harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
V(\textbf{0})=\dfrac{4\pi}{2^{n}a^{n}}\int_{0}^{+\infty}\left\lbrace \sum_{\textbf{k}\in \lbrace0,1\rbrace^{n}}(-1)^{\vert\textbf{k}\vert}\Theta (a\textbf{k}\vert v)\right\rbrace dv .
\frac{\partial v_\epsilon}{\partial t}\,-\,\triangle v_\epsilon\,+\,\frac 2{\epsilon^{2(1-\kappa)}}(v_\epsilon^2\,-\,1)v_\epsilon\,-\,\frac{\dot{\kappa}}{K(\epsilon)}(v_\epsilon^2\,-\,1)\log\frac{1+v_\epsilon}{1-v_\epsilon}\;=\;0
F_1(x ; z)=(F(x ; z)V_2)_1=\left(0, \mathcal{P}^{+}(\bar{r}_1(z) \mathrm{e}^{-2 i z x})\right), &F_2(x ; z)=(F(x ; z)V_1)_2=\left(\mathcal{P}^{-}\left(r_2(z) \mathrm{e}^{2 i z x}\right), 0\right).
\mathrm{KL}(\mathbb{P}_{B},\mathbb{P}_{B'}) =\dfrac{1}{2\,\sigma^{2}}\| B-B'\|^{2}_{2}\leq \dfrac{\tau^{2}\,n_1\,s^{2/q}}{\sigma^{2}}& \leq \tau^{2}\,n_1\,\log(1+n_2) &\leq \alpha \log\left \vert\Omega\right \vert
\frac{v_X (x,R)}{e^{\delta(\Gamma)R}}= \int_{{\cal D}}\frac{v_\Gamma (x,y,R)}{e^{\delta(\Gamma)R}}dy = \int_{{\cal K}}w(x,y,R) dy + \sum_{i=1}^m \int_{{\cal C}_i}w(x,y, R)dy
\begin{aligned}X^{\epsilon}(t) =x_{0}+\int^{t}_{0}b(X^{\epsilon}(s))ds-\lambda(\epsilon)\epsilon^{2}\int^{t}_{0}\sigma(X^{\epsilon}(s-))\theta_{\epsilon}(s-)\tilde{N}^{\epsilon^{-2}}(ds) &\ \ \ \ -\frac{\lambda(\epsilon)\epsilon^{2}}{2}\int_0^t\sigma(X^{\epsilon}(s-))d\theta_{\epsilon}(s).\end{aligned}
\sum_{n=1}^{\infty}f_n c_q(n) e^{2\pi i nz}= (-1)^k \xi(q) (Nq^2)^{-\frac{k}{2}}z^{-k}\sum_{n=1}^{\infty}g_n c_q(n) e^{2 \pi i \frac{-n}{Nq^2z}}.
D_{\theta_{\mu}}= ( 2 \pi \alpha^{\prime})^{2}( T_{1 1}T_{2 2}- T_{1 2}^{2}) + \frac{1}{4}\theta_{\mu}^{2}, \tilde{T}_{\theta_{\mu}}^{- 1}= \frac{2 \pi \alpha^{\prime}}{D_{\theta_{\mu}}}\left( \begin{matrix}{{T_{2 2}}}&{{- T_{1 2}}}\\{{- T_{1 2}}}&{{T_{1 1}}}\\ \end{matrix}\right) ~ .
P\{Y=1 \vert \textbf{X}=\textbf{x}\}=\dfrac{f_{\textbf{X}\vert Y=1}(\textbf{x}\vert 1) P\{Y=1\}}{f_\textbf{X}(\textbf{x})}= \dfrac{\dfrac{p}{\mu{(\mathcal{C}_{1})}}}{\dfrac{p}{\mu{(\mathcal{C}_{1})}}+ \dfrac{1-p}{\mu{(\mathcal{C}_{0})}}}
a_1^{36}=z_2^{16}=z_3^{17}=a_3^{57}=:x_3,a_3^{36}=a_1^{37}=:x_4, &z_1^{26}=z_2^{27}=:y_4,a_4^{37}=z_1^{27}=:y_5 &z_1^{16}=z_2^{17},a_4^{36}=a_2^{37}, &a_1^{37}-z_2^{17}=c^{13}=-a_3^{36}+z_1^{16},a_2^{37}-z_2^{27}=c^{23}=-a_4^{36}+z_1^{26}, &a_1^{36}-a_4^{36}=z_2^{35}=-a_2^{37}+a_3^{57},z_1^{35}=a_1^{37}-a_4^{37}.
x^{n/2}f_\nu = \begin{cases}x^{1-\nu}f_\nu = -(\sqrt{1/|C_1|}x^{n/2}f_{\nu-1})', n\leq 2 x^{1+\nu}f_\nu = (\sqrt{|1/C_1|}x^{n/2}f_{\nu+1})', n\geq 2\end{cases}.
g_1(X,Y)\xi-\omega \mathcal{A}_{X}Y+\mathcal{C}\mathcal{H}\nabla^{^{M_1}}_{X}Y =\mathcal{A}_{X}\mathcal{B}Y+\nabla^{^{M_1}}_{X}\mathcal{C}Y+\eta(Y)X, \phi \mathcal{A}_{X}Y+\mathcal{B}\mathcal{H}\nabla^{^{M_1}}_{X}Y =\mathcal{V}\nabla^{^{M_1}}_{X}\mathcal{B}Y+\mathcal{A}_{X}\mathcal{C}Y,
\langle T_{\mu}^{\mu}\rangle_{R}- ( T_{\mu}^{\mu})_{0}= 2 \pi m^{2}\int_{- \infty}^{+ \infty}\frac{d \theta}{2 \pi}\left[ \frac{e^{- m R \cosh \theta}}{1 + e^{- m R \cosh \theta}}+ 2 \frac{e^{- 2 m R \cosh \theta}}{( 1 + e^{- m R \cosh \theta})^{2}}+ \cdots \right] \, \, \, .
Mf_{1}(y)\lesssim\Gamma^{\gamma}\frac{(\Gamma e^{-\omega_{l}\tau})^{\omega}}{(Ke^{-\omega_{l}\tau})^{\omega+2}}=\Gamma^{\gamma+\omega}K^{-\omega-2}e^{2\omega_{l}\tau}=e^{\omega_{l}s_{0}(k{\theta}(\gamma+\omega)-k(\omega+2)+2)}e^{2\omega_{l}(\tau-s_{0})},
\varphi_{X}(u)(t):=E[e^{iu^{\prime}X_{t}}]=exp(u^{\prime}\gamma_{t}i-\frac{1}{2}u^{\prime}A_{t}u+\int_{\mathbb{R}^{d}}(e^{iu^{\prime}x}-1-iu^{\prime}xI_{|x|<1})\nu_{t}(dx)),
\frac{\frac{V_{+ 0 , + 0}}{V_{0 + , 0 +}}+ \frac{V_{+ - , 0 0}}{V_{- + , 0 0}}}{\frac{V_{0 + , 0 +}}{V_{+ 0 , + 0}}+ \frac{V_{- + , 0 0}}{V_{+ - , 0 0}}}= \frac{\left( V_{+ - , - +}- V_{+ + , + +}\right) \frac{V_{+ 0 , + 0}}{V_{0 + , 0 +}}+ V_{+ - , + -}}{\left( V_{+ - , - +}- V_{+ + , + +}\right) \frac{V_{0 + , 0 +}}{V_{+ 0 , + 0}}+ V_{- + , - +}}\; .
S_p(\mathbf{r},\mathbf{s})=\sideset{}{'}\sum_{\substack{\mathbf{d}|\mathbf{r},\mathbf{e}|\mathbf{s}d_i,e_i|p\forall i}}\frac{\mu(d) \mu(e) d e}{[\mathbf{d},\mathbf{e}]}=\begin{cases}p-1, p|(\mathbf{r},\mathbf{s}), -1, p|r,p|s, p\nmid (\mathbf{r},\mathbf{s}), 0, (p|rp\nmid s)(p|sp\nmid r).\end{cases}
\|U^{k+1}(t)-U^{k}(t)\|_{L_\delta^2}^2 \leq e^{C_2t}\left( \|U_0^{k+1}-U_0^k\|_{L^2_\delta}^2+ C_3\int_0^t \|U^{k}(\tau)-U^{k-1}(\tau)\|_{L_\delta^2}^2d\tau+ t6(2^{-k})\right).
u^{{m(k)}}\delta u^{{(n)}}_{{m}}(\Gamma_{{(k)(n)}})^{{a}}_{{b}}=D (v^{-{1}}\delta v)^{{a}}_{{b}}+(C\Gamma^{{(k)}}(v^{-{1}}\delta v)\Gamma_{{(k)}}C^{-{1}})^{{a}}_{{b}}\qquad
\rho(m_t^{-1})R_j = \epsilon([g_{\alpha}g_t^{-1}m^{-1}g_t g_j^{-1}]) &= \epsilon([g_{\alpha}g_t^{-1}mm^{-1}g_t g_j^{-1}]) + \epsilon([g_{\alpha}g_t^{-1}])\epsilon([m^{-1}g_t g_j^{-1}]) &= R_j + \epsilon([m^{-1}g_t g_j^{-1}])R_t.
\begin{aligned}\|\nabla^2u_0\|_{L^2(\Sigma_{p_2\varepsilon})}\leq \|\nabla^2 v\|_{L^2(\Sigma_{p_2\varepsilon})}+ \|\nabla^2 w\|_{L^2(\Sigma_{p_2\varepsilon})}&\leq C\Big\{\|F\|_{L^2(\Omega)}+\|h\|_{H^1(\Omega)}\Big\}+ \|\nabla^2 w\|_{L^2(\Sigma_{p_2\varepsilon})},\end{aligned}
S_{ni}= \sum \limits_{j = 1}^{i}\sqrt{4n}(i - j + 1)(\log(q_{2n, 2j}p_{2n, 2j}) + \log(q_{2n, 2j - 1}p_{2n, 2j - 1}) - 2 \log(\tfrac{1}{4})) ~, T_{ni}= \sqrt{4n}\sum \limits_{j = 1}^{i}(\log(q_{2n, 2j}) - \log(\tfrac{1}{2})) ~,
n_{+}\left( r,\frac{\textup{\textbf{w}}(p \otimes \tau) \textup{\textbf{w}}^{\ast}}{2}\right) = n_{+}\Big( r,p W \big( \mathcal{H}_{2}^{-1}\big) p \Big) & \leq n_{+}\left( r,\frac{p W (I) p}{\zeta}\right), r > 0.
\left. \begin{array}{l l l l r}{{a = \sqrt{2 1 5 3 3}+ 1 1 9 \,}}&{{\, \, \,}}&{{b}}&{{=}}&{{- 1 9 4}}\\{{c = \sqrt{2 1 5 3 3}- 1 1 9 \,}}&{{\, \, \,}}&{{d}}&{{=}}&{{1 9 4}}\\ \end{array}\right.
{S_{eff}}'=\int{d^{4}x[-\frac{1}{4}{F'}^{2}-\frac{1}{2}{F'}^{{\mu}{\nu}}S^{\lambda}_{{\mu}{\nu}}A_{\lambda}-2S^{{\mu}{\nu}}_{\lambda}A^{\lambda}S^{\beta}_{{\mu}{\nu}}A_{\beta}+\frac{1}{2}{m^{2}}_{\gamma}A^{2}]}
\limsup_{M\to\infty}\limsup_{n\to\infty}P\left(\|\hat{K}_{\lambda_n}^--K_Y^-\|_{\ell_1}>Ms_n\lambda_n\right)\leq\limsup_{L\to\infty}\limsup_{n\to\infty}P(\Omega_{n,L}^c)=0,
\mathcal{A}_4:=\{(j_1, j_2, j_3, j_4)\in \mathbb{Z}^4\setminus\{\textbf{0}\}\, :\, j_1+j_2+j_3+j_4=0, j_1^3+j_2^3+j_3^3+j_4^3\neq 0, & \mbox{and at most one among}\,\,j_1, j_2, j_3, j_4\,\,\mbox{outside}\,\,S\}.
L V(y, +1) = \left(\alpha^+ - \lambda^+(y) \left( 1- \exp(- 2 \beta^+) \right) \right) V(y,+1), LV(y,-1) = \left( - \alpha^+ + \lambda^-(y) \left( \exp(2 \beta^+) - 1 \right) \right)V(y,-1).
S = \int d^{4}x \sqrt{g}\left({\frac{m_{s}^{8}v_{1}v_{2}v_{3}}{( 2 \pi )^{7}\lambda^{2}}}R +{\frac{1}{4}}{\frac{m_{s}^{4}v_{2}v_{3}}{( 2 \pi )^{5}\lambda}}F^{2}+{\frac{1}{4}}{\frac{m_{s}^{4}v_{1}v_{3}}{( 2 \pi )^{5}\lambda}}\tilde{F}^{2}+ \dots \right)
\sum_{y_{n}}\log\Big(\frac{r_{n}(x_n|y^{n-1}_{n-M},y_n)}{\pi_{n}(x_n|y^{n-1}_{n-J})}\Big)q_n(y_n|y^{n-1}_{n-M},x_n)-s\gamma_n(x_n,y^{n-1}_{n-N})=1-\lambda_n(y^{n-1}_{n-J}),~\forall{x_n\in{\cal X}_n}.
\begin{tabular}{|r c l|}\hline Non-zero values of $\chi_{k}$ $\chi_{0}(\gamma^{*})=-q^{-1}$ $\chi_{1}(\alpha)=1 $ $\chi_{1}(\alpha^{*})=-q^{2}$ $\chi_{2}(\gamma)=-q $ All other values of $\chi_{k}$ on the generators are zero \hline\end{tabular}
\mathcal{L}= \frac{1}{2}\sum_{i,j=1}^3 \Lambda_{ij}\partial_{x_i x_j}+x_1 x_3 \partial_{x_1}- x_2 x_3 \partial_{x_2}+ (x_2^2 - x_1^2)\partial_{x_3}- x_2\sum_{j=1}^3 a_j \partial_{x_j}.
\mathcal{R}_{j}(t)= \sum_{k=1}^{c_{j}}\left(\nu_{k}^{j}-z_{k}^{j}\right)t^{k-1},\mathcal{R}_{s}^{N}(t)= \sum_{k=1}^{d_{s}^{N}}\left(\nu_{k}^{s,N}-z_{k}^{s,N}\right)t^{k-1}\qquad\mathcal{R}_{h}(t)= \sum_{n=1}^{m}\left(\nu_{n}+\nu_{n}^{N}-z_{n}\right)t^{n-1},
\partial_{0}L_{0}- \partial_{x}L_{1}= i ( \partial_{0}^{2}\vec{n}\wedge \vec{n}- \partial_{x}^{2}\vec{n}\wedge \vec{n}) \cdot \vec{\sigma}= i ( ( \partial_{0}^{2}\vec{n}- \partial_{x}^{2}\vec{n}) \wedge \vec{n}) \cdot \vec{\sigma}= 0
\begin{aligned}y\left[ i \right] = \sum\limits_{l = 1}^L{{\bf{h}}_l^H{\bf{d}}\left[{i -{n_l}}\right]}+ z\left[ i \right] = \sum\limits_{t = 0}^{{n_{{\rm{span}}}}}{{{\bf{g}}^H}\left[ t \right]{\bf{d}}\left[{i - t -{n_{\min}}}\right]}+ z\left[ i \right], \end{aligned}
\frac{\ddot{a}}{a}=\frac{-1}{3M_{Pl}^2\sqrt{1+4\alpha\Lambda_b/3M^5}}\bigg(\frac{\rho_{\rm vis}}{2}(1+\frac{2\rho_{\rm vis}}{\Lambda_2^+})+\frac{3p_{\rm vis}}{2}(1+\frac{\rho_{\rm vis}}{\Lambda_2^+})\bigg)>0
\begin{matrix}x_{n+1}=&x_{n}+1\frac{(1/f)(x_{n})}{(1/f)^{(1)}(x_{n})}\\ =&x_{n}+\frac{1}{f(x_{n})}\cdot(\frac{-f^{\prime}(x_{n})}{f(x_{n})^{2}})^{-1}\\ =&x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})}.\end{matrix}
L = \int d^4\theta \left( \sum_{i=1}^{N+1}\overline{\Phi}_i e^{2\sum_{a=1}^{N-1}Q_{ai}V_a}\Phi_i -\sum_{a=1}^{N-1}\frac{1}{2e_a^2}\overline{\Sigma}_a \Sigma_a \right) -\frac{1}{2}\left( \int d^2\widetilde{\theta}\sum_{a=1}^{N-1}t_a \Sigma_a +{\rm c.c.}\right),
\ell_0\ge\frac{r_1 (3 M_1+1)+r_2 (3 M_1+7)+3(r_1-r_2)^2}{12 (M_1+3)}=\frac{r_1 (-2-4k)+r_2 (4-4k)+3(r_1-r_2)^2}{8 (3-2k)},
h_{j^{\prime}}^{( B )}= \sum_{i = 1}^{k^{\prime}}| i \rangle_{j^{\prime}}e p s i l o n_{i}\_{j^{\prime}}\hspace *{- 1 p t}\langle i | + \sum_{\alpha = 1}^{k^{n_{s}}- k^{\prime}}| \alpha \rangle_{j^{\prime}}\ \epsilon_{\alpha}\_{j^{\prime}}\hspace *{- 1 p t}\langle \alpha |
\tilde{R}= \frac{\tilde{a}M^{2}( \gamma - 1 )^{2}e^{\phi_{0}}}{\rho^{4}}\; \left( 1 - \frac{\rho_{0}}{\rho}\right)^{- \frac{1 + 3 k^{2}- 2 l}{1 + k^{2}}}\; .
\tilde{f}_P(b,a,b) = \Big( I_a^{(q,r)}(b) - \frac{W^{(q+r)}(b)}{W^{(q+r) \prime}(b+)}(I_a^{(q,r)})'(b +) \Big) \tilde{f}_P(0,a,b) +r\Big(\frac{W^{(q+r)}(b)}{W^{(q+r)\prime}(b+)}\overline{W}^{(q+r)}(b)-\overline{\overline{W}}^{(q+r)}(b)\Big).
z(t)= \Phi_{12}(t)q(0) -\int_0^t \Phi_{12}(t-s) (I-\Gamma)^T Q\check z(s)ds, & q(t)= \Phi_{22}(t) q(0)-\int_0^t \Phi_{22}(t-s) (I-\Gamma)^T Q\check z(s)ds,
\begin{matrix}X_1+X_2+X_3 = \theta_1^2+\theta_2^2+\theta_3^2+\theta_4^2\ \ \ \frac{Y^2}{4}+X_1X_2X_3+(\theta_1^2-\theta_3^2)(\theta_2^2-\theta4^2)X_1+(\theta_2^2-\theta_3^2)(\theta_1^2-\theta4^2)X_2 =(\theta_1^2+\theta_2^2-\theta_3^2-\theta4^2)(\theta_1^2\theta_2^2-\theta_3^2\theta4^2)\end{matrix}
m^{[g(a), g(b)]}_{k}(j+1):=\dfrac{1}{{\rm i}k}({\rm e}^{{\rm i}kg(b)}(g(b))^{j+1}-{\rm e}^{{\rm i}kg(a)}(g(a))^{j+1})-\frac{j+1}{{\rm i}k}m^{[g(a), g(b)]}_{k}(j),
\tilde{B}_i:= \bigcap_{j \geq J}\tilde{B}_{n_j,i}\supset B_{1-\epsilon}(\tilde{x}_i) \ \ \mbox{and}\ \t\tilde{B}_i:= \bigcap_{j \geq J}t\tilde{B}_{n_j,i}\mbox{for}t \in \mathbb{R}^+.
\varrho_N^\varphi(x) = \frac 1N \sum_{j=1}^N \varphi (x-x_j)\,, \varrho_N^\varphi u_N^\varphi(x) = \frac 1N \sum_{j=1}^N \varphi (x-x_j) v_j\,, \varrho_N^\varphi T_N^\varphi(x) = \frac 1{Nd}\sum_{j=1}^N \varphi (x-x_j) |v_j-u_N^\varphi(x)|^2 \,,
D_{ij}= \delta_{ij}\frac{T_{ii}+ \left[ \left({\it \Gamma}T \right)^* T \right]_{ii}}{1 + \left[\left({\it \Gamma}T\right)^* + \left({\it \Gamma}T \right) + \left({\it \Gamma}T\right)^* \left({\it \Gamma}T \right)\right]_{ii}}\ .
S=\frac{1}{\gamma^2}\int_{-\infty}^{+\infty}d\tau \int_0^{2\pi}d\sigma\Bigl(\frac{1}{2}\partial_\alpha\varphi\cdot \partial^\alpha\varphi -\mu^2\sum_{a=1,2}{\rm e}^{\alpha^a\cdot\varphi}\Bigr) ~,
\omega_{\Xi,\Psi}:=d\sigma_{3}^{*}\Xi^{a}\wedge d\sigma_{3}^{*}\Psi_{a}=0, &\omega_{\Theta}:=d\sigma_{3}^{*}\Theta^{\alpha}\wedge d\sigma_{3}^{*}\Theta_{\alpha}=0, &\omega_{Q,P}:=d\sigma_{3}^{*}Q^{i}\wedge d\sigma_{3}^{*}P_{a}.
\tilde{n}_{o,i}=\mu\rho_i^2\sum_{j\in\mathcal{M}_i}\beta_j^{(i)}|\alpha_{ij}|^2+\sqrt{\mu\rho_i^2}\sum_{j\in\mathcal{M}_i}\beta_j^{(i)}|\alpha_{ij}|Z_{s,j}^{(i)}+\sum_{j\in\mathcal{M}_i}\beta_j^{(i)}Z_{0,j}^{(i)},
+ \frac{1}{2}\partial_+ f_2 \partial_- f_1 + \frac{5}{3}e^{-2 \rho}\Omega^{2/3}(e^{-\sqrt{2}f_1}F_{1 +-}^2 + e^{-\sqrt{2}f_1}F_{2 +-}^2+ e^{\sqrt{2}( f_1 + f_2 )}F_{+-}^2 ) = 0
\Big(A^{\frac{1}{2}}\big(A!_\lambda B\big)^{-1}A^{\frac{1}{2}}\Big)^{\frac{1}{2}}A^{-\frac{1}{2}}(A\nabla_\lambda B)A^{-\frac{1}{2}}\Big(A^{\frac{1}{2}}\big(A!_\lambda B\big)^{-1}A^{\frac{1}{2}}\Big)^{\frac{1}{2}}-I &=\lambda(1-\lambda)A^{-\frac{1}{2}}(A-B)B^{-1}(A-B)A^{-\frac{1}{2}}.
\begin{array}{rcl}\theta_2(1) = 2(1)^52 \theta_2(2) = 2(1)^32\end{array}\textrm{\; \; \; \; and\; \; \; \;}\begin{array}{rcl}\theta_{n+1}(1) = 211(21)^{b_n+1}12 \theta_{n+1}(2) = (21)^{b_n}2\end{array}\; \; \textrm{for\,}n > 1,
\begin{array}{l l}{T_{N}^{\mu \nu}}&{= \displaystyle{\frac{\partial{\cal L}}{\partial A_{\lambda , \mu}^{a}}A_{\lambda}^{a , \nu}- \eta^{\mu \nu}{\cal L}}}\\{{}}&{{}}\\{{}}&{= - \Pi^{a \mu \lambda}A_{\lambda}^{a , \nu}- \eta^{\mu \nu}{\cal L}}\\ \end{array}
S \approx \int d^{1 0}x \sqrt{| G |}\left[ e^{- \Phi}\left( R + | \nabla \Phi |^{2}- | d B_{2}^{( 1 )}|^{2}\right) - | d B_{2}^{( 2 )}|^{2}- | d D_{4}|^{2}\right] - \ldots
e(G') \leq \max\left\{\binom{k}{2},\binom{\frac{k+1}{2}}{2}+\frac{k-1}{2}\left(n-\frac{k+1}{2}\right)\right\}&=\max\left\{\binom{k}{2},\binom{\frac{k-1}{2}}{2}+\frac{k-1}{2}\left(n-\frac{k-1}{2}\right)\right\}.
\lim_{x\rightarrow x_0}(x-x_0)G_2(x,y,z;q) &=-2\cdot \frac{J_1^3J_2^3}{j(-q^{2n};q)j(y;q)j(z;q)}\cdot \frac{-(-1)^{n+1}q^{2\binom{n}{2}}q^{2n}}{J_2^3}\cdot\frac{j(-q^{2n}y;q^2)j(-q^{2n}z;q^2)j(yz;q^2)}{j(-y;q^2)j(-z;q^2)}&=- (-1)^{n}\frac{q^{n^2+2n}}{y^nz^n}\cdot \frac{j(yz;q^2)}{j(y;q)j(z;q)}\cdot \frac{J_1^4}{J_2^2},
S = \frac{1}{4 g_{Y M}^{2}}\mathrm{T r}\left( V \sum_{\alpha \beta}( F_{\alpha \beta}+ \phi_{\alpha \beta}\cdot{\bf 1}) g^{\alpha \mu}g^{\beta \nu}( F_{\mu \nu}+ \phi_{\mu \nu}\cdot{\bf 1}) \right) \, .
f_{-}( k , r ) = \sqrt{{\displaystyle{\frac{\pi}{2}}}}\, \sqrt{k r}\, H_{0}^{( 1 )}( k r ) + g \int_{r}^{\infty}d r^{\prime}\, \tilde{G}( k ; r , r^{\prime}) V ( r^{\prime}) f_{-}( k , r^{\prime}) .
g_{i j}= \left[ \begin{array}{l l l}{1}&{0}&{0}\\{0}&{g_{2}}&{0}\\{0}&{0}&{g_{3}}\\ \end{array}\right] \mathrm{a n d}h_{a b}= \left[ \begin{array}{l l}{h_{4}}&{0}\\{0}&{h_{5}}\\ \end{array}\right]
Z_{V^{E_{j0}}_{\mu, j}}* 1_{J_{E_{j0}}}= \sum_{\bar{\lambda}\in{\mathcal Wt}(\bar{\mu})^{+, \tau}_{E_j}}{\rm tr}(\tau \, | \, \mathbb H_{\mu, E_j}(\bar{\lambda})) \, \sum_{\bar{\nu}\in{\mathcal Wt}(\bar{\lambda})^{+, \tau}_{E_j}}P_{w_{\bar{\nu}}, w_{\bar{\lambda}}}(1) \, \Big(\sum_{\{\bar{\nu}' \}\subset W_{E_j}\bar{\nu}}z_{\bar{\bar{\nu}}'}\Big).
\hat{x}_{1 2}= x_{1 2}+{\frac{2 i}{( 1 2 )}}[ ( 1^{-}2 ) \theta_{1}^{+}\bar{\theta}_{1}^{+}- ( 1 2^{-}) \theta_{2}^{+}\bar{\theta}_{2}^{+}+ \theta_{1}^{+}\bar{\theta}_{2}^{+}+ \theta_{2}^{+}\bar{\theta}_{1}^{+}] \ .
|\langle j_p(x^*) - j_p(x), z-x^*\rangle| = \left|\left\langle j_p(x^*) - j_p(x_N) - \sum_{k=1}^N j_p(x_k) - j_p(x_{k-1}),z-x^*\right\rangle\right| = \langle j_p(x^*)-j_p(x_N), z-x^*\rangle \xrightarrow[]{N\to\infty}0,
d x_{j}^{\prime}= \left( \frac{\partial x_{j}^{\prime}}{\partial x_{k}}\right) d x_{k}\equiv J_{j k}\d x_{k}, A_{i}= A_{j}^{\prime}\left( \frac{\partial x_{j}^{\prime}}{\partial x_{i}}\right) \equiv J_{i j}^{t}A_{j}^{\prime},
A_{j}=\left\{t\in(t_{0},t_{0}+\delta):\left\vert \Psi_{k,t}(x)\right\vert >\tfrac{1}{4},\forall x\in\left[ \tfrac{j-1}{q},\tfrac{j}{q}\right] \right\}, B_{p}=\left\{t\in(t_{0},t_{0}+\delta):\left\vert \left(\Psi_{k,t}^{\ast},e^{i(2\pi p+t)x}\right) \right\vert >\tfrac{1}{2l}\right\}.
\begin{aligned}(f_1,f_2,\phi) = \sum_{j=0}^2 \sum_{k = 0}^2 w_{jk}(a_{1j}, a_{2k}, a_{1j}a_{2k}),\ \sum_{j = 0}^2 \sum_{k=0}^2 w_{jk}= 1,\w \geq 0, &\sum_{j=0}^2w_{j0}\leq \delta_1 ,\ \sum_{j=0}^2w_{j2}\leq 1-\delta_1,\ \delta_1 \in \{0,1\}, &\sum_{k=0}^2w_{0k}\leq \delta_2 ,\ \sum_{k=0}^2w_{2k}\leq 1-\delta_2,\ \delta_2 \in \{0,1\}.\end{aligned}
\int_{- \infty}^{+ \infty}d y \frac{e^{i k y}}{\cosh^{\mu}u}= \frac{2^{\mu - 1}}{\Gamma ( \mu )}\, \, \Gamma \left( \frac{\mu + i k}{2}\right) \Gamma \left( \frac{\mu - i k}{2}\right) ~ ~ \mathrm{f o r}~ ~ \mathrm{R e}\, \mu > 0 ,
S = \int_{E}d^{2}x \Bigr \{\mathrm{\frac{1}{4}}F^{\mu \nu}( A ) F_{\mu \nu}( A ) + \mathrm{\frac{1}{2}}\partial^{\mu}A_{\mu}\partial^{\nu}A_{\nu}+ \partial^{\mu}\bar{C}\partial_{\mu}C \Bigr \},
\left[ \, \, T_{+ +}- T_{--}\, \, \right]_{\sigma = 0 , \pi}= 0 , \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \left[ \, \, G_{+}^{1}- \eta_{1}G_{-}^{1}\, \, \right]_{\sigma = 0 , \pi}= 0 ,
\left[ p^{\mu}\partial_{\mu}+ \vec{Q}\cdot \vec{F}_{\mu \nu}p^{\mu}\partial_{p}^{\nu}+ p^{\mu}\vec{Q}\times \vec{A}_{\mu}\cdot{\frac{\partial}{\partial \vec{Q}}}\right] f ( x , p , \vec{Q}) = C ( x , p , \vec{Q}) + S ( x , p , \vec{Q})
\int_{0}^{\infty}p^{s-1}E_{\alpha, \beta}^{\gamma}\left(-\frac{p}{t(1-t)}dp\right) =\int_{0}^{\infty}u^{s-1}t^s(1-t)^sE_{\alpha, \beta}^{\gamma}\left(-u\right)du = t^s(1-t)^s \int_{0}^{\infty}u^{s-1}E_{\alpha, \beta}^{\gamma}\left(-u\right)du ={}^{M\!L\!}\Gamma(s) t^s(1-t)^s.
\begin{array}{r c l}{{Y}}&{{=}}&{{\frac{1}{2}z^{- 2}( x + y + 2 \prod_{i = 1}^{n}z_{i})}}\\{{X}}&{{=}}&{{\frac{1}{2}z^{- 1}( x - y )}}\\{{Z}}&{{=}}&{{z^{2}~ .}}\\ \end{array}
\begin{aligned}\log|p(z)| \leq \|z\|_{\infty}\sqrt{2}\sum_j |p_j(0)| + \|z\|_{\infty}^2((\sum_j |p_j(0)|)^2+ \sum |\Re [p_{jk}(0)]|) & \leq \|z\|_{\infty}\sqrt{2}\sum_j |a(e_j)| + \|z\|_{\infty}^2((\sum_j |a(e_j)|)^2+2 \sum |\Re [a(e_j+e_k)]|) &\leq \frac{1}{2}+ \|z\|_{\infty}^2(2(\sum_j |a(e_j)|)^2+2 \sum |\Re [a(e_j+e_k)]|)\end{aligned}
H^{I_{A}( \mu_{1}, \mu_{2}) J_{B}( \nu_{2}, \nu_{1})}= \mathrm{T r}\eta^{i_{A ( \mu_{1}+ 1 )}}\cdots \eta^{i_{A ( \mu_{2}- 1 )}}\eta^{j_{B ( \nu_{2}+ 1 )}}\cdots \eta^{j_{B ( \nu_{1}- 1 )}}
\left(\psi^L_{a;S}\right)_{0j}\colon f(x,y,z) \mapsto \eta^{-a \vert S \vert j}f(x, \eta^{-a}x,z) \left(\psi^R_{S;b}\right)_{i0}\colon f(x,y,z) \mapsto f(x, \eta^{b}z,z).
\left\{\Xi_{\mathrm{p},l}^{(\omega )}\left( t\right) \right\}_{k,q}:=\frac{1}{\left\vert \Lambda_{l}\right\vert}\underset{x,y\in \Lambda_{l}}{\sum}\sigma_{\mathrm{p}}^{(\omega )}\left( x+e_{q},x,y+e_{k},y,t\right)
\ll \frac{x(\log \log x)^2}{\log^2x}\max_{\substack{b\pmod*{q}(b,q)=1}}\sum_{\chi_0\neq\chi_1,\chi_2\pmod*{q}}\frac{\chi_1(b)\overline\chi_2(b)}{\varphi(q)^2}\sum_{n\leq \frac{x}{r(x)}}\frac{d_{\chi_1}(n)d_{\chi_2}(n)}{\varphi(n)}.
\begin{aligned}-I\geq \frac{1}{2}\int_{B_{R_i}\times\left\{t\right\}}\left(u-l_i\right)_-^2\eta_i^2\,dx-\frac{2^{2(i+1)}\theta_0^{\alpha_{0}}\left(2\theta_0\right)^2}{R_i^2}\int_{-\theta_0^{-\alpha_{0}}R_i^2}^{t}\int_{B_{R_i}}\chi_{\left[u\leq l_i\right]}\,dxd\tau\end{aligned}
\frac{\partial^{n + 1}f^{A}}{\partial z^{n + 1}}= \sum_{\alpha = 1}^{n}W_{\alpha}\frac{\partial^{\alpha}f^{A}}{\partial z^{\alpha}}, \quad \frac{\partial^{n + 1}\bar{f}^{A}}{\partial \bar{z}^{n + 1}}= \sum_{\alpha = 1}^{n}\bar{W}_{\alpha}\frac{\partial^{\alpha}\bar{f}^{A}}{\partial \bar{z}^{\alpha}}
{\cal L}=- \frac{1}{4}F_{\mu\nu}F^{\mu\nu}+ \frac{1}{2}m^2 (A_\mu + \partial_\mu \theta)^{2}+A^{\mu}\partial_\mu B+d_{2}\partial_{\mu}\bar{\cal C}\partial^{\mu}{\cal C}-\frac{1}{2}\alpha B^{2}.
\begin{aligned}C_1 : y^2 = (x-a_1)\cdots(x-a_r)(x-b_1) \cdots (x-b_{2g_1+2-r}), C_2 : y^2 = (x-a_1)\cdots(x-a_r)(x-c_1) \cdots (x-c_{2g_2+2-r}),\end{aligned}
\sum_{n=0}^{\infty}\frac{\pi_{n,q}(x)t^{n}}{(1;n)}=\sum_{n=0}^{\infty}\frac{(-1)^{n}q^{\frac{n(n-1)}{2}}t^{n}P_{n}(x)}{(1;n)[1-t]_{n+1}},
f(\alpha\mu_1+(1-\alpha)\mu_2) =\left(\alpha A+(1-\alpha)A\right)-\frac{C}{p^*}\sum_{i=1}^{n}\int_{B_{F^{*}}}\left|y^*(y_i)\right|^{p^*}\d(\alpha\mu_1+(1-\alpha)\mu_2)(y^*) &=\alpha f(\mu_1)+(1-\alpha)f(\mu_2)
\ \left\langle u^*(t),B^* \mathbb{T}_{\tau^*(M)-t}^* \eta \right\rangle_U= \displaystyle\max_{v\in U,\; \|v\|_U \leqslant M}\ \left\langle v,B^* \mathbb{T}_{\tau^*(M)-t}^* \eta \right\rangle_U.
M_{n}= n + \frac{1}{\pi}\, \operatorname{a r c t a n}\sqrt{\frac{\operatorname{t a n}^{2}( \omega \pi ) + t_{\rho}^{2}( \Lambda )}{1 + \operatorname{t a n}^{2}( \omega \pi ) \, t_{\rho}^{2}( \Lambda )}}
\sum_{k=0}^{\infty}\frac{(c)_k(d)_k(e)_k(1+a-b-c)_k(1+a-b-d)_{k}(1+a-b-e)_{k}}{(1+a-c)_{k}(1+a-d)_{k}(1+a-e)_{k}(1+2a-b-c-d-e)_{k}}begin{align*}1mm] \quad\times\frac{(-1)^k}{(1+a-b)_{2k}}\alpha_k(a,b,c,d,e) begin{align*}1mm] \:=\sum_{k=0}^{\infty}(a+2k)\frac{(b)_k(c)_k(d)_k(e)_k}{(1+a-b)_{k}(1+a-c)_{k}(1+a-d)_{k}(1+a-e)_{k}},
SINR^j_{1,k}= \frac{|\mathbf{v}_{1,k}^H\mathbf{h}_{1,1k}|^2 \alpha_{1,j}^2}{\sum^{j-1}_{l=1}|\mathbf{v}_{1,k}^H\mathbf{h}_{1,1k}|^2 \alpha_{1,l}^2 +|\mathbf{v}_{1,k}|^2\frac{1}{\rho}}.
B = \frac{\pi | \beta |^{2}}{4 ( 4 | \lambda |^{2}- | \beta |^{4})^{2}}[ | \beta |^{2}( | \beta |^{4}- 7 | \lambda |^{2}) + | \lambda |^{2}( 1 0 | \lambda |^{2}- | \beta |^{4}) g ( \lambda , \beta ) ]
\Delta(b)-\Delta^\sharp(b)=\sum\big([v_j^*, [v_j, b]]-[v_j, [v_j^*, b]]\big) \phantom{\Delta(b)-\Delta^\sharp(b)}=\sum\big([v_j^*, [v_j, b]]+[[v_j^*, b], v_j]\big)=\bigg[\sum [v_j^*, v_j], b\bigg],
\frac{q_{\nu_{1}\boxtimes\nu_{2}}(x)}{q_{\mu_{1}\boxtimes\mu_{2}}(x)}=\frac{|1-\eta_{\mu_{1}}(re^{if(r)})|^{2}}{2\beta}\int_{[0,+\infty]}\frac{1+t^{2}}{|t-\eta_{\mu_{1}}(re^{if(r)})|^{2}}\,d\sigma(t).
C^2 \geq \frac{3}{2}d + \mu_{p_{1}}- \sum_i m_{p_i}^2 \geq \frac{3}{2}d + \sum_j \nu_j(\nu_j-1) + 1 - r_{p_1}- \sum_i m_{p_i}^2 = \frac{3}{2}d +1 - r_{p_1}- \sum_j\nu_j + \underbrace{\Big[ \sum_j \nu_j^2 - \sum_i m_{p_i}^2 \Big]}_{(\dagger)}.
{\cal M}_{i i \rightarrow j j}\sim g^{2}\sum_{k}c_{i i k}c_{j j k}\int_{0}^{\infty}d t e^{- t ( p^{2}+ \lambda_{k})}= g^{2}\int_{0}^{\infty}d t e^{- t p^{2}}\sum_{k}c_{i i k}c_{j j k}e^{- t \lambda_{k}}
\widehat S_{\ell}(X_1 \times_{M_1}\dots \times_{M_{n-1}}X_n)=\coprod_{\ell_1+\dots+\ell_n = \ell}(\widehat S_{\ell_1}(X_1) \times_{M_1}\dots \times_{M_{n-1}}\widehat S_{\ell_n}(X_n)).
q \frac{d}{dq}E_{0}(q)= \sum_{k, r =1}^{\infty}\chi_{-3}(kr) r \left( \frac{1}{3}q^{\frac{kr}{3}}- q^{kr}\right) = \frac{1}{9}b(q^{\frac{1}{3}}) c(q) - \frac{1}{3}b(q) c(q^{3}) &= \frac{1}{9}\left( a(q)c(q) - c^{2}(q) - a(q)b(q) + b^{2}(q) \right) .
\mu_\Delta(\{(z,\ell)\in \hat Y:\varphi(z)>n\}) =\sum_{q=1}^\infty\mu_\Delta(\{(z,\ell)\in \hat Y:\varphi(z)>n,\,\rho(z)=q\}) & \le \sum_{q=1}^\infty q\mu_Z(\{z\in Z:\varphi(z)>n,\,\rho(z)=q\}).
\begin{pmatrix}1\!\!1_{k}0 0 t1\!\!1_{n-k}\end{pmatrix}\begin{pmatrix}B tb^{\top}b \beta+t\chi\end{pmatrix}-\begin{pmatrix}B tb^{\top}b \beta+t\chi\end{pmatrix}^{\top}\begin{pmatrix}1\!\!1_{k}0 0 t1\!\!1_{n-k}\end{pmatrix}=
W_H(c(a)) =n-\frac{1}{p^2}\sum_{y \in \mathbb{F}_p}\sum_{z \in \mathbb{F}_p}\sum_{x \in \mathbb{F}_q^*}\chi_m((z+ya)x) &=n-\frac{1}{p^2}(q-1+(p^2-1)(-1)) &=p^{m-1}-p^{m-2}
|X_{2}|^{2}=|X_{3}|^{2}=|X_{4}|^{2}=\frac{1}{2}\left(\frac{4}{3}\right)^{\frac{1}{4}}(r^{2}+1)^{\frac{1}{2}}(r^{2}+2)^{\frac{1}{4}}, &|X_{5}|^{2}=|X_{6}|^{2}=|X_{7}|^{2}=\frac{1}{2}\left(\frac{4}{3}\right)^{\frac{1}{4}}\frac{(r^{2}+2)^{\frac{1}{4}}(r+1)(r-1)}{(r^{2}+1)^{\frac{1}{2}}}.
[\begin{matrix}0&1&-3\\ 3&-13&23\\ -4&19&-36\end{matrix}]+4[\begin{matrix}0&0&1\\ -1&4&-6\\ 1&-5&10\end{matrix}]+[\begin{matrix}1&-1&-1\\ 1&-2&1\\ 0&1&-3\end{matrix}]+[\begin{matrix}-1&0&0\\ 0&-1&0\\ 0&0&-1\end{matrix}]=[\begin{matrix}0&0&0\\ 0&0&0\\ 0&0&0\end{matrix}]
\tilde S(\eta,\nu)= e^{ia\ln\left|\frac{\eta+\nu}{\eta-\nu}\right|}\left(A + 8B e^{2ia\ln|(\eta+\nu/2|}\frac{e^{i\beta (\eta+\nu)^3/8}}{(\eta+\nu)^3}\right)\left(\overline{A}- 8\overline{B}e^{-2ia\ln|(\eta-\nu)/2|}\frac{e^{i\beta (\eta-\nu)^3/8}}{(\eta-\nu)^3}\right)
y\left( \rho\right) :=y\left( \rho,x\right) =\frac{1}{\frac{1}{2}-i\rho}\sum_{j=0}^{\infty}a_{j}(x)z^{n}\left( \rho\right) ,\quad\sum_{j=0}^{\infty}\left\vert a_{j}(x)\right\vert^{2}<\infty.
\dfrac{e^{B_1-2}}{(t_0-\epsilon)^{n+1}}=e^{B_1-2-k(t_0)}\leq \psi_0(z_0,t_0)\leq C_5 (k'(t_0)+1)^n=C_5\left(\dfrac{t_0+n+1-\epsilon}{t_0-\epsilon}\right)^n.
\{\cdot , \cdot\}_{(\lambda)}\Omega = -\frac{1}{F(\lambda)}d\cdot \wedge\, d\cdot \wedge\big(\sigma_{(\lambda)}+ \frac{g_{(\lambda)}}{r-1}\omega\big) \wedge \frac{\omega^{r-2}}{(r-2)!}\wedge dF^1(\lambda)\wedge\ldots \wedge dF^k(\lambda),
|-\frac{1}{2\pi}\int_{t}^{\infty}ds \int_{B_{s-t}(0)}\frac{dA(y)}{\sqrt{(s-t)^{2}-|y|^{2}}}_{v_{4,1}}|\leq \frac{C}{t^{2}\log^{3b+2N-1}(t)}, r \leq \frac{t}{2}
W_y^{QFT}= E_y -E_{\infty}&= -\aleph_{\alpha,L}\frac{\alpha}{L^4}+ O(\alpha^4 \log(\alpha^{-1}))+ O\big( \alpha^2 L e^{-L/2}\big) + O \Big( \frac{\alpha^2}{L^5}\Big) + O\Big( \frac{\alpha^3}{L^2}\log(\alpha^{-1})\Big),
\xi^{j,*}_{0}=j,\xi^{j,*}_k=\sum_{i=1}^N i\mathbf{1}_{\{H^{\xi^{j,*}_{k-1}}(X_{\tau_k})=-c_{\xi^{j,*}_{k-1},i}(X_{\tau_k})+u^i({X_{\tau_k})}\}},k\ge1.
df(\vec{a},\vec{b})=(\frac{\partial(\vec{f}_{0}...\vec{f}_{k})}{\partial(\vec{a}_{0}...\vec{a}_{n})})_{\vec{b}}d\vec{a}+(\frac{\partial(\vec{f}_{0}...\vec{f}_{k})}{\partial(\vec{b}_{0}...\vec{b}_{m})})_{\vec{a}}d\vec{b}