 \end{matrix}\right) = \left(\begin{matrix}d_n 1 N 0 \end{matrix}\right)=M_{\sigma_n}.
a_h(v,v) = \int_{Q}(\partial_{t}v(x,t), v(x,t) + \theta h \partial_{t}v(x,t) ) + \nabla_x v(x,t) \nabla_x v(x,t) - \theta h \partial_{t}\nabla_x v(x,t) \nabla_x v(x,t) + \theta h \int_{\Sigma_T}n_t \nabla_x v(x,t) \nabla_x v(x,t)
P_{\sigma}(\chi^{(r,\varphi)})=(2\pi i)^{-\lceil\frac{n}{2}\rceil w_{0}}G_{\sigma}(\chi^{(r,\varphi)})^{r}a_{\sigma}^{*}(\chi^{(r,\varphi)})Q_{\sigma}(\chi^{(r,\varphi)})^{r-\lceil\frac{n}{2}\rceil}.
\left\|(u_h)_+-(u_{h'})_+\right\|_{L^\gamma(K_T)}begin{align*}2mm] \leq \left\|(u_h)_+-(u_h)_+^{(\ell)}\right\|_{L^\gamma(K_T)}+\left\|(u_h)_+^{(\ell)}-(u_{h'})_+^{(\ell)}\right\|_{L^\gamma(K_T)}+\left\|(u_{h'})_+^{(\ell)}-(u_{h'})_+\right\|_{L^\gamma(K_T)}begin{align*}2mm] :=\mathbf{I}_{h,\ell}+\mathbf{I}_{h,h',\ell}+\mathbf{I}_{h',\ell}.
A_{\sigma}(\Gamma) =\int_{\Gamma}(1+\sigma^2|\mathbb{I}_\Gamma|^2)d\mathrm{vol}_g=\int_{[0,\frac{L}{2}]\times S^1}(1+2\sigma^2+4\sigma^2\kappa^2(2t))dtd\theta &=(1+2\sigma^2)\pi E_{\sigma'}(\gamma),\sigma'=\frac{2\sigma}{\sqrt{1+2\sigma^2}}.
\limsup_{\ell \to \infty}N(0^+,\tilde u_{X_\circ,0,j_\ell}(X_\ell+\,\cdot\,)) \leq \inf_{\rho > 0}\limsup_{\ell \to \infty}N(\rho, \tilde u_{X_\circ,0,j_\ell}(X_\ell+\,\cdot\,)) &= \inf_{\rho > 0}N(\rho, \tilde u_{X_\circ,0}^{(\infty)}(X_\infty+\,\cdot\,)) &= N(0^+, \tilde u_{X_\circ,0}^{(\infty)}(X_\infty+\,\cdot\,)).
\frac{d}{dt}{\cal F}_k(t) =-\frac{1}{t^2}\int_{{\cal M}'_k}d_x \left(\omega e^{-\frac{1}{t}d_x\omega}\right) = =-\frac{1}{t^2}\int_{{\cal M}'_k}d \left(\omega e^{-\frac{1}{t}d_x\omega}\right).
{(\mathcal{L}_{X^n_t})\circ \phi_n^{-1}}(A)= P(X^n_t \in \phi_n^{-1}(A)) = P(\phi_n(X^n_t)\in A)=\mathcal{L}_{\phi_n (X^n_t)}(A)= \mathcal{L}_{X^n_{t\wedge \tau^n}}(A).
S = \int d^{4}x \sqrt{\hat{g}}\Bigl \{\hat{R}+ \frac{2 \hat{\nabla}_{\mu}\Psi \hat{\nabla}^{\mu}\Psi^{\dagger}}{( \Psi - \Psi^{\dagger})^{2}}+ \frac{1}{8}T r \bigl ({\cal L}\hat{\nabla}{\cal M}\bigr )^{2}+ \frac{i}{1 6}\Psi \hat{\cal F}_{\mu \nu}^{- T}{\cal M}\hat{\cal F}^{- \mu \nu}- \frac{i}{1 6}\Psi^{\dagger}\hat{\cal F}_{\mu \nu}^{+ T}{\cal M}\hat{\cal F}^{+ \mu \nu}\Bigr \}
= \sum_{n=1}^{\infty}\tfrac{1}{n^{1+\delta}}\int_{T}^{2T}\left({\tfrac{t}{2\pi}}\right)^{{1}/{4}+{\delta}/{2}+{it}/{2}}e^{-i ({t}/{2}+{\pi}/{8})}n^{-it}\left\{1+O({1}/{t})\right\}dt.
\partial_t P^\Sigma_{ij}= -(\nabla_i H) P_{\nu j \nu\nu\nu}-(\nabla_j H) P_{i \nu \nu\nu\nu}+(\nabla_k H) (P_{ijk \nu\nu}+P_{ij \nu k \nu}+P_{ij \nu \nu k}) & -HD_\nu P_{ij \nu \nu \nu}-Hh_{ip}P^\Sigma_{pj}-H h_{jp}P^\Sigma_{ip}.
I(W_{\sigma(1)};Y_1|X_{\pi(1)}) = I(W_{\sigma(1)};Y_1|X_{\pi(1)}) + I(X_{\pi(1)};Y_1) &= I(W_{\sigma(1)};Y_1) + I(X_{\pi(1)};Y_1|W_{\sigma(1)}) &= I(W_{\sigma(1)};Y_1),
\mathcal{L}(x,a)=\tfrac{1}{90a^{2}+2}\ln \left( x^{2}+x+\tfrac{3a+1}{3}\right) +\tfrac{45a^{2}}{90a^{2}+2}\ln \left( x^{2}+x+\tfrac{15a-1}{45a}\right) .
W^{\mu \nu}\equiv \frac{\partial^2 L}{\partial \dot x_\mu \partial \dot x_\nu}= - \frac{m}{\sqrt{\dot x^2}}\left( g^{\mu \nu}- \frac{\dot x^\mu \dot x^\nu}{\dot x^2}\right) = - m \left( g^{\mu \nu}- \dot x^\mu \dot x^\nu \right)\; ,
\begin{aligned}\left|\frac{v}{|v|}|w|-v\right|=\left|\frac{v}{|v|}|w|-\frac{v}{|v|}|v|\right|=\left|\frac{v}{|v|}\right|\cdot\left||w|-|v|\right| =\left||w|-|v|\right|\leq|v-w|.\end{aligned}
\pi = \tau_{r_{1}}^{-(v_{1}+1)}\cdot \tau_{r_{q_1}+1}^{v_{1}+1}\cdot \tau_{r_{q_1+1}}^{-(v_{q_1}+1)}\cdot \tau_{r_{q_2}+1}^{v_{q_1}+1}\cdots \tau_{r_{q_{z'}+1}}^{-(v_{q_{z'}}+1)}\cdot \tau_{r_{z}+1}^{v_{q_{z'}}+1}.
J\!=\!\!\int_{0}^{2\pi}\!\!\!\int_{0}^{\frac{R}{\gamma}}\frac{d\rho d\theta}{|\rho\sin\theta+1|^{\tau}\rho^{q-1}}\leq\int_{0}^{2\pi}\!\!\!\int_{0}^{\frac{R}{\gamma}}\frac{d\rho d\theta}{|1-\rho|^{\tau}\rho^{q-1}}\leq C(q,\tau)\left(\frac{R}{\gamma}\right)^{2-\tau-q}
\mathcal{D}(\mathbf{t}, \mathbf{f}) = \det \begin{bmatrix}t_0 t_1 \ldots t_{\frac{N+1}{2}-1}f_1 t_1 t_2 \ldots t_{\frac{N+1}{2}}f_2 \vdots \vdots \vdots \vdots \vdots t_{\frac{N+1}{2}}t_{\frac{N}{2}+1}\ldots t_{N}f_{\frac{N+1}{2}}\end{bmatrix}
g ( q ) = g_{i j}( q ) \, d q^{i}\, d q^{j}= 2 \int \, \frac{d x \, d y}{( 1 + r^{2})^{2}}\frac{\partial W}{\partial q^{i}}\frac{\partial \bar{W}}{\partial q^{j}}\, d q^{i}\, d q^{j}
\tilde A_{0}= - \frac{(1+g_{00})\sinh 2\alpha}{2\sqrt{2}\left [ 1+ (1+g_{00})\sinh^{2}\alpha\right ]}= -\frac{(1 - \left ( 1-\frac{2\eta}{r}\right )^{\frac{m+\sigma}{\eta}}) \sinh 2\alpha}{2\sqrt{2}\left [ 1+ (1- (1-\frac{2\eta}{r})^{\frac{m+\sigma}{\eta}}\right .\left .)\sinh^{2}\alpha \right ]}
S(t)u_0+\int_0^t S(t-s)\tilde f(v(s))\,\d s=S(t)u_0+\int_0^t S(t-s)f(AS(s)u_0+1)\,\d s &=S(t)u_0+\int_0^t S(t-s)\left(\frac{\tilde f(AS(s)u_0+1)}{AS(s)u_0+1}\right)(AS(s)u_0+1)\,\d s &\le S(t)u_0+\int_0^t S(t-s)\left\|\frac{\tilde f(AS(s)u_0+1)}{AS(s)u_0+1}\right\|_{L^\infty}(AS(s)u_0+1)\,\d s.
R^{pc}= 3q^{2}[2\epsilon_{\lambda\sigma}^{\gamma\delta}\epsilon_{\omega\nu}^{\sigma\eta}H_{\gamma\delta}^{\nu}H_{\eta}^{\lambda\omega}+ \epsilon_{\nu\lambda}^{\rho\eta}\epsilon_{\alpha\beta}^{\lambda\nu}H_{\sigma}^{\alpha\beta}H_{\rho\eta}^{\sigma}]
T r_{w}K_{H_{2}}={\frac{e^{-{\bar{s}/ 4}}}{( 4 \pi \bar{s})^{1 / 2}}}\int_{0}^{\infty}{\frac{d y \operatorname{c o s h}y e^{-{y^{2}/ \bar{s}}}}{( \operatorname{s i n h}^{2}y + \operatorname{s i n}^{2}{\frac{w}{2}})}}~ ~ .
\boldsymbol{\beta}(v)= \begin{cases}\frac{W(P_{u,v})\phi(A-P_{u,v},\lambda)}{\sqrt{\phi(A-u,\lambda)\phi^{'}(A,\lambda)}}\mathcal{P}(u,v)\neq \emptyset, 0 , \end{cases}|\boldsymbol{\gamma}(v)|=\sqrt{|\phi(A-v,\lambda)|}.
q^{-n(L+1)\left(1-H_q(1-\gamma)-r-\frac{1}{L+1}\right)}&\le q^{-n(L+1)\left(1-(1-\gamma+r)-r-\frac{1}{L+1}\right)}= q^{-n(L+1)\left(\gamma-2r-\frac{1}{L+1}\right)}\le q^{-\Omega(n/L)}
\sup_{\tau < t \le T}|B^n_0(t) - B^n_0(\tau)| \le \sup_{\tau < t \le T}|\zeta^n_0(t) - \zeta^n_0(\tau)| + 2 \sum_{k=1}^\infty |k-2|\sup_{\tau < t \le T}|B^n_k(t) - B^n_k(\tau)|.
f ( X ) \! = \! \frac{1}{\pi^{2}}\int_{S}\! f ( X^{\prime}) ( X^{\prime}\! - \! Q )^{- 3}[ 1 \! - \! ( X^{\prime}\! - \! Q )^{- 1}( X \! - \! Q ) ]^{- 3}d S_{\eta}^{\prime}.
\dot{\phi}({\bf k}-{\bf k}_1+{\bf k}_2,t) \,\phi ({\bf k}_1,t) \phi^* ({\bf k}_2,t) = \dot{\phi}({\bf k},t) \,\langle \phi ({\bf k}_1,t) \phi^* ({\bf k}_2,t) \rangle +{\rm fluctuations}\; ,
\hat{\cal W}_{\mu}\to \hat{\cal W}_{\mu}+ i [ \alpha^{a}T^{a}, \hat{\cal W}_{\mu}] + i [ \beta^{b}T^{b}( \hat{C}_{+}- \hat{C}_{-}) , \hat{\cal W}_{\mu}] - \partial_{\mu}\alpha^{a}T^{a}- \partial_{\mu}\beta^{b}T^{b}( \hat{C}_{+}- \hat{C}_{-}) .
\int_{\mathbb{R}^d}\biggl( \langle g(\mathbf{x})b(\mathbf{D})\mathbf{w}_k(\mathbf{x}),b(\mathbf{D})\boldsymbol{\eta}(\mathbf{x})\rangle+\sum \limits_{j=1}^d \langle a_j(\mathbf{x})^*\mathbf{e}_k ,D_j \boldsymbol{\eta}(\mathbf{x})\rangle \biggr) \,d\mathbf{x}=0.
H^{-1}_{jk}=\frac{1}{(j-1)!}\frac{\partial^{j-1}}{\partial x^{j-1}}\frac{1}{(k-1)!}\frac{\partial^{k-1}}{\partial y^{k-1}}\sum_{n=0}^{s-1}\frac{p_n(x)p_n(y)}{h_n}\bigg|_{x=y=0}.
\frac{\vec k^2}{m_f}\phi_\sigma(\vec k) + \sum_{\sigma'}\frac{m_f}{(2 \pi)^3}\int \frac{d^3k'}{\sqrt{E_k E_{k'}}}v_{OBE}(\sigma,\sigma',\vec k,\vec k')\phi_{\sigma'}(\vec k')=\frac{M^2_{full}-4m_f^2}{4m_f}\phi_\sigma(\vec{k})\;,
\Delta_p\hat u = \Delta \hat u |\nabla \hat u|^{p - 2}+ (p - 2)|\nabla \hat u|^{p - 4}\Delta_{\infty}\hat u \leq 0 \textrm{in}\left\{x : |x''| < 1\right\}\cap \Lambda^o_{\delta_c}\setminus \Lambda.
\sum_{I = 2}^{N}\delta_{J I}\Phi_{0}^{( I )}= \delta_{J 1}\left\{\frac{\sqrt{N}}{2 \sqrt{\pi}}\left[ \frac{\theta}{N}- \operatorname{a r c s i n}\left( \frac{e^{2}}{2 m c}\sqrt{\frac{N}{\pi \! + \! g N}}\Phi_{0}^{( 1 )}\right) \right] - \sqrt{\frac{\pi}{\pi \! + \! g N}}\Phi_{0}^{( 1 )}\right\}\; ,
A_{0}^{3}= 4g_{d}\left( 2\alpha^{\prime}\right)^{-\frac{1}{2}}(z_{1}-z_{2})^{1+ 2\alpha^{\prime}p_{1\cdot}p_{2}}(z_{1}-z_{3})^{1+ 2\alpha^{\prime}p_{1\cdot}p_{3}}(z_{2}-z_{3})^{1+ 2\alpha^{\prime}p_{2\cdot}p_{3}}e^{\sum \limits_{i=1}^{3}\alpha^{\prime}p_{i}^{2}\ln V_{i}^{\prime}(0)}
\vartheta_{1}\bigl ( \frac{w_{1}+ \bar{w}_{2}}{2 \pi i t}| \frac{i}{t}\bigr ) = \vartheta_{1}\bigl ( \frac{\tau_{1 2}- i \delta}{2 \pi t}- \frac{i}{2 t}| \frac{i}{t}\bigr ) = - i e^{\frac{\pi}{4 t}+ i \frac{\tau_{1 2}}{2 t}}\vartheta_{4}\bigl ( \frac{\tau_{1 2}- i \delta}{2 \pi t}| \frac{i}{t}\bigr )
(\partial_x \phi)^2-(\partial_t\phi)^2=Q_0(\phi,\phi)=2L\phi \underline{L}\phi, & \partial_x \phi \partial_x \tilde{\Lambda}-\partial_t \phi \partial_t \tilde{\Lambda}=Q_0(\phi,\tilde{\Lambda})=\dfrac{1}{2}L\phi \underline{L}\tilde{\Lambda}+\dfrac{1}{2}L\tilde{\Lambda}\underline{L}\phi.
u^N_t(x) = u^N_0(x) - \int_0^t \langle \mu_s^N, \nabla V^N (x-\cdot) \cdot F\big( K\ast u^N_s (\cdot)\big) \rangle \ds &- \frac{1}{N}\sum_{i=1}^N \int_0^t \nabla V^N (x-X_s^{i,N})\cdot dW^i_s + \int_0^t \Delta u^N_s(x)\ds.
\mathbf{E}( \mathbf{x}) = \frac{1}{4 \pi}\int \rho_{e}( \mathbf{x}^{\prime}) \frac{\mathbf{r}}{r^{3}}\, d^{3}x^{\prime}\, + \frac{1}{4 \pi}\int \mathbf{J}_{m}( \mathbf{x}^{\prime}) \,{\bf \times}\, \frac{\mathbf{r}}{r^{3}}\, d^{3}x
\lim_{n\to\infty}\int_0^\infty\exp_p(\mu|u_n|^{p'})r^{\alpha_0}\mathrm dr =\nu_0+\nu_\infty=\nu_\infty &=\lim_{R\to\infty}\lim_{n\to\infty}\dfrac{\mu^{p-1}}{\Gamma(p)}\|u_n\|^p_{L^p_{\alpha_0}(R,\infty)}\leq\dfrac{\mu^{p-1}}{\Gamma(p)}.
Q_{2}(x_{1},x_{2}) =x_{1}^{2}-x_{2}^{2}, Q_{2^{m}}(x_{1},\ldots ,x_{2^{m}}) =Q_{2^{m-1}}(x_{1},\ldots,x_{2^{m-1}})^{2}-Q_{2^{m-1}}(x_{2^{m-1}+1},\ldots ,x_{2^{m}})^{2}.
C_\Theta=\sup_{h\in P\cap\Omega_{2,0}}\sup_{A\in \mathcal{A}}\frac{\|Ah\|_{q/2,u}}{\|h\|_{p/2,v}}\le2\sup_{z>1}\sup_{A\in \mathcal{A}}\frac{\|A\omega_z\|_{q/2,u}}{\|\omega_z\|_{p/2,v}}=2C_\omega,
j(q^n x;q)=(-1)^nq^{-\binom{n}{2}}x^{-n}j(x;q), \ \n\in\mathbb{Z}, j(x;q)=j(q/x;q)=-xj(x^{-1};q), j(x;q)={J_1}j(x;q^2)j(xq;q^2)/{J_2^2}, j(x^2;q^2)={J_2}j(x;q)j(-x;q)/{J_1^1}.
y\left( t \right) = \sqrt{{{\rm{2}}\mathord{\left/{\vphantom{{\rm{2}}T}}\right.\kern-\nulldelimiterspace}T}}\sum\limits_{u \in \left\{{A,B}\right\}}{\left|{{h_u}}\right|{\mathop{\Re}\nolimits}\left({{x_u}\left( t \right){e^{j\left({2\pi f_u^{{\rm{RF}}}t + \varphi_u^{{\rm{RF}}}+{\varphi_{{h_u}}}}\right)}}}\right)}+ n(t)
= \left\langle \prod_{b = 1}^{N}e^{- 2 \sum_{b = 1}^{N}\beta^{( b )}\epsilon^{( b )}\int_{\Lambda}d^{2}x t ( x ) : \cos \big [ 2 \sqrt{\pi}\theta_{2}^{( b )}( x ) \big ] :_{C^{2}}}\right\rangle_{C^{2}},
\mathcal{D}_p^{\alpha}= \begin{cases}\mathcal{D}_p \alpha = 0 \mathcal{D}_p^{\alpha - 1}\cap \mathbb{C}T_pS_{\mathcal{D}^{\alpha - 1}}\alpha \hbox{is a successor ordinal}\bigcap_{\beta < \alpha}\mathcal{D}_p^{\beta}\alpha \hbox{is a limit ordinal.}\end{cases}
\widehat{S}_{p}:=\sum_{I \subset \{p, p+1, \ldots , r\}}\alpha^{r-p+1}c(I)\prod_{i=\nu_{p}}^{l-|I|}(q^{p-1}+\delta[p-1]\widehat{V}_{i})\cdot \widehat{Q}_{I}.
\rho'_\tau=\rho_{f'}R'=\rho_{f\widetilde{H}(RR')}R'=\rho_f\rho_{\widetilde{H}(RR')}R'=\rho_fRR\rho_{\widetilde{H}(RR')}R'=\rho_\tau(R\rho_{\widetilde{H}(RR')}R').
l_2 (d_1) = \min (r_5+r_4, d_1+r_3) - (r_5 - r_4, | d_1 - r_3 |) &=\min \big( 2r_4, r_5+r_4 - |d_1 - r_3 |, d_1+r_3- r_5 + r_4, 2\min(d_1, r_3) \big)
A_{b+1}(p) = \frac{b+1}{2}a_1^{p+1}+ \sum_{n=1}^b(-1)^na_1^{p-n}\sum_{b \geq \alpha_1 > \cdots > \alpha_n = 1}a_{b-\alpha_1+2}\prod_{i=1}^{n-1}a_{\alpha_i - \alpha_{i+1}+ 1}.
|v_{4,c}(t,r)| \leq C |\chi_{\geq 1}(\frac{2r}{\log^{N}(t)})| \begin{cases}\frac{1}{t^{2}r^{3}\log^{3b}(t)}, r \leq \frac{t}{2}\frac{\log(r)}{\log^{2b}(t) r^{4}|t-r|}+ \frac{\log^{2b\alpha}(t)}{t^{2}r^{3}\log^{3b+1}(t)}, r \geq \frac{t}{2}\end{cases}
\int_{\{-t'_3\le\Psi<-t'_4\}}|F_{t_0,\tilde{c}}|^2_h = \int_{\{z\in M:-\Psi(z)\in (t'_4,t'_3]\}}|\tilde F|^2_h+\int_{\{z\in M:-\Psi(z)\in (t'_4,t'_3]\backslash N\}}|F_{t_0,\tilde{c}}-\tilde F|^2_h &+\int_{\{z\in M: -\Psi(z)\in(t'_4,t'_3]\cap N\}}|F_{t_0,\tilde{c}}|^2_h.
\hbox{ad}(f_1\wedge\cdots\wedge f_{n-1})(f) =\hbox{det}\left( \begin{array}{cc}f_1\quad\cdots f D_1(f_1)\, \cdots \,D_1(f) \cdots\cdots\cdots\cdots\cdots\cdots \cdots D_{n-1}(f_1)\, \cdots \,D_{n-1}(f) \end{array}\right)=0,
T_{\boldsymbol\nu}(u)^M\Bigl(T_{\boldsymbol\nu}(u)^{-1}\sum_{\boldsymbol\mu}\binom{M}{\boldsymbol\mu}\mu_1(\mu_1+\mu_{1,2}) (u\nu^2)^{\mu_1}&\times (\nu(N-\nu))^{\mu_{1,2}}((N-\nu)^2)^{\mu_2}\Bigr)^{1/2}&=\Bigl(M\frac{u\nu^2}{u\nu^2+N(N-\nu)}+(M)_2\frac{u\nu^2(u\nu^2+\nu(N-\nu))}{(u\nu^2+N(N-\nu))^2}\Bigr)^{1/2},
\frac{1}{2}\frac{d}{dt}\|\sqrt{\rho}\Psi(t)\|^2+\nu\|\nabla\times \Psi\|^2 =(-\Delta F,\Psi)-([-\Delta,\rho]v_t,\Psi)-([-\Delta,\rho u\cdot\nabla]v,\Psi) &\leq C(\|F\|_2^2+\|\Psi\|^2)+\frac{\underline{\rho}}{4}\|\nabla\times v_t\|^2,
\int_{0}^{r}k\frac{q_{\alpha}q_{g}}{x^{2}}\cdot\frac{x^{3}}{r^{3}}\cdot dx=[k\frac{q_{\alpha}q_{g}x^{2}}{2r^{3}}]_{0}^{r}=1.27\times10^{-16}Joules
\partial_1 n \cdot A \partial_1 n = |\partial_1 n|^2 \left(|\partial_1 n|^2-|\partial_2 n|^2 \right)=0, \partial_2 n \cdot A \partial_2 n = |\partial_2 n|^2\left(|\partial_2 n|^2 - |\partial_1 n|^2 \right) = 0,
\frac{\partial}{\partial \theta}\Psi_{j}( \bar{x}) \left|_{\Psi_{j}( \bar{x})}\equiv 0 = \int d^{4}x \frac{\partial}{\partial \theta}S_{k}( \bar{y}) |_{\Psi_{j}}\frac{{\delta^{\prime}}^{2}\bar{Z}[ S ]}{\delta S_{k}( \bar{y}) \delta S_{j}( \bar{x})}\right.
\frac{(1-\alpha^2)(1-|z|^2)}{(1+\alpha|z|)^2}\frac{(1-|z|)^2}{(1+|z|)^6}\leq J_f(z)\leq\begin{cases}\displaystyle \frac{(1-\alpha^2)(1-|z|^2)}{(1-\alpha|z|)^2}\frac{(1+|z|)^2}{(1-|z|)^6}, |z|<\alpha, begin{align*}8pt]\displaystyle \frac{(1+|z|)^2}{(1-|z|)^6}, |z|\geq\alpha\end{cases}
[\frac{1}{n}z^n,l_k]=z^{n+k}\quad\mbox{and}[\frac{1}{n}z^n,m_k]=l_{n+k}, & [l_m,l_n]=(m-n)l_{m+n}+\frac{m^3-m}{12}\delta_{m+n,0}, & [l_n,m_k]=(2n-k)m_{n+k}+\frac{n^3-n}{12}z^{n+k}.
x(nP)-x(P)=\frac{\phi_n(x(P))}{\psi_n^2(x(P))}-x(P)=x(P)+\frac{\psi_{n+1}(P)\psi_{n-1}(P)}{\psi_n^2(x(P))}-x(P)=\frac{\psi_{n+1}(P)\psi_{n-1}(P)}{\psi_n^2(x(P))}.
dim\Bigg(\varphi^{-1}\big((a^1_{1}, \ldots, a^{k_1}_1), \ldots, (a^1_j,\ldots, a^{k_j}_j), \ldots, (a^1_r, \ldots, a^{k_r}_r)\big)\Bigg) = n+r-\sum\limits_{i,j}e^i_j =n+ \lvert F(\lambda)\rvert - \sum_{i}\lvert R(\lambda)_i\rvert
{\mathbb E}[W_1(\mu_n, \mu)]\leq\begin{cases}C (n^{-1/2}+ n^{-(p-1)/p}) C (n^{-1/2}\ln(1+n) + n^{-(p-1)/p}) C (n^{-1/\ell}+ n^{-(p-1)/p}) \end{cases}
{\rm Tr}\left(\phi^a \phi^{\dagger a}k^{u+m_1}\right)=\frac{1}{1-t^{u+m_1}}\prod_{i=1}^{a}\left(\frac{1-t^i}{1-t^{u+m_1+i}}\right)=\frac{1}{1-q t^{m_1}}\prod_{i=1}^{a}\left(\frac{1-t^i}{1-q t^{m_1+i}}\right),
\rho(g_\lambda^\alpha)=\exp_{G_{\lambda}}(\alpha\rho'(\mathfrak{g}))=\exp_{G_{\lambda}}\left(\alpha\sum_{i=1}^n c_i\,\rho'(\mathfrak{e}_i)\right)=:\exp_{G_{\lambda}}\left(\sum_{i=1}^n\alpha_i\,\rho'(\mathfrak{e}_i)\right)\;.
\frac{\sqrt{\widehat{w}_s^{(n)}(x_1^{(n)}) \widehat{w}_s^{(n)}(x_2^{(n)})}}{2n^2}\cdot \frac{p_1^{(n)}(x_1^{(n)}) p_1^{(n)}(x_2^{(n)})}{\widehat{h}_1^{(n)}}= \frac{(z_1z_2)^{\frac{s}{2}}}{(z_1 + u)(z_2+ u)}\cdot \frac{(2n^2)^{1-s}p_1^{(n)}(x_1^{(n)}) p_1^{(n)}(x_2^{(n)})}{\widehat{h}_1^{(n)}}.
R_i^{\left({\Delta t,\alpha ,\lambda}\right)}= \frac{{f_i^{\left({\Delta t,\alpha ,\lambda}\right)}- f_i^{\left({2\Delta t,\alpha ,\lambda}\right)}}}{{f_i^{\left({\Delta t/2,\alpha ,\lambda}\right)}- f_i^{\left({\Delta t,\alpha ,\lambda}\right)}}}={2^{p_i\left({\Delta t,\alpha ,\lambda}\right)}}.
s^{F_{n}\times F_{n-1}}=s_1(s_3s_2s_1)^{(n-3)/2}s_2s_3(s_1s_2s_3)^{(n-3)/2}s_1,\ s^{E_{n}\times E_{n-1}}=s_1(s_3s_2s_1)^{(n-3)/2}s_2s_1s_2s_3(s_1s_2s_3)^{(n-3)/2}s_1.
\sum_{j=1}^{m}\frac{s_{j+1}-s_{j}}{2\pi i}\big( \Psi_{j,11}\partial_{s_{k}}\Psi_{j,21}-\Psi_{j,21}\partial_{s_{k}}\Psi_{j,11}\big) = \sum_{j=1}^{m}\beta_{j}\partial_{s_{k}}\log \frac{\Gamma(1+\beta_{j})}{\Gamma(1-\beta_{j})}.
\int_0^\infty \mathrm{e}^{\beta t}t^\nu K_\nu(\alpha t)\,\mathrm{d}t=\frac{\sqrt{\pi}(2\alpha)^\nu}{(\alpha-\beta)^{2\nu+1}}\frac{\Gamma(2\nu+1)}{\Gamma(\nu+3/2)}{}_2F_1\bigg(2\nu+1,\nu+\frac{1}{2};\nu+\frac{3}{2};-\frac{\alpha+\beta}{\alpha-\beta}\bigg),
p(n)\sim\frac{1}{2\pi\sqrt{2}}\sum_{k=1}^{v}A_{k}(n)\sqrt{k}\cdot\frac{d}{dn}(\frac{1}{\sqrt{n-\frac{1}{24}}}exp[\frac{\pi}{k}\sqrt{\frac{2}{3}(n-\frac{1}{24})}]),
\|F(x)-F(y)\|_{l^\infty}\le\frac{3}{s}d(x,y)\left( \#(L(x)\setminus L(y)) + \#(L(y)\setminus L(x)) +\#(L(x)\cap L(y)) \right) (\frac{2}{3}+C)s \le 6N(\frac{2}{3}+C)d(x,y).
M^{\mathbb R^n_+}(\{F_1,F_2\}) \ge \lambda_n =\frac{(n - 1)v_{n - 1}\log 3}{16}\Biggl(\frac{[\Gamma(\frac{1}{2(n - 1)})]^2}{\Gamma(\frac{1}{n - 1})}\Biggr)^{1 - n},
\partial_{i}\hat{n}\cdot \partial_{i}\hat{n}= \left( \frac{2 k^{2}}{\lambda^{2 k}}\right) | z - z_{0}|^{2 k - 2}\frac{1}{\left[ 1 + \frac{1}{4 \lambda^{2 k}}| z - z_{0}|^{2 k}\right]^{2}}
\langle P[P,a][P,b]e_l,e_l\rangle_{L^2(S^1)}=-\langle (1-P)b_-e_l,(\overline{a})_-e_l\rangle_{L^2(S^1)}&=-\left\langle (1-P)\sum_{k=1}^\infty b_{-k}e_{l-k},\sum_{k=1}^\infty \overline{a_{k}}e_{l-k}\right\rangle_{L^2(S^1)}=-\sum_{k>l}a_{k}b_{-k}.
I-\mathcal{P}= \begin{bmatrix}I_r - \Lambda_1 S^\dagger 0 -\Lambda_1 S^\dagger B^T 0 I_{(c_1-1)r}0 -\Lambda_3B S^\dagger 0 I_{n-c_1 r}-\Lambda_3B S^\dagger B^T \end{bmatrix}.
\begin{cases}dx_i =( Ax_i +BR^{-1}B^Ty_i +G \bar m_i+\gamma \bar p_i) dt +D dW_i, dy_i=\left\{-A^T y_i +Q[x_i-(\Gamma\bar m_i+\eta)]\right\}dt +\zeta_i dW_i,\end{cases}